Differential Algebraic Groups o - A. Buium

Alexandru Buium Differential AlgebraicGroups of Finite Dimension

Author Alexandru Buium Institute ofMathematics of the Romanian Academy P.O.Box 1-764 RO-70700 Bucharest, RomaniaMathematics Subject Classification (1991):12H05, 14L17, 14K10, 14G05, 20F28 ISBN3-540-55181-6 Springer-Verlag BerlinHeidelberg New York ISBN 0-387-55181-6Springer-Verlag New York BerlinHeidelberg This work is subject tocopyright. All rights are reserved, whetherthe whole or part of the material isconcerned, specifically the rights oftranslation, reprinting, re-use of illustrations,recitation, broadcasting, reproduction onmicrofilms or in any other way, and storage

in data banks. Duplication of this publicationor parts thereof is permitted only under theprovisions of the Geman Copyright Law ofSeptember 9, 1965, in its current version,and permission for use must always beobtained from Springer-Verlag. Violationsare liable for prosecution under the GermanCopyright Law. � Springer-Verlag BerlinHeidelberg 1992 Printed in GermanyTypesetting: Camera ready by author Printingand binding: Druckhaus Beltz,Hemsbach/Bergslr. 46/3140-543210 -Printed on acid-free paperINTRODUCTION Differential algebraicgroups are defined roughly speaking as"groups of solutions of algebraic differentialequations" in the same way In whichalgebraic groups are defined as "groups of

solutions of algebraic equations". They wereintroduced in modern literature by Cassidy[C 1] and Kolchin [K2]l their pre-historygoes back however to classical work of S.Lie, E. Cartan and :J.F. Rltt. Let'scontemplate a few examples before givingthe formal definition (for which we send toSection 3 of this Introduction). Start with anylinear differential equation: (1) y(n). Sly(n-l). 0 ....anY= where the unknown y and thecoefficients a i are, say, meromorphicfunctions of the complex variable t. Thedifference of any two solutions of (1) isagain a solution of (l)l so the solutions of (1)form a group with respect to addition. Thisprovides a first example of *'differentialalgebraic group". Slmllarlly, consider thesystem xy- I =0 (2) yy.. (y,)2 + ayy' = 0

where the. unknowns x,y and the coefficient aare once again meromorphlc in t. This system(extracted from a paper of Cassidy [C]) hasthe property that the quotient (Xl/X 2, yl/Y2) of any two solutions (xlYl), (x2,Y 2) Isagain a solutionl so the solutions of (2) forma group with respect to the multipllcatlvegroup of the hyperbola xy- I = 0 and we areled to another example of "differentialalgebraic group". Examples of a more subtlenature are provided by the systems (3a)[y2.x(x= IXx-c)0 x"y -xy +ax'y=0 (3b) /y 2-x(x - l)(x - t) = 0 {. =y} - 2(2t - l)(x - t)2x'y+ 2t(t - l)(x - t)2(x'y - 2x'y') = 0 where theunknowns x y are still meromorphlcfunctions In t the coefficient a ismeromorphlc in t and c 0,1 is a constant in C.These systems (of which (3b) Is extracted

from a paper of Manin [Ma]) have theproperty that If (xl,Yl) and (x2,Y 2) aredietinct solutions of one of the systems, thentheir difference, In the group law of theelliptic curve defined by the first equationof that system, is again a solutionl so thesolutions of (3a), (3b] together with the point

vI at infinity (0: l: 0) of A form groups withrespect to the group law of A and lead toother examples of "differential algebraicgroups". A differential algebraic group willbe called of finite dimension if roughlyspeaking its elements "depend on finitalymany integration constants" rather than on"arbitrary functions". This Is the case withthe "differential algebraic groups" derivedfrom (1), (2), (3); on the contrary, for

instance, the group, say, of all matrices x 0 y)0 I z , xg0 0 0 I satisfying the system = ZXhas its elements depending on the "arbitraryfunction" y so It Is an "infinite dimensionaldifferential algebraic group". Differentialalgebraic groups are known today but to afew mathematicians; and this is becauseKolchin's language [K ITK 2 ] through whichthey are studied is known but to a fewmathematicians. Yet differential algebraicgroups certainly deserve a much broaderaudience, especially among algebraicgeometers. The scope of the present researchmonograph is twofold namely: 1) to providean algebraic geometer's introduction todifferential algebraic groups of finitedimension and 2) to develop a structure andclassification theory for these groups. Un]ess

otherwise stated, all results appearing in thisbook are due to the author and were neverpublished before. The original motivation forthe study o! differential algebraic groups in[C I] and [K 2] was undoubtedly theirintrinsic beauty and variety. Admittedly nosuch group appeared so far to play a role,say, Jn mathematical physics; but as we hopeto demonstrate in the present work,differential algebraic groups play a role Inalgebraic geometry. This Is alreadysuggested for instance by the implicitoccurence o! the differential algebraic group(3b) in Manln's paper [Ma] on the geometricMordell conjecture. Moreover as shown inthe author's papers [B6], [B 7] differentialalgebraic groups may be used along a linequite different from Manln's to prove the

geometric and infinitesimal analogues of adlophantine conjecture of S. Land (aboutinter- sections of subvarieties of abolishvarieties with finite rank subgroups). Thelatter application will be presented withoutproof ]n an appendix. Other applications andinteresting links with other topics ofalgebraic geometry (such as deformations ofalgebraic groups and thetr automorphlsms,toodull spaces of abelian varietlas theGrothendieck-Mazur-Messlng crystallinetheory of universal extensions of abolishvarieties [MM], a.s.o.) will appear in thebody of the text (and will also be touched inthis Introduction). vii 3o ForrealizationBefore explaining our strategy and results insome detail it will be convenient to providethe formal definition of differential algebraic

groups with which we will operate in thisbook. The frame in which this definition willbe given is that of "differential algebraicgeometry" by which we mean here theanalogue (due to Rltt and Kolchln) ofalgebraic geometry Jn which algebraicequations are replaced by algebraicdifferential equations. Let's quickly reviewthe basic concepts of this geometry, cf. [] [K1 J, [Cl]; for details we send to the firstsection of Chapter 5 of this book. One startswith a field of characteristic zero equJpedwith a derivation 6 .' -+ (i.e. with an additivemap satisfying 5(xy) = (6x)y + x(6y) for allx,y c,) and set = (x c(; 6 x = 0} the field ofconstants. We assume that , is "sufficientlybig" as to "contain" all "solutions ofalgebraic differential equations with

coefficients in it"; for the reader familiarwith [K 1 ] what we assume is that is"universal". Such a field is a quite artificialbeing; but since all problems related toalgebraic differential equations can be"embedded" into "problems over ," the useof this field appears to be extremely usefuland in any case it greatly simplifies!anguage. fext one considers the ring ofdifferential polynomials (shortly 6 -polynomials) 'fYl'"' Yn} which by definitionis the ring o:[ polynomials with coefficientsin J, in the indetermJnates Yi .... Yn'Y'i ......... yk), ,,(k) E.g. the expressions appearingin the left hand side of the equations (I), (2),(3) (where we assume now al,a,t t?., t' -- 1, ct) are A - polynomials. Now for any finite setFi,...,F m of i - polynomials we may consider

their set r. of common zeroes in the arlinespace An(l) =n; such a set will be called a -closed subset of n, _ closed sets form aNoetherian topology on ?n called the /, -topology which is stronger than the Zariskltopology. For instance the equation (1)defines an irreducible I - closed subset of thearline line AI() =? while the systems (2) and(3) define irreducible /, - closed subset ofthe arline plane A2() --7J 2. Any irreduciblea- closed subset r. in n has a natural sheaf(for the induced a - topology) of - valuedfunctions on Jt called the sheaf of A - regularfunctions; roughly speaking a function on a -open set of Z is called A - regular if, locallyin the I - topology, it is given by a quotient of/, - polynomials in the coordinates. r.together with this sheaf will be called an

arline differential algebraic manifold(shortly, an arline - manifold). Anirreducible ringed space locally isomorphicto an afine A - manifold will be called adifferential algebraic manifold (or simply a/, - manifold). Given a A - manifold r., the'direct limit, over all /, - open sets i2 of g, ofthe rings of A - regular functions on t is afield denoted by < g >. The transcendencedegree of .< : � over 2/ will be called thedimension of g and

VIII intuitively represents the number of"integration constants" of which the points of. depend. E.g. the A - closed sets given byequations (1), (2), (3) have dimensions n, 2,2 respectively. A map between two 6 -manifolds will be called 6 - regular if it is

continuous and pulls back A - re�uJarfunctions into A - regular functions. /e areprovided thus with a category which hasdirect products called the category of A-manifolds. Note that any algebraic .- varietyX (reAspectively any X- rayfiery X o) has anatural structure of A - manifold which � Awe denote by X (respectively by Xo); wehave dimX � = dim X � and dimX = o0provided dim X ) 0. Finally we definedifferential algebraic groups (or simply A -group) as being group objects in the categoryof A - manifolds, i.e. 6 - manifolds r whoseset of points is given a group law such thatthe multiplication r x - r and the inverse r ( Tare A - regular maps. Differential algebraicgroups of finite allmansion will be simplycalled 6o ' groups. The equations (l), (2)

clearly provide examples of 6o ' groups. Asfor equations (3) the o ' groups which can bederived are not the A - closed subsets ofA2() given by (3) but their A - closures in (p2) which contain an additional point (0: I =0), the origin of these groups. /e leave openthe question whether our - groups "are" thesame with KoJchin's [K2] one can show thatany 6 - group in our sense "is" a 6 - group inKolchin's sense. Moreover one can show thatour Ao- groups "are" precisely Kolchin'sirreducible "- groups of 6- type zero".Finally note that our concept of "dimension"corresponds to Rltt's concept of "order" [[]and also (in case we have "finitedimension") with KoichJn's concept of"typical 6 - dimension" [K l I2]. 4. StrategyCassidy and Kolchin developed their theory

of differential algebraic groups in analogywith the theory of algebraic group. Ourviewpoint will be quite different: we willbase our approach on investigating therelations, not the analogies, between the twotheories. Our strategy has two steps. Step Iwill consist in developing a theory of whatwe caJ] "algebraic D-groups"{ this will bedone in Chapters 1-4 of the book. Then Step2 will consist Jn applying the latter theory tothe study of do - groups; this will be done inChapter 5. Let's explain the concept ofalgebraic D-group; for convenience we willgive in this Introduction a rather restricteddefinition o! ]t (which will be "enlarged" inthe body of the text). Let 2, 6,: be as inSection 3 above and let .D =4[ 6] = i>0, 61(direct sum) be the - algebra of linear

differential operators generated by /, and .By an algebraic D-group we will understandan irreducible algebraic - group G whosestructure sheaf (C of regular functions isgiven a structure of sheaf of D - modulessuch that the multiplication, comultiplicatlon,antipode and co-unit are D-module maps; inother words if G x C -G, S .' G - C are thegroup multiplication and inverse and if e Gis the unit then for any regular tunctions ,defined on some open set of C we have theformulae IX () ( o s) = () � s ((e)) = ( Someple might Uke to call such a strture "angebralc group with nnection alonR we were[pired in our termilogy by the paper oNichols and eisteller [N]. Note however thatunlike in [N] we do t assume C is afine,imsin instead that C is o finite ty over !

Algebraic D-groups entirely belong to"algebraic geomry" (rather than to the Ritt-Kolchin "differential aJgebraIc geometry")hence Step I will inevitably performed In thefield o algebraic geometry) the task oclassifying algebraic D-grou wiiJ somimquite technlca] but In the end rewarding. Step2 wJJJ be based on the result that "thecategory o 6 - rou" is uivalent to "the ocategory o[ gebraJc D- groups"I or any o-group we shall denote by C() thecorresponding algebraic D-group. Note that(the underlying group o) appears as the groupall ints � G(X) or which the evaluation map(), + is a D-mule map) actuly appears as a 6- closed subRroup o G() Note that t untonfield (C()) identiias with < >. Tn Step 2 willdeal with the (sometim t so obvious)

translation o properti o algebraic D-groupsinto operties of -grou. In ord to get a 1ing autthe o corresodence G() l's !k at the examplwe gan with (equations (1), (2), (3)). I Isderived from tlon (1) then G() is thealgebraic vector oup C n = Sc[[o,[ 1, ... [n.l ]uid with t D-mole strte o{ its crdinate gebradefined by 6%0 = l (and by t condition that 6is a derivation of[[ o ..... l T is derived fromequation (2) then C() Js C m x C a = Sc[XI,the product the muttipHcatJve d additivegroups equid with the D-struure deid by 6X=X In case is derived {tom (3a) C(P) can beproved o oduc A x C while In e a P isderived rom (3b), C( is a n-trivIa] extensiono the elHpic curve A by in he Iater case G(r)does t descend o (because A doesn'O. 5.Results According to the preceedinR section

the c.[assification problem for A o - groupsreduces

x Xl to answering the following question.'given an algebraic - group G describe allstructures of algebraic D-groups on G l calltheir set P(G/6). Note that P(G/6) Js aprincipal homogeneous space for the - linearspace P(G/) of all ?,4- linear maps D: G ' Gsatisfying equations O) with D instead of 6.So we are faced with two problems herenamely: l) What irreducible algebraic [/,-groups G admit at least one structure ofalgebraic [3 - group ? 2) Describe P(G/?,)for any such G. Both these problems have adeformation- theoretic flavour: the first isrelated to deformations of the algebraicgroups themselves while the second Js

related to infinitesimal deformations ofautomorphlsms of our groups. Let's considerthese problems separately and start by statingour results on the first of them. THEOREM I.Let G be an affine algebraic ,-group. Then Gadmits a structure of algebraic D - group ifand only If G descends to The proof ofTheorem I will be done Jn Chapter 2 andwlU involve analytic arguments, specificallyresults of Mostow-Hochschildt �HI l] andHarem [Ha]. We already noted that TheoremI may fail in the non-arline case; we shalldescribe in what follows a complete answerto problem 1) in the commutative (non-affine) case. The formulation of the nextresults requires some familiarity with [Se],[KO]. Their proofs will be done in ChapterSo assume G is an irreducible commutative

algebraic .- group and make the followingnotations: L(G) = Lie agebra of G Xm(G) =Hom(G,G m) = group of multipllcativecharacters of G Xa(G) = Hom(G,G a) =group of additive characters of G. Recall[KO] that we dispose of the Gauss-Martinconnection: on the de Rham cohomologyspace MiR(A) of the abelian part A of G(where A -- GfB, B = linear part of G =maximum linear connected subgroup of WewiU introduce Jn Chapter 3 a "multlplicatlveanalogue" of the Gauss-Manln. connectionwhich is a ?,- linear map ,V .' Der , +MOmgr(H)R(A)m,MIDR(A)) , p ,?p whereM)R(A) m J the first hypercohomology groupof the complex * liA/ ' S /Zi ' ' ... View nowG as an extension of A by B, let S(G) a andS(G) m be the images of the natural maps

Xa(B) - HI(( A) and Xm(B) - Pjc�(A) [Se]and let SDR(G) a and SDR(G) m be theInverse images of S(G) a and S(G) m via the"edge morphisms" H)R(A) - HI( A) andHR(A) m - Plc�(A). THEOREM 2. Let Gbe an irreducible commutative algebraic ?,/,-group. Then G admits a structure of algebraicD - group if and only if Z?(SDR(G) m) �$DR(G)a V(SDR(G)a) � SDR(G) a and Inparticular if G is the universal extensionE(A) of an abelJan - variety A by a vectorgroup then G has a structure of algebraic D-group (recall that E(A) is an extension of Aby G8 a, g -- dim A, having no affinequotient)l this consequence of Theorem 2 isalso a consequence of the Grothendieck-Messing-lazur crystalline theory [11]. Notethat the algebraic D - group G(r) associated

to the A � - group derived from the equation(3b) above Is a special case of thisconstruction! Note also that Theorem 2 saysmore generally that any extension of E(A) bya torus G N has a structure of algebraic D -group ! m Theorem 2 will be deduced from ageneral duality theorem relating the Gauss-Manin connection and its multipllcativeanalogue to the "adjoint connection" on theLie algebra of commutatlve algebraic D-groups. Our duality theorem generalizescertain aspects of the theory in [MM] and[BBM] and our proof is quite "elementary"(although computational Let's pass todiscussing "problem 2)" of "describingP(C/g/,). It is "well known" that theautomorphlsm funclot Au_. C: { ?J,-schemes} + {groups} (Au.._t G)(S) =

Auts.grsch.(G x 5) of an algebraic ,- group Gis not representable in general lBS] (by theway we will prove in Chapter t that therestriction of Au% G to 'reduced [,-schemes} is representable by a locallyalgebraic t_ group Aut G this providing apositive answer to a question of Borel and5erre lBS] p. 12). Then P(G/?,/,) isobviously identified with the Lie algebraL(AU.t ' G) of the .functor Aut G, whichcontains the Lie algebra L(Aut G), but ingeneral exceeds it (due to"nonrepresentability of Au__!t over non-reduced schemes"). If G Is commutativeP(G//,) is easily identified with Xa(G)L(G)(where Der/,L0 G is identified with((G)L(G)). For noncommutetive G theanalysis of P(G/,) will be quite technical; we

will perform it in some detail for G arlineand get complete results in some specialcases (e.g. in case the radical of G isnilpotent or if the unipotent radical of G iscommutative), cf. Chapter 2. Let's discuss inwhat follows the "splitting" problem foralgebraic D-groups. If G o is any irreduciblealgebraic - group one can construct analgebraic D - group G-G � 3(, by letting Dact on (G via I ; any algebraic D - groupisomorphic to one obtained in this way willbe called split. A A o - group will be calledsplit if it is isomorphic to G o for somealgebraic

XII - roup Go this is equivalent to saying thatG(F ) is split. For instance the A o - group Tdefined by equation (l) above is split; for if

�1' * * * q'n e, is a fundamental system ofsolutions of (l) then we have an lsomorphismof A - group s (Ga,yC) I' (ci,... ,c n) Zcly iOn the contrary the A o - groups derivedfrom equations (2) and (3) are not split; for(3b)this is clear because G() does notdescend to while for (2) this followsbecause the D-submodule of �[X,x'l,x]generated by X is infinite dimensional. Nowfor any algebraic U- group the set of splitalgebraic 13 - group structures on G can beproved to be a principal homogenous spacefor L(AutG) so, at least in case G descendsto, the problem of determining whatalgebraic D- group structures on G are splitis equivalent to determining whatelementsKor L(Au__tG) actually lie in L(AutG). In particular if G descends to and Au__t

G is representable then any algebraic D -group structure on G is split; this is the caseif G is linear reductlve or unlpotent. Moregenerally we will prove in Chapter 4 thefollowing= THEOREM :. For an algebraicD-group G the following are equivalent= 1)G is split, 2) preserves the (ideal sheaf ofthe) unipotent radical U of the linear part ofG. 3) 6 preserves the (Ideal sheaf of). U/U n[G,G] in G/[G,G]. 4) 6 preserves themaximum semiahelian subfield of thefunction field 7.(G) (which by definition isthe function field of the maximumsemiabelian quotient of GI recall that"semlabellan" means "extension of anabelian variety by a torus"). Consequently aA o - group T is split iff T/[I',I'] is so, iff 6preserves the maximum semlahellan subfield

of < T >I if in addition I' is a A - closedsubgroup of GL N for some N _ J and we puti :={tdeal of all functions in ?J,{yij)d,d=det(Yij), vanising on I'} and {I':} := :=(yij)d/l then T is split [ff all group-likeelements of the Hopf algebra ( 1'} are killedby 6. V/e would like to close thispresentation of our main results with atheorem about A - subgroups of abel Janvarletiesl we need one more definition. V/esay thaLa 6 o - group T has no non-triviallinear representation if any A - regularhomomorphlsm r - GL N is trivial.THEOREM e. Let be an abel Jan -varietyof dimension g. Then there is a unique 6 o -subgroup A Its with the followingproperties= 1) it is Zariski dense in A , 2) ithas no non-trivial linear representation.

Moreover as A varies in the modului space gof principally polarized abellan - varieties ,nwith level n- structure, n _)3, the function AH. dim ^b varies lower semicontinuouslywith respect to the A - topoloLy of ,A-g,nand assumes all values between and 2. XlIIIn case g = I one may say more namely= a)either A descends toZ, say A = ^o )/,, andthen A4b= Ao(7) so dim A It: = I. b) or ^does not descend to and in this case dim =2. Note that the A - roup T defined by (3b) isnothing but the group Afil=above. On theother hand the A - group defined by (3a) isan extension of (Ca,) by A '4k above. 6.Amp!lficmtlmm Most definitions given so farin our Introduction will be "enlarged** in thebody of the text and most resuits will beproved in a generalize d form. For instance

the definitions related 'to - manifolds and A_groups will be given for "partial" rather than"ordinary" differential fields i.e. for fields ,equiped with several commuting derivations.Algebraic D - groups will be allowed to bereducible and will he defined for D = KiP]any "k-algebra of linear differentialoperators on a field extension K of k" whichis "built on a Lie K/k-algebra P" (cf. [NW]or Section 0 below this degree of generalityand abstractness might seem excessive, but itis motivated in many ways= l) our wish todeal with A. groups over partial differentialuniversal fields 2) our wish to "compute",for a given algebraic K-group C (Kalgebraically closed of characteristic zerocontaining a field k) the smallestalgebraically closed field of definition K C

of C between k and K; the existence of I Gwas proved in [B)] but we will reprove thisIn a different way in Chapter 4. 3) our wishto relate our topic to Deligne's "regular D-modules". 4) our wish to take a (shy) look at"algebraic D-groups in positivecharacteristic". The reader will appreciatethe usefulness of this more general conceptof algebraic D-group in the light of thearguments 1)-4) above. 7. Plan The bookopens with a preliminary section 0 whichfixes terminology and conventions. InChapter I we present the main conceptsrelated to algebraic D-groups. Chapters 2and 3 are devoted to affine, respectively tocommutative algebraic D-groups. Chapter 4deals with algebraic D-groups which are notnecessarily affine or commutative. Chapter

� opens with an introduction to - manifoldsand - groups and then deals with the"structure" and "moduli" of A � - groups.Internal references to facts not contained insection 0 will be given in the form (XW,z) orsimply (X,y) where X is the number of thechapter and y Is the number of the section.V/ithin the same chapter X we write (y,z)instead of (X,y,z). References to section 0will be given in the form (0,z). Each chapterwill begin with a brief account of Its contentsand of its specific conventions.

XlV 8, Prerequisites The reader is assumedto have only some basic knowledge ofalgebraic geometry [Har] and of algebraicgroups [H], [Ro], [Se]. The non-experts inthese fields might still appreciate the results

of this book via our comments and variousexamples. No knowledge of the Kolchln-Cassidy theory [Ki][C j] is assumeall theelements of tills theory which are relevantfor our approach will be quickly reviewed inthe book. Finally one should note that thepresent book is ideologically a continuationof our previous book iBIS, but it is logicallyindependent of it. 9, Aknowledgements Afew words cannot express the thanks I oweto Ellis Kolchin and the members o( hisColumbia Seminar: P. Cassidy, R. Cohn, L.Goldman, J. Johnson, S. Mortson, P.Landesman, /. Sit. Their continuousencouragment, constant support andnumerous suggestions played an importantrole Jn my fulfilling the task of writing thisbook. I would also like to aknowledge my

debt to H. Harem and O. Laudal for usefulconversations. Last but not least I would liketo thank Camelia Minculescu for herexcellent typing job. June 1989 Revised,June 1991 A. Buium CONTENTS 0.TERMINOLOGY AND CONVENTIONS............................ CHAPTER 1. FIRSTPROPERTIES ...................................... I.Basic spaces and maps....................................... 2. Prolongationsand embedclings .................................. 3.Local finlteness and splitting.................................. CHAPTER 2. AFFINED-GROUP SCHEIES ............................... 1.The analytic method......................................... 2. The algebraicmethod: direct products .......... ' ..................3. The algebraic method: semidirect

products ......................... CHAPTER 3.COMMUTATIVE ALGEBRAIC D-GROUPS...................... 1. Logarithmic Gauss-Maninconneotion ............................. 2. Dualitytheorem ............................................ 3.Descent. Regularity ................................ ?........ CHAPTER a. GENERALALGEBRAIC D-GROUPS ..........................I. Local finitness criterion...................................... 2. Representing theautomorphism functor ........................... 3.Products o! abellan varieties by arlinegroups ....................... CHAPTER 5.APPLICATIONS TO DIFFERENTIALALGEBRAIC GROUPS ......... 1. Ritt-Kolchin theory ......................................... 2.A o-groups versus algebraic D-groups............................. 3. Structure of Ao-groups

....................................... . Moduli of Ao-groups .........................................APPENDIX A. Link with movablesingularities ............................. APPENDIXB. Analogue of a diophantine conjecture ofS. Lang ................. APPENDIX C. Finalremarks and questions ................................REFERENCES.................................................... INDEX OFTERMINOLOGY .................................... ,....... INDEX OF NOTATIONS............................................. l 7 20 2a 60 668.t 87 91 99 106 116 125 139

O. TM]IOLOG� AND COIT!ONS (0.1)Unless otherwise stated rings, fields andalgebras are generally assumed associativecommutative with unit. This will not apply

however to Lie algebras, universalenveloping algebras or to Hopf algebras (thelatter are understood in the sense ofSweedler [Sw]). (0.2) Terminology ofalgebraic geometry is the standard one (cf.for instance [Har][DG]). Nevertheless wemake the convention that all schemesappearing are separated. For any morphismof schemes f X + � we generally denote thedefining sheaf morphism C� + f, by thesame letter f. If both X and Y are schemesover some field K then by a K- f-derivationwe mean a K-derivation of COy into theCOy-module f*x' By a K-variety (K a field )we will always mean a (separatedl)geometrically integral K-scheme of finitetype. By a (locally) algebraic K-group wewill mean a geometrically'reduced (locally)

algebraic K-group scheme. For any integralK-algebra A (respectively for any integralK-scheme X) we let K(A) (respectivelyK(X)) denote the quotient field of A(respectively of X). For any K-variety X andany field extension K i/K we denote by X(K1 ) the set HomK_sch(Spec K 1' X) of K l-points of X; more generally we write X(Y)instead of HomK_sch(Y,X) for any K-schemes X,Y. A morphism X-Y of K-varieties will be called surjectJve if the mapX(K) �(K) is surjective. For any K-schemeX, T X will denote the sheaf DerK X. If X =$pecA for some K-algebra A we sometimeswrite Der(A/K) instead,of OerKA. (0.3) Allarline K-group schemes C are tacitlyassumed to be such that the ring /(C) ofglobal regular functions is at most countably

generated as a K-algebra. This is a harmlessassumption in view of our applications inChapter . The unit in any K-group scheme Gwill be denoted by el If G is commutative wewill sometimes write 0 instead of e. For anylocally algebraic K-group G, G � willdenote the identity componentl Z(G) willdenote the center and we put Z�(C) .'=(Z(C)) �. (0.) For any algebraic C - varietyX (respectively algebraic C - group G) wedenote by X an (resp. C an) the associatedanalytic space (respectively the analytic Liegroup). For any analytic manifold we denoteby 1 the analytic tangent bundle. (0.5) Ourterminology of differential algebra is acombination of terminologies from [C 1] [Ki] [NW] and [Bi], In what follows we shallreview it in some detail and also introduce

some new concepts. Let K/k be a fieldextension. By a Lie K/k - algebra [NW] wemean a K-vector space P which is also a Liek-algebra, equipped with a K-linear map a: P- DerkK of Lie k-algebras such that [Pi,XP2]= a(pl)()p 2 + [pl,P2 ] for eK, pl,p 2 � P

Let's give some basic examples of Lie K/k -algebras. EXAMPLE 1. Start with aderivation d e DerkK; one can associate to ita Lie K/k - algebra P of dimension I over Kby letting P=Kp (Xp)=Xd, XeK [),lP, X2p] =(XldX 2 - X2dXl) p , ),l,X2 � K So we callthe attention on the fact that this P is far frombeing commutatJve! EXAMPLE 2. Start nowwith a family of derivations (di)ie l, d i eDerkK. Then one.can associate to it the "freeLie K/k-algebra" P built on this family: by

definition P has the property that it contains afamily of elements (Pi)lel with (pl)= d i suchthat for any K/k-algebra P' and any family(P)i�l' P e P' with l)'(p)--d i there Js aunique Lie K/k - algebra map f .' P + P' withf(pl ) = P'i' We leave to the reader the task ofconstructing this P. Note that if I consists ofone element this P Is the same with the one InExample 1. EXAMPLE 3o Assume we aregiven a family (di)ie I , d i � DerkK suchthat [di,d j] = 0 for all i,j � I. Then one canconstruct a new Lie K/k-algebra P asfollows: we let P have a K-basis (pl)ie] welet (pi) = d i and define the bracket [, ] by theformula [,pi,pPj] = X(dlp)pj - p(djX)pi , X,I� K, i,j � I This P can be called the "freelntegrabJe Lie K/k-algebra" built on ourfamily of derivations. Once again if I

consists of one element, this P coincides withthe one in Example I. EXAMPLE e. Let k �E c K be an intermediate field. Then P --DerEK together with its inclusion = DerEK -DerkK is an example of Lie K/k - algebra.Other remarkable examples of Lie K/k -algebras will appear in (I. 1). Given such a Pone' associates to it [NW] a k-algebra ofdifferential operators D -- by definition K[P]is the associative k-algebra generated by Kand P subject to the relations Xp = v(X,p) forX � K, p � P where v: K x P - P Is thevector space structure map p3.-Xp= (p)(X)lforXeK, peP PiP2 ' p2Pl --[pl,P2 ] f pl,P2 eP Ral[ [NW] tt D K[P] has a K-basisconsisting o all the momlaJs o the form = . ..Pin wre ej 0 and i I .., I n in some total orr ona basis (pi)i of P, For instance l-dimenslonal

P = Kp in Example I above then D = E Kp I(direct sum) i>0 is the ring of "linearordinary differential operators" on K"generated by K and p" multiplication in Dcorresponds to "composition of lineardifferential operators". This example is quitefamiliar: indeed, assume k = C, K = C(t) isthe field of rational functions and (p) -- d/dr;then D = C(t)[d/dt] is nothing but the"rational" Weyl algebra. Similarity one canobtain the "rational" Weyl algebra in severalvariables D--C(tl,... ,tn)[d/dtl,... ,d/dtn]starting with P as in Example 3 above. SinceD is a ring we may speak about D-modules(always assumed to be left modules). If V Isa D-module we define its set of P-constantsV D -- x px=0 for all p eP it is a vector spaceover the field KD= ), eK; px=0 for all peP},

We sometimes write also V P and K Pinstead of V D and K D and speak about P-modules instead of D-modules. Recall from[NW] that if V,W are D- modules then V �W V KW and HomK(V,W) hence inparticular �o = HomK(V,K), have naturalstructures of D - modules and the followin 8formulae hold p(x,y) = (px,py) for p � P,(x,y) c V � W p(xy) = pxey + xepy for peP,xy eVKW (pfXx) -- p(f(x)) - f(px) for p � P,x e V, f � HomK(�,W) By a D - algebra(respectively associative D - algebra, Lie D- algebra , Hopf D - algebra) we understanda D - module A which is also a K - algebra(respectively an associative K - algebra, LieK - algebra, Hopf K - algebra) in such a waythat all structure maps are D - module mapshere by "structure maps" we understand

multiplication AK - , unit KcomultiplicatJon A - A K A and co-unit A -K( K A and K are viewed with their naturalD - module structure). A D - algebra iscalled D - finitely generated if it is generatedas a K - algebra by some finitely 8erieratedD - submodule of it. Note that if P Is as inExample I (respectively 2, 3) a D - algebraIs simply a K - algebra A together with alifting of the derivation d e DerkK to aderivation l c DerkA (respectively withliftings of d i to i � DerkA in case ofExample 2 and with pairwise commutingliftings i � Derk in case of Example 3). AD - algebra which is a field will be called aD - field. Clearly K is a D - field. K will becalled D - algebraically closed if for any D -flnitely generated D - algebra A there exists

a D - algebra map A - K. Of course onewould like to "see" an example of D-algebraically closed field. Unfortunately wecan't "show" any (although one can prove theexistence of enough of them see (0.12)below). But one should not forget that with afew exceptions a similar situation occurswith algebraically closed fields. Note that If 1' A2 are D - algebras then A 1 )KA2

becomes a D - algebra. Clearly if is a D -algebra then Im(P- EndkA)C Derk . If noconfusion arrlses we denote the image of anyp � P in DerkA by the same letter p. (0.6)Following [151] we can define D -schemes= these are K - schemes X whosestructure sheaf (.0 X is given a structure ofsheaf of D - algebras (J.e. (x(U) is a D -algebra for all open sets U and restriction

maps (x(Ui)+ x(U2) for U2c U 1 are D-allebra maps). For any

D - algebra A, SpecA is a D - scheme.Clearly D - schemes form a category (whosemorpHisms will be called D- morphisms orD- maps: these are the morphisms f= X-Ysuch that {Yy + f. (. X IS a morphism ofsheaves of D - algebras). This category iseasily seen to have fibre products. Indeed ifX and Y are D - schemes over a D - schemeZ then the scheme fibre product X x ZY has anatural D - scheme structure cominig fromthe D - module structure of tensor products(0.)l if p =P then the correspondingderivation on the structure sheaf of X x zYwill be denoted by p � l + ] � p (thisconvention agrees with formula "p(x y) = px

y + x py" on products of arline pieces=XoXZoYo , Xo Yo' Zo affine open subsets ofX, Y Z). A D - scheme wUl be called' of D -finite type i! it can be covered with fin]relymany affine open sets U l whose coordinaterings are D - finlteiy generated. By a D -variety we understand a D - scheme X whichis a K - varietyl we shall usually write 1( <X > instead of K(X). Clearly K < X > is a D- field. (0.7) Let A be a D - algebra; by a D -ideal we mean an Ideal 3 which Is a D -submodule of A. For any D - ideal thequotient A/3 is a D - algebra. Let X be a D -scheme; by a closed D - subscheme we meana subscheme Y whose sheaf o! ideals is asheaf of D-submodules of (X (so Y Is inparticular a D-scheme). By a D-point on aD-scheme X we mean a (non necessarily

closed) point x �X such that the maximalideal of ('X,x is a D - ideal; we denote by XD the set of all D - points of X. l! X,Y are D- schemes we write XD(Y) instead ofHomD.sch(Y,X); If Y = SpecA for some D -algebra A we write XD(A) instead o!XD(Y). Elements of XD(A) are called D - A- points. (0.8) Now comes our main concept.By a D - group scheme we shall mean agroup object In the category of D - schemesi.e. a D - scheme C which is also a 1( -group scheme such that the multiplication G xKG -G, the antipods G G and the unit SpeckG are D - morphisms. By an algebraic D -group we will understand a D - groupscheme which is also an algebraic K- Igorup.(This is not the concept of algebraic D-group from [NW]!). One defines in an

obvious way the notions o! D - subgroupscheme and algebraic D - subgroup. (0.9) IfX and Y are two D - schemes then to give aD - map f X -Y is equivalent to giving forany D-algebra A a map fA "XD(A) 'YD (A)behaving functorially in A. If both X and Yare reduced it Is sufficient to define fA onlyfor reduced A. In particular any D - groupscheme C can be recaptured from the funclot{D - algebras} - (groups] A CD(A) (0.10) l!KI/K is a D - field extension (i.e. amorphism K -K l of D - fields) then oneeasily checks that Pl =1(IKP has a naturalstructure of Lie Kl/k-algebra and that D l -'=Ki[P i] identifies with K I K D. IUctreover,if V is a D-module then K l K V has a naturalstructure of Di-module. A similar assertionholds for algebras, associative algebras, Lie

algebras, Hopf algebras, schemesy groupschems, varieties, aligebraic groups (insteadof modules). If X is a D-variety we willoften write K 1 < X > instead of K 1 < K II2}KX >' (0.11) An important remark is'inorder. Namely the above D-schemes and D-varieties are objects quite close to the 6 -schemes and A -varieties considered in [Bi]more precisely A - schemes over a A - fieldK in the sense of [B 1] coinolde with D -schemes in the above sense where DK[P], P= free Lie K/k- algebra on fi, k--K A. Butour algebraic D-groups above are objectsquite different from (although, as we shallsee, deeply related to) the Cassldy-KoJchln6 - groups [Ci][Ki]. The relation between thelatter concepts will be discussed in Chapter5. Finally the algebraic D-groups from [NW]

are precisely our arline D-group schemes]One should also note that Kolchln's /, - fields[Ki]:K 2] correspond In our language tofields K equiped with a Lie K/k-algebra Pwhich has a finite commuting K-basis. (0.12)Kolchln proved In [K 3] that if P is a LieK/k-algebra having a finite commuting K-basis (char K = 0) and D = KiP] then thereexists a D-field extension Kl/1( such that K 1is D l-algebraicaUy closed, D l = K 1 K D'For the reader familiar with [!( 3] note that Kis D-algebraically closed iff (viewed as a A- field) it Is constrainsally closed inKolchin's sense. (0.13) Various otherstandard definitions can be given in thecategory of D-schemes. We shall need forinstance the concept of left (respectivelyright) D-action of a D-group scheme C on a

D-scheme X it is of course a D-map C x X-X (respectively X x C - X) satisfying theusual axioms of an action. Slmllarlly, onecan define D-principal homogeneous spaces(these are right D-actions X x G - X such thatthe induced D-map X x C - X x X is anlsomorphlsm). One can speak moreoverabout D-actions of a D-group. scheme C onanother D-group scheme H and about D-cocycles of C in H. It is useful to have somenotations also. We let PHSD(C/K) be the setof D-isomorphism classes of D-principalhomogeneous spaces for a D-group schemeG; it is a pointed set with distinguishedelement called I and represented by G itself.Any D-principal homogeneous space Vrepresenting I will be called trivial; V istrivia] Jf and only if VD(K) . (0.1) Let's

"recall" from [B 1 ] Chapter I some usefulelementary facts about D-schemes. Let X bea D-scheme, X noetherian. 1) If X Isreduced, its irreducible components are D-subschemes. 2) If Y is an Integral subscheme'of X its generic point is in X D If and only IfY Is a D-subscheme. 3) I! charK -- 0 thenXre d Is a D-subscheme of X. In particularby 1), 2) above X D . Assume moreover thatK 'Is D-algebraically closed and X IS of D-finite type. Then XD(K) t, t) If char K = 0and X is a D-variety then for any maximalelement x in XD the extension K(x)D/K D isalgebraic (where K(x) -- residue field of Xat x). 5) I! f.. X - Y Is a faithfully flatmorphism of integral 1(-schemes and K(Y)Is a D-subfield of K X ) then Y has a(unlquel) structure of D-scheme such that f is

a D-map. 6) If f X' - X is an tale morphism ofK-varieties then X' has a unique structure ofD-variety such that f is a D-map.

7) If f: X + Y is a D-map of D-schemes and yc YD then the fibre f-I(y) has a naturalstructure of D-scheme. g) Assume V is anormal K-variety such that K(V) has astructure of D-field, D = K[P]. Assumemoreover that dimKP < . Then the locus V O= [x � Vj D/V, x a /V,x } is ZarJski open (soit is a D-scheme) and V \ V o is the supportof some divisor on V. Recall also that wehave proved in lB l] p. 28 the following:(O.l) THEOREM. Assume K is algebraicallyclosed, and X is a projectire D-variety. ThenK D is a field of definition for X (i.e. X isK-isomorphic to XoKDK for some

projective KD-variety Xo). (0.16) We closeby introducing a concept which will appearseveral time in our work. Let P be Lie K/kalgebra, S m a group and 5 a a Lie K-algebraon which 5 m acts by Lie K-algebra *automorphisms. By a logarithmic P-connection on (Sm,S a) we mean a pair (?,?)of K-linear maps ? = P Derk(Sa,S a) P ?p g?-' P -' zl(Sm,Sa ) p such that Vp(),x)= ),pX +a(p)(X)x for all p � P, ), � K, x cS . NowZi(Sm,Sa ) has a natural a structure of Lie K-algebra induced from that of S a. We say that(?,?) is lntegrable if for any pi,p 2 � P thefollowing formulae hold; (0.16.1) ?[PI'P2 ] =[?Pl' ?P2 ] (0.16.2) g[Pl'P2 ] = Pl P2 P2 PI +['VpI' tVP2 ] Of course condition l) isequivalent to saying that there exists a(unique) D-module structure on S a (D =

K[P]) such that px = VpX for all p�P and xc5 a. A "trivial" example of integrahielogarithmic DerkK-connection is on (K*,K)where K* acts trivially on K and K isviewed as an aheiian K-algebra= it is givenby the formulae ?pX = px, lpy = y-lpy for allp cDerkK , x �K, y zK*. Various significantexamples o! logarithmic connections willappear in (I.1) and (IlL l). Note that if in ourdefinitions 5 m is trivial and $a is an abelfanLie algebra then a logarithmic P-connectionis simply a P-connection on S i.e. a K-linearmap ?: P ' Homk($,$) satisiying the usualLeibnitz rule. CHAPTER !. FIRSTPROPERTIES Everywhere in this chapterK/k is a field extension with K algebraicallyclosed (of arbitrary characteristic) P is a LieK/k-algebra and D = K[P] (cf. Section 0)o in

section ! of this chapter we associate to anyK-group scheme G a basic Lie K/k-algebraP(G) and investigate its "first properties".Sections 2 and 3 deal with basic concepts ofproJongations local finitehess and splitting.1. Basic spaces and maps (l.l) Let G be a K-group scheme. We shall denote by p::GxC -G, S.'G - G, : $pec K - C the multiplication,antipode and unitl we will also denote by P:CG" P*GxG 5; (G' S. G ,e the inducedcomultipllcatlon, antipode and co-unit (cf.(0.2)). Define DerkG -- !space of all k-derivations of (fig into itselfJ L(G) = v cDerkG v is K-linear and pv P(G) = p cDerkG; pK'K, IJP = (P I + 1 p)p, pS = Sp, p= p} Clearly DerkG is a K-linear space anda Lie k-algebra. Moreover L(G) is a Lie K-algebra and P(G) is a Lie K/k-algebra. Both

L(G) and P(G) are Lie k-subalgebras ofDerkG. Define D(G) = K[P(G)]. loreover forany p � P(G) let Kip] be the k-algebra ofdifferential operators associated to the l-dimensional Lie K/k-algebra Kp (clearlyK[p] c KIP]). Note that we made a slightabuse in our deflnitlons. For Instance theformula Pp = (p� 1 + lp)p in the definitionof P(G) means that the following diagram 'ofabelian sheaves on G is commutative.' (C P 'P* CxG Where p 1 + 1 p was defined in(0.6). On the other hand the formula pv = (vl)p in the

definition o! L(G) means that the diagram 7G P . W. GffGx G [ { W.(v, l) is commutativewhere v I is the K-derivation of GxG acin asv on the "firs coponent" and vanishln 8 on

:he "second component". Such abuses reatlysimplify our computations and will be doneYrom now on sys{emaYlcally. ote that ifchar K > 0 then DerkG = DerKG because Kls rfect. e can define more generally {or anyintermediate field K o between k and K thespaces er K G and P(G/K o) as consisting ofall Ko-linear embers of DerkG and P(G)}clearly P(G/K o) ls Lie K/Ko-algebra andP(G/K) = Ker( a = P(G) + DerkK). Aprlorlit ls t clear that L(G) can lden{lfled with theLie algebra Lle G o{ G as dellned in [DG](Lie G = tangent space of G at e). But ofcourse, i{ G is an algebraic group, L(G)coincides with the Lie algebra o{ rightinvariant derivations in erKG and hence withLie G. e wiU check below that L(G)-Lie G atleast in one ore case namely when G

integralmar{inc (see (1.8)). As {or oneinterpretation of P(G) note that there ls anatural bljectlon be{ween the set of D-grou pscheme structures on G d the se H�Li eK/k.alg(P, p(G)) In particul&r G has astructure o{ D(C)-group scheme and thisstructure is "universal" in an obvious sense.This remark is basic for our work. e wlUusuly investigate first this "universal" D(G)-strture and then "specialize" to =bltrary -structures. EMPLE. Assume P = Kp is 1-dimenalonal} then the set of D-group schemestructures on G is in bljection with the setP(G/a(p)) o{ all derivations } in P(G)inducing the derivation a (p) on K. So P(G/a(p)) is a prlncll homogeno sce {or P(G/K).Before computing P(G) for a few sci{lc G'swe nd to prove a series oY general prortles.

(1.2) LEMMA. [P(G), L(G)]c L(G) where tbracket Is taken }n erkG. P{. Let p � P(G),v L(G). We have: Upv=(pl + lp)pv=(pl + lp)(vl)p=(pvl +v pvp=(vl)pp=(vl)(pl + lmp)u=(vpl +v hence t[p,v] = ([p.v] l)l whichproves the lemma. We denote in whatfollows by G m, C a the multlplicative andadditive l-dimensional algebraic K-groups.Recall that G m -- spec K[X,X- 13 pX =x�x � K[X,X 'i ] � K[X,X' 13 SX= X '1and that G a =Spec e= 0 For any K-groupscheme we G put Xm(G) =HOmK_grsch(G,G m) Xa(G) = Horn K.grsch (G ,G a) Clearly Xm(G ) is asubgroup of (*(G) and Xa(G) is a K-linearsubspace of (G). (1.3) LEMMA. For any p� P(G), X � Xm(G), eXa(G ) and v �L(G) we have X-Ipx � Xa(G), peXa(G) and

v � K. Proof. The following holds: ppX =(p ) 1 + 1 p)p( = (p 1 + 1 p)(X�X) =pX�X + X� pX. Multiplying the aboveequality by X '1 )X '1 we get p(x-lpx) =X'Ipx 1 + I ex-lpx I.e. X-Ipx � Xa(G).Similarily one checks that peXa(G). Finallywe have pv = (v l)p = (v 1)( 1 + 1 ) = v aO 1Applying {) 1 to the above equality we ge v= (cO l)Vv = (e lXvao 1) = e(v) � K andthe lemma is proved.

10 (l.e) We have an integrable logarithmicP(G)-connectlon (9,[9) on ((*(G),(G))defined by the formulae Vpf -- Pf and [Vpg =g-lpg for all p � P(C), f iC(C), g i Ca(C).Here L(C) is viewed as an abelJan Lie K-algebra on which 7(G) acts trivially. By(1.3) above this logarithmic connection

induces an integrahie logarithmic P(C)-connection (?[V) on (Xm(G) , Xa(C)) whereXa(C) is viewed as an abelian Lie K-algebraon which Xm(C) acts trivially. In whatfollows we prepare ourselves to deal withL(C) for G affirm and with P(C/K) for Ccommutative and either affirm or algebraicover K. (1.5.) Assumed G is an affirm K-group scheme. Then we define a lineartopology on DerkG as follows. For anysystem of elements fl'" ' fn � (C) define Vfj,... 'fn = {p c DerkG Pfj =... = Pfn = 0} Thenwe take as a fundamental system of openneighbourhoods of the origin in DerkC theabove spaces Vf ... f as fl ... fn and n vary.Clearly DerkC with this topology isseparated and complete due to burconvention made in (0.3) this topolo[y has a

countable basis). (1.6) Recall from [DG] that(with our convention in (0.3)) any arline K-group scheme G is the projectire Jim Jr C =J._ G i of an inverse system u i ... 4-Ci+l--Gl4-... +G 1 such that each u i is a faithfully flatmorphism of arline algebraic K-groups. Inparticular LleC = lira LieC l = li_m._m L(Gi) (!.D Let � .. 4-Vi+ l 4-V i 4- .,. 4-V 1 be aprojective system of finite dJmenslonal K-vector spaces with surjectlve connectingmaps and let � = Jl.m_m V i be viewedwith its projectire limit topology. Then onecan find a sequence vl,v2,... I� (we agree toassume that the sequence is finite if V hasfinite dimension) such that for each index ithere exists an integer n i having the propertythat upon letting j be the image of vj in Vp l''" n i is a K-basis in V. and . = 0 for all j ) n

i. Call such a sequence (vi) I a pseudo-basisfor V. Clearly any element in � can bewritten uniquely as a convergent series Z),ivi with )'i � K. If dimkV < m a Pseudo-basisis simply a basis. (l.g) PROPOSITION. LetG be an affirm integral K-group scheme andlet G = lilieS_ram G i as in (1.6). Thens l)For any intermediate field K o between k andK the spaces DerKoG , P(G/Ko) and L(C)are closed in DerkG hence they arecomplete. 2) Let VlW2, ... be a Pseudo-basisin li,_m L(G i) (we may view each v i as anelement of 11 DerkC). Then any element inDerKG can be uniquely written as r. aiv iwith a i (G) (which is a convergent series inDerkC). 3) An element r. alv I as in 2)belongs to L(C) if and only if a I K for all lin particular L(C) - lJ?_m L(Ci) , hence L(C)

identifies wlth Lie G. a) Assume C iscornmurat[re. Then an element ]:alv i as in2) belongs to P(G/K) if and only if a i IXa(G) for all 1. Proof. 1) is easy and weomit proof. 2) Let fl "C 4- C l be the i-thcanonical projection. We have DerKC =l/+mDerK((Ci , fi2c)--lira Hrn(C,K,fi.(.0G)- - = lim H�(Gf,(fl.07 G) x L(Gi)) = lim((,0(G) x L(Gi)) which immediately impliesassertion 2). 3) Clearly 11mL(Gi)c L(C) sor.).ivicL(C) If ).jcK. Conversely assumev=]:ajvi, pv = (v ) l)p. Then w haver.P(ai)PV I = r.(a I � l)(v i e l)p = Z(a i l)pvi V/e shall be done if we check that anequality of the type r. mlPV i -- 0 with m i (Cx C) implies rn i = 0 for all i. For each i wemay find elements fl' ' '' fn. in the maximalideal m i of TGi,e 1 providing a K-basis In

mi/mi2 and such that Vpfq = 6pq (Krormckerdelta) for all I _( p,q _( nj. Then det(v f )._ -60*- V/e get P q l_Pq_n i i,e' ilm pv f 0 for I( q ( n i p--I p Pq - - Now der (pVpfq) --p(det (Vpfq)) OlXCl,ex e which implies thatmp= 0 for all I (_ p _( n I. ) If G iscommutative so are all Gi's so for all I wehave pv I = (v i l)p = (1 vi) p and SviS -- -vi Now for p = t. aiv I we have pp = p(r. aivi) = r.P(al)PV i (pO 1 + I � p)p -- Z(a i l)(Vl l)p + r.(l ai)(l �vi) p = r.(al � 1 + 1�ai)Pv i Sp = r.(Sal)Sv i

12 pS = :aiviS = -r..aiSv 1 p = rdal)V l p=OSo If a i c Xa(G) clearly p c P(G/K).Conversely, if p c P(G/K) then by the remarkmade at point 3) we have a I c Xa(G) and weare done. (1.9) PROPOSITION. Let G be an

irreducible commutative algebraic K-group.Then, under the identification DerKG C(G)L(G) we have P(G/K) = Xa(G)� L(G). (l.)).Proof. Same computation made in (1.8) (withthe "abusive" notatJonal conventions fromNow we are In a postion to compute P(G/K)for a few special commutatlve G's.EXAMPLE 0. Let G = G n = Spoc K[X 1' '' '(n ] be a torus (OX 1 = X i )Xi). ThenP(G/K) = 0. m EXAMPLE 1. Let C =-C n =SpecK[l,... 'in ] be an algebraic vector groupa (li = [1 � I + I [i ). Then a K-basis ofP(C/K) consists of the derivations [ a lj ' 1_< i,j<_n of K l .... ;n]. EXAMPLE 2. Let G -G m xGa= SpecK[x,X-I] (1 --XX, P[ -'[I +1[). Then a K-basis of ,(G/K) consists of thederivations [- and [X of K[}(,('I].EXAMPLE :. Let G=AxG =AxSpecK] (IJ[--

[l.+ l[)whereA is the elliptic a curve withK(A)= K(x,y), y2= x(x - l)(x -A), A c K.Then a K-basis of P(G/K) consists of thederivations and (l.10) Let G be a K-groupscheme. Then G(K) acts by adjoJntrepresentation on L(G): for any gc G(K) andvE: L(G) we put (AdgXv) = LlvLg where Lg(G* Lg(G is induced by left translation Lg:G* G defined by g. By (1.2) we dispose of aK-linear map ?: P(G)+ Derk(L(G),L(G)), p--p defined by pV = [p,v] (bracket in DerkG)for p c P(G), v c L(G). Then define a K-linear map J,?; P(G)+ ZI(G(K),L(G)), p J, Pby the formula J[Vpg -- LIpLg - p for pcP(G), gc G(K) The fact that J,?pg c L(G)follows from the following cornputat[om 13p(LglpLg - p) = pLgIpLg - pp = (L;I ) l)ppLg- pp = = (L 1 lp) 1 + 1 ) p)pLg - (p 1 + 1 p)p

= ((L 1 ) l)(p ) 1 + 1 p)(Lg< 1) - (p I + 1 )p))p = = ((LIpLg - p)O 1) It is easy to checkthat the maps V and I,V define an integrablelogarithmic P(G) - connection on (G(K),L(C))I in particular L(C) hcs a structure ofD(G)-module (in fact of Lie D(G)-algebra)such that for p c P(G), v c L(C) we have pv= [p,v] (pv is v multiplied by pin the D(G)-moduJe law there is some danger ofconfusion here with the composition p so oneshould be careful to distinguish between thetwo' possible meanings of the symbol pv;this Js why we sometimes write pV insteadof pv to denote The above structure of D(G)-module on L(G) will be sometimes calledthe "adjoint connection" on L(G). (I.11)PROPOSITION. Let G be an integral K-group scheme which is either afflne or

algebraic. Then the map � J, V .. P(C) +DerkK � Z I(G(K),L(G)) is injectlye.Moreover for any pc P(G) and g cG(K) wehave J,?pg= 0 if and only if g c CK[p](K)(recall that this means by definition that g:Spec K + G is a Kip]-map). Prooo Assumepc P(C/K) = Ker a is such that [Vpg = 0 forall g c C(K). Then of course (1.8) holds forLleft(C) := (v c DerKC; pv = (1 ev)p}; inparticular picking a pseudo-basis Wl,W2, ...in Ljeft(G) if G is arline (or simply a basis ifG is algebraic) we may write p = r. atw 1with a i c (C). Clearly LlwjLg = w l for all Iso the relation 0 = J[Vpg = LlpLg - p impliesr..ajwj = r.(Llai)w i hence by (1.8) a I = -ga ifor all i and g which implies a i c K for all 1,hence that p c Lieit(G) so pp = (1 p)p.Applying 1 ) to the latter equality we get p =

(l�p)p. But p = p = 0 so p = 0 andinjectlvity follows. Next, for a fixed g cC(K) we have J,?pg = 0 If and only if Lg: G+ G is a Kip]-map which happens if and onlyif g is a Kip]-map from Spec K to G(because Lg = p(l � g) and g = Lg�). Ourproposition is proved. (1.12) REMARK. LetG be an algebraic K-group and K o be afield of definition (say C; C O K K, G o aKo-group). For each p c Der K K let p*denote the trivial lifting of a(p) from Kto�G GoK (i.e. p* = I alp)). Then te mapG(K)* L(G) defined by g [Vpg coincideswith Kolchin's 1ogarlthmlc p-derivative [K1 ] p. 39t; an argument for this can be foundin [Bi]p. 25.

14 In what follows we shall use the

universal enveloping algebra of the Liealgebra of algebraic D-groups G in order toestablish a D-analogue of the classical Liecorrespondence between algebraicsubgroups of G and their Lie algebras. (1.13)Let L be a Lie D-algebra. Then the universalenveloping K-algebra U(L) has a naturalstructure of associative D-algebra inducedby the D-module structure of the tensoralgebra (0.5). 50 by (0.5) the dual U(L) �(which is a commutative algebra withconvolution product) becomes a D-algebra.Now for any K-group scheme G there Is anatural K-algebra map JG: 'G,e ' U(L(G))�defined as follows: for f � G,e' JG (f) =7where for any u � U(L(G)), ' (u)= cur (hereU(L(G)) acts on G,e by K-linearendomorphisms in the natural way). I! in

addition G has a structure of D-groupscheme, then by (1.10) L(G) has an inducedstructure of Lie D-algebra and one easilychecks that the map iG above is a map of D-algebras. (1.14) LEMMA. If char K = 0 andG is integral and either affine or algebraicthen the map jg :.CG,e- U(L(G)) � ininjectlye. In particular, if G is algebraicdlmKP(G/K) < e. REMARK. If char K > 0,P(G/K) is in general infinite dimensionaleven if G is algebraic, e.g. for G = G a(Indeed apply (1.9) and the fact thatdimKXa(Ga) -- ). Proof. First it is easy tocheck the Lumina for G algebraic. Then i! Gis arline write G = llmG. as in (l.t;)l we have(G = lira and a commutatlve diagram . I ,e .4.Li,e (G i,e JGi . U(L(Gi))� G,e JG 'U(L(G))� where JG. and u i are iniective

(for u i use (1.8), 3)). This impliesinjectivity of JG' The claim that dimKP(GK)< m for algebraic G follows form the factthat P(G/K) embeds into IerKL(G). (1.15)Recall from [DG] that if G is an (integral, tosimplify) afflne ,K-group scheme and H is anormal subgroup scheme of G, then thequotient G/H exists as an afine K-groupscheme. It can be constructed as follows: letG = 11L.mG 1 as in (l.t9, let 3 H be the idealof H in(G), 3 i = ((G i) n H (it is a HopfIdeal in (Gi)) and put H i = Spec/(Gi)/1111then G/H = 11_mGi/H i. Assume now H isintegral. By (l.g), 3) one can easily checkthat L(H) Identifies with { v v3 H c 11H}and by (1.8) we have an exact sequence 0 'L(H) ' L(G) ' L(G/H) ' 0 15 (1.1)PROPOSITION. Assume char K = 0 and H

is an integral K-subgroup scheme of anintegral D-group scheme G which is eitheraffine or algebraic. The following areeuivalent: 1) H is a D-subgroup scheme ofG. 2) L(H) is a D-submodule of L(G). Inparticular if G is algebraic then [G,G] andZ�(G) are algebraic D-subgroups of G.ProoL We have a commutative diagram G,eJc U(L(G))O ,e ' U(L())� J with r,ssurjective and JG' JH injectlye (cf. (l.lt) and(1.15)). If 2) holds then s is a D-algebra maphence its kernel is aD-ideal in U(L(G)) �.Since Kerr = jl(Ker s) we get that Kerr is aD-ideal which implies that H is a D-subscheme, hence a D-subgroup scheme ofG. Conversely, if the latter holds the idealsheaf 11FI of H in LO G is a D-ideal sheaf in(G hence if ve L(H), identifying v with an

element (still denoted by v) in L(G) such thatV3H� 3H, we have (for any pe P) pV)11 H= (p e v)(3H)+ (v � p)(gH)c 1114 , so ? pVL(H). The last assertion of the lemmafollows from the fact that L([G,G])=[L(G),L(G)] and L(Z�(G)) = Z(L(G)) (=center of L(G)) are clearly D-submodules ofL(G). (1.17) LEMMA. Assume char K -- 0,G is an integral D-group scheme which Iseither algebraic or arline and H is a normal(nonnecessary irreducible) K-subgroupscheme. The following are equivalent 1) H isa D-subgroup scheme 2) There is a structureof D-group scheme on G/H such that theprojection G -G/H is a D-map. (Clearly theD-group scheme structure on G/H in 2) mustbe unique I). Proof. 2)=1) is clear by (l.lt0,7). 1)=2) By (0. It0, 5) it is sufficient to

check that K(G/H) is a D-subfield o! K(G).I! G Is arline let G = Ii.mGi,G/H =11._mGi/FI 1 as in (1.15). If G Is algebraic,put G i = G, H i = H for all t. Letting i: Gi xH? G l be deduced from the multiplication i l-' G I x G i G i we have K(G/H) = l+imK(Gi/Hi) = lim( fe K(Gi); if = f � O wherewe still denoted (as usual) by i the Inducedmap CGi- i, (-0GlXH i' Now if f K(G i) is

16 such that lf -- f l and if p c P then f = fa 1(:G x H - G having the obvious meaning) sopf: (p! + 1 ) p)f = (pl + I p)(f )!) -- pf)l Butpf c K(Gj) for some j hence pf � K(G/H)and we are done. (1.18) pROPOSITION.Assume char K = 0, G is an irreduciblealgebralc D-group and H any algebraic D-subgroup of G. Then the homogenous space

G/H has a natural structure of D-variety suchthat the projection map G + G/H is a D-map.Preof. Same argument as in (1.17). (1.19)PROPOSITION. Let C = C l x C 2 be aproduct of K-group schemes. Then there is anatural Lie K/k-algebra map s -' P(Gl)XDerkKP(G2)' P(G) and a natural K-linearmap "over" DerkK: li: P(C) P(Gl)XDerkKP(C2 ) such that Rs = identity. Inparticular for any p � P(G) we have p-slip� P(C/K). Preof. Let E i: G + G l be thecanonical projections (and denote as usualalso by II 1 the induced maps (C. ' Ei.CG)'Then for p � P(C) define li(p) = ((l 2)PE l,(l l)pE 2) and s(PI'P2) = Pl � l + � P2where of course ti Is the co-unit of G l. Thenone checks easily all claims of theproposition. The next proposition shows that

the conditions defining the elements of P(C),cf. (l.l), can be weakened in case C Isalgebraic over K. This proposition will playan important role later. (1.20)PROPOSITION. Let char K = 0 and G be anirreducible algebraic K-group. Assume p cDerkK(C) is such that I) pKcK 2) Vp =(pel +Ip)v;K(G)*K(GxG) 3) Sp -- pS: K(G)-K(G). Then p P(G). Proof. Start with theremark that if K ilk Is a Kip]-field extension(0.10) then p extends to some derivation Pl eDerkKl(G) satisfying the correspondingproperties 1)-3) from the statement ot theproposition. Since the projection G oKi<i +G is faithfully flat it Is sufficient by (0.14),5) to check that Pl z P(G�KKi)' So we mayassume that K equipped with the derivationa(p)cDerkK is Kip]-algebraically closed. By

(0. it0 again, the locus (p)m = (x z G; p(60G,x) t O/G, x) is a divisor on C. Put U = G\(P)o; it is a I<[p]-variety.. 17 The set ofK[p]-K.-points UK[p](K) of U is dense inU(K). Take g � UK[p](K) and consider thecorresponding left translation :G gxl p Lg tGxG sG Its restriction (g x l)'l(p'l(U)n (U xU)) - g x I i.i(u) � n (UxU) " is a K[p]-morphism (where note that the sourcescheme is an open subset of U which isnonempty because U n 8'!U). Consequentlythe induced morphism Lg=K(C)-K(G)commutes with p: K(C) - K(C) this showingthat (p). Js Lg-invariant for all g � UK[p](K). By density this happens for all g c G(K)so (p)m must be empty, hence pC c (C'FinalJy, take any g � GK[p](K); since Sp --pS, the morphism -1 � : SPOcK gxg I GXG

G is a K[p]-morphism and we are done. Inwhat follows, by an lsogeny we mean asurjective morphism G'- G of irreduciblealgebraic groups having finite kernel. (1.21)PROPOSITION. Let f .. G'- G be a separableisogeny. There exists a canonical injectlye"lifting" map of Lie K/k-algebras f* :p(c)-P(G'). Moreover, if C and C' arecommutative and char K = 0 then f* is anisomorphism. Proof. Let p, S', be thestructure maps of G'. Since f is tale, any p �DerkG lifts uniquely to some p' I DerkG.Now If p I P(G), IJ'P'-(P' l + l�p')l' is a IJ-K-derivation K(G')- K(G' x G ) vanishing onK(G)I by separability this derivation mustvanish. Similarfly we get p'S' = S�' andp'�' = 'p'. Of course we put f*(p) = p' whichis well defined by the preceedIng remarks

and clearly is an injective Lie K/k-algebramap from P(C) to P(C'). Now assume char K= 0 and CG commutative. Then there is anisogeny g = C - G' such that gf .. C,- C+ C isthe multlpJicetion by some integer N _)lwhich we call XN' We get injective maps .g* f* X N: p(c,) , p(c) * p(c,) � is theidentity (this will close our proof). Thisfollows by noting that N is a We claim that Nmap of D(G')-varietlas hence for any p' cP(G') we have a commutatJve diagram (C') ,K(C') K(G')

18 showing that the canonical lifting of p' vial N is p itself. Our proposition is proved.(1.22) PROPOSmON. Let G be anIrreducible commutative algebraic D-group.Then the following hold: 1) Any torsion

point In G(K) of order prime to charK is aD-point (in other words if N is an integerprime to char K, N-TOrs(G(K))c GD(K)). 2)Any torus and any abellan variety containedin G is an algebraic D-subgroup of G. 3) Anynon,linear irreducible algebraic subgroup ofG contains a non-trivial irreduciblealgebraic D-subgroup of G. Proof. l)Let glN-Tors(G(K)). Then multiplication by N(call It IN:G-G) Is a separable isogeny and),1(0) is a D-subscheme of G, union ofreduced points. By (0.14), each of thesepoints is a D-point, in particular so Is g. 2)Let H be either a torus or an abellan varietycontained in G. Then the torsion points of Hwhose order is prime to char K is dense inHI so on any arline open set of G the definingideal 3 of H in G is an Intersection of D-

ideals so 3 Itself is a D-ideal and we aredone. 3) Any non-linear Irreduciblealgebraic subgroup H of G has Infinitelymany torsion points of order a power o! aprime , where I does not divide charK. Letbe the Zarlski closure [n H of the group ofthee torsion points. Then M is an algebraicD-subgroup of H and so will be itsconnected component. (1.23) Let's makesome general remarks on the link betweenP(G) and the automorphlsm functor of G (adeeper discussion will be done In Chapter4). For any K-group scheme G we have theautomorphlsm functor: Aut G: {K-schemes}-(groups) defined by (Aut GXS)=Aut$.gr.sch(G x S) (in particular (Aut G)(K)= AutG). This functor need not berepresentable even If char K = 0 and if G is

an algebraic group (cf. [BS], e.g. if G = G ax Gin). It is representable however if G isalgebraic reductive (and char K _) 0) cf.[GD]. Recall from [GD] that for anycontravariant functor A .' {K-scheme} +{groups} one can define the "Lie algebra"L(_) ,= Ker (A(n): _(SPOc Kid) * _(SpocK)) where K[ �] = K . K e 2 = 0 and themap II .. K[ �] - K is defined by � - 01L(_A) is a group but not a genuine Lie K-algebra 'In general. In our specific casehowevery L(Au_._!tC) is easily seen toidentify with P(C/K)= Ker( a :P(G)-DarkK),in particular It is a genuine Lie K-algebra;the identification is given of COurse by themap p +ld + ep (p � P(C/K)). There areremarkable cases when the restriction of ^utG to (reduced K-schemes} is representable

by a locally algebraic K groupy call it in thiscase ^utG (there is no danger of confusionwith the other meanlng of Aut C which is(^u____!t G)(K)!). This is the case If char K= 0 19 and G is linear algebraic [BS] (oreven char K = 0 and G is algebraic, non-necessarily linear, el. (IV.2) below). If theabove representability properties holds thenthere is an induced Lie K-algebra map I = IG: L(Aut G) L(Aut G) which is not anlsomorphism in general and which will bestudied in (IV.2). (1.2) For any K-groupscheme the group (^utGXK)=AutG acts onDerkG by the formula {o,p)-o'lp for anyoeAutGy pe DerkG (as usual we still denoteby the map (-G ' # (C induced by ). Underthis action P(C) and P(C/K ) are globallyinvariant. Let us call ? .' P(C)-'

Derk(P(G/K),P(G/K)) , p - V P the K-linearmap defined by pV = [p,v] (bracket taken inP(G)) for pe P(C), v P(C/K) and define theK-linear map ,!.V: P(G)- zl(AutG, P(G/K)),p P by the formida II.V o=o'lpo. p for pzP(G), oz AutG. One easily checks that(V,J.V) above P define an integrahielogarithmic P(G)-connection on (Aut G,P(G/K)). (1.25) !n way of our study of P(G)for commutative G we shall be lead toconsider one more example of logarithmicDerkK-connection (9lt9) whose "additivepart" is the Caussr -Manin connection [KO]on the first de Rham cohereology space ofabeltan varieties while the "multlpllcatlvepart" lt will be defined and used in Chapter3. (1.26) We close this section by noting thatif char K = 0 and If K o is a field of

definition for a K-group scheme C (i.e. if wehave a K-group scheme lsomorphism o: G aG o KoK , G o a Ko-grou p scheme) thenthere is a Lie K/k-algebra map Der(K/Ko)*P(G), p p* where p* =o(1 � p)o '1 Is thetrivial lifting of p to G (it depends on o !).We shall often identify Der(K/K ) with a LieK/k-subalgebra of P(C) (after having fixed oI). It is easy to check that P(G) K iscontained in any algebraically closed field ofdefinitlon for G containing k. If K p(G) is afield of definition for C (which is not alwaysthe casey cf. Chapter 3 below l) then wehave a split exact sequence of Lie K/k-algebras 0-' P(G/K) P(G) a Der(K/Kp(G)).,0 In particular P(G) is the semidirect productof Ker a and Im a . EXAMPLES Let'sconsider once again the Examples 1, 2, 3

after (1.9), where In Example 3 we assume

22 which we are briefly investigating inwhat follows. (2.2.) LEMMA. For any fieldextension K ilk the map (H�)I(K 1) ' H(K 1) is surjective. Proof. Any field extension K1 of K admits a D-field structure and we areimmediately done by adjunction. (2.3)pROPOSITION. Assume G is an irreduciblealgebraic D-group. Then: l) There exists anaturally associated D-cocycie G!. hA :(L(G)m) = (L(G)m) m such that themorphism G- G !� deduced from adjunctioninduces an isomorphlsm of D-group schemesbetween G and the reduced kernel (ker fi)red of , A. (Here L(G)= L(G !) is of courseidentified with its associated algebraicvector group an�. the action of G I� on

(L(G)m) � is induced by functorality fromthe adjolnt action of G on each factor ofL(G)m). 2) There Is a bijectionPHSD(G/I()= L(G)int/G(K) where L(G) lnt -- ? piVj - V pjv i . �vi,v j] = 0 for all i,j (cf.notations in (1.10)) and G(K) acts on L() ntby the "Loewy-typo" [C 1 ] formula: (g,(vl)i)+ (L;lviLg +V pig)i (g= G(K)) (where - p.g =L;IpiL 8 - Pi' cf. (1.10)). In particular, if G iscommutative PHSD(G/K) has a naturalstrLture of abelfan group. 3) There is anexact sequence of pointed sets (which Incase G is commutatlve is an exact sequenceof abolish groups): 1 - GD(K)+ G(N) - L(G)int ' PHSD(G/K) ' 1 In particular, if K is D-algebraically closed, , A above is surjective.REMARK. The last surjectivlty assertion isa generalisation of Kolchln's surjectivhy

theorem for the logarithmic derivative (cf.�K i] PP. 20-a21, or [NW it reduces to it incase G is split (cf. (3.11) below for thedefinition of "split") but our proof here isdifferent from those in [KI] , ProoL 1) Todefine . a it is suffieient by (0.9) to define forany reduced D-algebra A, coeyeles , A A: (GI )D(A ) = G!(A I). ((L(G)m) o )D(A ) =L(G)m � A behaving functorially in A. Weshall define. A A on each component to begiven by the map (*) g'" L;IpiLg- Pi (g� G!(A!)) where Pi denotes here the derivation of{GA corresponding to Pi P while Lg:CGSA* (GSA is deduced from the lefttranslation by g. Then L;lpiLg - pie L(G)�Abecause DerA(CG � A) 1 = H�mG A(GA/A' G A ) (DKG) � K A and because onecan transse mutatis mutandis he mpuation

made in (1.10) (note that for = K our map (a)Is ecisely t map gYP from (1.10)). To ove Ga (Kerfi)re d it is sufficient to chk that wehave for any reduced D-algebra A an exasequence of pointed sets 1 + HomD.h(SpecA,G)+ HOmK_sch(SP AG ) g 6A _,. L(G) mA This can done by adapting mutatismutandls the last part of the proo of (1.1 I)(which deals with the case A =KI). This closthe oo o 1). To prove 2) we may identiy (in an-canonlcal way I) the underlying varietiesof all D-priipal homogenous spaces or Gwith G itsel (since K is algebraically closed!) so to give a D-principal mogenous spaceor G i$ ulvalent to giving commutingderivations dl,... ,dm � DerkG tisying theormula wd i=(d il +lPi)p (where d i 1 + 1 $Pi is of course t derivation on G x G induced

by dj on the lrst factor and by Pi on tsecond). The ove ndition is ulvalent to w(d i- pi) = ((d I - pl) � l)p hence with d I - p[ eL(G) (once again we still denote by Pi tderivatio n in OerkG corrndlng to pi ). Nowtwo m-uples (all,... ,din), (d,... ,fire) giveisomorphic D-principal homogenous spas iand only i thee is isomorphism o K-varietl :G+ G making the ollowing diagramcommutative GxG GxG - +G Such that d='ldl for all 1. The above diagram shows thatmust be the left translation L g (g = (e)�G(K)) and so the latter relation becomesL;l(di ' Pi)Lg * 'V Pig -- el ' Pi which proves2). Assertion 3) follows directly from 1) and2).

24 . Local finitness and s91ittin Here we

come to one of our main concepts (and tools)namely to "local flnitness" and its relationwith "splitting" and "fields of definition";this is part of our ideology in [Bi]. We startwith some definitions (where we assume Kalgebraically closed as usual and of arbitrarycharacteristic while P is any Lie K/k-algebraand D = K[P]). (3.1) A D-module V is calledfinite dimensional if dimKV ( m. A D-module is called locally finite If it Is theunion (equivalently the sum) of its finitedimensional D-submodules. lote that if V andV/ are locally finite D-modules then V V/and V V/ are locally finite but HomK(V,W)won't be so in general and in fact not even V� = HOmK(VsK) will be locally finite ingeneral A D-module V is split [NW][IB 1] ifV has a K-basis contained in V D,

equivalently if the natural K-linear map KKDV D * V (which is always injectlye) isalso surjectlve. Of course if char K ) 0 thenV is split if and only if P acts trivially on Vso the concept of "splitting" will beinteresting only in characteristic zero. AnyD-submodule of a split D-module is easilyseen to be split. Moreover, any split D-Module is locally finite. Conversely, wehave the following result essentially provedin [BI], p. S5 (confer also with [T] for ageneraligation and also to (.2) [EMMA.Assume char K = 0 and V is locally finite D-module having at most a countable dimensionover K. Then there exists a D-field extensionK IlK such that KI D -- K D and such that K1 �K V is a split Dr-module (D 1 -- K 1 KD as In (0.10)). Moreover if we assume K is

D-algebraically closed we may take K 1 --K. Proof. For the sake o! completness wesketch the argument. It is clear that we mayassume N = dimKV ( m. Then the D-modulestructure of V induces a D-variety structureon GL N by just fixing a basis vt, .... v N inV.' if pv i = Zaij(P)e j with ajj(p) eK and ifGL N =Spec K[Xjj] d , d = der(Nil) then welet PXij = kaik(P)Xkj for all l,j,p Now let 8be a maximal element of (GLN) D and put K1 = K(g) (residue field at p it Is a D-fieldextension of K). By (0.1t)s we have Ki D --K D (because K D is algebraically closed).Viewing g as Kl-point of CL N, say g -- (glj)gij e KI it is easy to see that the elements fl''"fN D l defined by � i -- ?.gijfj form a K l-basis of K l V belonging to (K 1. � V) (3.3)A D-algebra A (respectively a Lie D-algebra

L, or a Hopf D-algebra H) is called 25locally finite if It Is so as a D-modulel A(respectively L,H) is called split [NW] Ifthere exists a D-algebra isomorphism A m KKDAO where A � is a KD-algebra and KKDAo is viewed as a D-algebra with Pacting via i) on K 1 and trlvlally'on 1 A o(similar definition for I.,H). Here is aremarkable (trivial) remark (3.) LEMMA.Let A be a D-algebra (respectively L a LieD-algebra of H a Hopf D-algebra). Then A(respectively L,H) is split as a D-algebra(respectively L is split as a Lie D-algebra orH is split as a Hopf D-algebra) if and only IfA (respectively L, H) is split as a D-module.Proof. Let's a give the proof for H forinstance. Assume H is split as a D-module.Then it is easy to check that H D is a KD-

subalgebra of H invariant under antiperleand mapped by H D comultiplication intoHD KD = (H K H) D so that H D has anaturally induced structure of HD. Hopf KD-algebra. Moreover the natural map KKD His clearly a Hopf D-algebra isomorphlsm,hence H is split as a Hopf D-algebra. Theconverse is obvious. (3.5) Let X be a D-scheme; an open set U of X will be called D-invariant if there is a closed D-subschemewhose support is X\ U..Now X will becalled locally finite if every point in X hasan affine D-invariant open neighbourhoodwhose coordinate algebra is 16rally finite.Note that if X is locally finite and quasi-compact then the D-algebra ((X) is locallyfinite; indeed if (Ui) l is a finite coveringwith afflne open sets such that C9(U i) is

locally finite then t(X) appears as a D-submodule of O((U i) which is locally finitehence it is locally finite. In particular thisshows that if X is an afflne D-scheme then XJs locally finite if and only if C/(X) islocally finite. (3.6) Recall lB 1] that a D-scheme X is called split if there is anJsomorphism of D-schemas X K DXo whereX o is a KD-scheme and K DXo is viewedas a D-scheme with P acting on K 1 via i)and acting.trivially on . Recll some basicfacts about split D-schemes o 0) Any split D-scheme is locally finite. l)If X � and Yo areKD-scheme and If X=K DXo , Y=K DYe areviewed with their natural structures of splitD-schemes then any D-morphJsm -'X Y hasthe form = K eo for some morphism o Xo*Yo' In particular XD(K)identifies with

Xo(KD). 2) Assume the D-scheme X iscovered by D-invariant open subsets U ieach of which is split; then X itself is split.3) Any closed D-subscheme of a split D-scheme is split. ) Any D-invariant open D-subscheme of a split D-scheme Is split. Theproof of the above assertions can be done asfollows.. first prove 1) for affine schemesusing the fact that D-algebra maps must carryP-constants into P-constants. Then prove 3)in the arline case using the fact that any D-submodule of a split D-module is split,

26 Now 4) In the arline case follows from 1)and 3) in the affine case. Next prove 2) byfirst noting that the arline case of 4) impliesthat for any set of indices 3={Ji,...,jN}, Ug =Ujl n ... n UjN is afflne and split (use

separation of X) hence by 1) in the affinecase again we may "glue together" thesplittings of the Ui's. Finally one uses 2) toreduce 1) 3) 4) to the afine case. (3.7)PROPOSITION. Assume char K = 0 and Xis a locally finite D-variety. Then: 1) Any D-invariant afflne open set of X is locallyfinite. D K D such that KIKX is a split 2)There exists a D-field extension Ki/K with K1 = D l-variety, where D 1 = K 1 K D.Moreover, If K is D-algebraically closedthen X itself is split. Proof. First prove 2).Cover X by arline D-invariant open subsetsU. such that (U i) Is locally finite. By (3.2)we may find an exter=ion of D-fields Ki/Kwith K? = K D (in case K is D-alebralcallyclosed we may take K 1 = K) such that eachK 1 KUi is split (one simply applies (3.2) to

V = � V I where V 1 is a finite dimensionalD-submndule of ((U l) generating ((U i) as aK-algebra). Clearly KiU i are Dl-lnvarlantopen subsets of Klex and we conclude byassertion 2) in (3.&) that K 1 X is split. Toprove 1) let Ki/K be as in 2) and let Uc X bean afflne open D-invariant subset. Then K 1� U is Dl-invarlant. By 4) in (3.6) K 1 U issplit, in particular ((K 1 U) IS locally finiteas a Dl-mndule. It follows that O(U) islocally finite as a D-module as well. Ourproposition is proved. (3.8) REMARK. LetX be a D-variety, char K = 0. It may happenthat there exists a D-field extension KI/Kwith K? = K D such that K 1 K X is a splitDl-variety (Di=KiKD) but X is not locallyfinite: take for lntance X to be an ellipticcurve over K viewed as a Kip]-variety

where p acts on X via a non-zero globalvector field v on X. In this case X contairzno non-empty Kip]-Invariant open set(because v is nowhere vanishing) but Xsplits over a "strongly normal" extension ofK by (.9) Let G be a K-group scheme. Wedenote by P(G,fln) the set of all p � P(G)such that G is.a locally finite Kip]-scheme (ifG is arline this is equivalent to saying thatC(G) is a locally finite K[p]-mndule, cf.(3.5)). We also put P(G/K,fln) = P(G/K) nP(G,fln) We will prove later (cf. (IV. .)) thatif G is algebraic (Irreducible) and char K =0 then P(G,fln) is a Lie K/k-subalgebra ofP(G). But note that if char K = q > 0 thenP(G,fln) (which coincides with P(G/K,fin))need not be a linear subspace of P(G) even IfG is algebraic. Let's give such an example In

what follows. First it Is easy to check that forq = 2,3,5 the following formula holds in thepolynomial ring A = (.) (( - [q) + [ - IA)q-l( )= (Presumably this formula holds for anyprime ql). Assuming formula (.) holds for agiven prime q (e.g. assuming qe {235)) let G= C a x G m = Spec[Xx'l] 1 = 1 + 1 1X = XXand define p = DerKC by the ormula p=(-q +5in -q and 1ong to Xa(G) and 1ong to L(G) Itollows by (1.9) that p � P(G). Now ormula(,) lmpli that p = and pq-l) = p-l) On theother hand clearly p = or all 2. Consequentlyp P(Gdin). On the other hand consir thederivation � DerKG = q. Clearly X = 0, -1)= 0, = (-)i+Z E P(G,fin). But + p P(G,fin)because (3.10) Here IS an example of an afi(non-gebralc) K-group scheme G char K = 0such that P(G/K,fin) IS t a linear subsco o

P(G). Take G =ScA, A =Ki i =i ] + 1 l' i = l'i= 0 (so G an "infinite vector group scheme").Define= P = il' 2i+1i +-') z = 2i+1 = + 2i- 1i) i1 - It easy o see using (1.8) that p,P(G/K,fin); but p P(G/K,in) since or y (p +)1[ 1) IS a linear orm in whichi+ 1 curs withcoefficient 1. EXAMPLES It is appropriatem examine here the Examples 123 given ter(1.9) from the viewpoint of local finitehess.Consider first G = G n viewed as analgebraic D-groups (D = KIP], P = Kp) viathe a derivation = * a p +r. aij[i.-. j as Jn(1.26). Then C is locally finite because foreach d _> I the K-space of all polynomials inK[ 1' "' n ] of degree _< d Js - stable.Consider now C =C m x C a viewed as analgebraic D-group via the derivation = p* +[XX. Then G is not locally finite because the

D-submodule of K[X,X-I]' generated byx isinfinite dimensional. Finally, consider G = Ax G a viewed as an algebraic D-group viathe derivation = p* +[ Yx ' Then C is notlocally finit for if Jt was then by (3.7) itwould split over a D-field extension of K,this forcing the subfield K(A)=K(x,y) of K(Ax G a) = K(x,y ) to

28 be preserved by . This is however not thecase since x = y K(A). (311) A D-groupscheme G will be called split if there is a D-group scheme isomorphism G = KKDG owhere G o is a KD-group scheme and KKDGo is viewed as a D-group scheme withP acting on K & I via a and acting triviallyon G ' By (3.6), l) a D-group scheme G is ospli:t if and only if it is so as a D-scheme. So

Proposition (3.7) holds if we replace "D-variety" by ,irreducible algebraic D-group".For the reader familiar with [K l] let's alsomake the remark that if P has a finitecommuting basis Pl' '" 'Pm and if G is a splitirreducible algebraic D-group (G = K �KDGo ) and V is a D-principal hornogenousspace for G then upon viewing the fields Kand K < V ) a A -fields with derivations(pl),... ,(Pm ) we have that the extension K <V )/K is G o - primitive in Kolchin's sense[K l] (cf. also [B i]p. 26). We close by ageneral remark on the link betweenP(G/K,fin) and the map G in (1.23) (for anextension of this result confer with (IV.2)below). (3.12) PROPOSITION. Assumechar K -- 0 and G is an arline algebraic K-group. Then the map X: L(Aut G)+ (Au.__t

G)= P(G/K) is injective and its image equalsP(G/K,fln). Proof. Start with a preparation.Assume W is a finite dimensional vectorsubspace o (G) generating (G) as a K-algebra and let Aut(G,'): {aJfine K-schemes}+ {groups} be the subunctor of Aut Gdefined by S = SpecB {f � Aut(G x S/S) f*=((G)B (C)B preserves W�B} We claimthat Aut(G,W) is representablel note that theaffine group scheme Aut(G,W) representingit will be reduced by [Sw] p. 280. To proveour claim let W I be the Intersection of allsub-K-coalgebras of (G) containing Wo= W+ SW by [Sw] W 1 is a finite dimensionalcoalgebra. Now define inductively theIncreasing sequence of subspaces W i of (G)be the formula Wi+ 1 = I(Wi W 1) for i _) Iand define functors A_o,AIA_2,... A i:

{affine K-schemes} {groups} as iollows.We let A_o(Spec B) be the group of all B-linear automorphisms o o of W o B such thatoSB=SBo where S B = SI B. For i_ 1, letAl(SpecB) be the group of all B-linearautomorphisms i of W i B such that Oi(gi_ 1B) = Wi_ 1 29 Vi I Wi-I BeAi'l(Spec B) andalp B -- pB(ai.l oi_ 1) where PB = p lB Wehave canonical restriction maps R_l * R_i ' 1for i _> 1. Now clearly all A_.i's arerepresentable by arline algebraic K-groupsA i hence we have a projectire system � .,AiAi_ 1 *...Ai*A o One checks thatAu.t(G',V/) -- lim_A i. ConsequentlyAu_t(G,V/) is represented by Spuc(lira (Ai))and our claim is proved. Let's prove that lm -- P(G/K,fin). The inclusion : "is clear.Conversely if p g P(G/K,fin) we may choose

V/ above such that pV/c W. Then id � gpAu(G,W) (Spec K[, hence we get amorphism f: 5pec K]* Aut(G,W) such that id.gp = faG,W where G,W Is the univerlAut(G,W)-automphism of G x Aut(G,W).Now since Aut(G,W) is reduced and sinceAu G restricted to (reduced K-schemes} isreesentable by a 1ocly gebraic K-group B5]AutG there exists a morphism h:Aut(G,) +AG such at -h* where G Is the . G,W - Gunivers Aut G-automorphsm of G x Aut G.Consequently id � p = (hf)aG hence p �lm. Finally, let's prove that is injtive. LetAut�G =Spec R (by [BS] Aut�G is fi I) wemay choose a finite dimensional subspace Wof (G) generating (G) as a K-algea such that(W R) W R (where : (G) R * (G) R isinduced by G ). Exaly ave there exists a

morphism h:Aut(GW) AetG such that h G:GW' There is also a nat morphism c:Aut�G* Aut(G,W) defined at the level of S-ints by (Aut�GNS) Aet(G,W)(S), s wheres* G Consider the affl group scheme A =Aut(G,W) x Aut G Aut�G note that theprojtion Pl: A + Aut(G,W) is a closedembedding. Now the map P c A AutOGAut(G,W) equals the map Pl A * Aut(C,W)for if (f,f,) A(S) is an S-point of A we havehf = if' (where i .' Aut�G ' Aut G is theinclusion) so the image of (f,f') via (cP2)(5)is a map s Aut(G,W)(S) such that s * = f'i* G= G,W = f*h* G = fG,V/; consequently s = fby universality of ^ut(G,W). We get that P2 isa closed embedding so A is an algebraicarline group. Since the map P2: A * Aut�Ginduces a bijectlon at the level of K-points

this map is an isomorphlsm. Consequently cis a closed embedding hence Aut�G is asubfunctor of Aut(G,V/) hence of Aunt G andinjectivity of follows.

CHAPTER 2. AFFINE D-GROUPSCHEMES Throughout this chapter K isalgebraically closed P is any Lie K/k-algebra and D = KIP]. I. The analytic methodOur first main result is: (1.1) THEOREM.Assume char K = 0 and let G be anirreducible affine algebraic K-group. Then KP(C) is a field of definition for G (hence itcoincides with KG). (1.2) The abovestatement fails for non-linear G, cf. Chapter3. Our proof of (1.1) will be analogue to thatof Theorem (1.1) in Chapter 2 of [B 1] in thesense that we are going to use "birational

quotients", a "Koalaira-Spencer map*' andan analytic ingredient. In [B l] the analyticingredient was the versa] deformation of acompact complex space. Here the analyticingredient is a combination of Theorems(1.3) and (1.#) below (due to Harem andHochschild- -Mostow respectively). To statethe first theorem let's fix some notations.Assume [: + is an analytic family ofconnected complex Lie groups, i.e. a map ofanalytic C-manifolds, having connectedfibres, such that one is given analytic - mapssatisfying the usual axioms of multiplication,inverse and unit. Assume moreover that a) issimply connected and Vl,... ,v m arecommuting vector fields on giving at eachpoint a basis of the tangent space and b) v 1,... v m can be lifted to commuting vector

fields wl,... ,wm on such that p,5� agreewith w 1, .. � w m in the sense that for eachw = w i we have: 1 ) (T(g t'g2 )pXw(gl)'w(g2)) = w(p(g l'g2 )) 2) (TgSXw(g)) =w(Sg) 3) (TxEXv(x)) = w(E(x)) Then wehave for any (gl,g2) � x, for any g C ,' forany x�, (1.3) THEOREM. (Harem, [Ha]).Under the assumptions above there exists ananalytic -lsomorphism *, o x * (where o issome fibre of T[) which above each point ofl, a 31 group isomorphism and such that uponletting v. be the trivial lifting of v i from to ox. we 1 have (T)(v?) = w i for all i. (l.t)THEOREM. (Hochschlld-Mostow [HMi]).Let Gl, G 2 be two irreducible affine an andan algebraic C-groups. If G i G 2 arelsomorphic (as analytic Lie groups) then G land G 2 are isomorphic (as algebraic

groups). (1.) REMARK. The abovestatement fails for non-linear groups (cf.[HMi[Se. Theorem (l.t) is a consequence ofthe theory developed in [HM 1 ] and no ideaof proof will be rlndicated here. We shallhowever include the proof (due to Harem) of(1.3) since it is completely elementary andfairly elegant: (1.6) Proof of (1.3) (Harem).By Frobenius, for any x o and any 8o l[ 'l(xo) = o there gg of go in o and an analyticmap exists a neighbourbood of x o in , aneighbourhood lg � o *: go x-- ' over suchthat '4) takes v (= trivial lifting of vi from . ,o*'go x 7go) into go ' w i. A triple (go' ?Jgo'Lb) as above will be called a "localsolution" at go' It is sufficient to show thatfor. a given Xo, the various ?gg *s appearingin the local solutions at various points 80 o o

can be chosen to contain a fixed openneighbourhood of x o. Let � o -- (x o) andconsider the set r. of aug, o such that thereexists a local solution,?gg,/g, t),,g with c .One easily checks that r. is an open subgroupof o (local solutions can be "multiplied" and"inverted" using p and 5) hence r. = o byconnectivity, which proves the theorem. Nextwe need some facts about isomorphism ofLie algebras. First "recall" the followingtrivial representability result: (1.7)LEMMA, Let R be a ring and L L two Lie R-algebras which are free and finitalygenerated as R-modules. Then the functor[SOL,L, from {R-algebras} to {sets} definedby ISOL,L,() = { - Lie algebra lsomorphismsR L = i R TM } is representable by a finitalygenerated !-algebra (call it lSOL,L). Exactly

as in [B I] pp. 3-36 the above Laminaimplies the following= (1.) LEMMA. Let Cbe an algebraically closed field, S an afflneC-variety and L a Lie ((S)-algebra which isfree and finitaly generated as an C(S)-module. Then there is a construc- tibia subsetZc S x S such that for any Sl,S 2 c S(C) wehave (Sl,S2) Z(C) if and only if the'Lie C-algebras L (s)C(Sl) and L )((s)C(s2) areisomorphic. Next we have= (1.9) LEMMA.Assume C, S, L are as in (1.8), let K be analgebraically closed extension of C(S) andassume KL( = smallest algebraically closedfield o! definition of L ((5)K between C andK, cf. (I. 1.27)) equals the algebraic closureof C($) in K. Then there exists an opensubset

32 S O c S such that for any s o � So(C) theset { s � S(C)t L C(s) L ) C(So)} is finite.Proof. By (l.S) and [Bi], (1.13), p, 36 thereexists an affine open set 51 c 5 and adominant morphism of afflne C-varietlas b .'Sl+ M such that for any s I � Sl(C) we have'lb(Sl) ={s Si(C)I LC(s) LeC(Sl)} if dim k =dims I we are done. Assume dim M ( dims 1.Then we use an argument similar to iV] p.576. Choose a closed subvariety Nc S 1 withdJmN = dim 1, let L N be the pull-back of Lon N, let L' be the pull-back of L N on theaffine scheme 31 = S 1 x MN and let L" bethe pull-back of L on 31 � Then for any C-point X$l(C) one checks thatL'eC(x)=L"�C(x). By representability ofISOL, L" (1.7) there is a generically finitedominant morphism of finite type of arline

schems 21 with Y integral such that the pull-backs of. L' and L" on Y are Y-isomorphic.Since Y+ S 1 is generically finite, one canembed C(Y) over C(S 1) = C(S) into K andwe get that C(N) is a field of definition for L�K between C and K contradicting ourhypothesis. The lamina is proved. Next weneed a Kedaira-Spencer map for irreduciblearline algebraic groups G over non-necessarily algebraically closed fields F ofcharacteristic zero containing k. Consider thefollowing complex (where A -- {(G))'. 0+DerF(A) ' , DerF(A,A�FA) 2 +DerF(A,AFA�FA ) where {) l(p) = pp-(p�l + I �p)p, pc DerF(A,A) {) 2(d) = (dl)p - (1 ) d)p + (p l)d - (1 p)d, de DerF(A,A� F A) and define Hi(C/F) for i = 1,2 as thei-th horn ology space of this complex (one

can identify Hi((;/F) wlth the correspondingHochschild cohomology spaces of theadjoint representation of G, cf. [D(;] p. 192,but we won't need this fact). Clearly Hi((;FFi/F 1)= Hi(C/F! )FFI for any fieldextension Fi/F. Moreover we can defineP(C) and P(C/F) exactly as in (I. l.l) in caseour ground field is not algebraically closed.(1.10) LEMMA. There Is an exact sequence0- P(G/F)+ P(G) � DerkF-H2(G/F) wherepis compatible with field extensions Fi/F.Moreover, we have an identification P(G/F)= HI(G/F) S (where HI(G/F) S Is the fixedpart of HI(G/F) under the involution v SvS otDerF(A,A) which is easily seen to preserveHI(G/F)). Proof. The assertion aboutHI(G/F) S is clear. Let's define p. Since G/Fis smooth and arline, any derivation p DerkF

can be lifted to a k-derivation o! A = ((G).33 Then one checks immediately that p - ( �! + l))p� Ker(a 2)c DerF(A,A FA) and theclass of this derivation in H2(G/F) does notdepend on the choice of the lifting call thisclass p(p). e must check that Keri0 =im(P(G)+ DerkF) The Inclusion ':" Is clear.Converse]y, if 19(p) = 0 then there is alifting of p, e DerkA such that p = ( I + I ))1.Then one immediately checks that .(ss) =[(ss)� l + i � (ss)]p Puttlng = (I/2X + SS)we see that ) lifts p and belongs to P(G) soour lemma is proved. (1.11) COROLLARY.If Fi/F is an exension of algebraically closedfields and C is an irreducible arlinealgebraic F-group them P(C FFi/Fi )aP(G/F) )FF ] and P(G O FFl/Fl,fin)P(G/F,fin)FF 1 . Proof. The first

isomorphism follows from (1.10). Thesecond follows from (1.3.12). (1.12)THEOREM. Assume char K = 0 and C is anirreducible aifine algebraic K-group. Let Cbe an intermediate field between k and K andlet K � be the smallest algebraically closedfield of definition for G between C and K (GCOKoK for some Ko-grou p Co). Then themap I o: DercKo + H2(Go/Ko ) is injectlye.REMARK, The field C in the abovestatement has only an auxiliary role in theproof. The interesting case of course is thatwhen C = k (then K o = K G. cf. (I. 1.27)).We shall give first the proof of (1.12) whenC is the field C of complex numbers. By (I.1.27) K � equals the smallest algebraicallyclosed field of definition for L(U) (U theunipotent radical of G) between C and K.

There exist group schemes C.s S and O.s S(S an arline (:-variety, 0 a closed subgroupscheme of C) such that K o is the algebraicclosure of K 1 = C(S), C )sKo = Go, 0 SKo= U � (Uounlpotent radical of Go) and thefibres of I.s S are the unipotent radicals ofthe fibres of + S. We may assume the relativeLie algebra L(O/S) is a free I(S)-module.Assume Po is not injectire. Then Pl"DercKI+ H2(CI/KI ) is not injectire (whereC 1 = C SKi ). By (l.10) there exists pDercK 1 , p 0 which lifts to a derivation �P(Ci). Now both p and can be viewed asrational vector fields on the C-varieties Sand C respectively. Shrinking S we mayassume both p and are regular everywhere.Now by (1.9) there exists a Zarisky open setSoc S such that for any Soe So(C) the set T. s

={s So(C);L(Us) L(U s )} o o

is finite where U s J x sC(s). Let be ananalytic disk in 5 an which is an integral = Osubmanifold for p and let := .an x '. Then isan integral submanifold of an for. By san(1.3) all fibers o -. are pairwise isomorphicas complex Lie Groups. By (l.q) the fibres o!+ S above the points In are pairwiseisomorphJc as algebraic groupsl inparticular all Lie algebras L(U s) with se:are pairwise isomorphic this contradictingthe finitness of r. So for s o � S(C). Thetheorem (l.12) is proved for C = C. (1.13)Proof of Theorem (1.1). It is sufficient toprove that for any Lie K/k-subalgebra P ofP(G), K p Is a field of definition of G. CaseI. K P is uncountable. Then we can assume

Cc K p. Let K o be the smallest algebraicallyclosed field of definition for G betveen Cand K, G = Go) K K. We may conclude o byInspecting the diagram with exact rows andcolomns (cf. (l. i 2), case C = C and (i. 10))0 Der K K 0 o P(G) d �Derc K P . H2(G/K)0 (DercKo)DKo K , H2(Go/Ko))K K PoK ,o o o that line = ImP/hence that K K p, i.e.that G is defined over K p. Case 2. K p iscountable. Then take an embedding KPc Cand conclude exactly as in [B 1] p. 3(instea.d of �B1], Lemma (1.19) use the factwhich we already know from [B] that the setof algebraically closed fields of definitionfor a linear algebraic group has a minimumelement, el. also (I. 1.27)). (1.1) REMARK.Using Theorem (l.l) one can immediatelyprove Theorem (i.12) for arbitrary C. Now

(1.1) and (I. 1.22) imply: (1.1)COROLLARY. Assume char K = 0 and let Gbe an irreducible affine .agebraic K-group.Then we have an exact split sequence of LieK/k-algebras= 0+ P(G/K)+ P(G) a _Der(K/KG). 0 EXAMPLE Assume P is a LieK/k-algebra, D = KiP] and K D = k K. Thenthe "sufficiently general" unipotent algebraicK-group G of sufficiently big dimensiondoesn't admit any structure o algebraic D-group. Indeed it is known that such a G is notdefined over k. We shall need in whatfoliow's a variation on Hamm's result (I.3)(1.16) LEMMA. Let be a connected Liegroup and v an analytic vector field on suchthat the multiplication x and the inverse - areequlvarJant (with respect to the vector field(v,v) on } x and v on ). Then there is a l-

parameter group of analytic groupautomorphisms C x + whose derivative at theorigin is v. Proof. Use Hamm's opensubgroup argument 0.3) once again to showthat therb is a disk 0Bc C such that for all gthere exists an analytic map g.'B -, g(0)=gwhose d tangent map TL0; TB + T takes (z acoordinate in C) into vl this JmmedIatalyimplies the lemma. Now recall from [HM 2]that the connected component of the groupAut(G an) of analytic group automorphismsof G an has a natural structure of affJnealgebraic group of which Aut�G is analgebraic subgroup (both being viewed asembedded into GL(L(G))). (1.17)COROLLARY. Let G be an irreduciblearline algebraic C-group. Then P(G/C)CL(Aut(Gan)) (as subspaces of H�(TGan)).

(1.18) THEOREM. Let char K = 0 and G bean irreducible arline algebraic K-group.Then a derivation in P(G) belongs toP(G,fin) if and only if it preserves the idealof the unipotent radical of G. In particularP(G,fJn) is a Lie K/k-subalgebra of P(G).We shall provide later (cf. (3.2a) of thisChapter) a purely algebraic proof of thisresult. Here we shall give a proof of it basedon the following analytic result: (1.19)THEOREM. (Hochschild-Mostow [HM2]).If G is a connected affine algebraic C-groupthen an analytic group automorphlsm �Aut(G an) belongs to Aut C if and only if itpreserves the unJpotent radical U of C.(1.20) Proof of (1.18). Start with the "onlyif" part so assume p e P(G,fin). Then by (I.3.11) there exists a Kip]-field extension

Ki/K such that KT[P]= K Kip] (call thisfield K o) and SUCh that K 1 )K C Js a splitalgebraic Kl[p]-grou p so K l K G" K 1 )KCo for some algebraic o Ko-grou p G o. LetU o be the unipotent radical of Co; then K 1KoUo must correspond via the abovelsomorphlsm with K l {)KU. But p clearlypreserves the ideal of K 1 K Uo so it wJJJ opreserve K 1 )K U hence U itself (byfaithfully flatness of the projection K 1 K GG). Let"s prove the "if part". It is easy toreduce the problem to the case K = C. Letnow P � P(G) preserve U hence also L(U)by (1. 1.16). Then by (1.15) we may write G= K $K Go o

36 (K � = KG, G O a Ko-group) so p = p" +v where p* acts via a on K and acts trivially

on G o while v P(G/C). Since p* clearlypreserves U so does v. By (l.l&) there is a l-parameter group of automorphisms C x Gan+Gan, (t)t C whose derivative at ! = 0 is v.Then for each t C, t preserves U. By (1.19)tEAutG for each t hence by (L 3.11)vE P(G/C,fin). But C(G)is locally finite both asDer(K/KG)-module and as P(G/K,fin) =L(Aut G)-module (I. 3.12). Since by (l.ll)P(G/K,fln)= P(Go/Ko,fin) K we have[Der(K/KG),P(G/K,fln)]c P(G/K,fin) itfollows that )(G) is locally finite also as aDer(K/K G) e)P(G/K,fin)-module inparticular as a Kip]-module which provesthe theorem. EXAMPLE Let's re-examineExample 2 given after (1.9) so let G = G m xGa. We have seen that the elements of P(G)have the form where pz DerkK, a,b z K.

According to Theorem (1.18) the elements ofP(G,fln) have the form p +-[ where pzDerkK, be K. Indeed the ideal of theunipotent radical of G is generated by X - IIn K[X, X-1] and the condition thatpreserves it is equivalent to a = 0. (1.21)The following concept will play a role inChapter 5. Assume G is a K-group schemeand assume P(G,fln) is a K-linear subspaceof P(G). An ideal In P(G) will be called arepresentative ideal if: a) it is a K-linearsubspace of P(G/K) b) it is a complement ofP(G,fin) in P(G) c) it is Aut G - invariant. Aswe shall see in Chapter 3, representativeideals may not exist even if G IS algebraicand char K = 0. We shall prove here (using[HM 2] once again): (1.22) PROPOSITION.Assume char K = 0 and G is an Irreducible

aflne algebraic K-group. Then P(G) containsat least one abelisa representative ideal. Ourbasic ingredient IS: (1.23) THEOREM(Hochschlld-Mostow [HM2]), Let G be anirreducible algebraic C-group. Then AutO(Gan) is the semidirect product of Aut�C bysome normal algebraic vector subgroup N ofAut�(can). We will also need the followingJemma whose proof is "obvious": 37 (l.210LEMMA. Let L be a Lie K-algebra ofdimension n, L 1 a Lie subalgebra ofdimension n I and A a locally algebraic K-group acting on L by Lie algebraautomorphisms. Assume K � Is analgebraically closed subfield of K overwhich all the above data are defined. Let Ydenote the subset of all K-points in theGrassmanlan X of (n - nl)-planes of L which

correspond to subspaces L' of L enjoying thefollowing properties: 1) L' is an abeliansubalgebra of L 2) L is an ideal in L 3) L 1 L'= L /$) L' is A-invariant, Then Y is locallyclosed in X(K) in the obvious Ko-tOpologyof X(K). (1.25) Proof of (1.22). Put K o = KG And let G -- K � K Go' cf. (1.l). It issufficient to find o an abel Jan ideal 3 o ofthe Lie Ko-algebra P(Co/Ko) complementaryto P(Co/Ko,fin) and ^UtCo-invariant becausethen P(G,fJn)=Der(K/Ko)eP(G/K,fin ) (cf.(1.15), (1.18)) hence 3 = 3o K K will be anabelJan representative ideal in P(C). Nowby (1.2/$) it IS su:[flclent to o find and ideal3 of P(G/K) which is abelJan complementaryto P(G/Kfin) and Aut G-invariant. By (1.2/$)again we may assume (after replacing K by afield extension of K or by a suitable subflald

of K) that K = C. But then (1.]7) and (I.23)show that viewing P(C/C) as a subalgebra ofL(Aut(can)) we have that 3 = P(G/C)n L(N)satisfies our requirements (N as in (I.17)).$o (1.22) is proved. EXAMPLE Assume G =G m x C a as in Example 2 (1.9), (1.26). Arepresentative ideal in P(G) is the K-space 3o! all derivations aXX, a K Indeed Aut G =G m acts on K[XX'[,[] by leaving ( fixed andmultiplying with non-zero scalars in K so 3is Aut G-lnvarlantl it is obviously acomplement of P(G,fln) in P(G) and it is anideal in P(G) because p � DerkK, a K 2.Algebraic method= direct The aim of thissection is to investigate by purely algebraicmeans the structure of P(G x H) where G,Hare arline (non-necessarily algebraic) K-group schemes (char K arbitrary) With H

linearly reductlve (i.e. all rational H-modules are completely reducible).lemarkably,

with only a minor extra effort we can makeour arguments work in the (non-commutative) Hopf algebra context [Sw].Therefore we shall place ourselves (In thissection only!) in this more general frame andfreely borrow from [$w] the Hopf algebraterminology. As an application we willprovide for instance a description of P(G xGL N) in case char K = 0 (recall that in thiscase GL N is linearly reductive). Let A be aHopf K-algebra (K/k as usual a fieldextension of arbitrary characteristic with Kalgebraically closed) and denote by 'p,S,�the comultiplicatlon, antipode and co-unit of

. Define ll( ) to be the Lie K/k-algebra ofas! k-derivations p of such that pKc K, pp= (p �) 1 + I �) p)p, p$ = Sp, pe = k-p (wewrote lI( ) and not P(A) to avoid confusionwith the space of primitive elements of ^[$w]). Clearly, if G is an arline K-groupscheme with Hopf K-algebra A = ((G) thenlI( ) = P(G). Our main result will describeII(A e B) where A, B are Hopf K-algebraswith B co-semisimple. Note that by a resultof Sweedler [$w] if B = CO(H) for someaffine K-group scheme H then B is co-semisimple if and only if H is linearlyreductlvel if moreover A = CO(G) for somearline K-group scheme G then of course lI(^B) = P(G x H). We start with a preparationon centers and cocenters of Hopf algebras.(2.1) Let C be a K-coalgebra; A coldeal I of

C is called cocentral if Tc(1)q)p = = (q )l)p:C+ (C/I))C where p is of course thecomultiplication, q: C * C/I is the canonicalsurjection and : C (D(C/I)+ (C/i))C is thetwist map (equivalently, If px - z px 1 o C forall x = C, where : C)C + C)C is the twistmap). If dim C < m, I is cocentra] if and onlyif the subalgebra (C/I) � of C � is centralOne easily sees that 1! dim C < then there isa minimum concentral coideal I C of Csindeed let A be the center of C � and put I C= Ker(C = cOO+ ^o). Now we claim that anyco-algebra C (possibly of infinite dimension)has a minimum co-central coldeal I C.Indeed, take any family (Ci) i ofsubcoalgebras such that C = T.C i and dim Cl < then I C = T. IC. Is easily seen to be theminimum cocentral coldeal in C. (2.2) ^

coalgebra C will be called cocentral if I C =Ker E (='= the counit). If dim C < = then C iscocentral i! and only if C � Is a centralalgebra. (2.3) Recall from [$w] p. 161 thefollowing basic properties of simplecoalgebras: if C is a coalgebra, C = r.C F C isubcoalgebras then any simple subcoalgebraof C lies in one of the Ci's. Recall that acoalgebra is called co-sesimple if it is thesum of Its simple subcoalgebras. The abovepropsty implies that if C is co-semlslmplethen any subcoalgebra of C has acomplementary coalgebra and is the sum ofits simple subcoalgebras. (2.) Let B be aHopf K-algebra. Hopf ideal :3 is calledcocentral if it Is so as a coldeal. Any Hopfalgebra B has a minimum cocentral Hopfideal 3B; indeed put 3 B = B(r. SnlB)B

where is the minimum cocentral coideal inB. The luotient Bc = B/2E is called thecocenter of B. 39 (2.) Assume in (2. a)above that B c is a group algebra and let B=)B be, he Xm(BC)-gradatlon correspondingto the coaction of B c on B on the right p: B+BB c (In analogy with (I. 1.3) we denoted byXm( ) rather than by G( ) as in [Sw] thegroup of group-like elements of ); lnce 3 BLs cocentral this gradation coincides withthe gradation Corresponding to the leftcoaction ,: B+ BCB. We claim that all B'sare subcoalgebras of B. Indeed p: B+ B)B isequivarlant with respect to the rightcoactions of B c on B and B)B (for the lattertake the coactlon p on the second factor)I inParticular p(B )c B B . Slmilarilyequivarlant with respect to the left coactlon

of B c X on B and BB (for the latter take thecoactlon ), on the first factor)and get p(BX)cB .B; consequently p()c B X )B and ourclaim is proved. Note also that B 0 for all XXm(BC indeed, if be B lies above X �Xm(BC) then bx, the X - homogeneous pieceof b, also lies above O hence b x 0. (2.6) ^Hopf algebra is called co-semlsJmple [Sw]If It is so as a coalgebra. In [Sw] p. 29aseveral characterizations of co-semlsJmpleHopf algebras are given; In particular asalready noted commutatlve co-semlsimpleHopf algebras correspond precisely tolinearly reductive affine group schemes.Note that if B is cosemistmple then thecocenter B c is a group algebra. Indeed writeB -- e) Ci,C l simple (cocentral)subcoalgebres. Then B/I B = e (Ci/Ict) -- =

(Ci/ker�Ci) is generated by IF'oup-llkeelements hence so will be ]C (being aquotient of /l). (2.7) Let's discuss the notionof center of Hop! algebra. A subset of a HopfK-algebra A will be called central If eachelement of it commutes with all elements ofA. Then contains a maximum central subHopf algebra cA (we let CA be the maximumelement of the family of all central, S-stable$ubcoalgebras of A). /e call CA the center ofAi It is contained (but apriori not equal to)the center of the underlying algebra of A.Note that tf Xa(A) respectively xa(cA)denote the spaces of primitive elements of Aand CA respectively, then Xa(CA) consistsprecisely of the central elements of Xa(A)(once again we used the notation Xa(A)instead P(A) as In [$w] In analogy with our

notations in (I. 1.3)). (2.g) Let A and B betwo Hopf K-algebras. Then A )B has anatural structure of Hopf algebra, As in (I,1.19) there is a K-linear projectionII(AB)+I(A)x Der K E(B) over DerkK,defined b y p + (pA,PB) where PA (IA)�B)PiA (where l A ., A+ AB iA( = a I) andPB defined slmllarlly. Our projection admitsa section (which is a Lie K/k-algebra map)defined by (Pt'P2) + Pl DI + l )P2' We shallusually identify II(A) x DerkKE(B) with itsimage in (2.) Next assume In (2.8) above thatthe cocenter B c is a group algebral by (2.6)this the case for Instance when B iscosemlsJmple. Then we shall define aremarkable Lie K-sub- algebra I(B: A) ofI[(A B) lying In the kernel of the projectionI[( B) + l( ) x DerkKT[(B). We proceed as

follows.. consider the Xm(BC)-gradeti_on B-- B on B defined by the coaction of BC on B(cf. (2.)) and consider also the space xa(cA)of prlmive elements of cA. Then for anygroup homomorphlsm aHom(Xm(BC),xa(cA)) define the K-linearmap Es-. B- A )B by

the formula Ea(x b) = r. xa(x) b where b = -bX is the decomposition of b B intohomogeneous pieces. Then clearly E a is anA-derivation and using the fac! that all Bsare subcoalgebras of 13 one checks the factthat E a � ](A B). /e defined a linear map E:Hom(Xm(BC)xa(CA)) + ](A B) whoseimage will be denoted by ](B = A). Since B(0 for all X (cf. (2.)) it follows thatHom(Xm(BC), xa(CA)) = I[(B '. A) is an

ahellan Lie K-subalgebra of ](A ) B). Ourmain result is the Then (2.10) THEOREM.Let A and B be Hopf K-algebras with Bcosemisimpie. Then T[(A B) = (l(A)XDerk!(T[(B)) � T[(B: A) (2.11) COROLLARY.Let G and H be affine !(-group schemes withH linearly reductive. P(G x H) =(P(G)XDerkKP(H)IHomgr(Xm(Z(H)),Xa(C)) (2.12) Proof of(2.10). We must show that every p Ker(][(A)13)+ ]I(A)XDerkKl[(B)) lies in ][(B .' A).Clearly p is K-linear. Step I. We show that pis an A-derivation. It is sufficient to checkthat p(A K)c A K. By (2.3) B--K )M forsome coalgebra M. Consequently for any a Awe may write p(at)a o)! +AM witha oeA.wegetthat (p(a ) 1)) iJ(ao ) l) � A )lVl )A)M, i(aol) � A � K )A K On the other hand

I(p(a[))=(Pl + lp)u(al)c (pl)! + I)p)(AK cA�KA K + AMAK +AKAM we get thatp((a � I)) = IJ(a o l) hence p(a e l) = a o �I and our claim is proved. Step 2. We showthat for any subcoalgebra C of B we have p(!< )C)c A )C. Indeed by (2.3) C has acomplementary coalsabra C'. For each b Cwe have p(l )b) � Xb + A C', XbeAC e et41 p(p(1 )b)) � p(x b) + A {)C' A e)C', p(xb) � A e C A C On the other hand p(p(lb))=(p)l+lp)p(lb)e(pl +I)pXKC+CK)CCABKC+KCA�B e get that p(p(l �b)) =p(x b) hence p(l b) = x b and we are done.Step . We show that for any simplesubcoalgebra C of 13 there exists a C �xa(CA) such that for all b �C we have p(l)b) = aC )b. Start with the following generalremark.' if C is any simple K-coalgebra and

L .' C + C is a K-linear map such that IL = (L) IC) p = (1C L)p '- C + C �C then L is ascalar multiple of the identity. Indeed chooset g K such that L - tl C Js not lnvertible. Theequalities I(L - tl C ) = ((L = tlc) elc)l =(Ic(L - tlc))l show that the Image V of L-tl Cis a subcoaJgebra of C since V is not thewhole of C it must be zero hence L = tl C andour remark is proved. Now let a eA � andLa. = (a* �) Ic)Pi C: C +A )C +A �C +Cwhere i C: C +A )C, ic(X) = 1 )x. We claimthat L . is a scalar multiple of 1C. To see thisnote first that a* = (a* ) eA)p -' A K whichimplies tat PC(a* gJ l C) = (a* 1C )e A e)IC)IJA�C = A )C * C C Now if x � C weget PC(La.X) = pc((a e l)p(l x)) = (a* 1C eA� IC)FA C p(I x) = = (a* 1C eA IcXP IA C+ IA C P)PA C (1 x) = = (((as Ic)P) eA �

IC)A C (1 � x) + (a IC)((e A Ic)P)p A e C(1x) The second term oJ the la*ter sumvanishes because p projects into 0 �A)XDerkK(B). Using the usual sigmanot=tion pC x = ZX(l ) x(2 ) we get PC(La.x) = ((a* C) c A lC)H x() � � x(2 )) = =La.x(j))x(2 ) = (La. JCXUC x) In otherwords cLa = (La IC)U.. Similarily startingwith the uality a* = (Aa*)U we get HCLa. =([CLa.)U � By the remark we made L . musta scalar multiple o the identity and our claimis proved.

42 Our claim impties that there exists a � Asuch that p(l b)= a)b for all b c C. All wehave to check now is that a � Xa(CA).Writing 0 = PA � B p(l ) b) - (p � l A ) B +l A ) B ) P)P B b = = (iA)IBX(PA a- a�l -

1 where =A)B+BOA is the twist map, we getacXa(A). Writing xa)b=p(xOb)= = p((! ) b)(x ])) = ax b for x A we 8at a � Xa(CA).Step 4. We show that whenever C and C' aretwo simple subcoalgebras of B contained inthe same B_, we have a C = ac,. IndeXed, by(2.1) and (2.) the ideal 3 B = Ker(B + B c) Isgenerated (as a two-sided ideal) by elementsof the form $nx where n _> 0 and x belongsto the union of all simple subcoalgebras ofB. This together with Step 3 shows that theideal ^3 B is stable under p In particular pinduces a derivation ](A )B c) and we have acommutatlve dia�ram Now clearly theprojection of in E(A) x DerkKlI(Be) is stillzero, hence by Step 3 applied to and B cInstead of C and B we get that (1 e X) = ae Xfor some a X � A. We have for y b C (pel +

I OXl e I el eq)p(lb) = (POt + IOlOblex )==acOb10 + lb On the oth hand (i ] l Oq)pp(lb) = irom hich e get a C = . 5imilaHly a C' =d e e done. St 5. !1 of . By (2.3) each B is asum of simple subcogebr hence by Step a weget a 1unction Xm(B ) XaA), X a) such tt p(lb) = a()b for 1 b B � (notethatB B.0foralJxi, x2becauseby(2.5)0B cB ' 112 ).Xl X2 X2 (2.13) REMARK. The oo o Step 2above shows that or any co-semlslmple HopK-algebra B, any subcoalgebra C o B and ypt () we have pC c .C. In particular B is alocally finite D-module where D =EXAMPLE The above remark plus (I. 3.2)show that if char K = 0 and G Is a reductlvelinear algebraic K-group then P(G/K) =L(^ut G) = L(Int G) = L(G)/L(Z(G)) Inparticular for Instance if G = CL N =

SpecK[y,-y-y ], y--(Yij), then P(G/K) has aK-basis consisting of the derivations Pb=i(m bimYmj. m b B m jYim)'Yij where b =(bij) � gIN(R ). Consequently for G =CLNP(C) consists of all derivations of theform p* + Pb where e I p e DerkK and p#DerkK[Y,d-e], lifts p and kills y. K 1 Assumenow G=GnxCLN=$pec [[,y,ey], [=([1'' ,in ).Then by (2.10) P(G) consists of allderivations of the form: =p*+pa+Pb+Pcwhere p e DerkK , Pb is given by the formulaabove, Pa = ' a = %) � gin(K). PC =(m�m;mXi Yij ']j )' c= (�l .... ,On), c i .K Itworths noIng that P(G,fin) consists of allderivations of the form p* + Pa + Pb' On theother hand t space o derivations of the ormPc is a reesentative ideal in P(C). .Alsebrale m!l: $eml-t IXXluets Let M be an

afflne K-group scheme, T a diagonalizableefflne K-group scheme [DG] and 0: M x T -'M (13(m,t) = m t) a right action of T on Ml tosimplify discussion we always assume inwhat follows that O(M) is an integral domainand Xm(M)= 1. Our aim here is to study thestructure of P(M x pT) where M x pT Is thesemidirect product of M and T constructedwith the help of p. This will enable us inparticular to re-prove algebraically (lolg),get new information about the map [V: P(G) -Hom(Xm(G) , Xa(C)) and compute P(C) incase the radical of G Is nilpotent or theunlpotent radical of G is commutative. Forsake of simplicity we agree from now on thewrite W(G) instead of Hom(Xm(G) , Xa(G))for any K-group scheme G. 0.1,.) We shallthink of M x pT as having its underlying

scheme equal to M x T while

multiplication is given by the formula t 2(ml,tl)(m2, t2) = (m I m2,tlt2) , m I � M(K),t I � T(K) hence by the composition (3.1.1)p:MxTxMxT' TxT lXxl TxMxTxM 'TxMxTxM PT x Pm lxxl TxTxMxM TxM,MxT where : M x T + M x T, (m,t) =(P(rn,t),t) = (mr,t) and T is the twist map.The antipode is given by (m,t) + ((m-l)t' l,t-])hence by the composition Sl x s T (3.1.2)S.'M xT -M xT ' M xT and the unit is givenby �MXCT -MxT (3.1.3) : Speck Let r.denote the group Xm(T) (hence ((T) -- K[Z]_- group algebra on Z). To P: M x T + therecorresponds an algebra map still called P:d(M) + ((M) )((T) hence a Z -gradation on((M);, if we denote by fx :(M)+i(M))( c ((M)

the projection onto the X-component thenPMfx X "X"--) (3.1.a) (a ex) = r-Lx,(a)�X% a(9(), where ;C(M)�((T)+((M)iT) isinduced by . we denote by (T, (M)) the set o!all weights o! T in d)(M) i.e. (T,(M)) = {X Zd(MLx 0} -- {x � z x 0} Since ((M) isintegral, e(T,(M)) is a semigroup. For each X� r. put (Oerk) x _- {p � OerkMp(<9())X,cC(M) or ali ' s Z} _- = {P DerkM; fX-Ix, p =Pfx' for all X' � Z} o cour, e the sum in Uerkof all (UerkM) X is direct. We "so t [(M) x --[()n (UorkM)X P(M) X --P(M)n (DerkM) x.Now if p � Derk(L9(M)((T))= Derk(M xpT) is such that pKc K we define for each X� r. endomorphisms p)( � Endk)(M ) by theformula pxa = (, eex)P(a e l), 45 where e( .'((T) -- K)(' e K( - K is the natural projectiononto the y-component. In other words the

following formula holds: (3.1.) p(a� 1) =y(pxa)e x for all a ed)(M). One checks easilythat Pt � Derk(M) and PX � DerK(M) forall X J. We put pO = PX E Derk(U)Moreover for any X we put Xa(M) X = Xa()n (M) X. It is easy to check that Xa(M)=eXa(M) and that we have Xa(M x pT) --Xa(M) i � I c (() ((T). Moreover sinceXm(M) = I one chec{<s that Xm.(M x pT) --i � Xm(T) c (M) � ((T) so we have V/(M xpT) = Hom(Xm(T), Xa(M) 1) (.2) LEMMA.Let p Derk(M x pT) such that pK c K as in(3.1). Then p � P(M x T) H and only if thefollowing conditions are satisfied: 1) Forany X � r. we have p(l X) = aX�X forsome a X � Xa(M) 1 . 2) For any X � r. wehave (fx ) l)PM p I = (p I fx � 1 + r. fxax)PM 3) For any X,X' � r., X I we have

(fX,X_ 1 I)pMP X = (pxfy, I)P M a) poe M =cup � ) For any X r. we have r'SMP I SMf-+ axf X--'r'f I P Proof. 1) is equivalent to theact that pp and (p i + ! p)p agree on ] �r.(use(3.1)). We claim that 2)+3) Js equivalentto the fact that these maps agree on (M) I.Indeed, for a � )(M) we have (using (3.1.1),(3.1.)): = r.x,((pxa)(l)) � x � (pa)()� (p J+ ! p)p(a l) = r. px,(fxa(l)) X' a(2)X + + r.f7a(I) 1 e Px,a(2) XX' + r. fxa(l) I a(2)axe XUsing the fact that 0)(M)<(T)C(M)(P(T) is afree d)(M)CP(M)-module with basis i ( � !.}( and identifying coefficients we get ourclaim.

48 Condition ) is equivalent to pc= ep.Finally using formula (3. h2) and acomputation similar to the one above we see

that condition 2) is equivalent to p$ = 0.3)COROLLARY. The map assigning to each p� P(M x iT) the function (pa) (3.1), (3.2)gives a hi}action between P(M x pT) and theset of all functions T = Xm(T) + (DerkM) x(Xa(M)l) X (Pxa) satisfying the followingconditions: a) The map ' a X is a grouphomomorphlsm b) plKc K PX L() or 1 X 1and or y a () here are only finiSely many swhich p 0. c) The relations 2), 3), ), ) rom(3.2) are salsied (wlh = . p. Les check firs hali p � P( x T) hen PX L(M) I. IndeedsumminR up he relations (3.2), S) or 1 � wege p = (p 1) m which means ha p L().Inruclng laer ormula In (3.2), 3) again andapplying l we gel .lp X = P., whl is wha wewanted. To conclude R i sufflcien o e t orany function satisfying a) b) c) above he

{ormula p(a X) = (Zpa X) + aa x a � (), X� Z defines elemt p P( x T). (.q) COROLY.Aume in afions ave ha p � P( x T). Then thelollowing relations ld f pO d a a) ppo = (pOe I + 1 pO + tX a PI. c) Is (3.2), ). a) lollowsby summing up (3.2) 2) ov 1 � d adding othe obtained relation relations Mpx = (pl)pM f X I (cf. (3.3)). b) follows by summingup (3.2) ) over 1 X � E. ( There e threeremarkable ma from P(xT) namely P( x )+L(), p p+ = X I p x + x p + vp, 0. P( x T)*Derk , p *p�=p++pl , Of course he secondmap assigns o any p h map a cf. our previousnotations in (3.3) and 47 where we identifyV(M x pT) with Hom(T.,Xa(M)i). We havethe following properties of these maps.' 1)pOe p(M)epO is SM-invariant =[?p vanisheson (T,{(M)) (the latter simply means that p

kills the weights, viewed as elements of (MxiT )3 we shah repeatedly us this expressionJn what follows). 2) If p+ = 0 then pO(DerkM)l 3) We have an exact sequence 0 +P(M) 1 + P(M x pT) + L(M)eW(M x pT) *)We have {p � INIVl x pT)Ip + = 0}{(p�,a) e (DerkM) 1 x Hom(T.,Xa(M)l)p�,a satisfy a), b), c) In (3. t)} 2) The imageof JL? .. p (M x iT)+ W(M x pT) contains thespace of those maps a: T.+ Xa(M)i whichvanish on e(T,((M)). So clearly equalityholds provided P(M x pT) kills the weights.Proof. 1) follows directly from (3. t). To get2) note that pO = Pl so introducing (3.#), a)in (3.2), 2) and appying 1 )gM we get fxp �= p�f X for all xwhlch Is what we want. Toget 3) note that the inclusion P(M)lc P(M xpT) Is given by associating to any Ple INM)I

the function I +(pt,O), X (0,0) for X 1, cf. theidentification In (3.3) hence clearly P(M) 1Is contained in the kernel of P( x pT) [,(M)�W(M x pT).' The converse inclusion is alsoeasily seen. Finally t0, ) are easily checkedthrough the Identification (3.3)-' for 2) put aX = a00 and PX = 0 for all X. (3.6)EEM^EK. We will give at the end of (�.2)an example of a situation when P(M x pT)contarns a derivation not killing the weightsl.This shows that P(M x pT) can be rathercomplicated. In our example T will be G m(hence T will he algebraic) and M will beunlpotent (but non-algebraic)[ we dont knowwhether it is possible to find such anexample with both T and M algebraic. On theother hand we have the following.. (.7)[,EMMA. !n notations above assume ts a

vector group scheme (i.e. ((M) Is thesymmetric algebra over its subspace Xa(M)of primitive elements). Then P(M x pT) killsthe weights of T In d)(M)o Equivalently, theimage of the map J[.'P(MxpT)+W(MxpT)= =Hom(Z,Xa(M) l) is precisely the space of allmaps in Hom(T.,Xa(M)i) vanishing on the setof weights (T,t(M)). Proof. Applying the tvistmap .. (IVl)m((M)+ ((M))(M) to formula (3.t0, a) and using the fact that TI M = i M weget IM'p � = (1 pO + pO I)l +

hence using (3.#), a) once again we get that(r. fX & ax)p M = (r.a( Let x e Xa(M) X, x 0be a primitive element of weight X Iiapplying the latter equality to x we getXeaX: ax�X. Ve claim that ax:O; for if it isnot so the latter equality im plies x -- Xa X

� Xa(M) 1 which is impossible since )r hSo a X = 0 for all weights of T in Xa(M)(i.e. for all X such that Xa() X 0). But theweights of T in (M) are products of weightsof T in Xa(M) and hence a( = 0 for all ( �(T,{(M)). (3.8) LEMMA. Assume L(M)contains a finite dimensional Lie subalgebraL with the following properties-' a) L(/)( c Lfor all ( I and [P(M),L] c L in DerkM. b) (M)is locally nilpotent as an L-module. c) ((M)is locally finite as a P(M)-m odule. Put P:Ker(V: P(U x pT) g(M x pT)). Then ((M xpT) is locally finite as a P-module; inparticular Ker J c P(M x pT,fin). Proof. Firstnote that since dmKL< the set =(}(�r.L(M)X0, 1) of all non-trivial weights of T inL(M) is finite. Next we claim that for any a(), the K[P]-subm o- dule of (() (T)

generated by a I has finite dimension. Indeed,by (3.a) for p � P we have that pOp(M)(because a X=0 fgr all () hence Pl--P�'P+P()L and PX L for ' I. Since (() is alocally finite both as a P(M)-m odule and asan L-module and since [P(M),L] c L itfollows that (() is locally finite as a P(M))L-module. Choose a finite dimensional vectorspace V of (() containing a � (() and stableunder P(M)eL. Hypothesis b) says that for Vthere exists an integer N _ I such that ele 2 ...e N � = 0 for all e l, ... ,e N � L. Then It iseasy to check using [P(M),L] c L once againthat we have ele 2 . .. en� -- 0 for all e 1, ...,{a n � P(M)e L provided card (ie i � L)>_I. Now let (N) be the K-span in (T) of allproducts of the form XiX 2 ... X N with X i� u 1) -- . For pl,...,pn � p we have

plp2...pn(a)l)= I 2 ...P;na " r'Px I PX2 XnXn-1 the sum bein taken for all n-uples (Xl,...Jn)with XinU. As we have seen, 1 2 n = 0whenever card {1;' i � *} > N.Consequently K[P](a 1) c � (N) and our PXIPx2 ' ' ' P)'n a - claim is proved. Now weclaim (and this will close the proof) that forany x e)� (T) we have-dimKK[P]x ( .Indeed write x --r.a I i and let � be a finitedimensional K[P]-submodule of {(IU)�((T) containing all ai's. Then r.�(l a)Xl) is afinite dimensional KiP]-module containin xand our lemma is proved. (3.9) REMARK. Ifin (3.8) above we replace the condition c) by"c') P(M) = P(l,fin) (in 49 other words (M)is locally finite as Kip]-module for anypeP(M))" then we still get the second part ofthe conclusion namely that "Ker J[?c P(M x

pT,fin)". This follows just by inspecting theproof of (3.10) LEMMA. Let G be anirreducible unipotent arline algebraic K-group. Then: l) Assume char K ) 0 and G iscommutative. Then P(G) -- P(G,fin). 2)Assume char K = 0. Then (C)'Is locally finiteas a D(C)-module. Proof. l) By [H] pp. 2 and63-6, ((G) is locally nilpotent as an L(G)-module i.e. for all y ((G) there exists anInteger N such that 012... aNY=0 for all 01..... e N eL(.G). If Xx End K (G) denotes themultiplication in (G) by some element x e(C) then for any O eL(G) we have [0,Xx] =).; so by (I. 1.3) if x Xa(C) then [0,] Is themultiplication by some scalar In K. Nowpick and element p = r. a 6 l = rnie (eXa(G)eL(C)= P(C/K)= P(G), cf. (I. 1.9)where ()i is a K-basis of L(G) and a Xa(G).

Then it is easy to see using the aboveremarks that K[p]y is contained in the K-linear span in (G) of the set (iai2 ... ain%l2... In particular dlmKK[p]y < for all yC(G)and assertion 1) is proved. 2) By [H] p.231 the image of the map (I. hi3) G(G)+G, eJG , U(L(c))O coincides with the algebraB(L(G)) of nilpotent representative functionson U(L(G)) (recall that by definition B(L(G))consists of all functionsis on U(L(C))annihilating some power of the ideal 3 =L(G)U(L(C)). Now B(L(G)) is a locallyfinite D(G)-submodule of U(L(G)) since it isthe union o the finite dimensional D(G)-submodules B n = {! .U(L(G))�; f vanisheson 3 n } Since Jc is a map of D(G)-modulas,d)(G) will be locally finite and we are done.(3.11) REMARKS. l)(G) is not a locally

finhe D(G)-module even in the case G = G a,i d char K = q > 0 (if G a = SpecK[F,], Pi =q � P(G) then pi = q hence dimK(P(G)D =.) so the conclusion in (3.10), 1) cannot bereplaced by "((G) Is locally finite as a D(C)-moduJe". 2) It would be interesting to knowif in (3.10), 1) one can drop thecommutattvity assumption; one can formulatethe conjecture that P(G) = P(G,fin) for anyirreducible arline unlpotent algebraic K-group G (char K arbitrary). (3.12) Let C bean irreducible arline algebraic K-group, letT be a maximal torus of the radical o! G andlet (T,�(C))c Xm(T ) be the set of weightsof the action of T on ((G) be Innerautomorphisms. Moreover let (G) be thesubset of Xm(G) of all characters of Gwhose

5O restriction to T belongs to (T,(C)); theelements of (G) will be called the weights ofG. Since the maximal tori of the radical areall coniugate, (G) does not depend on thechoice of T. We let Wo(G) be the subspace ofW(G) = Hom(Xm(G), Xa(G)) consisting ofall maps vanishing on (G). Here is our mainresult in arbitrary characteristic: (3.13)COROLLARY. Let G be a solvableirreducible arline algebraic.K-group. 1)Assume the unipotent radical of G iscommutaive. Then the kernel of is containedin P(G,fln). 2) Assume the unipotent radicalof G is a vector group. Then P(G) kills theweights of G. So the image of J[V: P(G)W(G) equals Wo(G). Proof. By standardstructure theory [H], G -- MTwith U theunipotent radtcal of G and T a maximal torus.

Applying Lemma (3.8) and Remark (3.9) forL -- L() together with Lemma (3.10) we getassertion l) of the Corollary. To check asser,tJon 2) note that the weights of G as definedin (3.12) can be identified with the weightsof T on ((M) as defined in (3.1) so we mayconclude by (3.7). (3.1e) REMARK. Innotations o! (3.13) it is not true that P(G,fln)is contained in Ker (?: P(G) +W(G)). Indeedtake G -- G a x G m as in (I. 3.9); then withnotations from loc. cit. this phenomenon doesnot occur in characteristic zero as shown bythe following: (3.15) LEMMA. Assume charK = 0 and G is an arline integral K-groupscheme. Then P(G,fin) = Ker( ?: P(G)+W(G)) Proof. Let p P(G,fln) and assume lpX-' a 0 for some X EXm(G)' By (I. 1.3),Xa(G) is stable under P(G). By [H] p. $g the

symmetric algebra $(Xa(G)) embeds into(G) and then of course each homogeneouscomponent $n(Xa(G)) is a P(G)-submoduleof ((G). By induction we get that pax =(an.)X lorn_)0 with [n� ) Si(Xa(G)). By ourassumption the K-span of the family (pn n isfinite i<n- 1 dimensional' this implies that thesame holds for the family (an+ )n which isimpossible because a n e$n(Xa(G)). Thelemma Is proved. In what follows we devoteourselves to affine algebraic group incharacteristic zero. (3.16) THEOREM. Letchar K = 0 and let G be an irreducible ainealgebraic K-group. Then: l) P(G,in) = Ker(JT: D(G) +/(G)) and ((G) is locally finite asa P(G,fin)-module. 51 2) The image of J[?:P(G)+ W(G) contains Wo(G). H. oreprecisely there exists a K-linear map E:

Wo(G)e P(G/K), a E a such that 9) � E isthe natural inclusion Wo(G)c W(G) andhaving the following properties: a) Im E isan abelian ideal in R(G/K,fJn)elm F(cD(G/K)). b) For any� AutG and a� Wo(G)we haveo'lEa o = Eoa where we still denoteby the induced automorphisms of ((G) andWo(G). c) For any pe Der(K/K G) uponletting G = (G o a KG-grou p) and letting p*be GOKGK the trivial llftings of p from K toC(G) = K KG(G o) and to Wo(G) = KKGWo(Go ) we have [p*,F a] = ERa a forall a REMARK. Assertion l) can be easilydeduced in fact from our previous result (I.1.18), cf. (].2) belowl but our proof here for(3.16) will be purely algebraic. We startwith some preparations (cf. also (3.17)LEMMA. Assume char K = 0 and g is a Lie

D-algebra of finite dimension. Then theradical r of g is a Lie D-subalgebra. Proof.By (l. 3.2) and (I. 3. a) g splits over some D-field extension K1/K with K o := KiD = K Dso gK l = goK K] for some Lie Ko-algebrago' Let r � be the radical of go' o BothroKoK l and rKK l coincide with the radicalof gKKl . But r=(r)KKl)n g= = (ro) K Kl}n gand the latter space is preserved by D. o(3.18) COROLLARY. Let char K = 0 and Gbe an irreducible affine algebraic K-group.Then the radical R of G is an algebraicD(G)-subgroup of G. In particular there is anatural Lie K/k-algebra restriction mapP(G)+ P(R). Proo:L Combine (I. 1.1) and(3.17) above. (3.19) COROLLARY. Assumechar K -- 0 and M is an irreducible arlinealgebraic K-group whose radical is

unipotent. Then ((M) is locally finite as aD(M)-module. Proof. By (I. 3.2) and (I. 3.#)we may assume that g = L(G) ts a split LieD-algebra so g = goK K (K � = KD go = gDD = D(G)). Let go -- ro + So be adecamposltlon of go with r e its o radicaland s o a complementary semisimpie Liealgebra. Then by [H] p. ll2, s--SoK is analgebraic Lie subalgebra o g,s = L(S), 5c G.By (l. 1.1;) both 5 and the radical R of G arealgebraic D-subgroups of G; therefore themultiplication map R x 5+ G is a D-maphence (G) identifies with a D-submodule of((R) (S) so we are reduced to proving thatboth (R) and (($) are locally finite D-modules. The assertion about ((R) followsfrom (3.10) because R is unlpotent. Theassertion or ((5) ollows for instance from

(2.13) (it also follows from representabilityof Au.tS [GDI but the latter fact involves thewhole structure theory of reductive groupschemes over non-reduced basis so ourargument (2. ! l) should be viewed as "the

52 elementary" argument for the localflnitness of C(S); for another "elementary"argument see ge shall repeatedly use thefollowing rernarks (3.20) Let G = G I)G 2 bea semidirect product of irreducible atfinealgebraic K-groups, char K = 0 and identify((G) with (Gi)I(G 2) via the multiplicationmap G 1 x G2+ G. If Xm(G 1) = I thenXm(G) identifies with Xrn(G2). I! Xa(G 2) =0 then Xa(G) = Xa(Gi )G2. Now assumeG2* G 1 is an isogeny. Then the map Xrn(Gl)Xm(G 2) is injectlve with finite �okernel

and the map Xa(G 1 ) Xa(G 2) is anisornorphlsrn. In particular there is aninduced lsornorphism V(G l ) = V(G2).(3.21) To prove Theorem (3.16) we fix somenotations let U be the unipotent radical of G,let H be a maximal reductive subgroup of Gand T the radical of H. By a theorem ofMostow [H] p. 117, G = UH{ moreover T isof course a maximal torus of the radical R ofG. Put S = [H,H], U = US and := T = Us4(S xT). The isogeny S xT- H induces an isogeny (, G. g/e write I: = Xm(T). Note thatrestriction map Xm(G) r. is injectlye and hasfinite cokernell Indeed this map is easilyseen to identify with the map Xm(G) =Xm(H) + Xm(S x T) = Xrn(T) cf. (3.20)above and we are done also by (3.20). By(3.15) Ker (V = P(G)+ W(G)) contains

P(G,fln); let's prove it actually equalsP(G,fin). By (I. 1.21) and (3.20) we have acommutative square P(G) ---W(G) I [I P() ---,w() so it is sufiicient to prove that (() is alocally finite P-module where P -- Ker(l?:P() + g()). Since T centralizes S, L(M) X --(L(U) L(S)) -- L(U)(c L(U) for all X � r., y1. Due to (3.19) we may apply Lemma (3.8)to our situation (with L -- L(U)I note that[P(M), L(U)] c L(U) by (3.18) and (I. 1.16))so we get that (() Is locally finite as a P-module and assertion 1) in (3. i6) is proved.To prove assertion 2) let's define the map EWo(G) - P(G), a + E a as follows. The actionof T on H by the left (or, which is the sameby the right) translations gives a Z-gradationd() � d)(.)X X Let f(:((H)-((H)Yc((H) bethe corresponding projections. Restriction

provides an identification W(G) -Hom(r.,X,au)l'{). Then, any a : /o(G) �Horn(Xm(G), Xa(G)) can be viewed as ahomomorphisrn r. Xa(U ) vanishing on theweights (T,C0(G)) e(T,((M)). Moreover,identify ((G) with ('(U)((H) via themultiplication map U x H- G and deitne theK-linear endomorphisrn ! of ((G) by thefornula; Ea(xOy )-_ r. x a()r)fY(y), 'x �((U),y d(H). X Clearly E a Is an ((U)-derlvatton.To check that E a E P(G) it is sufficient tocheck that Its lifting a to d() -- C(U) � (S xT) belongs to P(). But we have Ea(Xz)--i:xa(x)''X(y), x ed'(U), z E(SxT) X where'Xg((S x T)*(S x T)Xc ((S x T) is thecorresponding projection and d)(S x T)X_- -- d)(S)KX. Now a P() by (3.3) (in notationsof (3.3) put a X -- a() and PX -- 0 for all So

our map E Is well-defined and clearly (V)�E Is the inclusion /o(G)c /(G) so Wo(G)cIs(L?). It is also clear that Im(g) is an abellansubalgebra of P(G/K). To prove theremaining assertions in clalrns a) and b) of(3.16) note that ^ut G is generated by lnt Gand the group Aut(G,H) of allautornorphisms of G preserving H.Consequently by (I. 3.12) P(G/K,fin)-- _-L(AutG) is generated by L(lnt G) andL(Aut(G,H)).' So it is sufficient to check thatthe following hold for all a Wo(G): (3.21.1)o' IEao -- E a for all o e !rn(M Int G)(3.21.2) [d,E a] 0 for all d Im(L(U) L(Int G))(3.21.3) o' IEao-- Ea for all o � ^ut(G,H))(3.21./) [d,E a] Irn E for all d z L(Aut(G,H))To prove the assertions above we mayassume G = .. Start with (3.21.1) and

(3.21.2). The action of M on G by innerautornorphlsrns is given by the corn positionof maps MxMxT a MxMxT b xMxT C)MxT(m,x,t) : c (mx,m,t) , (rnx,trn'lt'l,t)-+(mxlrn'lt'l,t) where a and c are induced byrnultiplications while b is induced by takingfirst the antipode on the middle factor andthen applying the action MxT M of T on Mby inner autornorphisms. This irnmediatelyimplies that for any z � (M) we have that theimage of z 1 via the map a' b + c a: (M) )(9(T) - ((ikl) ((IVI) ((T) belongs to )(M)d)(M)< {(G)> where < {(G)>c d)(T) Is the K-span of the group generated by {(G)= e(T,({)(G)). This Shows that if oand d are as In(3.21.1) and (3.21.2) we have o(z 1), d(z 1)� (ikl))< (G) >. This immediately impliesthat the derivations o-IEao- E a and [d,E a]

Vanish on ((M) 1. On the other hand (since ois the identity on 1 {3(T) and on Xa(G) and dis the zero map on these spaces), the abovederivations also vanish on 1 ) (T) hence theyvanish on )(M))d)(T) and (3.21.1), (3.21.2)are proved. To prove (3.21.3) let o stilldenote the

54 restriction of a to M and T. Then for x((M),) � Xm(T)C ((T) we have - IEao(X�X) = o'lEa(Ox eoX) = 'l(a(oX)oX oX) = =o'l(a())x X = To prove (3.21.) let d stilldenote the derivation induced on d(M');noting that d kills Xm(T) we have for x andX as above: [d,Ea](x eX) = xd(a(x)) e =Eda(X X) 5o assertions a) and b) areproved. Assertion c) can be provedsimilarily. (3,22) COROLIAR�. Assume In

(3. i6) that either the unipotent radical of Gis commutative or the radical of G isnilpotent. Then Im(J[V: P(G)+ W(G)) =Wo(G) and ]m E is an abelian representativeideal in P(C). Moreover each element of ImE kills Xa(G). Proof. Assume the uniputentradical of G Is commutative. By (3.18) wehave a commutative diagram Jc ? P(C) +w(c) p(l) ,,, w(R) Moreover the rightvertical arrow is injectle and viewing it asan inclusion we have W(G)n Wo(R) =Wo(G). By (3.7) Im(J[RV)C Wo(R) so weobtain that Im(ZGV)C Wo(G). A similarargument shows that the latter holds If isnilpent. This and (3.16) show that K P(G) =G and P(G) = Pl P2 P3 with Pl = Der(/Kc)'P2 = P(C/K,fin), P3 =Im E and [PI'PI ]c Pl'[PI'P2 ]c P2' [PI'P3 ]c P3 [P2'P2 ]c P2'

[P2'P3 ]c P3 [P3,P3] -- 0. This and theAutG-lnvarlance of ImE proved in (3.16)show that ImE is an abelJan representativeideal. It clearly kills Xa(G) so we are done.EXAMPLE 1 Assume char K = 0, H is areductlve Irreducible algebraic K-group withradical T, V is a K-linear space of finitedimension and let = H GL(V) be arepresentation. Put G = VxpH. Then (3.16)and (3.22) say that P(G) has an abelianrepresentative ideal which naturallyidentifies with the space Wo(G) of allhomomorphisms Xm(H) +(V�) H whichvanish on all those y e Xm(H) for which ylTbelongs to the set (TV) of all weights of T inV . So if [ is the subgroup of Xm(T)generated by e(T,V) then dim Wo(G) -- (rankH - rank )dim((V�) H) S/here recall that

rank H -- dim T by definition. Consequentlywe get: dim L(Aut G) - dim L(Aut C)= (rankH - rank )dim((V�) H) In particular P(G) =P(G,fin) (equivalently L(Aut G) = L(Aut G))in each of the following cases a) rank 1' =rank H b) (v�) H = 0 EXAMPLE 2 Wespecialize the above example and then saymore. Start with a representation I: GL NGL(V) and put G = VxpGLN; then P(G)behaves quite differently according towhether p factors through PGL N or not.Indeed= 2e) if p doesn't factor through PGLN then P(C) = P(C,fin) (indeed in notationsof the previous example rank 1' = rank H =ll). 213) If p factors through PGL N thenP(C) has an abelian representative ideal ofdimension equal to the dimension of thespace (��) GLN of fixed elements of p in

V �. We may describe this ideal explicitelyas follows. Let G = 5pecK[[,y,d---] , [= ([1.... 'In )' y = (Yij)I_<i,j_<N where [ is abasis of V �. Then 3 consists of allderivations of the form: Pf -- f(D( Yij y-S-,),f-- f(D �(V�) CLN ZK[ i i,j lj By (3.16)we have seen that Wo(G) c Im(J[). In whatfollows we use the (analltlcally proved)result (1.1) to "aproximate lm([?) fromabove" more precisely to show that Im(J[?)is contained in the kernel of a natural (Liealgebra theoretically defined) map W(C) +H2(L(U),L(U)). So fix an irreducible arlinealgebraic K-group G (char K = 0) withradical R and unlpotent radical U, pick amaximal torus T of R and put r -- L(R), u =L(U), t -- L(T). (3.23) We start with theremark that W(C) naturally identifies with

Hom(u/[g,g] n u,r/u). indeed, identify bothL(G m) and L(G a) with the field K via theidentifications G a = SpecK[t], Grn =SpecKit,t-1]. Moreover write G/[G,C;] -- G (a) xC (m) with C (a) a'vector group and G (m) a

torusl clearly the natural map U/U n [G,G] +G (a) is an isomorphlsm while the map T +G (m) is an isoseny. We have identificationsL(G(a)) � Hom(L(G(a)), L(Ga))Hom(G(a),Ga ) -- Xa(G) and L(G(m)) � =Hom(L(G(m)), L(Gm)) -- Xm(G))K. Hencewe have an identification: W(G) --Hom(Xm(G) K, Xa(G)) = Hom(L(G(m)) �,L(G(a)) �) = = Hom(L(G(a)), L(G(m))) --Hom(u/u n [8,g],r/u) (3.24) Next let's notethat for p P(G) the image of JtVp in Hom(u/un [8,g], r/u) (still denoted so) has the

following particularly simple description:upon letting l[:r + flu be the naturalprojection we claim that JVp = l[(px) for allx = u where c denotes the image of x in u/u n[g,g: In particular Vp = 0 If and only if pu c u(this together with (3.16) and (I. 1.16)provide a purely algebraic proof of (l.lg)).To check our claim note that under theIdentification in (3.23) we have thefollowing formula (d x) o .Vp = d(x'lp)) forall character X:R/U-G m (where dx:L(R/U)-L(Gm)= K is the tangent map of X, -1 andsimllarily d(,' lp) is its X PX Xa(G) isviewed as an additive character U/U n[G,G]- G a tangent map). Identifying nowL(U/U n [G,G]) (respectively L(R/U)) with asubspace of I(U/U n [G,G]) � (respectivelyC(R/U) �) the above formula reads ^

(VpxXx) = (x- lpx) for all x L(U). But x(xPX) coincides with the image of ( under themap (R/U)((!)=C(U))C(T) PC(U)aC,0(T)IET, (U) x ,K and our claim is proved. (3.2)Now we have the following Lie algebratheoretic construction. Let r be any Lie -algebra, u an ideal in r containing [r,r] and s:r/u r a Lie algebra section of the projectionr/u. For each f � Hom(u/[r,r], r/u), thebilinear alternating form b(f) -' uxu + udefined by b(f)(x,y) = [sf,y] - [sf,x] for x,y �u (where , are the images of x,y in u/[r,r]) iseasily seen to be a 2-cocycle of u In u so wecan consider the linear map induced by b: 1:Horn(u/Jr,r], r/u) + H2(u,u) (3.2) Comingback to our specific situation where u =L(U)) r = L(R) and noting that 57 Hom(u/u n[g,g], r/u) is a sub, pace of Horn(u/Jr,r], r/u)

we get a map (which we still call lB: W(G)-H2(u,u) by composing the map lB from(3.2.s) with the identification isomorphismW(G) = Hom(u/u n [g,g], r/u) from (3.23).Then we have: (3.27) PROPOSITION. Letchar K = 0 and C be an irreducible affinealgebraic K-group. Then there is a complexexact in the first two terms: 0 + P(C,fln) -*P(C) J[V, W(G) -H2(L(U),L(U)) where U tsthe unlpotent radical of G. Proof. By (3.16)the only thing to prove is that lm([V)c Ker ISo assume a = J, Vp W(G) fo' some p P(G).By (l.l) K � = K Kip] is a field of definitionfor G. 'riting G = Go)Ko K (G o a Ko-group) let's denote by p* the trivial lifting of pfrom K to G. Moreover let � l :r +r, e 2: r+r be the projections onto u and trespectively. Then by (3.2), (3.2) we have

(*) b(VpXx,y) = [e2Px,y] - [e2py,x] for allx,y u. Projecting the equality p[x,y] = [px,y]+ [x,py] on u and using the relation () we get(.) elP[X,y] -- [elPx,y] + [x,elpy] + b([?pXx,y) Now clearly p+ maps u into u and wehave (...) p#[x,y] = [p*x,y + [x,p*y]Substracting (a,) from (,) and putting v = elP- p eEndKU we get b([?pXX,y) -- v[x,y] -[vx,y] - [x,vy] which shows that b(J[?p) is acoboundary and we are done. (3.28)REMARK. Note that in the abstract frame(3.25) one can easily check that for any !ellore(u/it,r], r/u) the form b(f) defines infact a (new) Lie algebra multiplication on ulComing back to our specific situation (3.2(;)we see that for any p � P(C) we get a "newLie algebra multiplication" b(J[ 9p):uxu'+uon u. It would be interesting to

understand what infor- mation about pcarries this "new Lie algebra multiplication".We end this section by recording some moreconsequences of our results (for ageneralisation to the non-linear case, see (IV.1.2)): (3.2) COROLLARY. Let char K = 0and G be an irreducible arline algebraic K-group with radical R. Then: 1) The naturalrestriction map P(C)/P(G,fin) +P(R)/P(R,fin) is tnJective. 2) !f i: G +G' isan isogeny and i#P(G ')- P(G) is the naturallifting map, then

i (P(G',fin)) = P(G,/in) n i* (P(G')). 3) If thecenter of G ts finite then P(G) = P(G,fin). ) IfXm(G) -- I or Xa(G) = 0 then P(G) = Proof.1) follows from (3.16) and the fact that themap W(G)* W(lq) is injectire, cf. (3.21). t)

follows from (3.16). 2) follows from (:).It;)and the fact that the map W(G')* W(G) is anisomorphism, cf. (:).20). To check :)) startwith a preparation. Assume V is an N-dimensional E)-module. Then the coordinatealgebra )(GL(V)) of GL(�) has a naturalstructure of Ftopf D-algebra define byidentifying C(GL(V)) with $(gl(�)�)[d -1]where S = "symmetric algebra" and d �$N(gl(�)�) is the "determinant". We claimthat C(GL(V)) is locally finite; indeedS(gl(V) �) clearly is so and we are done bynoting that d is killed by P (to check thisreplace K by some D-field extension of Itsuch that V splits so V will have a K-basiscontained in V D. Associated to this basisthere is a P-constant basis Xij of gl(�)l nowd is a polynomial in the Xij's with integer

coefficients so is killed by PI). Coming backto our group G, let Ad: G+ GL(g), g = L(C)be Its adjolnt representation. Using thedescription of Ad in [H] p. 51 one checksthat Acl : C(GL(g))+ C(G) is a D(G)-algebramap. Consequently if Z is the center of G,((G/Z) is a locally finite D(G)-module(being identified with ((GL(g))/Ker Ad*).By assertion 2), ((G) must be locally finiteas a Kip]-module for all p P(G) and we aredone. EXAMPLE The example 2x) in (:).22)can be arranged to provide an example ofarline algebraic K-group G for which: i)P(G) -- P(G,fin). il) G has a positivedimensional center. iii) Xm(G) 1. iv) Xa(G)0. Indeed it is sufficient to choose 10 p I P2where Pl: GLN* GL(�1) is trivial, dim� 1) 0 P2: GLN* GL(V2) does not factor

through PGL N . We can rephrase part of ourresults in terms of algebraic D-groups:(3.30) COROLLARY. Let G be anirreducible arline algebraic D-group (char Karbitrary, D KIP]). Then 1) Assume dim P --l G solvable and the unipotent radical of G iscommutative. If the group like elements of(P(G) are.P-constants then G is locally finite.2) Assume char K -- 0. Then the radical R ofG is an algebraic D-subgroup of G. loreoverG is locally finite i! and only if R is so, ifand only if all group like elements of (P(C)are P- constants. 3) Assume char K -- 0 andi: G * G' is an isogeny of algebraic D-groups. Then G is locally finite if and only ifG' is so. ) Assume char K -- 0 and the centerof G is finite. Then G is locally finite. )Assume char K = 0 and either Xm(C) = I or

Xa(G) = 0. Then G is locally finite.

CHAPTER 3. COMMUTAllVEALGEBRAIC D-GROUPS In the firstsection of this chapter we introduce the"logarithmic Gauss-Yanin connection" andthe "total Gauss-llanin connection"associated to it. In the second section weprove a "duality theorem" laying that the"total Gauss-Mania connection" on the "totalde Rham space" H)R(A) t of an abelianvariety A is isomorphlc as a D-module withthe continuous dual of the inverse limit of theLie algebras of the "relative projectivehulls" of A, viewed with its "adjolnt" D-module structure. This implies a precisedescription of P(C) for any irreducible cornm utative algebraic K-group C; note that K

P(C) need not be a field of definition for C!11e also get from the above duality thefollowing regularity theorem: let C be anirreducible algebraic D-group withC(C) = K(D = K[P], P = DerkK , tr. dug. K/k < 41 thenL(C) is a regular D-module in Deligne'ssense. This together with a "descentcriterion" will be proved in section 3.Everywhere in this,chapter K isalgebraically closed of characteristic zeroand if not otherwise specified P is any LieK/k-algebra and D = K[P]L 1. Logarithmic G- Manin cmmectlen Let V be a smoothprojective K-variety. Recall from [Ka], [K0]that there is a natural integrable connection v,DerkK p Vp called the "Gauss-Manln"connectionl here HR(V) is the first de Rhamcohomology space of V (see (1.1) for a quick

review o! this). Recall from lillY] that thereis a "m ultipllcative analogue" of HR(V)a :=HR(� ) which is an abellan group calledhere HIDR(V)m , see (1.2)I we shall definea K-linear map v: DerkK +OmgrmR(V) m,Ve shall call the pair (?,9) the logarithmicGauss-Mania DerkK-connection on (H)R(V)m , H)R(V)a); HIDR(�) a will be viewedas an abelian Lie K-algebra on which HR(V)m acts trivially. 61 (1.1) For convenience ofthe reader and also for computationalpurposes we recall briefly the constructionof ? ci. [K0]. Let = (Ui) l be an a/finecovering o V, let C; ts = (C a ), o t cts=c;s(zO= H% l'v/K ) a io<11...<I s o'" s bethe standard "ech-de Rham" double complexof V and let C= C() be the simple complexassociated to C. Recall that the differential

C:+ C 1 Is ilven by the formula a (fl) + (d/i,fj - fl ), fie H�(Ui,/ V) . while thedifferential C 1+ C 2 Is given by the formulaa a (u i,xij) + (dr0 t,u j - i - dxij'Xjk - Xik +xij) where m 1 � H�(Ui,/K ), Xlj ydefinition, the l-st de Rham cohomologyspace HR(V) (called here also HR(V) a) isthe l-st hypercohomology K-linear space ofthe de Rham complex (called DR(V)a)=Recall that HR(V) a = HI(c). Recall alsofrom [Dell] that the spectral sequenceErrS(a) associated to C degenerates in E I. Inparticular we have an exact sequence.'(1.1.1) 0 ' H 2V/K) - H R(V)a (V) + 0 Theconnection V is defined as follows. LeteiDer k (U i) be any lifting of p DerkK. Then Vp takes a class in HIDR(�)a represented by(m i' xij)� CI Into the class represented by

Here Li : H�(UiP,/K)+ H�(Uifi/K) is the"Lie derivative" in the directlone i (recallthat H�(Ui2 l V/}= 3/32 where 3=Ker(/(Ui))d7(Ui)* i(Ui)) so H�(Ui /K)hasanatural structure of K)i].module then Ltd. isby definition the multiplication with 1 in thismodule structure)l moreover is the con'actionbetween K-derivations and l-forms withvalues regular functions. (1.2) Consider nowthe complex DR(V) m of abellan sheaves ldlo_., l 2 ,,, '"V/K " V/K and let HiDR(V)mdenote the l-st hypercohomology group ofthis complex cf. for instance [MM] I). 31. Soif C Is the direct limit (over all coverlngs,)of the double complexes C() associated toDR(Y) m and and if is the simple complexassociated to the double complex

62 C" then we have nl Now let ErrS(m) bethe spectral sequence associated to C; it doesnot degenerate in E l like in the "additivecase" but one can easily check that El(m) =Pic(V) and st E S(m) = !..I (nV/K) for t_) l,s_>0 Consequently E201(m) = PicT(V) (seerG]) and by degeneration in E l of ES(a),E220(m) = H�("2V/K ) It is known(essentially by [MM]) that dl(m): El(m) '*'E220(m) is the zero map. For conveniencewe check this; it is sufficient to check thatdl(m) is a morphism of algebraic groups(because PicT(V) Is an extension of a finitegroup by an abelian variety while H�(i2/K)is an algebraic vector group so no algebraicnonzero homomorphJsm can exist betweenthem). To check algebraicJty of dl(m)consider for any noetherian reduced K-

algebra R the complex dlog, nl d 2 I +( te R"V e R/R ---'"V eR/R '" Let ctS(m) R be theassociated double complex and ErtS(m)R thecorresponding spectral sequence, Exactly asabove we have: EI(m)R = Plc(Ve R) =H (2VR/R)= s t (where we identified H (lV/K)with its associated algebraic vector groupSpec(S(Hs(t/K))). Clearly dl(m)R: Pic(V �R) + Hl(a R/R) factors through PJc(V eR)/Pic R = PiCV/K(I ) = (R-points ofPic�/K}. Moreover, since R is reduced thekernel of Pic�/K(R ) HI(y ( R/i ) equalsPic/K(R ). Analogously we claim thatdl(m)R EI(m)R E0(m)R 0 2 � ' + = H vmR/R ) factors through EI(m)R/Pic R =Pic//K(R). Indeed El(m) R is a quotient of {aC10(m)R + C01(m)R; da C20(m)R} 01 andd 2 (m) R is defined by takin the class o a

into the cl of da. ow J a = (u W.) then -1 -1 'cocycle and :- ,=y, dy,,. So l a da=(dui' mj'ui-Yij dYij' YjkYikYlj ) so (Yij) s a ; 1; ucomes from Pic R we have (after suitablyrefining the covering) that Ylj = ulu viisuitable ul's with vii belongin 8 to some ringof quotients of RI we get (becau dvjj = 0)hce dj = d(u i + u]duj) But [ + ulduj tchtogether to give a 81obal m the c]ass o[ davanishes and our claim Js proved. We getgroup momphisms behaving functoriaUy in R1 reded, hence we get a morphism oalgebraic K-grou Pica/K+ H�(/K ) (causeth these grou are retired). 5o the claim thatdl(m)= 0 follows. !n partJiar we get an exactsequence Now we define p Homgr(HR(V)m,H(V) a) for p DkK Jet =(U i) be an afinecovering of V let e I Derk(U i) Uftings of p

Jet C() be the simple mmp!ex ociated to theubJe mmplex C() and define a grouphomomorphism C()+ C() by the formula (i,Yij)+ (Lieei i,(ej - e 1) Kuj + y[jieiYij) m iH�(Ui'/K )' Yij O"(Ui n Uj). One chks byhand compation that this morphism is well-defined, sses to homm 1ogy and agrees with"refining "so we e a Uoup homomphlsm [Vp'HR(V) m + HR(V> a which dends K-linearlyon p P. The ir (VEV) defines a logarithmicDerkK.conntJon on (HR(V)m, R(V)a) calledIn what follows the logarithmic Gauss-aninconnection. e will prove later that It islntegrable f V an abelJan variety (i.e. that(0.16.2) holds).

(1.]) REMARK. Recall that if K = (: thenHR(� ) = H l(van, (:) and HR(V) m = HI(V

an, (:*). Moreover these two groups can beinterpreted in term of differentials of thesecond raspact]rely of the third kind on , cf.for instances N. Katz, invent. Math. ]8, 1-2(]972) p. 108. We won't need thisinterpretation in what follows. (!.4)REMARK. From the very definitions wehave (in notations of (1.1) and (1.2)) that forany p c DerkK the following diagram iscommutatlve H�=tV/K ),- , p (1.)REMARK. it follows from [K0], [Ka] (andindeed from the very definitions) that for anypc P the K-linear map Ho( tv/K ) Vcoincides with the cup product with theKoalaira-Spencer class p(p)c HI(T )(where p: DerkK+ HI(T ) denotes theKoalaira-Spencer map). (1.6) In provinglntegrability of the logarithmic Gauss-Manin

connection (and also in formulating our"duality theorem" In the next section) we areled to make a certain construction of linearalgebra whose abstract version we nowdescribe. Let's consider the following data:an abelian group Hrn, a K-linear space H aand a K-linear space H o equipped with agroup homomorphism Jm.; Ho+ Hm and K-linear map ja Ho+ H a (both jm,Ja injectire).By a total space for these data we will meana K-linear space H t equipped with a grouphomomorphism i m: Hm+ H t and with a K-linear map 1 a Ha+ H t such that thefollowing diagram is commutative Jm H o mH a --' H t a and such that for any other K-linear space H', any group homomorphism 1': Hm+ H' and any m K-linear 'map i' H' ' ' aHa+ with I m Jm = ia. Ja there is a uniqbe K-

linear map f Ht+ H' such that f i m i m and f ia-" Ot course (H t, ir, i a) is then unique up toa canonical 65 isomorphisml it is a sort of"fibred sum" of H m and H a over H o. totalspace always exists? it is constructed asfollows. First put Hm, K = (H m<zK)/Ker(Ho Z K+ HoKK = H o) and notethat we have an exact sequence of K-linearspaces 0+ Ho+ Hm,K+ (Hm/Ho) K+ 0 Thendefine H t as the cokernel of the map below0+ Ho--6 Hm, KHa- Ht+ 0 where (x) = (x,-x) (we view Jm' Ja as inclusions!). Note thati a is always injectire and we have dlmKH t= dlmKH a + rank(Hm/H o) (1.7) Nowassume in situation above that we dispose ofa logarithmic P-connection (.) on (Hm,Ha)where H a is viewed as an ahellan Liealgebra on which H m acts trivially, such that

for any pc P we have a cornmurat]re diagramJm H H o m Ja Ha Ha P Then we can definein a canonical way a P-connection on the K-linear space (H m K)oH a by the formula Vp(y e ,,x) = (y (p)), ( ? py) , + pX), p P, y �H m, x c H a, . � K . This P-connectioninduces a P-connection '. P+ HOmk(HtH t)which will be called the total connectionassociated to our logarithmic connection(relative to H o !). The basic' (trivial) factabout this construction is the following: (1.S)LEMMA. In notations above, the logarithmicP-connection (VV) on (HmH a) is integrahieif and only if the associated total connection? on H t is Integrahie. (1.9) Coming back toour logarithmic Gauss-Mantn connection(with H m = HR(V)m, H a = H R(V)a, H �= H GV/K), see Remark (l.t)) we may form

the total de Rham space Ht = HR(V) tviewed with the ota] Gauss-Maniaconnection on it cf. (1.7). This object willplay a key role in the next section.

2. Duality iimmern (2.1) Let C be anirreducible commutative algebraic K-group,let B its maximum linear connected subgroup(which will be called the linear part of C)and consider A = C/B (which will be calledthe abelian part of C). By [$e] there arenaturally associated map m Xm(B)+Plc�(A) a .' Xa(B)+ HI(A ) with m a grouphomomorphism and a a K-linear map.(Recall that they are defined as foUows: onetakes a covering (Ul) i of A and sections s i '.Ui- C of the projection C * A; then for X �Xm(B) and [ � Xa(B) we let re(X)�

Pic�(A) be represented by the cocycle X �(sj - s i) and a([) HI((? A) be represented bythe cocycle [ � (sj - sl). lecall also that themaps (re,a) uniquely determine the extensionclass of C as an extension of A by Finallyrecall that given an abelian variety A, anirTeduclbl.commutatlve algebraic group Band maps (re,a) as above there exists anextension C of A by B whose "associatecP'maps are (m,a). Consider $(C) m -- Im(m)cPic�(A), $(C} a = Im(a)c Hl(( A) and, with1I a: Hi(A)a + Hi(( A) and E ; HE(A) m*Pic�(A) as in (1.l) and (1.2), define mSDR(C) m -- l(S(C)m)C HR(A) m SDR(C) a= nl(s(C)a o 1 Both 5DR(C)m and SDR(C)acontain H (fiA/K) and by (1.(;) we mayconsider the total space ,,Ote l 5DR(C)t ofthe data consisting of 5DR(C)m, 5DR(C) a

and the inclusions Jm o I ' 5DR(C)a. Clearly5DR(C) t is a K-linear subspace of the totalde + 5DR(C)m' Ja" H (A/K) Rham spaceHR(A) t of (1.9). One more definition= wesay that C is a relative prt)jective hull of A ffthe map m is injectlye and the map a is anisomorphism (the terminology Is motivatedby 5erre's consideration of projeotive hullsof abelian varieties In.' "GroupesProalgebriques", Publ. Math. IHE5, 7 (1960)our "relative projectire hulls" are'truncatlons" of the projective hull of A,enjoying properties similar to those of aprojectlye hull but relative to a given fixedsubgroup Pic�(A). We will not go intodetails since they are irrelevant for ourpurposes). As we shall see in (2.5) below,(!I(C) -- K if C is a relative projectlye hull

of A. Here is our main result= (2.2)THEOREM. Fix a Lie K/k-algebra P and D -- K[P]. 1) Let A be an abellan K-variety.Then the logarithmic Causs-Manln DerkK-Connection (1.2) is integrahie. In particularthe total de Rham space H)R(A) t has aninduced structure of D-module. 2) Let C bean irreducible commutative algebraic K-group with ((C)= K and abeltan Dart A. ThenC has at most one structure of algebraic D-group. It has a structure of algebraic 67 D-group if and only if SDE(C)t Is a D-submodule of HR(A) t. In particular if C is arelative projectire hull of A, then C has aunique structure of algebraic D-group. 3) LetC be an irreducible algebraic D-group with(C)= K. Then there is an isomorphism of D-modules 5DR(C) t L(C) � where SDR(C) t

is a D-module via the total Gauss-Maninconnection, (cf. assertion 2) above) and L(C)� is a D-module via adjolnt connection (cf.(I. 1.10)). a)Let Ci, C 2 be irreduciblealgebraic D-groups with C(Ci)= K, i= 1,2.Then any morphism of algebraic K-groupsCi+ C 2 is automatically a morphism ofalgebraic D-groups. Moreover If both C i arerelative projectlye hulls o A then we have acommutative diagram 5DR(C2) t : L(C2)�5DR(CI) ! L(Cl)� Consequently HR(A) t isisomorphic as a D-module with thecontinuous dual of L(A hull) == lira L(C)where C runs through the set of relaiiveprojection hulls of A (note that these C's donot form a filtered projective system but theirLie algebras do I). (2.3) REMARK. Recallthat for any abelian K-variety A there exists

a "universal extension" of A by a vectorgroup called E(A) (see [MM]); E(A) issimply the extension.of A by . ._dlmA,KdimA HI(i27A). Clearly E(A) is a cdim Adefined by any linear lsomorphism 2a a j = arelative projectlye hull of A (in fact it is the"smallest" one i.e. a quotient of any relative Iprojectire hull). Note that $DR(E(A))m =HoA/K) and $DR(E(A))a--HR(A) a --HR(A) so $DE(E(A))t = HR(A). Then theabove theorem says that E(A) has a uniquestructure of algebraic D-group and we havean isomorphism of D-modules HR(A)=L(E(A)) � the left hand side being endowedwith the usual Causs-Manin connection,(1.1). Now (.) is easily seen to be aconsequence of the crystalline theorydeveloped In ("Crothendleck's Theorem")

and of the duality theorem proved in the lastchapter of [BBM]. But proving Theorem(2.2) along the linear oi [MM] and [BBM]seems to us much harder. In any case ourproof of Theorem (2.2) (and hence of (.)) ismuch less sophlstJcated Jt is Just amanipulation of ech cocycles on a Zarlskiopen covering of A (no crystallinebackground being necessary). It has theadvantage of constructing explicitely thelifting of the operators pep to E(A) andindeed to any relative projectire hull o A I Inparticular the D-module

lsomorphlsm SDR(C) T L(C) � from (2.2)should be viewed as a "dual" generalisationof the D-module counterpart o!Grothendleck's theorem cited above. It also

suggests that the following might hold (?)Any relative projectlye hull of an abelJanvariety has a crystalline nature. (??)For anysuch relative projective hull C theisomorphism SDR(C)t a L(C) o is "induced"be the crystalline structures. Note also thatour proof of (2.2) is mainly based on certaincohomologicai properties of A making verylittle use of the group law of A. This permitsto pass from A to other varieties; this will beexplained in a subsequent paper (and makesan essential diffeence between our approachand that in [MM]). (2.e) V/e make apreparation. Assume C is as in (2.1) andwrite B -- B m x B a, B m a torus ofdimension M, B a an algebraic vector groupo! dimension N. The projections B + B mand B + B a induce extensions of A by B m

and B a respectively (call them C m and C a)which are described by the maps m = (m,0)and a = (0,a) respectively (in fact C m = C/Ba and C a = C/BIn). l! E' is a linear subspaceof E := Xa(B) then we have E' = Xa(B') for awell defined quotient B' of B a and the mapB a + .B induces an extension C' of A by B(C' -- Ca/Ker(B a + B9) whose definingelement a' � Hom(E',H (A)) is therestriction of a to E'. Clearly, if E -- E' �E", upon letting 1�', C": a" be thecorresponding objects for E", the naturalmap C + C'XAC"XAC m Js an isomorphism.From now on we fix E'--Ker(a:E + HI(J)A))and fix E" an arbitrary complement of E' inE. Since a' = 0 we have C' = B' x A so wemay write C=B'xC 1 C 1 = C" x ACre It wiUbe useful to recall how one obtains C via a

giueing procedure starting from the maps(re,a). Start by lifting the map m to ahomomorphlsm (mij)ij from r. == Xm(B ) tothe ech group Cl(?,) = itH�(U. n U.,I.)corresponding to a covering = (Ui) i, U i =SpecA i. /% i I J Slmilarlly lift a to a{nearmap (a .),. from E = X.(B) to CI(,,(). V/rlte Ui. = Uin Uj and put lj ,j Z Aij =((Uij). Let'Rbe any ring. Letting R[ ] be the group R-algebra on and R[E] be the symmetric R-algebra of E and putting R[T',E] = R[T*]R][E] we have that C is obtained by giuelngSpec(Ai[r-,E]) via the Atj - isomorphisms ij:Atj[T"E]' Alj[r*'E] (2. a. i) eli(X) --mijCX)X for X r r. (2.a.2) ij() = � aij() forall E Clearly C m is described by glueingSpec(Ai[r']) via (2..1) while C*, C" aredescribed by gluelng Spec (Ai[F']), Spec

(Ai[�"]) via (2..2). (2.) LEMMA. In theabove notations we have: 69 1) (C) = K ifand only if the maps m and a are injectlye. 2)Xa(C) = 0. REMARK. Condition {(C)= K isequivalent to saying that C has no non-trivialarline quotient and also equivalent to sayingthat C has no non-trivial linearrepresentation (cf. [Ro] [DG]). ProoJ. Notefirst that any additive character C i G avanishes on B m hence factors through C/Bm= C" so 2) follows from i)l let's prove 1).Assume first that a is not Jnjective so innotations of (2. a) B' is non-trivial whichimplies C(C) K in view of thedecomposition C = B' x C . Similarily, if mis not injectlye, put r. l = Ker (m). Thenletting T l = Spec (K[r.i]) we have asurjective homomorphism B T 1 to which

there corresponds a surjectivehomomorphism C + C i where C! is a trivialextension of A by T 1. This implies againd7(C) K. Conversely, assume m and a areinjective and let's prove that )(C)= K. Wedispose of a commutative diagram with exactrows= O Bo B* B/Bo* 0 0 C O C Spec(C) 0where ((C o) = K and B o is the linear partof C o (cf. [DG], p. 358). From the exactsequence 0 Ker(u) Co/B � C/B Coker (u) 0and from the fact that Ker (u) and Coker (u)are arline while A � = Co/B o and A = C/Bare abeltan varieties we get that A o A is anlsogeny. Now look at the commutativesquares fa, Xa(Bo) f Xa(B ) Xm(B ) m �Xrn(Bo ) H]({A)'--HI(IAo ) Plc�(A)isogeny ' Pic�(Ao) Since a and m areinjective while fa and fm are surjectJve it

follows that fa and fm are isomorphismshence B � -- B which implies Spec )(C) =Spec K and we are done (2.6) LEMMA. LetG be an irreducible algebraic K-group with((G) = K, Then the map B:P(C)DerkK isinjectlye and DarkG = L(G)$P(G), Inparticular there is at most one structure ofalgebraic D-grou p on G. ProoJ. By [Ro] Cis commutative. By (L 1.9) ker(8) = P(C/K)= Xa(C)iIJL(G ) = 0. To prove that DerkC _--L(C)P(C;), let p �DerkK since d)(C)= Kwe have pK c K. Let m e be the

7O maxim-' idea, of C,e and let e, PC,e + K -- PC,e/me be the cou.it as usual. Then ep-p�:G,e +K is a K-e-derivation hence it is ofthe form ev for a unique K-derivation v: G +G' v L(G). The equality �(p - v) = sws that

(p - vXme)� m e. e claim that d = p - vP(G). Indeed he map p G + GxG is anisomorphism (because p = where t G x G +G x G (xy) = (xxy) and 2 = G x G + G, 2(x,y)= y and e Knneth mula to get p2.GxG = Gwhich implies P- GxG = G )' Then Since theabove derivation (identified with a vectorfield in L(G)) vanish at e (recall d(me)C me)it must be zero so =(di + ld)p Similgily oneprov Sd = dS nsuently de P(G) and we arene. (2.7) LEMMA. Let Gi, G 2 beirreducible gebraic D-grou with (G i) = K, i= 1,2. Then (with notations rom (I. 2.1)) Pg.Let fGieG 2 a morphism og algebraic K-grouand pg P. Then the mphism pg - fp = G2 +gG1 is a K-g-derivation (where we stilldenoted by p the rivatio induced on Gi andG2 respectively) hence 1ongs to . (c2 ) e

�(c2, . ci). L(%) H�(C i, Ci) = Menverupon identHylng p - fp with a vector field onG2, It must vanish at e G 2 so it vanisheverywhere and we are done. (g) In order tororm comarations with coccles, assumeagain we e In the situation o (2.) so that wedispose o maps 71 lifting the maps m and awhich define a given commutative algebraicK-group C. Moreover r = aij(r ) choose abasis Xl , ... ,X M of the Z-module r., a K-basis l '" 'N of E and define i and miSj =mij(Xs). Upon refining ?,we may assume thatthe elements mij lift to some elementsnotations being as in (1,2); indeed this ispossible because IIm = HR(A) m + Plc�(A)is surjective (1.2). On the other hand ajautomatically lift to elements (, ai), K er (Cal(/,). Ca2()) so recall that the following

relations hold: r r r (2'8'1)a datj = nj ' ni m s-Idms _- s_ s (2'8'1)m ( ij ) ij 0j 0 i Fcrconvenience, let's call a collection (a , mij)as above a system of adapted cocycles for C.Then C is obtained by gluelng the affineschemes SpecAi[{l, ... ,q,Xl,Xi 1, ... ,XM,yl) (A i = (Ui)) via the isomorphLsm % r(2.s.2) a %%)-- r + (2.8.2) m %(y) = mjy(2.) LEMMA. Assume C is a relativeprojectlye' hull of A and assume (a? mj)Is asystem of adapted cocycles for C (2.g). Thenfor any p e DorkK there exist' vector fields vk � H�(TA ), 1 _< k _< N, liftings �DerkA 1 of p to i = ((Ui) .and regularfunctions r s $ 3k' ' jk' "i =)(Oi ) satisfyingthe following system of equations N (2.9.1)rl a[jvr -- ej- 1 for all i,j (2.9.2)a r r r vkaij= ek - Ck for all l,j and 1 _< k, r _( n s -I s s

s for alli, jandl <k<N, I <s<M (2'9'2)m (mij)vkmij = Pjk ' Pik .... Nv r k r_ r (2'9'3)a airj+ kl �kaij = c a l for all i,j and I _< r _< N,ms,-I m s N s a k s foraUi,jand I <s<M

72 Proof. Since the cup product u: HI(A) eH�(T A) + HI(T A) is an isomorphism onecan find v l, ... ,v n � H�(TA ) and liftingsof p to derivations of ((U i) such that (2.9. l)holds. By degeneration in E 1 of E"(a), themap d: HI(7 A) +HI(gJA/K ) is zero. Hencethe map HI(A)- HI(( ) defined by applying vk to cocycies is zero, in particular there ~re(Ui) satisfying exist ik r -_ r r for all l,j,k,rvkaij jk' ik Similarily, since the map dlog:.l(O I) + "l(iI/K) sends the classes of(mj)into zero, one Can find iik (Ui) such thats,-I s Now recall that we dispose of relations

(2.g.l) a and (2.8.1) m. To conclude the proofit would be sufficient to check that r v-r ak -sak are cocycles. Now we know from (1.1)and (1.2) that ' + ar (( ) wi +(mill mij)ij arecocycles so what we have to check is thatare cocycles. Now taking contraction with vk in (2,8.1) a we get Vk' Vk = vkaij = jk Oikik)l glue together to give a constt eK.SimilarHy (2.8.1) m implies that ml Vk - Pik= Ck e get which Is a cocycle and simUarily- a k k s - ak= caikj ' = ffaijvkmi - Uik ijwhich is also a cocycle. Our lemma isproved. (2.10) [EMMA. Assume C is anyirreducible commutative algebraic K-group(aijmij) is a 73 system of adapted cocyclesfor C and assume we have a solution _ r r ss, Vk# i jkj 'Pjk'Pj ) of the system (2.9.1)-(2.9.3). Define for each i a derivation Pi �

Derk(Ai[E: ]) by the formula Pt=O i ?kVk .!(r r. r, 0 s Then the Pi'S glue together to givea derivation PG � DerkG' Preef. An easycomputation using (2.8.2). (2.11)COROLLARY. Let C be a relative projectivehull of A. Then the map : P(C)* DerkK is anisomorphism. In particular given any D, wehave that C has a unique structure ofalgebraic D-group. Proof. By (2.5) I!)(C) =K hence by (2.6) 8 is injective. To checksurjectivity of ) take pc DerkK. By (2.9) and(2.10), p lifts to some derivation PC" DerkG'By (2.6) again l+ Pl with pie L(G), pie P(G);clearly Pl lifts p and we are done. PG = PG(2.12) LEMMA. Let C be an irreduciblecommutative algebraic K-group, ((C)= K andr s (aij,mij) be a system of adapted cocyclesfor C. Consider the K-linear space S of all

systems *r s r Oj,pj)l_<r_<N,l_<s_< Msuch that 'e L(A)= H�(T ) and aj, pe 0(U i)satisfy the following system.. r r r (2.12)a eaij = aj - a i , I _< r _< N (2.12) m (mtj)-imj s s =uj '"i' l <_s_< Moreover, for anysuch {'0, ,,r& :s)� S consider thederivations j � Der(Aj[E, ]) defined by Jthe formula =%. v&r Then: l) The j's gluetogether to give a derivation p E L(C) (wewrite ) = ('e, .r,..s}); 2) The map assigning toeach element of S the correspondingderivation in L(C) is a K-linearisomorphlsm. Proof. 1) is again animmediate computatlon using (2.g.2). 2)follows by noting that the map S L(C) dimKs= dimKL(A) + N + M = dim C = dimKL(C).is injectlye and that (2.13) To formulate thenext Lemma let's make the following

definition. Assume P is a Lie K/k-algebraand V, W are two finite dimensional K-linearspaces with P-connections. By a

74 p-duality between V and W we will meana bilinear non-degenerate form <,):VxW Ksuch that p < x, > -- <px,y> + (x,py> for allpc P, x c V, y W. We will speak aboutorthogonals in the usual sense. Note that a K-linear subspace �1 of � is stable under P ifand only if its orthogonal �[ � W is stableunder P, Moreover the connection on V isintegrable if and only if the connection on Wis so, Here is our main step in provingTheorem (2.2); (2.111) LEMMA. Assume Cis a relative projective hull of A, P is any LieK/k-algebra and D = K[P]. Then there is a P-duality <, > = SDR(C)t x L(C) K where

SDR(C)t has a P-connection induced fromthat of H!(A) t (of. (1.g)) and L(C) has a P-connection defined by the adjolnt connection(cf. (l.l.10) where C is viewed with itsunique structure of algebraic D-group, cf.(2.11)). Moreover, for any unipotent linearalgebraic subgroup B l of C we have L(Bi )1= SDR(C/Bi)t hence we have an induced P-duality <, > .' SDR(C/Bi) t x L(C/B 1) + K rs Proof. Consider a system of adaptedcocycles (ail,mi) for C as in (2.8) fromwhere we borrow notations. We also usenotations from (l.l), (1.2) Le ]m() be thesubgroup of Zlm(?Z) := = Ker(C()' C2m())of those elements (t i,yij ) � C lm(tO suchthat n = (Fl(m.S.) S)z.z7 l (2' lt' l)m Yij s U I1 for some integers n s and some zj ((Uj).Clearly the map ]m() SD](C) m ls. surjective

(due to our assumption that our cocycles are"adapted"). On the other its kernel is easilyseen to coincide with B()= Im(C�(?Z)'C())so after all we have SDR(C) m = 1()/Blm(9A) m Of course since the map a .Xa(B) HI(A) was assumed to be anisomorphlsm we have SDR(C)a--H)R(A)a--Zla()/Bla(9) where Zat(l,) = Ker(C 1(20 *C2()), B l(?z) = Im(C(Z) - C(?)). We willdefine a Z- bilinear map a a a 75 Let (t l,yij)''7Z), (tli,xij). Zla(?)and *note by CO,;[,? .L<C). derivation defined M in (2.12). e have= Xa + uj- u i (2'la' l)a xij k ' f someXk Kand uj � (Uj). Tn define (using tatls from(2.1.1) a and (2.1*.l)m) I k K J O course wemust ove that the right hand side o the aboveequality is independent o[ j and nce beJonssto K. But indeed uj - ui =(uj ' u[) =(YjJdYiJ)

= = + zj 0zj Consequently Slmilarly we5ummJn 8 up the lt two equalitas we get thatthe right hand side member o (2.1.2) is]ndendent o j required. ]t s easy to check thatour Z-bJlinear map nduces a non-degenateK-bilinear map (, >: SDR(C) t x L(C) + K its so ey to check that L(B] = SDR(C/]) t yJtent B] c C: Jrst one shows that ourdefinition o <, > does t dend on choosing thebasis 1 .. � 'N o Xa(B) in (2.8) next wechoose a basis with the prorty that Xa(B/ ])admits 1''" 'N (Ni N) as a sis. Sncedimensions o L(i and SDR(C/]) t are equal JtJs sfJcient to chck that (SDR(C/]) t, L(Bi)> =0 Whi follows rom definitions using ourspecial basis. k remains to prove that <, > isa P-du]ty. This can be done by a tedious (butnot Stralght[orwd) computation with

�ocyclas we 8tve t outline o this mputat]onsJn we [elt it would have en unfair t0 just"leave it to the reader". Pick any p � pchoose a solution

76 r r s sx Vk Ja Jk ai'pjk'pjl of the system(2.9.1)-(2.9.3) with the 9j's lifting (p)eDerkK and pick closed l-forms 1 and m jsatisfyin8 k k (mi)-ldm. = s (2.1a.3) da =j -j,j- mj The whaz we must prove is that theexpression s r r r * equals the sum o (2.1.5)and (2.1g.6) below= where { } was us todenote closes in SDE(C) t and we made tobvious abrevJations e = ( ), .... Now (2.Jb.b) equals Next (2.1a.5) becomes s . r s(2.1q.8) (( 0,0,Li.u j + Lj, ej -ei)( j j) +eiaij+ (mij) J To compute (2.1.6) note irst thatsince Derk = L(C) � P(G), c. (2.6) we have

,as P } = (Lie bracket in DkC) where PG isgiven by the ormula in (2.10) and p Isdefined by the ormula in (2.12). A Ion[computation yields: [pCp]={[6j ] k e rk r so(2.1q.6) com Finally let k e K and uj � (Uj)sh that s r r , s,-I ems Xka +uj u i (2. le. 10)ej-ei)(u j+j)+01aij+[mlj ilj = ' Then by thevery definition (2.1.8) comes j k " ' 77 Bywell known properties of Lie derivative wehave ,, ,, +} j[(Liee ) eje ) -- [j ] .) for any l-form . In view of this, in order to prove that(2.1.7) equals (2.1q.9) + (2.1.il) it is s rsufficient to check the following (with p = e0j + j): (2. l.12) (vk'- jk-Pjk-Xkxj uj) k Nowintroducing (2.9. l), (2.9.3)a, (2.9.3) m Jn(2.1.J0) we get k Exactly as in the proof of(2.9) we get that s s Vkj-Pjk eK vkTl r -OjkeK hence we set (usJn the act that the

cohomoJoy classes o a orm a sJs in HJ(A))that= Vk' jk'Pj'Xk =� uj-a s ui r pK - pj = -oi - which clearly implies (2. t q. 12) and ourJemma is proved. (2. J) Let's ss to t proo ofthe dualJt theorem (2.2). By (2.]q) or anrelative projectire huh C O A we have anisomorphJsm of K-linear spaces with P-connection 5D(C)t L(C) �. Assume now thatCj and C 2 e two relative ojtJve hulls o sathat c[ is bier than C 2 J 5(C1) m eq containsS(C2) m a natural Lie aJ[ebra map L(Ci)L(C2) defined as oJlows. Choose an integern I such that nS(C2)mC S(Cj) m (assub[roups PJc�(A)) and Jet C 3 be therelative projectire huh e A for which S(C3)m = nS(C2) m. Tn there is an [so[en C2+ C aLie algebra map L(Cj) + L(C3) L(C2) onecan easily check that this map does not dend

on the choice o the inte[er n. Moreover onechecks that the induced square o sces with P-connection SDE(C]) t' L(C )o SDE(C2) t(C2)� is commutatJve. PsJn to direct limitwe = = ? [(c) o = c c

78 Since the P-connection on L(A hull) isobviously integrahie o will be the P-connection on HDR(A)t which provesassertion 1) in (2.2). Next let C be anyirreducible commutative algebraic K-groupwith abelJan part A and with C9(C) = K.Then by (2.5) the maps m: Xm(B)* Pic�(A)and Xa(B)+ Hl(7 A) (B = linear part of C)are injectlye so if we let C 1 be the relativeprojective hull of A for which S(Ci) m --S(C) m there will be a surjective morphismCi* C whose kernel B I is unipotent.

Consider the P-duality from (2.10) < , > .'SDR(Ci)t x L(Ci)+ K By (2.1) SDR(C)t =SDR(Ci/Bi)t=L(Bi )1 so SDR(C)t is a D-submodule of SDR(Ci)t (equivalently ofHR(A) t) if and only if L(B 1) is a D-submodule of L(Ci). By (1.1.16) the latterhappens if and only if B l is an algebraic D-subgroup of C 1 which by (I. 1.17) happensif and only if C has an algebraic D-groupstructure such that Ci* C is a D-map. By(2.7) and (2.11) the latter condition isequivalent to C having an algebraic D-groupstructure. This together with (2.6) and (2.11)proves asertion 2) in Theorem (2.2) as wellas the assertlon 5). Assertion 4) followsfrom (2.7) and from the above discussioninvolving L(Ahull). Our Theorem (2.2) isproved. Now we pass to our applications.

(2.16) For any irreducible commutativealgebraic K-group C we let DerSk(C)K bethe Lie K/k-subalgebra of DerkK consistingof all derivations p � DerkK for which �%(5DR(C)m) c SDR(C)a V p(SDR(C)a) cSDR(C)a where (JLV,?) is the logarithmicDerkK-connectlon on (HR(A)m,HR(A) a) cf.(I.). Equivalently Der(C)K is the Lie K/k-subalgebra of P = DerkK consisting of all p� P which send SDR(C) t into Itself (wherewe view SDR(C)t as a subspace of the D-module H)R(A) t, D = KiP])- (2.17)COROLLARY. Let C be an irreduciblecommutative algebraic K-group. Then thereis a split exact sequence of Lie K/k-algebrasa (C)K * 0 0 Xa(C)L(C)' P(C) Der Proof.Let as in (2.t) E'= Ker(a: f ' HI(iA)) and E" cE be a complement of E' in 15. Moreover, let

r., = Ker(m: r. + Pic�(A)), Sm= lm(m: r..PicO(A)), choose a basis of theZ-moduleSm/TorS(Sm) , lift it to a subset of r. and r.,,be the subgroup generated in r. by this subset;hence Z' nr.,, = 0, [Z: :,OZ"] < eo and therestriction of m to :" is injectlye. Theinclusion r.,r.,, c g induces an isogeny B m =Spec K[r-] ' Bn x B'i where Bn = SpecKIT"]. Hence we get an isogeny: where B' --SpecKlE'l, B"= SpecK[l"]. To this isogenythere corresponds an lsogeny C- C l 7g = B'x B' x C 2 and the maps m 2 and a 2corresponding to C 2 are injectire. By overA where C l a m (2.5) (C 2) = K. ClearlySDR(Ci) t = SDR(C2) t. By (1.1.19) wehave Im(P(Cl)- DerkK) = = Im(P(C2)-DerkK) and there is a Lie K/k-algebra mapfrom the latter space to P(C l) (use the S(C

2) fact that Bn x B is defined over QY). By(2.2) P(C 2) Der k K so we get a split exactsequence of Lie K/k-algebras S(C l ) 0+P(Ci/K) P(C l) 3 Der k K + 0 Finally notethat SDR(C) t -- SDR(C l)t and we concludeby (LI.21). (2.18) COROLLARY. Let C be anirreducible commutative algebraic K-group,P a.Lie K/k-algebra and D: KIP]. Then C hasa structure of algebralc D-group If and onlyif a (p) Der(C)K for all p z P. If 9(C) -- K,this structure is unique and is given by theformula in (2.10). (2.19) REMARK. One cangive an analytic interpretation of the uniquestructure of algebraic D-group on theuniversal extension 5(A) of an abelianvariety A (cf. (2.3)) in case k: C and A -- x XSpec K where L. X is an abelJan scheme andK is the algebraic closure of k(X). Indeed in

this cae (upon replacing X by an etale openset of it) we get that E(A) -- x XSpecKwhere + X is analytically (but notalgebraically) isomorphic over X an with ananalytically trivial bundle with fibre (C)2dim (A). Then any analytic vector field v onX lifts to an analytic vector field on ()anwhich agrees with multiplication and inversemaps. The remarkable fact is that If v isalgebraic then v is also algebraic this is ofcourse a corollary of (2.2) and indeed ofGroethendJeck's theorem []. Our method hasthe advantage o! giving a formula for interms of v cf. (2.9) and (2.10). (2.20) /eclose by illustrating (2.18) in the simplestcase. EXAMPLE Let A be an elliptic curveover K, with j-invariant j z K, let C -- E(A)be Its universal extension and let p t DerkK ,

pj O. Ve will indicate what is the explicitlifting of p to P(C), cf. (2.18), (2.10). CoverA by two arline open sets Ui, U2, choose anyliftings e 1 � Derk((U1) , e 2 Derk(J)(U 2)of p and choose a global vector field e on A.Then by (2.10) there exist a t )(U 1 n U2) ,al,l 1 )(Ui) , e2,132 e ((U2) such that thefollowing hold: (*) Define: ae =e 2 -e l ea --a. 2 - a 1 e2a + ala -' f2 -

Pl = el + [O- (all+ Il)-�Derk(Ul)[F P2 = e2+ e-(02+ B2)--� Derk((U2)[ and view C asobid by glueing UixAI and U2xA1, AI=spK[via the (U 1 n U2)-gebra map :(U 1 nU2IU+U1 n U2 Then Pl and P2 !u together m ive ourliffin PC of p to P(C). So he problem otcomputing "explicitely" PC for a given A andp amounts o finding a, 1' BI' satisfying he

system () ave. If A is he smooh projecfiwmel U 1: SpecK[xy]/(y 2 - x(x- INx- ) henthis computation can don explicitely. Raherhan perforin his computation (whose finalrult is after all irrelevant) we shall indicatehe seps which should be followed and henleave i to he re,der who enjoyscontemplating formulae. A can covered by U1 and U 2: 5cK[uv]/(u - v(v - uNv - u)) luedvia the formulae U= V=y e may take o coursee = yl solving the system () reduces to twooblems= a) lilting explicitely p to Derk(U]),Derk(U2). b) finding (in explicit way) or any, e(U n U 2) a relation o the form ]{ + {2 =a2 - al where (Xl, ) eK 2 {(0,0)}, al e(Ul), a2 e(U2). Indeed by a) we find el, explicite]yand put a = ( - e])/y. Then by b) we writeexpHcJtely = 2 - ;1' where ;] e(U), ;2 e(U2)'

Finally by b) write. expHcJte]y.e2a + ;]a + =- 1 or me � K, e(Ui), 82 e(U2) and deineLet's deal sepiarely wlh problems a) and b).For problem a) let p denote e ]JtJn p toK[x,y] which kJUs xWi then one candetermine, explicitely by a standardprocedure, polynomials M,N, e K[xw] suchthat M + a where = y2. x(x - ]Xx - and notethat the derJvati sends the ideal () into itselLThis derivation induces a Hting o p toDerkU]) constructed analogously. 81 To dealwith 'oblem b) note that (upon denoting by ,the classes oi x, y modulo (f)) we have ((U ln U 2)=K[,, '1]--(' K n)+(' K n)+(' K2 n) neZneZ neZ Now note that 1) for n_>0 n e KI/,I/]= 19(U 2) for n < 0 --n d(Ul) for n > 0 xy �_ .n .(U2 ) for n < 0 2n e((Ul ) for n _> 0 / 2ne (9(U 2) for n < -1 Consequently any

element o! K[,, -1 ] can be explicitelywritten as a sum l2[ + a 2 - a 1 where l e K,all: K[,], a2l: K[/,I/] and we are done (notethat 2/ appears as a "canonical" echrepresentative for HI(dA)I). Similarcomputations can be performed providing theexplicit liftin 8 of a derivation pc DerkK toP(C) where C is the extension of A(=projectire smooth model 2 x G a with (C)= K corresponding to a given line y -x(x-IXx-X)--0, Xl:K, pX0) by C m bundle ((Q -Qo ) on A (Q'Qo c A(K)), Descent.Regularity We continue with our applicationsof the preceeding theory. Start with apreparation. (3.1) Let V be a smoothprojectire K-variety and K � analgebraically closed field o! definition for V.Then there is an Integrahie Jogarithmic Der

K K-connection on (HI(dV), HI(dV)) definedas follows. Fix a descent isomorphism V =Vo)K�K, V o a smooth projectlve Ko-variety , view V as a split K[Po]-variety , Po= Dark K and for PPPo define maps-' o vp,v) by the formulae Vp(falj]) = Epaij]

82 JLV p([mij]) = [mjlPmij] where Exjj]denotes Cech cohomology class of thecocycie (xij). The link between the abovelogarithmic connection and the logarithmicCauss-Manin connection is given by thefollowing: (3.2) LEMMA. With notationsfrom (1.2) and (3.1), for any p � DerKoKwe have the following commutatlvediagrams: ?p Proof. Use only definitions.The above lemma shows in partlcular thatthe restriction of (V,V) to a logarithmic Der

K K-connection on (Plc�(�), Hl(�)) doesnot depend on the choice of the "descentlsomophJsm" � �OKoK we have fixed.The following lemma provides anotheruseful description of our map JV defined in(3.1): (3.3) LEMMA. With notations fromC3.1) identify HI(;) with the group of K-points of the (locally algebraic) Ko-grou pscheme A = PicVo/K � and Identify Hl(�)with the group of K-points of'the Lie algebraL(A). Then the map [Vp:HI(;)- Hl((�)defined for each P e Der K KJn (3.1),restricted to ^�(K) (A�= PiC/o/Ko)identifies with Kolchin's logarithmic Oderivative .p: A�(K)+ L(A�), see (LI.12).Proof. To check the above statement it isuseful to adopt a functorial viewpoint. So letA_: {locally noethertan Ko-schemes} +

{groups} be a contraviarint funclot. Thenone can define for any p e Dar K K the"logarithmic derivative" O J[A p: A_(K)LK(A-) (where LK(A_ ) Is the kernel of themap A_(Spec Kiel)- A_(SpecK) induced bye 0) by sending any element ge A_(SpecK)intog-l(A_(l + p))ge A(SpecK[]), where I -ep: K[e]+ Kiel is the 83 obvious Ko-aigebraautomorphism of K[ ] and where we stilldenoted by g the image of g in A(Spec K ]).Next note that any morphism of functors asabove A A' "agrees" with J, ^p and j[ A, p.Finally, it is not hard to check that: 1) If A--Is the funclot of points of an aigebrlc Ko-grou p then J. Ap coincides with KoJchlffslogarithmic derivative[9 (to see this, use forinstance the formalism in rB i P 2) If A_' isthe functor S+ Hl(� o x S( ) then [A p

coincides with our [9 P defined Jn Thediscussion above plus the fact that Arepresents the relative Picard functor [G]closes our prooL The foliowlng theoremestablishes a description of P(C/K C) for Ccommutative (notations as in (l.2), (l.),(2.1)). Intuitively it says under whatconditions on pe P(C) we have that Cdescends to the constant field of p. (3.e)THEOREII. Let C be an irreduciblecommutative algebraic K-group with abelJanpart A and let pc P(C). The following areequivalent: l) p vanishes on K C(equivalently, C is defined over KK[P]), 2)The image of J[9 pe Hom(HIDR(A)m , HIDR(A)a ) in Hom(SDI(C)m, S(C) a)vanishes 3) p vanishes on K A and the imageof the map [gpe Hom(Hl(), HI((A)) in

Hom(S(C)m, S(C) a) vanishes. (Of courseJ,V P in 2) is that defined in (1.2) while[ P in3) above is the one defined in (3.1)). Proof.2) =:3). By 2) and (1,#) we get ? p(HO/K)) co 1 H A/K ) hence by (i.) the image (p) of pvia the Koalaira-Spencer map vanishes. So plifts to some derivation of )A hence by (0.i5) p vanishes on K A. Now 3) follows byapplying (3.2) to � = A, Ko= K A. Write inwhat follows S a = S(C) a. 3)==)1). SinceVp(Sa)C Sa, S a is a K[p]-submodule of thesplit Kip]-module HI((^o))KoK (where =Ao) K K for some abelJan Ko-variety , K o -- K A) hence S a itself is O split: Sa=So K Kfor some Ko-Subspace SoC HI(iA ). On theother hand, upon Jetting o o � A�(K)(where A � = PiCo/K o) be the K-pointcorresponding to m() (notation as in (2.1)),

our hypothesis 3) together with Lemma (3.3)and with (l.l.ll) and (1.3.6), l) show thatXXe A�(Ko). This clearly implies that C isdefined over K o i.e. KcC K o so .pK C 0.Implications i):=3)::=)2) can be provedusing similar arguments. (.) REMARKS. LetC be an irreducible commutative algebraicK=group. Then (with usual notsions of thischapter)=

1) It may happen that neither C nor A ( =abelfan part of C) are defined over K D(C)The simplest example of this kind is that ot C= E(A), cf. (2.3). 2) It may happen that A isdefined over K D(C) but C is not. Anexample can be obtained as follows. We letK be a universal (ordinary) 6-field withconstant field k, we let A be an elliptic

curver over K admitting k as a field ofdefinition and we let B = G a x G m. Then leta: Xa(B) = K+ HI(A ) be any linearisomorphism and let m: Xm(B) --Z-Pic�(A) be such that re(l) is taken viaKolchin's logarithmic derivative J,?p.:Pic�A + HI(C0 A) into a non-zero element(this is possible slnce by Kolchin's theorem[? p. Is surjectlve!). Then using (3.3) and(3.0) one can check that the extension C of Aby B described by the maps (re,a) satisfiesour requirements. 3) If B ( -- linear part ofC) is unlpotent and A is defined over K D(C)then C itself is defined over K D(C). Thisfollows directly from (3.t). t) If B is a torusthen C is defined over K D(C). Indeed, by(3.0) it is sufficient to check that is definedover K D(C) But by (1.1.22) B is a D(C)-

subgroup of C hence by (I. 1.17) A becomesan algebraic D(C)-group. /e conclude by(0.15). /e close by proving a regularitycriterion (in Deligne's sense) for the Liealgebra of an irreducible commutativealgebraic D-group. (3.6) In the rest of thissection (only) we assume (in addition to thefact that K is algebralcaUy closed) that k isalso algebraically closed, tr.deg. K/k= m (, P= DerkK, D = KIP]. Let W be a finitedimensional D-module: by a model of V wewill understand the following data: a smootharline k-variety X of dimension m, anembedding (over k) of the function field k(X)of X into K, a vector bundle V x on X, anintegrable connection [Del 2] and a K-isomorphism V VX(xK such that for anyvector field v on X we have a commutative

diagram v Vx ---" �X � sV V p(v) wherep(v)� DerkK is the unique extension of vand V p(v) is the multiplication with p(v) inthe D-module V. Any (finite dimensional) D-module admits a model due to the following:(3.7) RE!dARK. In the situation of (3.6) IfPI'" "Pro is a commuting K-basis of P and ifxl,...,XnC K then there exists a flnitelygenerated k-subalgebra A of K preserved byPI' '" 'Pro and containing x j, ... ,x n (for aproof see lB3]). (3.8) Coming back to (3.6)we say that a finite dimensional D-module Vis regular if it has a model (X,Vx, V) withregular in DelJgne's sense [Dal 2] p. 90. ByJoc.cit. it is easy to check that if V is regularthen for any other model (X',V(,'), ' must beregular (indeed by [Del 2] regularity is abirational concept and does not depend on

passing to finite coverings). Using the latterremark and the theory In [DeJ 2] p. 90 itfollows that: l) K viewed as a D-module isregular. 2) If �' *V *�" is an exactsequence of D-modules with V', �" regularthen %/ is regular. 3) If %/1 and V 2 areregular D-modules then V 1 oV 2, V l %/2'H�m(%/l'%/2) are regular, in particular Vis regular. By 1) above it follows that anysplit D-module (of finite dimension) isregular, since It is a sum of copies of theD=module K. The "regularity theorem" [DeJ2] p. IlS implies in particular that if V is asmooth projectire K-variety then the spaceHiE(V) is a regular D-module, with its D-module structue given by the Gauss-Maninconnection (l.l). Now here Is the mainconsequence of our Theorem (2.2). (3.9)

THEOREM. Let C be an Irreduciblecommutative algebraic D-group (D = K[P], P= DerkK, tr.deg. K/k ( o. Then the D-moduleL(C) is regular if and only if the D-moduleXa(C) is regular. Proof. Let U: = Spec d)(C).By [DC] p. 398, C is a surjective morphismof algebraic groups and. its kernel C has theproperty that ((G) = K. Clearly has aninduced structure of algebraic D-group and C+U: is a D-map so C is an algebraic D-subgroup and we have an exact sequence ofD-odules. (3.9.1) 0 +L(G) *L(C) +L( ) +0Now by (I.1.22) the maximal torus ' of is analgebraic D=subgroup of is an algebraicvector group whose space of additivecharacters Xa(/) identifies with Xa(C). It istrivial to check that the nondegeneratebilinear map Xa(/?) x L(/?) * K, <x,v> = vx

defined in (I. 1.3) is a D-module duality (i.e.that p(x,v> = (px,v)+ (x,pv) for x � Xa(/�),v �L(:/')) so we have a D-moduleisomorphism Xa(C)� = L(/) hence an exactsequence (3.9.2) 0 +L(') ' L() 'Xa(C)� '0Finally let T be the maximal torus of thelinear part of G; by (1.1.22) it is an algebraicD-subgroup of C and the quotient [ := C/T isan alebralc D-group (I. 1.17) with ;() -- Kand With unipotent linear part. Let A denotethe abel Jan part of G; then is a quotient ofthe Universal extension E(A) hence L(I:) is aD-module quotient of L(E(A)). By (3.8)HR(A ) is a regular D-module hence so is itsdual HIDR(A) � which by (2.3) islsomorphic to L(E(A)). By

86 again we conclude that L(,) is a regular

D-module and we have an exact sequence ofD-modules (3.9.3) 0 + L(T)+ L(C)+ L()+ 0Now T and being tori are split algebraic D-groups (cf. (I. 1.9), (I. 1.26)) hence their Liealgebras L(T), L(') are split D-modules,hence are regular. Then the exact sequences(3.9.2), (3.9.3) show that L(C) is regular ifand only if Xa(C) is so. (:1.10) REMARK. ]fwe consider the theory which we developedin the present chapter for K a perfect field ofpositive characteristic then all the interestingphenomena we encountered dissappearbecause DarkK = 0. But they reappear and infact multiply in case K is non-perfect. Weshall come back to this situation in asubsequent work. CHAPTER If. GENERALALGEBRAIC D-GROUPS In this chapter wemake a synthesis of the affine and

commutative cases which were studiedseparately in Chapters 2 and 3 respectively.We get some results for general (non-affine,non-commutative) algebraic groupsconcerning local finitness (cf. section 1),automorphism functor (cf. section 2),representative ideals (cf. section 3).Throughout this chapter we assume withoutany explicit mention (as we did in theprevious chapter) that K Js algebraicallyclosed of characteristic zero. 1, Localfinitehess crlterlo (1.1) Let G be anirreducible algebraic K-group. Denote by Lits linear part ( = maximum linear connectedsubgroup of G), A its abelian part (A :=G/L), Z the center of C. lEecall [Ro] that G =LZ so any normal subgroup of L is normal inG. Let LJ be the unipotent radical of L

(which will also be called "unipotent radicalof G") H a maximal reductive subgroup of L,T the radical of H and S = [H,H]. Note thatG/US does not depend on the choice of H.' itis the maximum semiabelJan quotient of Gand consequently the field K(C/US) will becalled the maximum semiabelian subfield ofK(C). We denote by g, J., u, h, t, s the Liealgebras corresponding to the groups definedabove. loreover we put 7 := C/[C,C], :=g/[g,g], :-- u/u n [g,g]c I note that is the Liealgebra of the unipotent radical of . Here isour main result (notations as above): (1.2)THEOREM. Let C be an irreduciblealgebraic K-group. The following areequivalent for p E P(G): 1) p � P(G, fin) 2)p preserves

88 3) p preserves u ) p preserves U 5) ppreserves the subfield K(G/U) o! K(G) 6) ppreserves the subfield K(G/US) of K{G). b)P(G,fin) Is a Lie K/k-subaigebra of P(G) andG is 1ocnlly finite as a Dfin=varJety whereDfin = K[P(C,fin)]. c) K P(G'fin) = K G d)The natural map P(G)/P(G,fin) P(,)/P(,fin) isinjective. e) lf i=G*G' is an isogeny and i*:P(G')P(G) is the lifting map then i*(P(G',fin)) = P(G,fin)n i*(P(G')). Proof.Assertions d) and e) follow from a).Assertion c) follows from b) and from (I.3.7) and 0.1.26). That P(G,fin) is a Lie K/k-subalgebra of P(G) follows from a). Let'scheck 1)------)) in a). Indeed, by (1.3.11)upon replacing K by a Kip]-extension of Kwe may suppose G is a split algebraic Kip]-group. Then one easily checks that p

preserves U. Now #)::53) in a) follows from(I. 1.16) while 3)2) is clear. tdoreover a)=5)follows from (L I. 17) while 5) =) followsfrom (L 1.17) and (0. I ), 5). To prove 2) ::1)and the remaining assertion in b) it issufficient to prove that for any Lie K/k-subalgebra P of P(G) such that P13c U (orequivalently Puc u + [g,g]) we have that G islocally finite as a D-variety, D = KIP].Claim 1. u + [g,g] = u + s. Indeed s = Is,s]being semisimpie hence sc [g,g] hence '5" :[ollows. Conversely since G/US issemiabellan, hence commutative, Iu + s) iscommutative so [g,g]c u + s and 'E" follows.Claim 2. JL is a D-subaigebra of g. Indeed,by Claim I and by Puc u + [g,g] we get that u+ s is a D-subaigebra of g. By (Ll.16) US isan algebraic D-subgroup of G. By (I.1.17)

the semiabelian variety G/US becomes analgebraic D-group. By (1.1.22) the torusL/US is an algebraic D-subgroup of G/UShence L = R-I(L/US) is an algebraic D-subgroup of G (where ]: G +G/US is theprojection map) and we are done by (I.1.16).Claim 3. u is a D-subalgebra of L Indeed, byClaim I we have Puc u + s. On the other handby (II.3.17) the radical of (which equals u +t) is a D-ideal in 4. Hence Puc (u + s) n n (u+ t) = u and our Claim is proved. Now by(11.1.15) and (I.I.16)(L) is a locally finite D-module while by (1.1.17), 0.1.9) and (0.15)A -- G/L is a split algebraic D-group. Claime. In order to prove that G is a locally finiteD-variety it is sufficient to prove it afterreplacing K by some D-field extension of itK i/K. Indeed since A is locally finite we

can cover it by D-invariant open subsets A 1.Then Gi, the preimages of A i in G, will beD-invariant arline open subsets hence by(1.3.7) (GjNK) will be locally finiteKl)KD.modules hence ((G i) will be locallyfinite D-modules and the claim follows. 89Now we put a "D-structure" on "Chevalleysconstruction" and on our construction from�B 1] p. 97 of an equivarlant completion ofG (cf. also [B 5] for group actions on thesame construction). Let us quickly recallthese constructions. (1.3) One starts bychoosing a finite dimensional K-subspace Eof O(L) containing a set of generators of themaximal ideal m of {G and such that E Jsinvariant under left translations withelements in L(K). Then put d = dim(E nm), dQ = projectlye space associated to/ E d qo =

point in Q corresponding to (E n m) andconsider the induced action o:LxQ'eQ,o(b,q)=bq (the isotropy of qo is triviail).This construction will be refered to asChevalley's construction. Next we defined in[B 1 ] p. 97 two commuting actions . L x (G xQ)+ C x Q, (b,(g,q)) = (gb- l,bq) e: G x (G xQ) + G x Q, e(x,(g,q)) = (xg,q) and provedthere is a cartesian diagram u GxQ v with Zquasl-projectlve, I[ 1 the projection onto thefirst factor, v the natural projection onto theabelJan part and u is a principal fibre bundlewith group L for the action . Moreover Odescends to an action = G x Z Z and thelsotropy of x o ;= u(e,qo) Z with respect to isstill trivial. Flnally let 'o c Z be the closureof the orbit Gx o of x � under the action .We call e: G + Z the map (g) = (g,Xo) then w

* e = v. (l.a) Let,s Introduce D-varietystructures on the objects and maps occurlngin (1.3). First replacing K by some D-fieldextension (which is allowed by Claim ) wemay assume L is split, L = L � OK andtaking E = EoK with Eo= (Lo) big enoughand Lo-invariant we may assume in (1.'3)that E is a D-submodule of )(L). Then Q willbe a D-variety, qo will be a D-point of Q, o,and {) will be D-maps. Since Z is ageometric quotient of G x Q by T thearguments in the proo:[ of 0.1.17) show thatZ has a natural structure of D-variety and u isa D-map, Since � � G is a D-point, (e,q o)is a D-point of G x Q hence x � will be a D-point o:[ Z. Clearly is a D-map, hence so ishence '?o will be a D-subvariety of Z. Weclaim that Gx o is a D-invariant open

90 91 subset of G- o. To check this note thatu'l(Cxo ) = G x Lqo and u-l(Go ): G x L'owhere L'o = closure of the -orbit Lq o of qoin G x Q. Hence u' 1(' o \ Gx o) = GX(o' Lqo) Since L is split so will be E and Q henceL- o \ Lq o is a D-subscheme of C. Since u isfaithfully flat and arline the ideal of G- o \Gx o in (Z is the intersection of (1Z with theideal of Gx(o \ Lqo) in u, OGxQ hence is aD-ideal and so Gx o is D-invariant. Now Go is a projectlye D-variety hence by (0.15)upon replacing K by a D-field extension of it(cf. Claim b) we may assume G- o is split.Since Gx o is D-invariant it follows from(1.3.6), b) that Gx o (hence G which is D-isomorphic to it) is locally finite so the proofof 2) :::)1) in (1.2), a) is finished. (1.) Toconclude the proof of (1.2) we must check

that a)=6). Exactly as in the proof of ) :::5) itis sufficient to check that for p c P(G) wehave pu c u if and only if p(u + s) c u + s. Butthis can be checked via Claim I and thearguments used to prove Claim 3 in the proofabove. (1.6) REMARKS. 1) If In (1.2), a)we assume p c P(G/K) then condition 6) canbe replaced by: "6') p vanishes on K(G/US)"This follows from (I. 1.9). 2) Note that weimplicitely reproved in (1.2) above theexistence of a minimum algebraically closedfield of definition for C between k and K(this was shown in [B]). (1.7)COROLLARY. Let C C t be an isogeny. ThenK G -- KC,. Proof. Let N -- Ker(C + G'). Tocheck that KC, c KG it is sufficient to checkthat N and the inclusion N c G are definedover K G. Now N Is central hence N lies in

the kernel of the multiplication map n" Z(G)+ Z(G), n x = nx for some n )_ 1. Since Z(G)is clearly defined over K G so is Ker I n(which is finite!) hence so are N and theinclusion N c G. Conversely to check that KC c KC ' it is sufficient by (1.2} to check thatfor any p' � P(C',fin) its lifting p P(G)belongs to P(G,fln); but this follows from(1.2), a), 1)3). (1.8) COROLLARY. Let P bea Lie K/k-algebras and D = KIP]. Moreoverlet G be an irreducible algebraic D-group.Then 1) G is locally finite if and only ifG/[C,G] is locally finite, if and only if theunJpotent radical of G is an algebraic D-subgroup of G, if and only if the maximumsemiaboliSh subfield of K<C> is a D-subfield of 2) If G +G' Is an isogeny ofalgebraic D-roups then G is locally finite if

and only if C' is SO. 2. Representin theautomorphism functor Our aim in this sectionis to prove the following result (cf. (I. 1.23)for notations).' (2,1) THEOREM. Let G bean irreducible algebraic K-group and letAu__t G .' { K-schemes} + { groups} be itsautomorphism funclot. Then: 1) Therestriction of Au._tG to (reduced K-schemes} is representable by a locallyalgebraic K-group (call it Aut C). 2) Theidentity component Aut�C is offinc. 3) Thekernel of the natural map AutC AutL x Aura(L = linear part of C, A = C/L) is finite. ) Thenatural map G .' L(Aut G)- L(Au.._t C)=P(G/K) is inJectire and its Image equalsP(G/K,fin). (2.2) REMARK. According tothe general assumption of this chapter, K inthe above statement is algebraically closed

of characteristic zero. But if we assume K ofcharacteristic zero not necessarilyalgebraically closed and also that G is notnecessarily irreducible then our proof stillleads to conclusions l) and 2) in (2.1), thisproviding a positive answer to a question ofBorel and Serra [BS] p. l2. Note that in [BS]assertion J) was proved for linear G butmuch more was proved in this case namelythat AutC is of type ALA i.e. an extension ofan arithmetic group by a linear algebraicgroup. It was also observed there that thereexist examples of non-linear (non-connected!) Cts for which (Au.._tC) (K) is"discrete" but not an extension of anarithmetic group by a finite group. In anycase assertion 3) in our Theorem shows thatif Aut L and Aut A are algebraic so is Aut G.

(2.3) Conclusions 1) and 2) in (2.1) are quiteeasy to obtain by inspecting our construction(1.3) and will be derived now while the restof the conclusions require further analysis.As we shall see, in way of provingassertions 3) and a) we will get a quiteprecise description of all objects involved incase G is commutatlve. Start with any locallyalgebraic reduced K-scheme Y acting on Ginthe sense of (i.e. with a "family" ofautomorphlsms of G with "parameter space"Y). Then Y will also act on L and since L islinear, by [BS] Au__.! L restricted to {reduced K-schemes} is representable bysome locally algebraic K-group Aut L sothere is an induced map � Y+ Aut L.Assume now �(Y)c Aut�L. Under thishypothesis we can put an Y-action on the

objects in (1.3). Indeed, in notations from(1.3) the space E can be chosen to be inaddition Aut�L-invarlant (because thenatural semidirect product of L by Aut�Lacts rationally on ((L)) so Y acts naturally on. One sees that our varieties L,G,Q,Z,, := Go, [} := [\ (G) are provided then with Y-action, qo and x o are fixed points underthese actions and all our mapso,,e,ll l,V,W;,eare Y-equivarlant. (2.) Let us prove assertion1) in the theorem. Since we want to apply therepresentability criterion in [BS] p. 1#0 wefirst construct a certain connected algebraicgroup H � as follows. Let I' c G x G x G bethe graph of the multiplication and be theclosure of

92 in x x . By [G] the functor S+

{acAuts(xS);o(DxS)=DxS, (oxaxaXxS)= xS}is immJately seen to be reprentabie on thecategory o locally algebraic K-schemes by aiocly gebraic K-group H we iet H � be asusual the identity component o H. There is anatural action q: H � x G e G which isaithul and hence effective in the sense o [BS]p. Le now Y be y connoted reduced K-scheme o finite type acting on G and Yo bech that the corresponding automorphism o Gis given by some ao e H�l in order or Au tobe representable on {reduced K-schemes}by a locally algebraic K-group Aut G (withAutOG = Ho) it is sufficient by [BS] p. ]a0to prove that or any y e Y(K) the correspodinautomorphism o G is given by some point oH�(K). Now both H and Y act on G henceon L by representability o AuL we et

mphisms : H+AutL y:Y*AutL Since yo ) =$(0 o) e Aut�L we get that Y) Aut�L $oour discussion in (2.3) applie. In particul Yacta on letting and globally fixed o there i aninduced orphism Since (yo ) = (o e H � weget that (Y) c H � and we are done. Toprove assertion 2) in the theorem, we mustshow that H � is linear. Let ( + , be a H�-equivariant resolution of ; then the map r: ( +A is nothing but the Albanese map of C. andis H�-equivariant (with respect to thetrivial H�-action on A). So H � c Ker(Aut� '* Aut�(Alb())) which is linear by[Li] and we are done. In fact assertion 2) canbe proved more "elementary" (i.e. usingneither [Li] nor "equivariant resolution"), cf.proof of assertion 3) below. To proveassertions 3) and 0) in (2.1) we need a

preparation. (2,5) Let C be an irreduciblecommutative algebraic K-group and let'sborrow notations from (lII.2.0). Recall that C= B' x C 'L with B' a vector group andXa(C/) = 0 moreover B" i$ the unipotentradical of C 1. We claim that P(C/K) =(Xa(B') )L(C/)) g P(B'/K) P(C/K ,fin) =(Xa(B') $ L(B")) e P(B'/K) where weidentify K(C) with K(K(B')eK(CI)). Indeedthe formula for P(C/K) follows from (I. 1.9).To check the second formula note that by(1.6) P(C/K,fin) consists Of all pe P(C/K)93 vanishing on K(Cm). Now since C Z =C"XAC m we have L(C/) _-L(C")XL(A)L(C m) Let us choose a basis ofL(C t) as follows: first take a basis (ai) i ofthe image of L(C/) eL(A) and lift it to afamily t(3 i,a i eL(C")XL(A)L(Cm); next let

(()j be a basis of L(B") -- Ker(L(C") + L(A))and (ekm)k be a basis of L(B m) =Ker(L(Cm) e I.(A)). Then: is a basis ofL(CI). Identifying K(C) withK(K(B')OK(C")$K(A)K(Cm)) we may writeany p � P(C/K) as where d � P(B'/K) andfigjh k e Xa(B') c K(B'), Clearly, if all fi'sand hk'S vanish then p vanishes on K(Cm),Conversely assume p vanishes on K(Cm),There exis. ts a family (xj)j xj � K(A) withdet(oxj) 0, e set 0 = pxj = je(oix j) so all j'svanish, Finely there exist elements yq eK(Cm) with de, (myq) 0. e et 0 = pyq = k(myq)hence hk=0 or aH k and our claim is proved,(2.6) LEMMA. In notations of (III.2.) wehave 1) The naral map Aut C" Aut A isinjectire. 2) The natural map Aut C m * AutB m x Aut A 1s injtive. 3) The natural map

Aut C Aut B m x Aut A is injectire. Prf. Toprove 1) let oeAutC" induce the identity onA. Viewing C" as obtained by glueing 5pAi[[i,...,] , ([1,...,) a basis of E", we get thato induces A-automorphlsms of these rings.Let oi = Pik([). Compatibility with gluelagsgives (.) Pik([ + aij) = Pjk(0 + a I t where [ =([1 ..... )' aikj = ajj(), aij = (aij ..... aij). Soall Pik'S (or fixed k) have the me dree n k. Ifn k 2 for me k then Taylor's formula appliedto (e) gives (writing fi instead of Pik and ninstead of nk): .(n) + ? -.t_ (Da k + n-l)(o +.... (**) f(in)( 0 k a ij = ,Jn)(o + ,n-1)(0 + ,,,+ a where t dots stand for terms of, degree atmt n - 2. Looking at degree n terms in () getfn)= fn)so the coefficients of (fn)) i gluetogether to give everywhere defin regular (n)_ f(n) f(n) eK[[],...,Et]. Looking at terms of

de(tee n - ] in functions on AI consequently fi- j =

94 95 (**) we get a f(n) k ' fi () [._5k ()aijf(n- 1)() (n-l) n Upon making a base changein I:, we may assume that f(n) contains 1 withcoefficient 1. . . n-I ,(n*l) we get (identifyingcoefficients of Letting i be the coehctent of 1in [ I + k = ej a i nai k2 Xkajj - wih X k E K.Bu the let hand side equs aij(nl + 2Xkk ) so kk - : Xkrr + bik for Xkr K, bik A i- all k. Theors of degree I in (Pik)i glue together Pik rEqua:ion(,) above impli r Injtivity oE"+HI(A) iplies Xkr=O or r k and Xkk 1.Hence in particular bjk = bik 81ue toEetherto 8ive a constant b k K. Consuently Pik: k +bk or all i and k. Since ust have a ixed pointwe get b k: 0 for 1 k so identiy. To prove 2)

ake C inducin 8 the identiy on th Bm and A.Then induces automorphisms oi of Ai[E] forall i. Sie for X hve oi aiX [ or e Yi � E andsome invertible element a i o Aj.Compatibility with gluein gives aiij(j)Xi:aij(X) j so Xi j or all i,j. Since restriction toB is the identity i X so a i = aj or all i,j.Consequently the ai*s 81ue together givin 8 aglobal section is so on Bm nd we concludethat ff = ldentiy. Assertion 3) ollows from 1)and 2). (2.7) LEMMA. In notations of (III.2.a) we have 1) The map AutC + Aut B x Aurais injectire. 2) Aut�C = Horn(B',B")aAut B'where the right hand side acts on C = B*x C1 by the formula (,aXx,y) = (o(x),y + (x)), eHorn(B',B"), ( e Aut�B ,, x I: B*, y C I. Inparticular Theorem (2.1) holds forcornmutative G. Xa(C 1) _- C 1 C 1 C 1

Proof. Since 0 any JsornorphJsm u of C = B'x must send onto hence sends B" onto B" andBrn onto Bm in particular o inducesautomorphisrns of C"= cl/Brn, Crn = CI/B"and B' = C/C I (call them (', rn' d). If oinduces the identity on A and B then by (2.6)a" and Orn must be the identity, while d isclearly the identity. The isornorphism C :' B'x C" x ACrn shows that o itself must be theidentity and assertion 1) is proved. Assertion2) follows immediately from (2.5), (2.6) andthe remarks above by viewingautornorphlsms of B' x C / as 2 x 2 matriceswhose entries are group homornorphisms. Asfor the assertion that (2.1) holds in thecornmutatJve case, all we have to note is thatfrom the assertion 2), from (2.5) and from(1.3.12), (II.3.10) we get: L(Aut�C):

Horn(L(B'), L(B"))� L(Aut�B ') = =(Xa(B') � L(B") � P(B'/K) = -- P(C/K,fin)Our ]emma is proved. As a by product of theproof we get.' (2.8) COROLLARY. For C anirreducible cornmutative algebraic K-groupthe cokernel of the map X C: L(^ut C) +L(Au C) identifies with Xa(B') L(Crn)(notations as above). (2.9) Let's proveamertions 3) and a) in (2.1). Put C = Z�(G)so G -- LC. If B is the linear part of C then15 is of finite index in L nC and L/C nL -'-' .Now by (2.7) the map ^ut G + ^ut L x^ut(C/B) is injectlye and assertion 3) ollowsbecause C/B + is an isogeny. To prove ) weproceed in several steps: Claim 1. Anyelement of ^ut�L preserves C n L. IndeedAut�L acts on the finite group Z(L)/Z�(L)hence acts rivially on it. Now C n L is an

intermediate group between Z�(L) and Z(L)which implies our claim. Claim 2. Anyelement of Aut�C preserves C n L. IndeedAut�C preserves B and acts trivially onC/B hence acts trivially on C n L/B hencepreserves C n L. Claim 3. L(^ut G) = L(^utL) x L(^ut (C n L)) L(^u! C) Indeed thisfollows frorn the identification Aut�G =Aut�L x AutO(c n L) Aut�C which in itsturn follows from Claims I and 2. Claim t.P(G/K,fin) =' P(L/K,fin) x P(C L/K,fin)P(C/K'fin)' Indeed, any p P(G/K,fin)preserves L and C (use a splitting argument,cf. (L3. I 1)) hence induces an element(pL,PC) of the right hand side of the latterformula. Clearly the map p +(pL,Pc ) isinjectlve; to prove surjectlvity view G = (L xC)/(L n C) and apply (LLI7).

96 Now we conclude the proof of z) in ourTheorem. By (I.3.12) the maps XL; L(AutL)e P(L/K,fin) and : L(Aut(L n C))e P(L nC/K,fin) XLn C are lsomorphlsm (note th L nC is not irreducible in general but in (I.3.12)we do not assume irreducibilityl) and by(2.7) so Is XC: L(Aut C)e P(C/K,fin) Claims3 and t show that lG: L(Aut G)+ P(G/K,fin)must also be an isomorphism and we aredone. 3. Products of abelien varieties byaffine 8roups As we have seen in Chapter 3the results on P(G) do not extend from thearline to the non-arline G at alil But there isstill a case when such an extension ispossible namely when G is the productbetween its linear and its abelian part asshown by the following: O.!) THEOREM.Let G = L x A, A an abellan E-variety and L

an irreducible arline algebraic K-group.Then: 1) K ID(G) -- K G and K G is thealgebraic closure of the composltum KAK L.In particular there is a split exact sequenceof Lie K/k-algebras 0 e P(G/K) e P(G) eDer(K/KAK L) + 0 2) P(G/K) P(L/K) �(Xa(L) L(A)) 3) P(G/K,fin) = P(L/K,fin) a)P(G) contains a representative ideal. Moreprecisely, if � is a representative ideal inP(L) (which always exists by (]I.1.22)) then� � (Xa(L) L(A)) is a representative idealin P(G). ]n particular if the radical of L isunipotent then Xa(L)� L(A) is arepresentative ideal in P(G). 5) Uponidentifying AutG with Aut L x Aut A theisomorphism in 2) is equivartant (with Aut Gactin 8 on the left hand side and Aut L x AutA acting on the right hand side naturally). 6)

Upon fixing "alascent" isomorphisms A =AOKoK, L = LoKo K, Ko--K G, lettin Po =Dark K and D o --KiP o] and viewing bothmembers in assertion 2) as IDo-modules inthe o natural way (in the right hand side wetake the Do-mOdule structure induced by thenatural one on each factor and term) thelsomorphlsm in 2) is an isomorphism of Do-mOdules. Proof. 1) follows from (].1.19),(0.15), (II.i.l) and (].1.25). To prove 2) notethat for the projection map :G--L xA eL wehave gC G =L so K(L) is a D(G) - subfieldof K(G). The restriction map P(G/K)+P(L/K) which is therefore well defined hasan obvious section given by trivial liftin Po+ Po Gl' Let p � P(G/K) restrict 97 to 0 cP(L/K). Since p is a vector field on Gtangent to the fibres of ] we have p = r.f i ) e

i where (el) i is a K-basis of H�(T A) andfietiT(L). The equality pcp=(p)J+ l)P)pC isequivalent then to r'(PLfi ' fi � I - 1 � fi)PL�PAOi = 0 Choose (xj)j, xj I((A) such thatdet(eixj) 0. Applying the above equality to lxj c K(K(L)�K(A)) we get PLfi- fil - lfi=Ofor alii hence fi t Xa(L)' Conversely, if fi�Xa(L) one immediately gets r-f i( 0 i zP(G/K); this proves 2). 3) Clearly P(L/K,fin)injects via trivial lifting Po e po I intoP(G/K,fin). Conversely, assume p = dl +ZfiO i z P(G/K,fin) with d c P(L/K), (el) l abasis of H�(T A) and fi Xa(L)' By (1.2) and(1.6) (and with notations from 1oc. cit.) pvanishes on K((L x A)/US) cK(K(L/US)K(A)); this shows that d vanisheson K(L/US) hence dP(L/K,fin). On the otherhand Jet xj �K(A) be such that det(eixj) 0

then 0 = p(l �xj) = Zfi�eix j from whichwe get fi = 0 for all i and 3) is proved. a) LetV be a representative ideal in P(L). Thenputting Po = IDer(K/KG) Pl = P(L/K,fin)P2= P3 = Xa(L)$ L(A) we have P(C) = PoPt � P2 e P3 and the following commutationrelations hold: [Pi,Pj] c Pj for all 0_( i_( j_(3 Indeed to check [Po,P3] c P3 take p �Der(K/Ko), let p, be the trivial lifting of pfrom K to C = G O)KOK (G O = L O g Ao, LO an arline Ko-group, A � an abelJan Ko-variety) so PG = PL I � I �p and let v = r-fl6i e Xa(L) L(A). Then Pc i A i = = r'fi)VpOi + ZVp[fiO I z Xa(L) �L(A) Thiscomputation already proves assertion 6).

98 To check [Pl,P3] � P3' [P2'P3 ]c P3 takev = r.f i O01 as above and d � P(L/K)

(identifying it with dO 1). Then one checksthat [dO l,v] c Zdf i �e i = r. Vdf i 0 i �Xa(L)O L(A) .The rest of the commutationrelations are clear. So P2OP3 is an ideal inP(G) contained in P(G/K) and it is a K-Linear complement of P(G,fin) = Poa)PI inP(G). To conclude the proof of assertion 0)and to prove assertion 5) of the theorem it issufficient to note that for any o = a L x a A �Aut L x Aut A = Aut G the followingformulae hold: OLVO 1 = V and for v = r.f i0 i as above, a vo=(OLOA)(ZfiOOi)(o lo1)= = r. fOLf i) ) OAeio l) Xa(L) � L(A)(where we denote as usual byaoAOl. theinduced field automorphisms of K(G), K(A),K(L)). (3.2) COROLLARY. Assume G is anirreducible algebraic K-group such that O =GiG 2 with G i normal irreducible

subgroups, G 1 arline and G 2 an abelianvariety. Then K D(G) = K G. In particularthere is an algebraic D-group structure on Gif and only If K D is a field of definition forG. Proof. We have an isogeny C 1 x C2+ Cand apply (I. 1.21), (3.1) and (1.7). EXAMPLE We already considered in (1.1.9) and (I.1.26) the example of a product G = A x Ga.With notations as in 1oc. cit. we have thatwhile P(C/K,fin) = !<[j.. Klyx x is arepresentative ideal. CHAPTER 5.APPLICATIONS TO DIFFERENTIALALGEBRAIC GROUPS I. Ritt-Kolchintheory The theory and examples of thissection are classical. (1.1) By a f-field wewill understand in what follows a field 'ofcharacteristic zero equiped with a set A ={l,,..,m} of pairwise commuting derivations

a I a m a =l,...am)Nm we will writex insteadof 1 '" m X for anylt:', If m= I we say isordinary and we write X',X",V",...,X (k)instead of 6),, 2X, 6 3X ,..., 6 kX. We definethe constant field of ' to be the field 9:6 = {X� iX = 0, l_<i_(m}. EXAMPLES 1) Let (,61,...,6 m ) = (Q(Zl,...,Zm), a/a Zl,...,a/a Zn')be the field of rational functions. Here 9 :A =Q. More generally one may take 9' anyfinitely generated field extension of Q and 6 i= /a z I where Zl,...,z m is a transcendencebasis of /Q. Note that in this case 9 A/Q isalgebraic. This is of course the most"geometric" example one can think of{indeed one may think 9: as the function fieldof some smooth arline Q-variety V and 61,...,6 rn as corresponding to pairwisecommuting vector fields on V generating its

tangent bundle. 2) Let (,6 1,...,6 m ) =(Q(Zl,...,zm,exp Zl,...,exp zm), a/a Zl,...,a/aZm). Here .A = O too. This should also beviewed as a "geometric" example, in thesense that it "comes from" m commutingvqctor fields on a variety. V which is amodel o! '/Q{ we can be very specific hereby taking V = SpecO[zl.,...,zm,tl,...,t m] andletting the vector fields be 3) Let :=Q<y>=Q(yo,Yl,Y2,...) be the fJeJd of rationalfunctions in infinitely many variablesyo,Yl,Y2,... equiped with the derivation : '- rdefined by 6 Yi = Yi+I' i_> 0. This exampleIs very dlilerent in nature from Examples 1,2 because Yo and its derivatives arealgebraically independent over A =O itshould be viewed as very "non-geometric".Nevertheless it will be crucial to include this

kind of examples in the theory! #) Let (9:,61,...,6 m ) = (Met(R),8/8 Zl,...,a/8 z m)where Mer(R) is the field of meromorphicfunctions on the domain Rc Cm. Here A = C.5) Let 9r= Met(cS,0) be the quotient field ofthe ring {S an of germs o! holornorphlcfunctions at (cs,0) and 61 = a/a z i, I < i _<m. Cs'0 (1.2) By a morphism of A - fields wewill understand a field homomorphism a:such that

lOO lol for all X � r' and I _( 1 _( m. If o isan inclusion we say that is a A -extension of"and that ' is a A -subfield of . We say that isA - finitaly generated over ' if there exist Xi,...X n e such that is generated as a fieldextension ofo' by the family ($�'Xi)l_<i_<n,oeNm � EXAMPLES We have the

following natural morphisms of A -fields o Io2 Q , Q(zi,...,z m) sQ(Zl,...,Zm,eXPZl,...,expz m) + leer((m) +Mar(cm,0) �3 di/dz i Q s Q<y> - Mar(C,0),Yi where e Mar(C,0) has the property thatthe family consisting of ? and its derivativescf arbitrary order are algebraicallyindependent over Q (such functions arecalled transcendentally transcendental andtheir existence is easily established). Of theabove A -morphismsOle2e 3 are the oneswhich provide A -finitaly generated A -extensions. Other remarkable examples ofmorphisms of A -fields are provided byrestrictions lEer(R) + lEer(R 1) where R 1is a subdomain of R. (1.3) Let be a A-extension of ' one says that is semiuniversalover 'if for any morphism of A -fields o :+ '1

with 1 A -finitaly generated over ' thereexists a morphism I:: r' 1 + making thefollowing diagram commutatlve A basicresult of Seldenberg [Sell says that Mar(R)is a semiuniversal extension of Q if Rs say,is a disk in C. (l.tt) Let be a A -extension of 'one says is universal over ' if issemiuniversal over any A -finitaly generatedextension of r contained in Such an is alwaysalgebraically closed hence so is its constantfield. A -field will be called universal if Itis universal over Q. basic result of KoJchin[Kl] says that any A -field ' has a universalA-extension (which is countable if is so). Onthe other hand Seidenberg's results in [Sellimply that any countable A-field can be A-embedded into Met(cm,0). In particularMar(cm,0) contains A-subfields 2/, which

are universal; admittedly no explicitdescription of such an 9 seems to beavailable. But at least we dispose of"theoretic" examples of universal A-fieldswhich are realized as A-fields ofmeromorphic functions. (1.) By a A--a!gebraover a A-field 'one understands an '-algebraequiped with m commuting derivations onlifting 61'"" m (and still denoted by 81,..., m). By a A-ideal in . one understands an ideal3c such that 6i(3)c 3, I i m. If Sc is a subsetone denotes by ($) the smallest ideal ofcontaining S, by iS] the smallest A-ideal ofcontaining 5 and by {S} the smallest perfectA-ideal of containing S. Note that (S} = {r t:n such that r n t:[S]). There is an ambiguity ofnotation here: if Xl,...,x p E then {Xl,...,x p}will generally denote the smallest perfect A-

ideal of containing Xl,...,x p rather than theset whose elements are x.,.,x (unless thecontrary is obvious from context). p A A- '-algebra is called A-finitely generated if thereexist Xl,...,x n E such th'at 6 o is generated asan algebra by the family ( xi)l_4i_(n,aeNm.By a A--algebra map we understand an '-aJgebra map which commutes with thederivations in the obvious way. EXAMPLE .(a) Let := '{yl,...,yn ) denote the :'-algebra ofpolynomials in the indeterminates Yi where I_( i _( n, o e N m viewed as a A- '-algebrawith the derivations j: + satisfying (a) (a + I)Yi = Yi (o) and if m = I (ordinary case) wewrite for all I <i<ns {8eN m. We write Yiinstead of Yi y',y",y"' instead ofy(l)y(2),y(3). So for instance if m = 1, n= 2and if A� grwe have in '{yl,Y2} the

equality.' (yly . (y)2) -- X,yly + .y,jy + Xyly _2yy' One calls :'{yl...yn} the ring of A-polynomials in the A-indetermlnates yl,...y n.To illustrate the concept of A-ideal start witha A-polynomial F e e'{yl...Wn } where is,say, ordinary. Then the ideal iF] can bedescribed of course as iF] = (F, F, 2F,...,kF,...) but the ideal {F} does not have ingeneral such an easy description bygenerators. We make the important remarkthat even if F is irreducible (as apolynomial) the ideal {F} may fail to beprime. E.g. take n= 1, Fe{y} F=(y")2-y, [Ri]p. 2tq then the ideal {F} is not primebecause combining 6 2F and 6 3F we get y,,(y(3)y(5). 12(y())2 + gy(q). 1) � while noneof the factors above belongs to {Fl.Concerning A-ideals of A-polynomials we

have the following basic results due to Ritt:

102 103 (1.6) THEOREM ([Ri] p. 10). Forany perfect A -ideal 3 in {yl,...,yn } there Is afinite subset Sc 3 such that 3 = {S}. (1.7)THEOREM ([Ri] p. i3). Any perfect A -ideal in {yj,...,yn } is a finite intersection ofprime A -ideals. The problem of making(1.6) and (1.7) as "explicit" as possible wastreated in detail by Ritt [Ri]. (l.g) From nowon we fix throughout this Chapter a universalA -field {, and we denote by =,A its constantfield. Call the set ,n the n-arline space. Forany subset $ of 5J[{ yl,...,y n} we define thezero locus of S in ,n by the formula Z(S) = {ae ,n; F() = 0 for all Fe S} The subsets of /,nof the form Z(S) as above will be called A -closed subsets. Conversely for any subset .

of ?jn we define the ideal of - as follows: I )= { fe /,{ yl,...,yn } F() = 0 for alia e }Clearly I ) is a perfect A -ideal in { yl,...,yn} . Due to universality of the correspondenceS* Z(S) establishes a bijection between A -closed subsets of ,n and perfect A -ideals in{ yl,...,yn ] . The A -closed subsets of /n forma topology called the A -topologyl thistopology is stronger than the Zariski topologyand by (1.6) it is a .Noetherian topology. AA-closed subset . is irreducible iff l ) isprime. EXAMPLES l) }n Is an irreducibleA-closed subset of /,n; indeed (n= Z(yls...,y)and i(x. 2) The A -closed subset Zy") 2 - y)of , is reducible. 3) The A -closed subset Z(y2 - y") of is irreducible. (1.9) Let . be anirreducible A -closed subset of /,n; then . hasthe topology induced from the A -topology on

,n which we call the A -topology of r- . Letbe aA -open subset ore (i.e. an open set forrhea -topology); a function ! :l- ?if, will becalled A -regular at Xoe D if there exist a A-open neighbourhood o of x o in and A -polynomials F,G { yl,...,y with G nowherezero on o and f(x) = for all x f2ol f will becalled A-regular on 12 if it is A-regular atany point of [. Any A -regular function 1 gCis continuous with respect to the A-topologies on 17 and . Define (i) := {f: i ; fis A-regular on } It is a A- ,-subalgebra ofthe A--algebra of all functions +. Inparticular we dispose of the A--algebra ((.)of all A-regular functions . *. Note on theother hand that we dispose of the algebra{r.} := 9{y 1 ..... yn}/l(T) which can becalled the A-coordinate algebra of r. and that

we dispose of an injective A- t/,-algebramap This map may not be surjective (and thisis a crucial contrast with algebraicgeometry!). Nevertheless it is easily seen toinduce an isomorphism to the quotient fields.The common quotient field of ,[r.] and {(r.)is denoted by ,<r.>. Define dim r. = tr.deg.EXAMPLES 1) If = ?j,n then ,(T.} + (T.) iseasily seen to be an lsomorphism; in otherwords ((,n) ={yl,...,yn} ' Note that dim,n = .More generally (but this is more subtle) wehave shown in lbs] that {.}- ((.) is anisomorphism if . is Zarlski closed In t/, nand, as an algebraic varlety Js smooth. Wewill not use this result in what follows. 2)Let .=--Z(y')C,. Then if ,, Xlwe have that 1/(-,)](), l/(rl - ) Z{} where I is the image of y in{}. Note that dim() = i. (1.10) Let F be any

field. By a na've ringed space over F wewill understand a pair (X ) consisting of atopological space X and of a subsheaf () ofF-algebras of the sheaf F X of all F-valuedfunctions on the open sets of X. By amorphism (X, GX)- (Y,y) of naYve ringedspaces over F we will understand acontinuous map f .' X + Y with the propertythat for any open set V c y and any : (y(V) wehave ef (X(f' I(V)). We obtained a category{na've ringed spaces over F} (l.tl) By anarline A-manifold we wJiJ understand anobject In the category of naive ringed spacesover , isomorphic to (.s .) where . is anirreducible A-closed subset of some ,n andT. is the sheaf of A-regular functions,*defined in (1.9). By a A-manifold we willunderstand a na've ringed space (T.) over ff,

which is

104 105 Irreducible and can be covered byfinitely many open sets r. l,...:p such that each(i, CjE i) is an arline A -manifold. For any A-manifold ]: we define > as the direct limitof all C7(fl) for c E non-empty A-open sets;then > is a A-extension of , (and in case ]: isarline it coincides with the one defined at(1.9)). We define the dimension of E bydim]: := tr.deg. By a morphism of Weobtained a category -manifolds} We leave tothe reader the verification that this categoryhas direct products. As expected if r.c%(, n,TcL m are A-closed irreducible subsets thenr. xTc?,(, n+m is A-closed and irreducibleand so it has a structure of arline A -manifold; this construction is "invariant" and

"globalizes" via "lueing" to get the existenceo! products in the expected way. A groupobject in the category of A-manifolds will becalled a fi-group; so a A-group is a A-manifold T equiped with morphisms andwith an element eer satisfying the obviousgroup axioms. With the obvious notion ofmorphisms of A -groups we obtain acategory { A -groups} A A -manifold offinite dimension will be called a A o-manifold. 6-group of finite dimension willbe called a A o-group. Any A-group in oursense "provides" an "Irreducible A -group"in Kolchin's sense [1(2]; It is not clearwhether the converse holds. One can provethat our 6 o-groups "are" precisely Kolchin'sirreducible A -groups of type zero [K2].lefore giving a few examples of A-manifolds

and A-groups it is convenient to discussmore generalities. Let F be an algebraicallyclosed field. In the rest of this section (only)F-varieties will always be viewed as naiveringed spaces (rather than schemes) over F;so thcy will be identified with their sets ofF-points and their structure sheaf will be thesheaf of (F-valued) regular functions. F-varieties will always be assumedirreducible. (J.J2) Let X be an arline R-variety and take an embedding i: X - A n =.n;then X being Zariski closed in ?.n is inparticular A -closed and by a basic result ofKochin [K l] is irreducible in the A -topology so it has by (1.9), (I.11) a structureof arline A -manifold which we cat] . 'eleave to the reader the task of checking thatthis structure does not depend on the

embedding i. Note that dim ( = o if dim X >O. Now if X Is any -variety, cover X byarline Zariski open sets X. and note that the IA-manifolds i glue together to give a A-manifold structure on X which we call ; willbe called the A-manifold produced from X.As sets we have = X but the topology on(called the A,-topology) is stronger than the(Zariski) topology of X. The A-manifolddoes not depend on the arline covering of Xand is "naturally" associated to X. Actuallyone may check that X gives a functor {?Z-varieties} { A -manifolds} It worths notingthat if X, � are ,-varietles then the mapHomiL_var(X,Y) + Horn A-man () isinjectlye but not surjective in general. Indeedif X =Y =A 1 then Homa.man( 1, 1) containsthe map f(y)=(y,)2. y3, say, which does not

belong to Homg t_var(AI,Al). The functorabove commutes with products so induces afunctor G + {irreducible algebraic 21-gr0ups} - { A-groups} So we alreadydispose of a series of examples of A-manifolds and A-groups: EXAMPLES Forany ?,C-variety X we dispose of a A-manifolds (; in particular n, n are A-manifolds. Any irreducible A-closed subsetr. o a A-manifold ( has a structure of A-manifold (the way of seeing this ls rathertrivial and we leave it to the reader). Inparticular the A-closed sets below havenatural structures of A-manifolds-' l) Z(y" +ay' � by)c l/o= J, a,b , y a A-indeterminate.2) Z(xy - [, yy" - (y,)2 + ayy9 c /, 2 _- 2, a ,x,y two A-indeterminates. 3a) Z(y 2 - x(x -IXx - c), x"y - x'y' + ax'y) c ?,L 2 = 2, c , a e

,, x,y as above. 3b) Z(y 2 - x(x - lXx - t),.y3.2(2t - l)(x - t)2x'y + 2t(t - lXx - t)2(x"y -2x'y'))c 2L 2 = 2, t EL, t' = l, x,y as above. A) Z(YI2'Y21'Y31'Y32'Y22 ' 1'Y33 'I'Y3'Y'11 ' Y23Yll )c GL3' Yij 6-indeterminates l/i,j I . The irreducibiJity ofthese sets is easily established directly in allcases except 3a) and 3b); the latter cases aremore subtle but nevertheless they can betreated using lItts theory and we leave themto the reader, too. Examples l), 2), 3) havefinite dimension while ) has infinitedimension. Now for any irreduciblealgebraic i-group G we dispose oi a 6-group; in particular Gin, CLN, are A-groups (ofinfinite dimension), where A is any non-zeroabelfan ,-variety. It is easy to check that anyirreducible A-closed subgroup o! has a

natural Structure of A-group. On the otherhand if I: is one of the A-closed subsetsdefined in 1)-#)

106 107 above then the following hold. If 11is given by I) then r. a A-closed subgroup of1 = a' If r. is given by 2) then Ii is a A-closedsubgroup of Z(xy - 1) = m c 2. If r. is g'ivenby t) then r. is a A-closed subgroup of CL 3.Finally if r. is given by 3a) respectively 3b)then the A-closure ]' of r. in 2 is a subgroupof c 2 where A c p2 is the elliptic curveobtained by taking the Za. riskJ closure ofZ(y 2 - x(x - l)(x - c)) respectively Z(y 2 -x(x - l)(x - t)) in p2. All these statements canbe checked by direct computation! Weconclude that in examples 1) 2) r. is a Ao-group, in example #) I: is a A-group while in

examples 3a), 3b) ]' is a Ao-group. Asremarked in the Introduction examples 1) 2)3a) #) are due to Cassidy and Kochin whileexample 3b) is implicit in Manin's paper[Ma]. Here is another remarkable class ofexamples of A-groups of infinite dimensiondue to Cassidy an related to work of litt: (i).(j), c . 5) I' = 2, ,: I' x I' + T, F((Ul,U2),(Vl,V2)) = (u I + Vl,U 2 + v 2 + i<j aiju I�1 ' aij Many other examples may be foundin Casidy's papers [Ci]. (1.13) In (1.12) weconstructed A-manifolds starting with -varleties. Now we shall construct A-manifolds starting with -varieties. Let X �be an arline X-variety, choose an embeddin[l � -' Xo-,ln and compose it with the naturalinclusion n+ 9,l,n; then X o appears as a A-closed subset of ,n (indeed if X O is the zero

set in ,n of Fi,...,Fp[yl,...,y n] then X o is thezero set in ,n of FI,...,F p, y'l,...,y' n e {,{yl,...,yn}). So X O has a natural structure ofarline A-manifold, cf. (1.9), (1.11) which wecall o and which is easily seen to beindependent of the embedding i o. Now if XO is any (-variety we may cover X o witharline Zariski open sets Xoi and note v -manifold with dimX o=dimX o. As that theA-manifolds Xoj glue together to give a A oo ~ � . topological spaces X = X but thestructure sheaves are drastcally different(e.g..If X O = J{ we already saw that e(Xo )c�ontalns the function f :X', f(l) = 1/( - 1)where t 7, t l; but of course f is't-valued, not -valued so f I ((Xo). v We obtained a functorX o s Xo {-varieties} + { Ao-manifolds}inducing a functor C o o {Irreducible

algebraic -groups} + { a o-groups} Any Ao-grou p isomorphic to C o for some C o asabove will be called split. Finding criteriaior a Ao-grou p to be split will be one of ourmain concerns. 2. Ao-roups versus algebraicD-roups (2.1) lecall that we have fixed in thepreceeding section a universal A-tield //withconstant field 1. Then let P .be the freeIntegrahie Lie ./ff1-algebra built on 6 l,...,Am(0.)1 recall that P = J& 1 "' � m and for alli,j and ,la � ?L. Moreover let D = [P] recallthat D has an -bis consisting of monomialsEN TM, i.e. D = y m6a with the "obvious"-8ea structure. ith these notations remark thatthe concept of a-extension of in (1.2) is thesame with the concept of D-field in (0.). Thencept of 6--gebra in (l.) is tbe same with thatof D-algebra in (0.). T ncept of A-ideal in a

6--alEebra in (1.) is the same with the con,ptof D-ideal in a D-alEebra , cf. (0.7). A 6--gebra is A-finitely 8enerated in the sense of(1.) iff the corresnding D-algebra Js D-finitely 8erierated Jn the sense of (0.}). By(1.6) and (1.7) is D-algebraically closed inthe sense of (0.}). By the ave discussion, anyintegral D-scheme of finite ty can be coveredby D-schemes of the form Sc {yl,...,yn}/3where 3 is some ime a-ideal. (2.2) In orderto relate the Ritt-Kolchin frame (cf. Section Iof this Chapter) to our frame in Chapter 1- itwill be convenit to introduce a bic functor{integral D-schemes of D-finite type} { a-manifolds}, V V This is done follows.Assume V is an integral D-scheme of D-finite type then a set we put V A = HornD_sch(Spac ,V) Clely V6 identifi with the

subset of V consisting of aJJ p E V for whichthe maximal ideal mp of V,p is a A-ideal andt resid field V,p/mp is a trivial extension of .In tations oi (0.7) we actually have V =VD().The subsets of Va of the iorm Ua = =HOmD.sch(5cU) where U is any ZariskJopen subset of V are the open sets o atopology on V a which we call the a-tology.For any U as ave we t peU Any element f in(U&) can be Jdent[fied with a function f =U& + via the formula f(p) = (f rood top) . 5owe obtained a subsheaf to of the sheaf of 1 -valued functions on V . Cl this subsheaf Vthen we leave to the reader the tk to eck that(Y&, V ) is a -maniold and that actually welot a functor [rom {integral D-schemes of D-finite type} to { &-manifolds}. The key int tobo checked is actual]y the fact that it V =

5pec{yl...yn}]3 for some ime &-ideal 3 then(V&, V& ) (Z Z) as naive ringed spaces over

108 109 where r. = Z(3) � t,n is the zero setof 3 and (r. is the sheaf o:f A-regularfunctions on I: (this shows in particular thatany affine A-manifold has the form (SpecJR) A where R is an integral A-:finitelygenerated A- -algebra. It also worths beingnoted that the stalk V A, p of V A at p � V Aidenti:fles with and that the quotient field /.(V) of V identi:fies wlth ,(r.:. Consequentlythe :functor V V induces a functor {D-varieties} + { A -manifolds} o Now the:fupctor V V A commutes with products so itinduces funclots G G A {integral D-groupschemes o:f D-:finite type} + { A-groups}{irreducible algebraic D-groups} - { Ao-

groups} few easy remarks about the functorV V A are in order 1) If X is a D-variety thenX 6 is ZarJski dense in X. 2) I:f X - Y is asurjective D-map of D-varieties then the mapX A e Y A is surjective. 3) l:f X is a D-variety and Y c X is a closed D-subvarietythen X A n Y = Y A (as subsets o:f x). (2.3)]t will be use:fuJ to express the functors X +and X � + o defined in (1.12) and (].13)respectively Jn terms.of the funclot � V6above. Start by recalling that we introducedin (l. 2.1) a functor {reduced ,-schemes} +{reduced D-schemes} X k X e, which is anadjoint for the forgetful funclot � V I. Nownote that if X = $pec/J.[yl,...,yn]/3 then X m=Spec {yl,...,yn}/{3}; this can be checked bysJm ply checking that HomTj,_sch(V!,Spec ,[yj,...,yn3) =

HomD_sch(V,Spec{Yl,...,yn}/{3}) for anyreduced D-scheme �. But by Kolchins resultquoted in (l.]2) if 3 is prime then so is {3}.C:onsequently if X is an /,-variety then X isan integral D-scheme of D-finite type so wedispose of a funclot {/,-varietles} +{integral D-schemes of D-finite type} X + Xo� We leave to the reader the task to checkthat the composition of the above functerwith the funclot in (2.2): {integral D-schemes of D-finite type:} - { A-manifolds}rV H �A iS isomorphic to the :functor in ( I.12) & (J, -varietie - { A -mani:foida} X ,- XIn other words we have a naturalIsomorphisms 4 X' (Xm)A (2.) Similarly wedisuse o a unctor { -varJie { D-varietie deJnby X o Xo where Xoe is viewed with thesplit D-scheme structure (L 3.6). [t is ey to

check that the comsJtJon o this unor with thefunclot { D-varietJ + { A o-manifold, V V Ais isomorphJc to the functor in (]. J 3): { -rJetie * { o-manifoJd, Xo o In other wds ehave natural JsomorphJsms Xo= (Xo) A ()'Similily y irreducible klgeaic -group G wehave a natur isomorphism o A -groups (G)Aand for any ieducible gebraic -group G �we ve a natur isomphism o A o-grou - (c o e) Next e epare ourselves to ove that thete8ory o irreducible algebraic D-roups isequivalent to the.category of & o-rou. TheJrst step is the olJowin: (2.6} LEMMA. L a& -group. The oilowJn[ are equivalent= t) isa & 2) = C& some irreducible gebraic D-roup G. 2)]) Js obvious. J) 2) Pick any ainePiece = (SpecR)& o T with R a & -finitaly8enated integral 6 --algebra. It is a fact [B3]

that any such R whose quotient field s finitetranscenden

110 111 degree over contains an element f �R, f 0 such that A := If is fin{rely generatedas a (non-differential) l,-algebra. Performinga translation we may assume that (Spec )&contains the unit e of I'. So in particular themultiplication and inverse of I' induce on (V= SpecA,e) a structure of germ of algebraicgroup ("groupuscule algebrique" in the senseof [U] in fact we have rational maps V x �---,� and V---V defined at (e,e) respectivelyat e enjoying the usual properties of grouplaw with e es a unit whenever compositionof maps makes sense). Now there exists analgebraic group G whose germ at the identityis isomorphic to the germ (V,e) (cf. [U]; this

can be seen as follows.' V has in particular anormal law of composition in Well's sensehence by Weil's theorem there is an algebraicgroup G and a birational map f: �--+G suchthat the transported law fore V to G via fcoincides with the law of C. Now onechecks that f must be defined at e, f(e) = eand i commutes with multiplication andinverse arround e, so f gives an lsomorphismof germs (V,e) (G,e)). Transporting the A-structure from V to G via the iso- morphism(V,e) e (G,e) we get m commutingderivations pl,...,Pm on (C) which agree withcomultlplication ?.I(G) ' 2(G x G) andantipode (G) + ?(G) (preserving in fact both(G,e and its maximal ideal). By (L 1.20) PieP(G) so by (I. l.l) G becomes an algebraicD-group. We see that we have a A-rational

map r--,G& (i.e. a morphism of 6-manifoldsfrom and open subset of r to G&) whichagrees with multiplication wheneveroperations make sense. Exactly as in the caseof algebraic groups such a map must beeverywhere defined on r and our ]emma isproved. (2.7) LEMMA. Assume G and G'are irreducible algebraic I-groups. Then thenatural map Homalg. D.gr(G,G,) + Horn A.gr(G A ,G'&) is bijective. Proof. Themorphism G& + G'i induces a commutativediagram of local &-rings = %,e, xG,exe--This extends uniquely to a commutativediagram in which the horizontal arrows areD.rationai maps of D-varieties (i.e. [-rnapsdefined on open subsets of the sourse).' GxG---+G' x G' G ..... ,G' The bottom horizontalarrow must then be an (everywhere defined)

morphism of algebraic D-groups and ourlemma is proved. (2.g) COROLLARY. Thefunctor G ' G A {irreducible algebraic D-groups} -+ { Ao-groups } is an equivalenceof categories. Proot. It is just (2.6) + (2.7).REMARK. We shall fix from now on aquasi-inverse I' ,.G(T) of the functor in (2.8).have Horn &_gr (r,r') HOmalgD.gr(G(r),G(r')) Moreover <r> identifieswith t//,<G(l'>) and (r,e identifies withOc(r),e' (2.9) Clearly a do-grou p I' is split ifand only if G(l') is a split algebraic D-group.But by (I. 3.7) the latter happens if and onlyif C.(I') is locally finite. So by (IV, 1.2) uponletting G = G(r) ! we have r is split iff theImages of l,...,m 6 c p in P(G) belong toP(G,fln) (recall that any algebraic D-groupstructure on G is determined by a Lie /[-

algebra map P * P(C), cf. (i. 1.1)). (2.10)LEMMA. For any do-grou p r and anyirreducible algebric -group H we have anatural bljectJon Horn &.gr (l',lq) = HOmalg.gr(G(T)!,H) Proof. By (I. 2.1) we haveHOmalg ql. gr(G(r)l,H) HOmD.g r sch(C(r),H ) Since I' = G(I')& there is an obviousmap HOmD_g r sch(G( I'),H ) - Horn fi-gr(r,lq) Let's construct an inverse to it{ startwith f e Horn &_gr(l',Jq). Then f provides a&-ring map Composing this map with thenatural map (H ,e '" (.,0H .,,e we get a mapH,e '+ (G(I'),e

112 113 hence a rational map G(I')I+ Hwhich is easily seen to agree withmultiplication generically and hence is an(everywhere defined) homomorphism. This

homomorphlsm provides a morphism G(T)+Hm and our inverse is constructed. (2.11)LEMMA. For any irreducible algebraic ]'-groups G o and H o we have a naturalbijection v v Homalg_gr(Go,H o) = Horn A -gr(Go'Ho ) ProoK By (I. 3.6) we have abijectlon HOmalg .gr(Go,Ho) = HomalgD.gr(Go , H o /,) and we conclude by (2.7).(2.12) One can define in the obvious wayactions of A-groups on A-manifolds (and onfi-groups) so one can define A-cocycles of aA-group I' in a A-group I" on which T acts.Some more definitions: a morphism y -+ I"of A-groups will be called an embedding ifit is injectlye. A A-group y will be calledlinear (respectively abelian) if there is anembedding I' '+ GL N for some N > I(respectively I' '+ for some abelfan t,-variety

). A A -group I' will be said to have no non-trivial linear representation if for any N > Ithe only morphism y + GLiN is the trivialone. By (2.10) T has no non-trivial linearrepresentation if! G(y) has the same propetyi.e. iff (G(I')) =q. Let 1' be a A-group. Twoembedclings o': T'- I', o" = I'"- l' are calledequivalent if there is an isomorphism : T' -T" such that o"o = o'. A class of equivalenceof such erabeddings will be called a fi-subgroup of y. By abuse we will sometimessay (given an embedding I"+ Y)that l" is a A-subgroup of I'. (2.13) PROPOSITION. Let 1'be a Ao-group and let G = G(y) {. Then thereis an embedding I' + C and a A-cocycle C +(L(G)m) inducing an exact sequence ofpointed sets I - y + G('[{,) + L(G) int -+ 1where the action of C on (L(G)m)^= (L(C)' m

is induced by adjoint action (L(G) is viewedas usual as an algebraic vector group) andL(G) int = ((e I ..... o m) c L(G)m; ? 6i �) -? 6je i * [ei,e j] = 0 for all i,j} where V hereis the adjoint connection (I. 1.10). Proof. It'sjust a translation of (I. 2.3) via the yoga ofthe present section. REMffkRK. The aboveproposition answers (in the finitedimensional case) ]<ochJn'S question fromthe Introduction of [K2] whether differentialalgebraic groups can be embedded intoalgebraic groups. (2.1) In what follows by aHopf fi -/d,-algebra we mean a Hopf 9-al�ebra H which is a fi -,/,-algebra suchthat comultiplication, antipode and unit are fi-algebra maps; this concept is equivalent ocourse to that of Hopf D-algebra. There is afunctor induced by that in (1.2); { integral A

-finitely generated commutative Hopf A--algebras}- { arline fi -groups} H (Spec H)fiAny fi -group isomorphic to an objectcontained in the image of the above functorwill be called strictly arline. It can be shown(but won't be done or' used here) that ourstrictly arline A-groups "are" preciselyCassidy's -algebraic groups with A-polynomial law [C 1] and consequently byCassidy's results [Cl] any strictly arline A -group is linear. The latter result for fi o-groups will be reproved below. Thefollowing result is due to Cassidy [C3] butwe provide for it a direct proof,' (2.1)LEMMA. For any two integral fi -finJteJygenera/ted commutative Hop! fi - ,/,-alge-bras H and H' there is a bijectlon.' HomHopffi. .alg(H',H) = Homfi _gr((Spec H) ,(Spec

H') A ) Proof. The natural map from left toright is clearly injectlye. To checksurjectivity take a morphism f: (Spec H) +(Spec H')fi; it induces a morphism Writingboth H and H' as direct limits of fJnitelygenerated Hopf algebras [DC] oneimmediately sees that the latter morphism isinduced by a morphism of arline groupschemes Spec H- Spec H' hence by a Hopf fi- t-algebra map H'- H and we are done.(2.16) Due to (2.15) we may define correctly(up to an isomorphlsm) for any strictly arlinef -group 1' a f -finitely generated Hop! fi --algebra { I'} such that I' - (Spec?Z( I'} )fi {this { 1 } can be called the f-coordinateHopf algebra of I' (and coincides withCassidy's { I'} in [Cl]D. It does not coincidehowever with (I' ) in general{ (2.17)

LEMM/, [B3]. Let I' be a strictly arline fi o-group. Then ( I'} is finitely generated as a(non-differential){,-algebra. In particularG(I')=Spec{I'} and ,{l'} =O(G(]')).Moreover I' is split if and only if /,{ I'} is alocally finite D-module. Proof. We alreadyused in (2.6) the fact (proved in B3]) that if l'=(Spec H)fi then the scheme Spec H containsa non-empty open subset of finite type over .A translation argument shows that one cancover Spec H by arline open sets of finitetype over ,, hence Spec H is of finitey typeover ?d,, hence H is finiteJy generated over .The remaining assertions follow

114 115 immediately from (2.8)and (2.10).(2.18) COROLLARY. A A o-group [' Jsstrictly arline if and only if G([' )! is afine, if

and only if T is linear. Proof. The firstequivalence follows from (2.17). The secondone follows from (2.17), (2.13) and (2.10).(2.19) REMARK. By (1.25) we see thatthere exist arline Ao-groups T which are not� strictly arline (e.g. I' = A o with A � anabe]ian '-variety; this example was firstdiscovered by Cassidy). Let's analyse a fewEXAMPLES It worths describing explicitelythe algebraic D-groups associated to the A o-groups introduced in Examples 1), 2), 3)given after (1.12). 1) Assume first thatZ =Z(y". ay'. by)c a =' a,b. Then G) = G_ 2 =Spec[ j2 ] viewed as an algebraic D-groupvia the derivation c P(G) defined by [1 =[2and [ 2 = -I 1 ' a['2 i.e. by where 6 * is thelifting of 6 to %1 1,[ 2] which kills [ 1 2.This structure is of course a special case of

the corresponding example in (1.9), (1.26).To check that G 2 viewed as an algebraic aD-group as above is indeed isomorphic to Zwe must check that (Ga2)fi = r.. But (Ga2)fiis by delinition the set of all /,-points p = 1 2) c G 2 for which the evaluation map ?Z 1 ;23* ' is a a D-algebra map hence for which a1 --a 2 a = -t l - 2 i.e. for which a' l =a 2 Wechecked that (Ga2) E as sets; the reader maycheck that this is an isomorphism oi -manifolds. Note that e<Z)=?(y,y') where y isthe class of y.{ y} rnodry,. ay'4 by]. Underthe identification J.(y,y') = 1 2) we have y = [I' y' = [ 2' 2) Similarfly if we assume E=Z(xy- l,yy"- (y,)2 + ayy')c G m=Z(xy- l) 6 2as in fl.12) then one can check as above thatG([)= G m x G a = Spec [X'I[] as In (1.9),(1.26) viewed as an algebraic D-group via

the derivation � P(G m x G a) defined by )t=rX and i.e. by Once again we have t< ) =.(y,y,). Under the identification t(y,y,) =(X[)we have = y and [ = y'/y. 3a) Finely if we letbe the fi -closure in 2 of Z(y 2 - x(x - l)(x -c),x"y - x'y' + ax'y)c c2=A 2 then one can eckas ave tha G()=AXGa=AxSpec] as in (1.9)(with A = Zariski closure of Z(y 2 - x(x - l)(x- c)) c 2 in p2) viewed as an algebraic D-group via.the derivation � P(A x G a)defined by x = [ y and [ = - hence by =,* *y-We have ) =(x,y,x') under t identification(x,y,x') =(x,y)we have [ = Note tt in 1 threeexampl o fi o-grou appear embedded in l-dimensional algebraic -groups (me ecisely ina'' and e given by second order difentlequations. Thee emddings are entirelydifferent from the emddings provided by

(2.13) above (whi should view as the"natural" embedclings, in which t ambient fi-roup should b, x, x,Zx). Conere,ely, inexample I)Z is given in x by the first ordersystem =0 In example 2). is given in m x a bythe first order system X' -X = 0 ['+a[ =0 Inexample 3)I' is given Inkx a by the first ordersystem x' -[y = 0 ['+a[ =0 We are not yetreedy to compute G([') for T as in Example3b) o! (l.i2). See (3.20) for a treatment of it.But we are ready to compute the & -coordinate Hopf algebra =?J.{Z} of the & -group g = t I , � GL3();c'--O,a'=ca 0

116 117 in Example t)of loc.cit. Indeed, as a/[-algebra R ={a,b,C}a/[C',a' - ca] =[a,a'l,c,b,b',b",b"',...] The computation ai I I 0 I 2= I c I � c 2 0 0 0 0 I gives the following

formulae for the comultiplication p '. I + R� R'. a=aa pc=cl+lc pb =a�b + b ] p(b9 --cab + ab + b 1 p(b") = c2ab + 2ca b' + a b" +b" 1 Let R n =[a,a'l,c,b,b',...,b(n)]l it is aHopf subalgebra of R so we may considerthe arline algebraic group G n = SpecR n aldthe arline group scheme G = Specie. Put T --[a,a-l]. Then each G n contains T and has aprojection onto T. Moreover ker (Gn+ 1 + Cn) is a vector group for all n. Consequentlyeach G n is solvable hence a semidirectproduct of T by the unlpotent group , b(n) Mn -- Ker(G n + T) = 5pec[c,b,b ,,.., j 5o (3itself Is a semidirect product M x p T, M =im M n. Now note the remarkable fact thai ae Xm(T) is a weight for p but is not killed by6; this is the example promissed in (I].3.6)!3. Stntttte e! be-'oups Since we proposed

ourselves not to use Kolchln's powerfultheory [K 2] we are faced ,ith the problem ofconstructing a '.ie correspondence" and atheory of subgroups and quotients for b-groups. We are able to do this only if werestrict ourselves to o-groups. We begin withsketching such a theory. (3.1) Let 1" be anembedding of &o-groups. We shall define ab-manifold structure on the set T/T' asfollows. By (2.2) G(r')()* G(rX) is injectiveso G(')+ G() is a closed immersion hence by(I. hiS) C(I')/G(I"') has a natural structure o!D-variety. One checks using (2.2) that theunderlying set of the ao-maniloJd (G(r)/(G(r')) A coincides with r/r*. So we giveT/T the above Ao-manifold structure. If inaddition T' Is normal in T then by a densityargument (c. (2.2)) we get that C(I")(/) is

normal in C(T)(,) hence G(T') ls normal inG(r) hence C(I')/G(T') will be an algebraicD-group so I'/T' will be a Ao-grou p. Thereader can check that in act the above r/r,satisfies the usual universal property ohomogenous spaces (respectively quotientgroups). (3.2) Clearly (I. 1.16) provides thedesired *'Lie correspondence" for Ao-groups; we leave its formulation to thereader. (3.3) Let's discuss commutators. By(I. 1.6) If r is a Ao-grou p then [G(I'),G(I')]is an irreducible algebraic D-subgroup o!G(T). Then using (2.2) one checks that theunderlying groups of the Ao-grou p[c(r),G(r)] A coincides with the closure of[P,T] in r. So we denote the above Ao-grou pby [r,r]cr (3.e) Let's discuss centers. By (L1.6) i! 1' Is a &o-group then ZO(G(p)) ts an

irreducible algebraic D-subgroup of G(P).Using (2.2) one checks that the underlyinggroup o the Ao-grou p z�((r)) coincideswith the connected component Z�(l') of thecenter of I'. 5o we give Z�(r) the above o-group structure. (3.) Some words aboutisogenJes. A morphism of o-groups l' + I"will be called an isogeny if it Is surjectlvewith inite kernel. By a density argumentagain of. (2.2) the homomorphlsm G(I')(L)+G(T')(,) Is surjective; moreover, if H isthe Identity component of K er(C(l') +G(I"))by (2.2) we have H = H(/,) n l' is dense InH(,), this showing that H() consists of oneelement only. Hence G(I') +G(l") must alsobe an isogeny. Here is one of the mainapplications; (3.6) THEOREM. Let I' be Ao-grou p. Put G = G(r) and identify ,4 l'> with

(G>. Then 1) l' is split if and only if I'/[ I'1']is split. 2) is split if and only if the maximumsemlahellan subfield of Z(I') Is a A-subfieldof 1< r>. 3) If r * r is an isogeny then r issplit if and only if I is split. t) If Z�(r) ={e}then r is split. ProeL 1), 2) tollow from(2.9), (3.3) and (IV. hg). 3) follows from(2.9), (3.5) and (IV. hS). #) follows from(2.9), (3'.t) and ([L 3.30).

118 119 (3.7) A linear A-group r will becalled solvabie (respectively nilpotent) if itis so as an abstract group. Note that a linear&o-group Y is solvable (respectivelynilpotent) if and only if is so (this :[ollowsfrom "density", cf. (2.2) and from (2.13)). AA &-group will be called unlpotent If thereexists an embedding r +GL N whose image

:onsists ot unipotent matrices (this definitionagrees with Cassidy's [C2]). Note that a &o-group r is unlpotent if and only if (3(T) is so;indeed if G(r) is unipotent so is F by (2.13).Conversely if y embeds into GL N and rconsists of unipotent matrices of GLN(/,)then one can easily show that the action of rvia translations on the algebra /,.{GLN}=(GLN m) is locally unipotent hence so willbe the action of r on /,{ T} = ((G(r)) henceso will be the action of G(r)() on ((G(T)) (by"density" again) and we are done (see [H] p.64 for background of locally unipotentactions). Let T be a linear &o-group. By(2.18) G(r) is arline and by (1I. 3.30) itsradical R(G(T)) is an irreducible algebraicD-subgroup of G(r). Ve define the radical ofT by the formula R(r) := R(G(r)) We have an

embedding R(r)+ r and it is easy to checkusing (2.8) that for any embeddin I - r oflinear &o-groups with r' soJvabJe andnormal Jnr there is a commutative diaEram Ia ,r R(r) with (unique. Here is anotherconsequence of our theory. (3.8)THEOREM. linear Ao-group r is split i!and only if its radical R(T) is split, if andonly if all group like elements of the. Hop!A-U-algebra {F} are A-constants. Proof. Itfollows directly from (3.7), (2.9), (1I. 3.30).In what follows we briefly discuss abelianAo-groups: (3.9) I. EMMA. Let C be anirreducible algebraic D-group, H anirreducible'algebraiC ,-group and C - H e C) + H two morphisms which correspond toeachother under adjunction (I. 2.1). Thefollowing are equivalent.' 1) The kernel of G

- H eo is trivial. 2) The kernel of G I +Hcontains no non-trivial algebraic D-subgroupof G. 3) The morphism G + fl is anembedding. Proof. 1) :O3) is clear. 2)==H)follows since Ker(G-H is an algebraic D-subgroup of G contained in Ker(G ! - H). 1)-------------------:2) Assume M is analgebraic D-subgroup o! G contained inKer(G !+ H). Then both the trivial D-morphism and the D-morphism :M+G+H mcomposed with (H m)I + H give the trivialmorphism. By universality, -- gJ so M istrivial. (3.10) LED. MA. Let r be a Ao-group. The following are equivalent: I) r is abelJan. 2) There is a D- morphism G(r) + A mwith trivial kernel with A an abelfan variety.3) The linear pert of G(Y)! contains no non-trivial algebraic D-subgroup of G(r). ) Any

morphism from a linear Ao-grou p r' to r istrivial. Moreover t if the above conditionshold then the linear part of G(T) ! Isunipotent. Proof. Let B be the linear part ofG ! (G = G(F)). 3)==)2) follows from (3.9)applied to the projection C ! G!/B. 2):=H)follows from (2.10) and (3.9). 2) =3). Wehave a commutatIve diagram G "A providinga commutatlve diagram G $ :A' (c/s) - Sincehas a trivial kernel, so has Applying (3.) towe get our conclusion. Note hat if 2) or 3)hold G is commutative so by (L 1.22) themaximal torus B m of B is an algebraic b-subgroup of G hence is trivial so B isunipotent. 3) ) If 1 '+ r is as in ) then theimage of G( I +G( l ls an algebraic D-subgroup of B hence is trivial So G( r +G( Dis trivial so P'+ ris trivial. 4) ==)3) Since [

I'Tcl IS linear It Is trivial so r iscommutatlve. By (L 1.22) once again B m isan algebraic D-subgroup of G. By 4) B m istrivial. Now assume 3) does not hold hencethere exists an algebraic D-subgroup I o! Ccontained in B with H 0. Since H is unfpotentIt Js irreducible. Letting r= H 6 we get acontradiction. Our lemma is proved.

120 121 (3.11) COROLLARY. 1) Any Ao-grou p ]' possess a linear Ao-subgroup r suchthat is an abel Jan go-group. 2) Any ^o-groupI' possesses a Ao-subgrou p I' 1 with no non-trivial linear representation such that ['/I' I iSa linear Ao-grou p. Proof. l) Take I'*= C' bwhere C' is the largest irreducible arlinealgebraic D-subgroup G(1') and use (3.10).2) Take I' 1 -- (G l ) g where G [ = Ker(G -

Spec d7(G)) and use [DG] p. (:3.12)REMARK. The first of the above assertionsanswers, in the particular case of ^o-groups ,Kolchin's question in the Introduction to [!<2] whether a "ChevalJey structure theorem"[Ro] holds for differential algebraic groups.In the case of %-groups one sees that the roleof abelian varieties among algebraic groupsis played by abelian go-groups. This justi-fies a special study of the latter. Note that wehave the following inclusions of sets moduloisomorphisms reTresentation moduloisomorphlsm corem I = commutatlve &o-groups modulo isomorphism It is easy toshow that qab� qli,4F(tak e the preimagevia logarithmic derivative A(L) -L(A),where A Js an abelian -variety defined over:, of a A -subgroup of L(A)) and o that qe�

q1ab (take a split 6o-grou p l' such that((G(l')) = and G(l') Is not an abelian variety).In what follows we concentrate on objects ofthe set of isomorphism classes of abelian&o-groups with no non-trivial linearrepresentation. (3.13) LEMMA. Let F be a&o-subgroup of I with H an irreduciblealgebraic U-group. The following areequivalent.' 1) Flu Zariski dense in H. 2) Themorphism G([") ! -H (cf. (2.10)) issurjectJve. Proof. Use definitions. (3.1)LEMMA. Let A be an abelian i,-variety andl' +2 a morphism from a ^o-group Thefollowing are equivalent: l) I'ls a Zariskidense b-subgroup o!/. 2) l'is a -subgroupof/(' containing the torsion subgroup of A. 3)The morphism G(l) I - (cf. (2.10)) issurjective, its kernel coincides with the

linear part B of G(T )I and B contains nonon-trivial algebraic D-subgroup of G(I' ).O. 15) COROLLARY. Any abelian Ao-group I has a ZarJski dense embedding I' bintosome produced abelian variety . Moreoverfor any two such embeddings l: I' 1, 2: I' 2there exists an Jsomorphlsm o: A I + A2 suchthat Preef of (3.1tO. In view of (3.9), (3.10)and (3.13) in order to check that 1):3) itsufficient to check that under the assumptionthat contains no non-trivial algebraic D-subgroup of G(I') it follows that iV equalsthe linear part B of G(I'). Indeed by (I. 1.22)M � must be linear hence Mo= B (actuallyby 1oc.cit. B must be an algebraic vectorgroup). We want to prove M = B; assumeiV/B 0 and look for a contradiction. Take xiV, x B, Jet N = Iu/BI hence Nx B; choose (B

being divlslblel) be B such that Nb = Nx.Then x - b Tors(G(I')) hence by (I. 1.22)again the (finite) subgroup of G(I') generatedby x - b is a non-trivial algebraic D-subgroup of G(I') contained in M,contradiction. So the equivalence 1) 4=)3) isproved. The implication 2)=1) is obvious.Conversely to check that 1)::)2) it issutficient to prove that any torsion point xA() lifts to a D-point of G(T). But any x asabove lifts to a torsion point of G(I') whichis automatically a D-point due to (I. 1.22)again. (3. I&) Let I' be an abelian Ao-grou p.By its embedding dimension we willunderstand the dimension of any abelJan -variety A such that there exists a ZarJskJdense ba ]' . Clearly the embeddingdimension of I' equals the dimension of the

abelian part of G(I') [ (cf. (.l), (.l)). (3.17)Recall from (IlL l.l), (IlL 1.3) cf. [KO] thatfor any abeltan -variety A, H1DR(A) has aD-module structure induced by Gauss-ManJnconnection. Call DH�({21A/) the D-submodule of HR(A) generated byH�(i]lA/). Moreover recall from (III. 2.15)that the universal extension E(A) has a(unique) structure of algebraic D-grouphence we dispose of a naturally associatedAo-grou p E(A)g, c. (1.9). Here is one of ourmain applications of the theory in (1II. 2):(3.1) THEOREM. Let A be an abeltan 2-variety. Then contains a unique Zariski dense6o-Subgroup with no non-trivial linearrepresentation (call it ) and we have theformula Moreover =W= is a quotient ofE( ) A. ProeL Let I' be any Zariski dense

Ao-subgrou p of , Such l"s correspondbljecttvely to equivalence classes ofsurjectlons C ! +A (G an irreduciblealgebraic D-group) whose kernel is thelinear part of C ] and contains no non-trivialalgebraic D-subgroups of G, cf. (3.1a), (2.8)two such surjectlons C[+A, G- areequivalent if there Is an algebraic D-groupiSomorphism (= (;i + (;2 making the followlndiagram commutative',

122 123 (3. is.l) d Now assume in additionthat equivalent to G( [')! having the sameproperty). By (3.10) and (III. 2.5) G(I') ! isan extension of A by a vector group B suchthat the map Xa(B) +HI(( A) is iniective. $oG(T) ! is lsomorphic over A with a quotientof the universal extension E(A) be some

linear group. By Oil. 2.2) the $urjectionE(A)+G([') is a D-morphism. In view of ourequivalence relation (3.18.1) we mayassume G([')= EGA)/H for some linearalgebraic D-subgroup H of E(A). e claimthat o 1 SDR(E(A)/H)a=DH (fl/tL); indeedSDR(E( )/H)a contains H�(fllA/,) andsince it is a D-submodule of cf. (m. 2.2), itmust contain Assume o l DH (O./,) is strictlycontained in SDR(E(A)/H) a. Then by (UL2.14) we have o I .L where = Ker(E(A)+A).Now DH (/,U.) = L(Hi) for someintermediate algebraic vector group H lbetween H and I. By (I. 1.16) H l is a D-subgroup of E(A). But this leads to acontradiction since we would get a non-trivial linear Ao-Subgroup (HI/H) Aembedded .into the abeltan Ao-group

(E(A)/H)A = = DH�(alA/%) from whichwe get that dim r. For the existence of aZariskl dense ouf> [' in with no non-triviallinear representation we simply let ]'_-(E(A)/H) A where H is the maximum linearagebraiC D-subgroup of E(A) and use (3. l).(3.19) COROLLARY. Let A be an abellan q,-varJety. Then the following are equivalent(notations as in (3. J7) (3..lg)): I) E(A) . isan abeltan Ao-gr�up' 2) ' = E( ) 3) dim' --2dim A (3.20) EXAMPLE. Assume dim A =! (A an elliptic curve). Then only twoposslbilities may occur for DH�(/) namely:a)DHO(f//,)= o 1 (/). In case a) we get by(ill. 3.#) that A is defined over , say A =^o ,A � an abeltan (,-variety. Then clearly A= Ao !n case b) we get by (3.19) that A -- E(A)A and A does not descend to . , Note that in

the latter case A q is the unique Ao-SUbgroup of dimension 2 of (lndeed by (3.14) for anyAo-subgrou p T o! of dimension 2, G(l') isan extension of A by Ga; it cannot be a trivialextension by'(l�. 3.1) so tt must be theuniversal extension E(A)). As an appicationof this unicity remark consider once again theAo-Subgroup l' of (where A - Zariski closurein p2 of .Z(y 2 - x(x - l)(x - t))c /,2 t'= l)obtained by taking the A-closure Jn 2 of Z(y2 - x(x - i)(x - t),-y 3 - 2(2t - i)(x - t)2x'y +2t(t - l)(x - t)2(x"y - 2x'y')) as In Example3b) in (1.12). Since dim ['--2, by the unicityremark above we must have [' = A i' = E(A)A so G([')--E(A) viewed as an algebraic D-group via the unique Jilting of 6 � Der/, toP(E(A)) whose explicit formula can beobtained as explained In the Example after

(IlL 2.19). It would be nice to "see" theequality [' = A directly, without using theabove "unlcity remark". In particular sincer(l') =(x,y,x') and ,(E(A)) =(x,y,[) (with [ asin (Ill. 2.1)) it would be nice to "see" the:formula expressing as a rational function inxyx . (3.21) REMARK. More generally Itwould be interesting to disPOse for anyabeltan -varlety A defined explicitely bysome algebraic equations in the projectivespace pn of a method to get algebraicdl:fferential equations defining A inside A.(3.22) REMARK. If we :fix an abelian -variety A there may exist plenty of Zariskidense Ao-subgroups of A; they may be ofarbitrary dimension. For example take A anelliptic curve defined over put G = G N x Aand assuming for simplicity that In ordinary

(A -- {6}) a . let 6 # be the trivia] lifting of 6from to (3 and e � H�(T ) be a non-zeroglobal vector field. Then letting [l,...,[ N be a-besls of Xa(Ga N) and T 1 ..... be thecorresponding basis of a De.(G). Then by (I.1.9) � P(G)so L(CaN) , put -- 6#+[10+[2[1+...+[1 � defines a structure ofalgebraic D-group on G. It is esy to checkthat there 1s no intermediate field between ,(A) and (G) preserved by . Then G A is anabelian Ao-grou p embedded in . e. Moduli oho-Sroupa We propose ourselves to finddescriptions of the set q = ( Ao-groupsmodulo isomorphisms} or o:f remarkablesubsets of It. We start with "set-theoreticdescriptions". /e shah continue by putting 6-manifold structures on such sets. By (2.8) theset q/t can be written as a disjoint union rt =

J I qR(G) c

124 125 where G runs through the set ofisomorphism classes of irreducible algebraic-groups and for any such G we denote ql(G)the set of isomorphism classes of Ao-groupsI' such that G(I') ! = G. So one reasonablething to do (although as we shall see not theonly one) is to fix G and try to describe I(G).Here are some easy consequences of ourtheory: (.1.) LEMMA 1) Assume G is affine.Then (G) is non-empty if and only i! G isdefined over . 2) Assume G is commutative.Then 11,(G) is non-empty if and only if _S(C), (i) uer t.A, for all i. 3) Assume G hasno non-trivial linear representation. Then9R(G) has at most one element. Proet. Use(II. 3.31) and (IlL 2.18). Now we claim that

for any irreducible algebraic -group G wehave.' IL(G) - P(c)int/Aut G where P(G) intis the set of m-uples (pl,...,pm) of pairwisecommuting elements in P(C) such that (pi ) =(6 i) for I _( i _ m, while AutG acts on P(C)tnt by the formula (o,(p l,...,pm) ) (o-i -I +plo,..,O PmO), oe AutG. whereo stands alsofor the induced automorphlsm of .(G). Tocheck our claim note that by (2.8)q(G)identifies with the set of AutC - orbits in theset of algebraic D-group structures on G; butnow we are done by (L i.l). The abovedescription of qqq,(C) is of course notsatisfactory. To get a better picture assumefor a moment that is a field of definition forG so where G:ls some algebraic -group. * *acts trivially on (G)). Then Let 8i be thetrivial liftln 8of a(6i) from to G (as usual 6i

the map P(G) m + P(G/,) m (Pl ..... Pm) "(h '- Induces a bijection qL(G) = P(G/Z)lnt/AutC where 1) P(G/) Int is the set of m-uplesl,...Om ) of elements els P(G/ )such that V,Oj - V ?i + iej ] = 0 for all l,j 2) Aut G actsby the "Loewy-type" action [Cl]: He (VV) isthe logarithmic P(G)-connection on (AutG,P(G/%)) defined in (I. 1.2) so recall thatThis dcription is unsatisfactory the first onecause everything is "hidden" by the "-algebraic action" of Aut C; what one shouldbe looking is a description involving analgebraic (rather that a " -algebraic") action.Such a scription will be given in theTheorem below in a scial case. (.2)EOREM. L G an irreducible algealc -8roup,G = G and sume (G) ntains a representativeideal V. Then we have an identification

where �1L(G) = �lnt/Aut C, l) V int = {(al,...,a m) $ vml V - ,a. - V 6; ai + [ai'aj ] =0 for all l,j}. 6j i ' 2) Aut G acts on V jnt bythe formula b,(a l,...,am)) + b' la lO,... 'JarnO) Proof. The inclusion vint+ p(G/%) intinduces a map b: vint/Aut G P(G/,)int/Aut GWe must check b Js a bJjection. To checkinjectivity assumee = (e l,...em ) and 0' --j,...en) are ^ut C-conjugate elements inP(CJ,)int, hence ior some element� Aut G.Since V is an Aut C-invariant subspace ofP(G/) we have o--]op v

126 Since P(G,fin) is an Aut G-invariantsubspace of P(G) too, we have o'16 7o - 67� P(G,fin) Since V n P(G,fin) -- 0 we geta}e =v'lef for all i I' :o'lo for all i. From b)and (I. 3.0, 1) we get v c Aut G). By a) 0 and

0' are Am -conjugate so, is injectlye. To eckthat is rjecive ake0 = l,...m) P(G/) ira. Wemay write 01 = d i + a i with di P(G/,fin), aiV. We claim that (dl,...,dm) P(G/) nt. Indeedwe have in P(G)the equaliies: Since V is anideal In P(G) the lt 3 terms of t6e right handside of the above formula belong to V whilehe first erm 1ongs to P(G,fin), becauseP(G,fin) is a Lie /-subalgebra of P(G), (IV.1.2). We conclude above that the first termvanlshes which prov our claim. By this claiwe may consider the gebraic D-groupswucmre on G efine by he map P , P(G) cL{I. I.D. By {2.) thi* algebraic D-grouptructur on G i ,plit hnc by (2.8) ther exi,ts o� Aut G such that We get that hence 0 is AutG-conjugate to O' lalo,...' lamo)� V lnt andour Theorem is proved. (.3) The above

theorem giv a rather satisfactory answer tothe classification problem f linear fi o-grou.Indeed for any linear fi o-group Y, G = G(F)! is alline by (2.18). By (II. 1.1) G isdefined over and by (II. 1.22) P(G) containsa reprentative ideal. So Theorem (0.2)applim to GI We ge he following bijtion lin== { linear f o-groups modulo isomorphism}= (V(G)int/Aut ) G 127 where the disjointunion is taken over all isomorphism classesG oi irreducible arline algebraic -groupsdefined over J and V(G) is a representativeideal in P(G). Moreover note that if G issuch that all its weights are killed by P(G)(e.g. if either the radical of C is nilpotent orthe unipotent radical of G is commutative)we have by (IL 3.16) and (II. 3.22) thatV(G)int/Aut G;)(, = Wo(G)int/Aut Gij where

Wo(G)int = {(al,...,a m) c Wo(G)i 61*aj =6;a i for all i,j} (67 being here the triviallifting 1 of (61) from to W.o(G)--Wo(Gj) andAutG acts on Wo(G)int via its action onWo(G( ' ). (.) As already noted, by (2.10) ago-group r has' no non-trivial linearrepresentation If and only 1! G(r I) has thesame property. Then (4.1) implies that wehave a bijection go-groups with no non-trivial linear representation modulo isomorphisin s(c) C(G) =[/.sucl'that )(6 i) �Der qfor all i, modulo isomorphism (it.5) AssumeG = I. x A where L is an irreducible arlinealgebraic group and A is an abelJan variety.Then by (IV. 3.1)(G) is non-empty if and onlyif is a field of definition for both L and A,say A A.j L L3y ' )q. Assume this is the caseand assume in addition for simpllcJt'y that

P(L) -- P(L,fin) (e.g. the radical of L isunipotent). Then we have by (IV. 3.1) and(4.2) that rlt,(G) = (Xa(L) � L(A))int/Aut Lx Aut where (Xa(L) � L(A)) int = {(al,...,am) � Xa(L) ) L(A)I 67 aj = 6 a i} ( 67 beingthe trivial lifting of (6 i) from , to Xa(L)L(A) = (Xa(Lv ) �;, L(A)) 1 and Au! L, xAut A acts via its natural action on Xa(Lj1 )� L(A: ). (4.6) Let's give a description oftab, := 71,ab n fil?= [ abellan go'gr�upswith n� non-trivial linear representationmodulo lsomorphism Indeed by (3.19) wehave a decomposition into a disjoint union Il_<g<_d_<2g ,g where

128 qLd,g = i Ao-groups in 11%ab'= ofdimension d and t embedding dimension gMoreover by (3.19) we have a bijection f

abelJan -varieties A of dimension g with theproperty t d,g" ) = d modulo lsomorphlsm{t.7) Up to now we discussed only "set-theoretic" descriptions of "toodull sets". Butthere is an interesting possibility that some ofthese moduli sets (or some natural"coverings" or "pieces" of them) carrynatural structures of A-manifolds. Forinstance if G is as in Theorem (3.3) (withordinary, to simplify discussion) then qr(G)is of the form where is an arline locallyalgebraic ,-group acting on n via arepresentation () P , CLn()c GLn(.) where pis locally algebraic. Now has infinitelymany components in general so our task of:HndJng A-manifold structures is hopelessunless we pass from Ln/ () to its "covering"q,n/^o() (the fibres of the proiection /n/A�()

- n/A() are then at most countable). The studyof such quotients Ln/A�() was part of aprogram initiated by B. Weisfeiler; somegeneral theorems were proved by him andlots of conjectures can be made at least or A� reductive. In any case what one shouldexpect is that, exactly as in the "non-dl:ferential" invariant theory only an "opensubset" of ,n/^o() can be equipped with a A-manHol d structure. Fortunately for manylinear algebraic groups G we dispose of aquite explicit description of the action of^utC on a representative ideal (cf. (a. 3)) thisleading to a direct easy description of (G) =V/^ut�C;g the latter turns out aposter[ori tocontain a "big" piece which is a 6-manifold.We [lustrate this with one example (thereader can consider more complicated

examples and try to apply our theory to dealwith them): (t.8) THEOREM. We have anatural bijection t,(Ga N x GNm) ,glN(,)/(GLN(Z ) x GLN()) Moreover thesubset 1 � = GLN()/CLN() of the set 129identifies (via logarithmic derivative J'?6"GLN() * glN( )) with glN(). 5o has a naturalstructure of A-arline space , N2. Poog. Innotations of (II. 3.1)) (II. 3.12) we clearlyhave: Wo(G) -- W(G) = Horn( 2: N, ,N) =gin () while AUt(Ga NxGNm)=GLN(Z)XGLN(). By (#.3)we getour formula for [(G xCNm ). The restfollows from KoJchln's surjectivity of [76(cf. (I. 2.3)). (t.9) We close our search for"6-manifold structure on moduJJ sets" byconsidering abel Jan 6o-groups with no non-trivial linear representation i.e. the set 4%ab'

cf. (#.6). Once again the example of abelJanvarieties from algebraic geometry teaches usthat if one expects toodull one has tointroduce polarizations. In what follows by apolarized abel Jan Ao-grou p with level nstructure we will understand the equivalenceclass of a triple ( r A ,,,) where r is anabelian Ao=group T +A is a Zarlski denseembedding (A an abelian -variety), 1 is apolarization on A and rt Js a level n structureon A; the triples (l: ri +A l,).i,), i = 1,2 arecalled equivalent if l' l --r 2 = r and there isan isomorphism o.'A 1 +A 2 such that 1 -- c2'� )'2 = )'1 and o(1) -- 2' A triple as abovewill be said to be associated to r. There is anobvious notion of lsomorphism of suchobjects. Moreover a triple as above will becalled principally polarized if ). is a

principal DoJarization (see [/u] forbackgraound). Note that given an abel[an 6o-group r there are at most countably manytriples as above sssociated to r, moduloisomorphism, cf. (3.1). Let ld,g,n be the setof lsomorphism classes of principallypolarized abelian Ao-groups with level n-structure having the following properties.' 1)they have no non-trivial linearrepresentation, 2) they have dimension d, 3)they have embedding dimension g. In whatfollows we shall prove tht these ld,g,n (for g_ 2 and n _ 3) have natural structures of A-manifolds. They are non-empty only if g _( d_( 2g. The case g = I was left aside justbecause Jt is too easyl we leave it to thereader cf. (3.20). On the contrary thepresence of level n-structures (n _ 2) Js a

technical condition which we could not getrid of. Finally we expect that the case of non-principal polarizations can be worked withalong the same lines as we do for principalonas principal polarizations simplifydiscussion at a few points. We will prove infact much more namely that dg,n can beidentified with certain subsets of thecorresponding moduli spaces of abel[anvarieties which are locally closed in the Gtopology o these moduli spaces (o! coursethey will be not locally closed in the

130 131 usual ZariskJ topology). To state ourmain result let's make some notations.Denote by ug,n b the toodull space ofprincipally polarized abelJan .-varietieswith level n structure [Muj tn tn O o a -

varlety. Then for g < d < 2g the natural mapstld,g,n + g,n (I' + ,,,n) ' (A,X,rO are injectiveby (3. I8) and their images d,g,n are disjoinerand described as polnts in g,n whichc9rrespond to abel Jan d,g,n = [varieties Asuch that dim.,.DH�(P./q4) = d t cf. (3. IS),(a.6); note that %,g,n and �d,8, n can beIdentified sets. Here is our main result. (.10)EOREM. In tations ave, i g 2 and n 3 then: l)F any g d 2g the sets dgn are n-empty and thesets ,g,n u g+J,g,n u "' u d,g,n are -ci in gn' 2)e have ,g,n - d' ,g,n u ... u 2g,g,n = ,n (.11)Note that the ua]ity =o oUows rom (]I. .).The rest o the g,g,n g,n aerlons require aprerafion. (a. 12) Let Y be any -varietydenote by TY V(Ty) Spec(S(/)). It co[ndeswith the langent bundle o Y (vewed avariety) in case Y [s smooth. Assume or

simplicity this is t case. Aume morver that Ys defined over and [x an JsomorphJsm Y Yo� � Te claim that or each p E P there is anatural map o A-manolds To define it it issufficient to define a D-map Vp Y +(TY)hence by (0.9) and (l. 2.12)' It is sufficient todefine for any (reduced) D-algebra R mapsVp .' Y(R) +(TY)(R) behaving- functoriallyin R. We define them as follows. For anymorphiSm f $pec R +Y still denote by f themap induced between the correspondingsheaves of algebraS, denote by PRc Derl)t(R) the multiplication by p c P and by p#Derl (Cy) the trivial Jilting of a(p) from ( toY = Yo ( then the difference pR f - fp* willbe an ,-derivation from (fly to f, (flSpecRhence will induce an y-linear map hence amorphism SpecR +TY which we 11 pf. More

generall for .any faroill pl,P2,...,P we have amphism of A-manifolds (pl, 9p2,...) Y *(TYx yTY x ...). (.13) Clearly to prove (.10) weed a meted to study the "variation" of the D-module structure of HR(A) A varies In amulJ space. Remarkably this can done bytranslating t D-module orti of HR(A) interms of the Gauss-Manin connection of any"deformation of A which is defined over ".This is what we do next. +Y be a smoothprojectlve morphism of smooth (-varieties,assume (.l) Let L '* Xo o its fibres areconnected and let II: X *Y be obtained fromX o +Yo by tensorlzatJon with Let moreovery ��() be any 9-point of Y and X be thefibre of X +Y at y. Y By (III. 1.1) we disposeof a Gauss-Manin connection which is an -lJnear map. X X � R(Xy)'H)R(Xy))' Y (O.

la.l) V y: De --'*Hom>(,(H d * d On theother hand one can consider the relative deRham sheaf HLi(X/Y):-- = Rl]I((/y)l recallfrom [Ka] that thls Is a locally free (y-module and its formation is compatible withany'base change, in particular HR(X/Y)e (y)H)R(XyX(y) -- residue field of Y at y).Denote by HR(X/Y)y the stalk at y and by ry,HR(X/Y)y * HR(Xy) the reduction modulothe maximal ideal my of (y,y. By [Ka] wealso dispose of a Gauss-Manin connection v(defined in the same way as the one in (III.1.1)) inducing at y an (flyby - linear map(.]4.3) ?X/Y,y, DerrbL (,y) +HomtL(H)R(X/�)y ' HR(X/Y)y), o !+ ?/Y'Y The map (4.lq.l) should be thought of as an "internal"Gauss-Manin connection while (q. lq. 2) asan "external" one. Note that we can see T Y

as a subset of TY (cf. (4.12)) it Is of coursethe fibre of the Y projection TY+Y at y. Inparticular denoting by fy-'$pac+Y themorphism defining Y �Y((.) and al,o theinduced morphIsm fy (yy ' we may considerby (4.12) for any p P the -point ?pfy TyY cTY dellned by (.la.a) ?ply = a(p)fy. fyp#eDer(y,y,() _- TyY

132 133 where p* is the trivial lifting of B(p) from /, to Y, viewed here as an elementof Der (y,y). Choose moreover anyderivation e � Der (y,y) "lifting" ?ply i.e.with the property that (.lt.5) V f = f e py yFinally, let p## be the trivial lifting of a(p)from , to X. It induces an obvious -1inearendomorphism ?p# of HR(X/Y)(anyderivation of (X lifting a derivation of )�

does!). The following lemma shows that the"internal" data can be recaptured from the"external" ones. Philosophically thls ispossible because of our hypothesis that the"deformation" X/Y of X is defined over ,hence is "sufficiently rich". Y (4.15) I.EMMA. The following diagram iscommutative vp., </�'y , H.(X/Y)y ry I r XY HR(Xy ) ? (Yp) HI__(X ). DR y Praof. PutC/ = $pec(y,y, = X x y , cover by open affinesubsets i = SpecAi' lift 0 to derivations 6 i� DerA i' consider the cocycle (6j - e l) �C l(( i)i, T / ) and reduce it modulo my. toget a cocycle (el - 6 i) � CI(Xy,i,T X ) Y nand the upper bar means "reduction modulom "(note that 0j cannot be where Xy,i = Xy iY reduced modulo my, so one cannot speakabout j, because 0j may not preserve my;

only the differences ej - 0 i preserve it,indeed they vanish on it). By (t. it.t) and(t.l.5) we get the equality a (p)fy = fy(p* + 6)which shows in particular that p#+ epreserves my. Hence its liftings p'e+ eiDeiAi preserve my so they can be reduced modulomy to get derivations p*# + O i � Der ' iwhere l )(Xy,i)' We get the equality ej - e i =(p** + el) - (p** + e i) Now 'choose arepresentative (oi,xij)of a class �HR(X/Y)y-- l-fdrms on i and xij R( are H/C/ ) where ,i :C( i n j). Then ry(VX/Y'Y)Tis represented by (.) (Lie0i 0 i,(0j. el) , J +Oixij ) x Moreover ( ap))ry is represented by(p## + 0i) )mj + (p*# OiXlj)) (**) (Liep.# +01t0i,((p ## + el). + while ry?p. 1] isrepresented by (***) (Liep** i' P**Xij) andour lemma follows. Now for any Pi in a

commuting basis pl,...,p m of P let's fix a 0 ias in (t.l) I.e. with f = Ply fYOi' (4.16)LFMM . In notations of (t. lt) and (3.17) let Ibe the subring of End. (HI,u(X/Y).) --. dl '-"Y generated by . and the maps 7i TM ?p? + ?/Y'y' I < i < m./non we have the equality ~el21 DHO(t ./i,) = ry(D H (X/Y)y) Y Proof.Let D N be the L-linear subspace of Dgenerated by {, and the products ... (k < N)and let [N be the -1inear subspace of Igenerated by and the PilPi 2 Pi k - productsof the form VilV12 ... Vik (k< N) We checkby induction on N that o 1 (#)N DNH (i]X/%1, ) = r. (I_ H�(l,,.,,). ) y N ^1 y y For N= 0 this is just the base change theorem.Assume equality holds for N - I and let o 1 e[and i any index between I and m. Moreoverlet's agree to denote eH ( X/Y)y , N-I X g Y

again by V i (no confusion will m'rise withthe previous notatlons{). Then we have bya(p i) (.l): ry(?ic = (?j)(ryC ) But byinduction hypothesis (?iXry Z ) eV.D...H�(i]. 1. ... ) CDNH�(i]<y/ ) so weget e o 1 ry( ?lC r DN H which proves theinclusion ":' in (*)N' The inclusion ,,o,follows by an entirely similar argument.

134 135 (.17) To formulate the next Jemmalet's Introduce some new constructions. Innotations from (.1) let Y' be an open subsetof Y whose tangent bundle is trivial and fix atrivialisation T�, Ny,. Associated to thistriviaJisation consider the natural map : TY'+ H�(Y',Ty) (:[or each vector v in someTyY consider the vector field on Y' obtainedby displacing v in the other points of Y' via

the trivialisation of Ty,). Then for any point(y,v)=(Y,Vl,...,v re) sTY x...x TY, y �Y', v i�TyY t Y . Y , m factors we can consider the-endormorphism V l(Y,V),...,Vm(Y,V) ofH)R(X/Y)y defined by the formula .X/Y,y I<i< m Vi(Y,V ) = Vp? + v x(Y,Vl) - _ anddefine {y,v to be the subring ofEnd(HIDR(X/Y)y) generated by , andVi(Y,v), I _<1< r. Then we have (t.18)LEMMA. !n notations from (#.17) and (.la)for any integer d the set � .. HO; 1 ' d} Disd = {(y,v) � TY x x TY; y � Y', dimry( y,vX/Y'y' - Y Y is A.closed in TY x ... x TY. YY Proof.' We may assume Y = Y' shrinking Y(the question is local) we may assumeH�(t.l.,. ) is free and a direct sumsand inHR(X/Y) which is also free. We also maywrite /'/ N TY = Y x , and choose a frame of

vector fields Wl,...,w N on Y. The map : TY'= Y' x %N + Ho(y,,Ty) has then the form �(y,Xl,...,X N) = ZXiw i The elements of TY x... x TY are then pairs (y,X) where X = (Xij)is an m x N matrix with Y Y elemetns in , andVi(y,X ) = Vp + j!iXijVw) Finally, fix abasis el,...,e n of H�(fi(/y) as an (y-moduleel,...,en,...,e k of HR(X/Y); with respect tothis basis we may write Vp= p+ M i V wi+Ni W. which extends to a basis where Mi, Njare matrices with entries in ((Y). All wehave to check is that for any fixed matrix (ai(of elements in ((Y) and any integer r thefollowing set Is A-closed: (,,) {(y,X) ey x2/,raN{ dim<ryde(Zaiaei) where < > means"-1inear span", (x = (Ol,O2,...) aresequences of integers of length {(x I and d.-=(p . + + '-V�l V�2 "' I I jalj(Wj + NJ))

(P2 + M�2 jX�2j(wj + Nj)) ... Now thecoefficients of the ei's in do(Zaioej) arepolynomials Jn (pi 1) ... )(Pik)). whosecoeffcients are(fixed) functions in ((Y). Theconditions defining (#) are obtained by thevanishing of certain determinants whoseentries are the coefficients of the ej's Jnd((.ajoj) evaluated at various points y � Y'.This closes the proof. (e.l) Let's checkassertion 1) in Theorem (q. lO). As wellknown the moduli space Y = dg is smooth (n> 3!) and there is an abelJan scheme E: X +Y (the "universal amily"). Moreover X,Y, E.are defined over the prime field Jn particularover so we are Jn the situation of (.1). Thestatement of (/.10), 1) is local on Y so wemay replace Y by some aftins subset wherethe tangent bundle is trivial. Noting that in

notations of (/.17), 1ywi) ' H�(Ty)lifts (byits.very definition) the tangent vector v i �TyY we get by (.16) that Y o1 UH�(l X ./q)= ry([(y, Vply ..... Vpmfy)H (X/Y)y) itfollows by (/.9) that Cd,g,n is the inverseimage of Dis d rom (.18) via the map.. Y() +(TY x ... x TYX[) Y Y y + ('? f ,..., ?pmfy)Ply The latter map being a map of A-manifolds (/.12), closedness o 0d,g,n in theA. topology :[oliows from (.l To concludethe proof of (.10) Jt remains to produce foreach d with g_< d_< 2g an abel Jan variety Ad belonging to rd, g,n. For this let E be anelliptic curve not defined over the field o!constants of 61, say, let F be an elliptic curvedefined over 3, and put A d = E d-g x F 2g-d.(.20) It would be interesting to know whetherTheorem (/.i0) holds for i (n (2 (especially

for n = i since "level l-structure" meanssimple "no level structure"l). So assume n _<2. Then assertion 3) in (. i0) still holds forIdentical reasons. Moreover the sets (.)Og,g,n u UJ, g+l,g,n u ...u d,g,n are theprojections via ,3 '+ Lg of the correspondingsets for n= 3. The latter are known to

136 137 be A-closed by (t.10). But in spiteof the fact that the map tg,3 ' tg,n is finite it isnot clear to us whether it takes A.closed setsinto A-closed sets. Note that a "Chevalleyconstructibility theorem" holds Jn our context[B2]B 3] so the sets (,) are at leastconstructlble in the A -topology! (4.21) Theloci d,g,n deserve further study. They stratify(in the A-topology) the moduli space in anon-trivial way and bare a formal

resemblance with the degeneracy loci in,,Brill-Noether theory". It would beinteresting to compute their A-dimensionpolynomials [K 1 ], to investigate theirirreduclbllity or speciallsation properties(e.g. is td,g,n contained in the A-closure ofLd+l,g,n?). APPENDIX A. LINK WITHMOVABLE SINGULARITIES In thisappendix we discuss the relation betweenalgebrlc D-groups and A-function fields withno movable singularity (NMS) in the senseof [B 1] (recall that by the latter we mean anextension of D-fields F c E, where D = F[P],P a Lie F/k-algebra, such that E = F<V>where V is some wojective D-variety. Forthe understanding of the discussion below, acertain familiarity with [B 1] is preferable.We shall assume from now on that all Lie

K/k-algebras occuring have a finitecommuting basis and all fields havecharacteristic zero; K will be alwaysalgebraically closed. There are easyexamples showing that if P is a Lie K/k-algebra, D = KiP] and G is an algebraic D-group then the extension K c K<G> need nothave (NUS); for instance P. Cassidy provedthat this is the case with G =Spec(K{y}y/[yy" - (y,)2]) (note that G ! = Ga x Gin). On the other hand we can wove:(R.I) PROPOSrrloN. Let G be an irreduciblealgebraic D-group all of whose weights areP-constant. Then there exist a Picard-Vessiotextension K i/K and an intermediate D-fieldK cEcK <G) such that E/K is split (ie E-K(EP)) and KI(G>/E is regular (l.e. E is I I 1 ''- 1 algebraically closed in K i<C>) and

generated by exponential elements (i.e.elements of K whose logarithmic derivativesbelong to E). For the concept of Picard-�essiot extension see [K 1 ], Chapter 5. Wewill also wove the following Woposltlon(where we recall that an element x of a D-field extension E of F Is called a Picard-Vesslot element if x is contained in a finitedimensional D-submodule of E). (A.2)PROPOSITION. Any D-field extension E/F(with F algebraically closed in D = F[P], P aLie F/k-algebra) is generated by Picard-�essiot elements has (NMS). So we get by[B 1] p. 103= (A.3) COROLLARY. Innotations of (A. 1) all three extensions K cK1 cE cKi<G> have (NM5). (A.4) Proo o!(A.I). Denote by 61,...,6 m a commutingbasis of P and let Pi be the image of 6i in

P(G). Since we assumed all weights of G areP-constant we get (in notations of (II. 3.16))that a i := l?i � w�(G) so by 1oc.cit. wemay write Pi = 1 +di+E a i wre P(G) is thetrlvl lifting oi (i ) from K to G=Go K K(Ko:KD , Goa Ko-group), d i P(G/K,fin)and'a Ea is the map rom Wo(G) to P(G)appearing in (11. 3.1). Exactly as in theproof of (V. .2) the derivatlo (&[ � di) i arerirwise commuting so by (L 1.1) they definean gebralc D-group Structure on G. By theproof of (I. 3.11) we can find an automphismc � Rut(C K i ) where K I/K is a Picard-Vessiot extension such that So we mayassume the images of i in P(G K 1 ) have theform ] + Ei Let U and H as in (IL 3.21) and tE = KI(U); clearly E is eserved by D 1 =KiKD and E 1 is split. oreover the extension

K i<C) is genoated by elements of the torre 1y with y � O(Ho) x (where H = Ho K K, Ho a Ko-group, notations in (!. 3.21) again)and any such ly is an exponential element inthe extension Ki<G)/E cause its logarithmicderivative with respect to Pi is ai(x) e 1 E (iy G(Ho)X). Our prosition is proved. .5) P.2). Let E/F a regular extension of D-ieldsgenerated by Picd-Vessiot elemts. Thenclearly a finite set of them suffices so E =F(X) where X = ScA d A is some D-subalgebra of E, finitely generated an F-algebra and locally iinite as a D-mule. Let Vc A be a inite dimension D-submodulegeneratin 8 A as an F-algebra. Then thesymmetric algebra SV h a natural structure ofD-algebra and the closed embedding X + Y= Sc SV is a D-map. Give the lynomi algebra

(SV)[t] a structure of D-algebra extension o5V by lcttJn 8 = 0 for J p � P. Since Dprerves the gradation on (SVt] tt follows thatq = Proj((SV)[t becom a D-variety and theopen embeddin 8 Y + is a D-map. Let X bethe Zariski clure of X in . By (0.1), 2) X is aD-subvariety of O and this closes the proof.Usin 8 (IV. 3.1) one can prove (in a waysimilar to (A.I) above)-' .6) PROSmON. LetG be an algebraic D-group and assume itsunderlying K-group G is the product of aunittent linear group by an abelJan variety.Then there exist a Plcard-Yessiot extensionKi/K (which by [B 1] has (NMS)) and anintermediate D-field

138 139 Ki c E c Ki<G> such that E/K 1 isregular and split (hence by [B 1] has (NMS))

and K i<G)/E has (N4S). We close thisAppendix by briefly inspecting the relationwith Painleve's property [GS], [Ida]). Weneed an analytic preparation: (A.7) Let f: X+ Y be a map of analytic complex manifoldsand suppose we are given analyticcommuting vector fields 81,...,6m on Ylifting to some commuting vector fields 6 onX. We say that ! has the Painleve property(with respect to these vector fields) if, uponJetting + Y be the universal covering of Y,there is an analytic -isomorphism XXvC/Zx'7 Z f'l(yo)' Yo �Y' sending into 6. whereis the unique lifting of . from X to X x = i '~ J' and 6# is the trivial lifting of 6. from Y to Zx Y � Note that if dimY = m and if we throwaway from Y the locus where 61,.,6 m do notgenerate the tangent space and from X the

preimage of this locus (we assume that weare left with some non-empty opensubmanifolds!) then we get a foliation on Xtransverse to f and bein[ � x ire a"feuilletage de Pamleve de espece" in thesense of [GS], i.e. having the property thatany path on Y starting from Yo � Y can belifted in the leaf passing through any point off'l(yo). (A.S) Let's assume in what followsthat K is an algebraically closed fieldcontaining C with tr.deg. K/C = m <-- andput P -- Der cK, D =K[P]. Let pl,...,pm be acommuting basis of P. Let moreover V be aD-variety. We say that V is a Palnlev D-variety if there is a cartesian diagram of C -schemes.' S . Spec K with U and S smooth C- varieties and j dominant and if there arecommuting vector fields 6 6 m I ""' m I '"" on

S lifting to commuting vector fields on Usuch that 1) Each (pi) � Der C K lifts 6i andthe image of each Pi in Der C (V lifts i' / 6 mand 2) fan uan + san has the PainIcrcproperty (with respect to 61 "'" I ""' mviewed as analytic vector fields on S an,uan). Finally let's say that a D-field extensionE/K is a PainIcrc extension if E -- K(V)where Is some painlevY'D-variety. Thefollowing hold: a) Any smooth projectlye D-variety is a Pamleve D-varety in particular,any extensio" E/K with (NM$) is aPainievextension. This Is a trivialconsequence o! Ehresmann's theorem [3]. b)A consequence of the above remark and ofour result in [B 1] p. 103 is that any stronglynormal extension of K is a Pmnleveextension (as it has (NMS)). c) Let G be an

irreducible algebraic D-group. Then K(G)/Kis a Patrdeve extension (more precisely C isa Painlev D-varietyl). This follows fromHamm's result (11. 1.3). cf. [Ha]. d) Painlevextensions need not have (NMS) for theymay have "movable singularities hidden atinfinity". For instance K. Okamoto's work (inSero. F. Norguet Fvrier 1977) shows that thefamous Palnleve second order equations"lead" to Pamleve extensions; these are notnecessarily NMS extensions (as proved byNishioka iN]). APPENDIX B. ANALOGUEOF A DIOPNANTINE CON3ECTURE OFS. LANG The following result was provedin [B6] and illustrates the "diophantineflavour" of Ao-groups: (B.I) THEOREM.Let A be an abelfan /,-variety with /-tracezero, X c A a subvari- ety and Y c a Ao-

subgrou p. Then there exist abelfansubvarieties Bi,...,B m of A and points a I ,...,a m � A such that m 1' nXcL./(B i.a i) cXi=l The proof relies on analytic arguments(namely the Palnlevff property in (A.8), c)plus'a "Big Picard Theorem" due toKobayashl-Ochial). Using (B.I) we deducedin [B 6] the following (B.2) COROLLARY..Let be an abelfan variety over analgebraically closed field F of characteristiczero, assume A has F/k-trace zero for somealgebraically closed subfield k c F, let X c Abe a subvariety, T c A a fJnitely generatedsubgroup and Ydiv = {x eA; ] n such that nx� �} Then there exist abelJan subvarietiesB 1 .,. B m c A and points a I .,. a m � Asuch that m 'Ydlv n X c i=kl (Bi + ai) c XThe above statement without the trace

condition was conjectured by Serge LanE;recently C. Galtings proved a weaker formS. Lang's conjecture in which Ydiv isreplaced by y. Note also that in [B?] weproved an "infinitesimal analogue" of (B.I).APPENDIX C. FINAL REMARKS ANDQUESTIONS Ve start with some intriguingshort questions:

140 141 (C.I) Does there exist non-affineAo-groups? (compare with (V. 2. la)). Wethink there should be; a candidate wouldperhaps be E(A) A where A is an ellipticcurve over whlch Js not defined over v.(C.2) Let G be an irreducible algebraic D-group (D =K[P], char K = 0). Is the linearpart of G defined over KD? This is crucialfor understanding P(G) for arbitrary (non-

linear, non-corn muratlye) G. (C.3) Let G bean arline algebraic D-group (D =K[P],charK = 0). Are the weights of G necessarilyP-constants? (see (fl. 3.12) for background).This is quite significant since we dispose ofa very explicit description o! affJnealgebraic D-groups all of whose weights areP-constant, cf. (IL 3.16), (V. .3). (C.e) A hardtask is to extend our theory from Chapters 1-by shifting from Spec K to a more general k-scheme, in particular a k-variety 5 (k = C)and build a bridge between our theory andDeligne's theory of linear differentialequations with regular singular points[DeJ2]. Vector bundles on S withconnections are in fact group schemes over S(with fibres vector groups) on which thealgebra of differential operator D S on S acts

compatibly with muJtJpHcatJon, inverse andunit. So the theory we expect is an analogueof Deligne's in which algebraic vectorgroups Ga N are replaced by more generalalgebraic groups. More precisely, Deligne'stheory in [Del 2] shows in particular thatthere is an equivalence between the"analytically defined" category (#)top (localsystems of vector spaces on S an } where SJs a fixed smooth algebraic C - variety andthe "algebraically defined" category (*)alg{regular D S - modules on S } (here vectorspaces are assumed finite dimensional andDS-mOdules are assumed 09s-coherenthence locally free). One is then tempted toinspect relations between the "analyticallydefined" category: (,e)top (local systems ofLie groups on S an } and the "algebraically

defined" category: (e e)alg {"regular"algebraic Ds-groups } Here an algebraic Ds-group should be of course a group schemeover S on which D S acts compatibly withmultiplication, inverse and unit, while"regular" should mean at least that the DS-module o! relative Lie algebras is regular inDeligne's sense (compare with our regularitycriterion (lIl, 3.9)). Note that local systemsof Lie groups were studied in [Hal. We posehere the problem of their algebralsation. It isnatural to start with the simplest examples ofLie groups hich are not vector spaces. Let'stake for instance local systems on S anwhose fibres are (Ce) M (these areequivalent to representations of ]{j(S an) inCLM(Z)). What local systems of this typecome from algebraic Ds-groups? Of course,

group shemes which become tori afterpassing to a finite etaJe covering of S lead tofinite monodromy groups. But at least foreven M = 2g one can get infinite monodromygroups by taking non-trivial abel Janschemes f: A S of relative dimension g andthen consider their extension E(A)'+S by(Rife (A)v; there is natural structure ofalgebraic Ds-grou p on E(A)/S whosemonodromy is that of Rlf#anZ. Of course thefibres of E(A)/S are analytically Jsomorphicto (Ce) {W. There are {owever topologicalrestrictions :[or I an R fe Z (e.g. if S Js acurve the latter should be quasi-unipotent atinfinity). This shows that "algebraisatJon"should not be always possible. It is certainlynot unique either. So the taks is to replace(ee)to p and (ee)alg by some other related

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INDEX OF TERYlNOLOGY Lie K/k-algebra ................................................. (0.5)algebra of differential operators..................................... (0.5) D-module, D-algebra, P-module .....................................(0.5) D-scheme..................................................... (0.6) of D-finite type ................................................(0.6) D-variety..................................................... (0.6) D-ideal, D-point ................................................(0.?) D-group scheme, algebraic D-group.................................. (0.8) D-actions, D-cocycles ........................................... (0.13)

logarithm ic connection........................................... (0.16) adjointconnection .............................................. (I.1.10) automorphism functor............................................ (1.1.23) field ofdefinition ............................................... (I.1.27) locally finite, split.............................................. (I.3)representative Ideal............................................. (I. 1.21)weights .......................... : ....................(I].3.1), (11.3.12) killing the weights.............................................. (11.3.5) linearpart, abellan part ..........................................(IV.i.i) logarithmic Gauss-Manln connection................................. (111.1.2) Koalaira-Spencer map ............................................(111.1,4) total space, total connection

................................ (I11.1.6), (111.1.7)relative projective hull........................................... (III.2.1)universal extension.............................................. (III.2.3)regular D-module............................................... (111.3.6)maximal semiabelian subfield........................... : ........... (IV. 1.1) A-manifold ...................................................(V.I.I 1) A -group, A o_group , dimension..................................... (V. 1.11) split Ao.group ................................................. (V.1.13) produced A -group............................................... (V. I. 12)embedding dimension............................................ (V.3.16) modulisets ................................................... (V.O

NMS extensions................................................ Appendix ^Pa,nleve extensions.............................................. Appendix .145 K<X>........................................................ (0.6)XD(�)' XD(A).................................................. (0.7) KI<X>........................................................ (0.]0)PHSD(G/K).................................................... (0.13)(Vfi[V) .......................................................(0.16) L(G),P(G),P(G/K),DerkG,Lie G (I.1.1 ) KIp]......................................................... (1. i.i)Xm(C)' Xa (c).................................................. (I. 1.3)U(L), JG ......................................................

(I.].13) Au.._t G, Aut G, )'G............................................... (I. 1.23) KG'KL ......................................................(1.1.27) V !, X....................................................... (1.2.1)P(G,fin), P(G/K,fin)............................................. (1.3.9) ]I(A).before (11.2.1) II(B A)...................................................... (II.2.9)W(G) ........................................................(II..1) (T, ( (lU)).................................................... (II.. 1)(G)'Wo(G) ' ' (tl.3.]2) E a................................................... (11.2.9),(II.3.16) H l V H l 1 DR ( )a'DR(V)m,HDR(V) t ............................. (11I.l.l), (Iii. 1.2), (1II. 1.9) E(A)........................................................ - S(C) K

.... (III.2.3) uerk

................................................. (Iii. 2.16) /,<X> .......................................................(V.I.11) V fi,G fi...................................................... (V.2.2)Xo' o ............................................ (V. i. 13)X,G ........................................................(V.l.12) r/r,........................................................ (v.3. i)[r,r] ........................................................ (v..)z�(r) .......................................................(v..) R(r) ................... D o f]l..................................... (V.3.7) H (A/i,),E(A)a ............................ (V.3.17) A4 k........................................................ � .(V..i8) , m(C) ....................................... (V.Ob r ab,{ .............. d,g' d,g,n....................................... (V.t0 INDEX OF

NOTATIONS K[P],V D V P K D lp................................................... (0.5) (0.6)