Top Banner
Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than it answers, will be to study natural operations acting on the cohomology of various types of algebras. It will contain a lot of very surprising partial results and examples. 1
11

Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

Sep 24, 2020

Download

Documents

dariahiddleston
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

Cohomology operations and the Deligne conjecture

Martin Markl

The aim of the talk, which raises more questions than it answers, will be tostudy natural operations acting on the cohomology of various types of algebras.It will contain a lot of very surprising partial results and examples.

1

Page 2: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

1..1111111111-" ~l . '. }

. M .0 JVlA R K L~M~ .\f4,~f.,~ ~~1:.ort\,) ~ftt,~8Q.4\1l.(.L. t.~ 'r'~, '~A.."':l:llrJ.~;(A. ~",1,

~~~ 01 c.o!f.M,~~, t(..o)~(mffp~~ ":~ S. Ih'l"'r1...~'\f~~~. r~~: .,."..t,,(,~:

'---

.) "C~~?ia.t(' tk:~pfJ.8 ( kAJ~\~&t"\rJ,, ~: .

.) ~QdJA,e.: ~p. \)! R. ~1J,~\frAJ..LS '

.) QAlt.,Wfp\,. . \j j'\J.L" ~u~.~ =.:.~ .\~(,-j)'1.i. (1 II).,.,~ I/~ (\" t:"",f /):(.\ ii,

,) ru..p'\I. ~~t"J \. ~ ~a ol~ L- ~ 1.Ct..\ : 'rf ~I .. ,

1.: IJJ ,,) 'II I. till.. '.~t 1,(. (~~,(,j""', ,',,'I ;' ' ( 1 ,j.(, \

- - . ;,~.. I". ~.J,'})1J, a.. .1':;\~',11.A,I'1 "'c""J.]

..) ~~~.A)w..t. "'3. .~('t'l fftt. t{~~ie4 ('QR.om\.oft,~\t-f' ctl.~f'... ;),(1(. \)11 l)t., I

l~tCt+1~ C'.,()()\4,pflX J\\t\J..1-.S ~ L ~ !r\l., ,i f)\.rr)-,. (_o.f. f~! t, ~ ~ .- .

:1!',.~.,,~--~ CA. ~t.rr\AA.oJA,~~l\.o(\\' (WirJ.},(. fA 8"'~hj(.C':.' ~., ,:t" ~ ,.,)).~...

8) H~eo\t~ 'PROP ~tJOJi~ b\\ ~t1MC~wu.,~ -t.\.c. (~I:.t.t . I:' (,..,,;f))1'):.1 l/, .

(~1~r~ (;()~r~t~ o~ tftt ~()\('y)1. t rtE.~~I"fi - (. -~..~"J). 1~l.f I:~. ryj:j,,(,'!/( ~ 1':!

.b~l ~~-~~-- '

~ 1;-\~~c,..(1~ t.~ ,i,""\r-<lk~(AJ\1"b ()~~~,.f)~ 1A'f." (~i~(f ~ ~wC)tU.':;lJ1 (:)!).('.~~

+,.() (~~D1)~~ ~ r ( E.) ~ ) . f

I ~ Jo J l "...o_.~A-cl) TRiJf tuIJ,:(!/JJJ.'{ ('r'J,~'. :t!:'/I/~..t 'lU- Ulr.J>\.L~ .'N~, .'!! "

. d- \~ l~~_!~ ~~~--~~--~~~~~-~~~~~ .

~~(\)~ p\,. htio(\\ t \"\.4~. (v~~. ~tt ~). CoO'\o'~o#;.61J'-t ()(),_. () "1~l'()'lN:'.\'!A')!!,J.,

I _..II~I~~I~~j(l at crJlCJAe~k1..C\ V~rt}wr-,XJ.(., £J' If'{'~'!i, It j\I"\,,li\'t.'.(J)l.J... Cfr'..L-vh.-~~ U (, . -- -",t " '

LI

Page 3: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

,.

~~~~~--~~~ ~ w. ~ c..o ("m, p ~ ?. ~ -= (OJ"') f())

~ '=. ~~"r'1;(~/L, wrtrtlbi/rIAtilO~s o~ 1'L/A.r~S ~t

~11 ~~ Q,- 0 '. ~ 8 ~(;4..ck~JJt III : . ) ~ tW W, t t:o (\1, Q j \t1), ~ u. ". ~ ~ I\A xt =' ~ - ()UIt ~tl\M;6tli,u

(] v~..", ~-/ : I

op.s ~ *' ii~ ~ PROfs

.) oo_eo~~~. . .) C}vUlct~OM oj 1d~{s} LW . ,

. + ~ a.SIl4,(Vk,p~OM8: .) 0 01, " CQ~/&()":1J"i.L~. .) (Yt,O cw¥IA.d S J .too.

..~ Mt8 ~~'- 'J'

d . I ~ L:::t {I. 3 ~1l,

~t

~o ~ cL.t~~~~ C~~ ~ts om, 1J(A,tLCt~

~ 'RL~3~ht(,) ptOpfA, (\(,aR,,;ll.d eol oj ~1..(A.p~ CiJ('rYj,~~ {.!

~ (-Q. ("P)) d) } 'r. fA. IrAMc\ b~ 0pfl'l,Cl~ Q1. 'P'RO'P J

_Cl-~-) Q. 3~tC'\,bf<, bQJ\, CoC),.,lh.

~Vtt"L~ t. j ~ ~ 1II.0l~~ tLc & 1<01 ~e (\r1.L s a r ~ l S .

t 1. """;1;,'4 """

Page 4: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

~~ D~ iliJ t~pl. 1&l (\4..Q,t~ t,k'Q..fm,pk tk ()y~ @, 01 ~~ b~Q,e~(,b1,~ ~'RO-P ) 1~t.A.~~ lo(M.L Jc,rn.o~ bt!O1L

b~ mol 1.A..unU}(tJ. a.s ~(..(.cj.,. EUtrt1/ jo" {~1t. ()'nJ,t:.~cl t

~~i~ (fmI., ~Vow('nt>V') H~(P'r) ..) t

~M ~~lS Wt s~w f~~l ~1..obfAIIWII G~ }'~iJ'I,1,~ ("h;Q,tw~

. ~~Q.ti,0("r\! oCr\ c,o~o~~~ ~as.f" ~. c. o~ ('\;O('t\ - c.o~QI\.,

baFt. ~'Q ~fWIII &~orw R(f-.,) ~RA, ~~ ~O\ b! ~.c. 1k8~jQ.u..~ .l~ t.A./t1.,tIl~l.,C~ 8~~Q.tvorns_, /

--- ~ - - ~ ~ -- )( ---~=--"'--

~~.2~~~~,: +-\ l-{ ~~ o,1S. Ctf.«"Atb\Q. A ~ CN(,~. iIM, .t,t~t.e~..

'. ~ t '

+\+to~ ~ A) A) =- ~'* \ C +loci. (A) A) )J-I-1o~)~ ~OC dIIIOr /

'. 'C('f1, l R J A) ~ ~cn ( A~~fn~1) I A)

~o~ '<fn..,WS ~. .

~t Q/l;(. ~~~~~e '1~~~t~~~ ocyv C "', ( f) , C1 ) .

~t 1 e~ ~ ~ (Y)\AAttl~~M. (Y)a,~~ f)(t\ Q. t p(A.~

~!~ OM ~SIO~~\'11{, ("mM,e,~pe,;catW()\f ~

~) 0", ", ~flAQ.tl8(k1, "°1, 1 - . 11 ahId. tLfAP-,lt.

.) (YY\.At1ttfA"f~c.Q.t~o~

'2f" l ~ V'. 8, ,,~ aRt Crt - (YnAA. t t~~~ ~t,t.(lta..t '

() pVt,Ot -lL 0 (".-\ t I . rr1, ~ O.

Page 5: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

:~~~g~ ~ECAtAJA)J~~C~(A)PJ\ W

=-"') lIA \ ~)~) clA.~rt\..(,rA ~d ~ t1) ~) (a.)'o, C)~ t.) ~ ~ J (() b) # ~ tc.) d,)Q.)

~~ rn.~w"'a1. f)p~t\,Ofr'\, I CU e. ~(~) #

. ,---

I ~!i.caR1: ,aMq ~ flA(~)~) = j ~,

Fo\, C«. {'tII.~&O~l1--: .~t1)~)-:'(olA°-t~)o1J :~(aAAo1~)'O3~.

. II,l ~= h ' I

. . crf..ls is ~ ~ tn\h\Q ..l cuJL) ~ 0 f.., 1M ~ /:aM tt., ,

3 I .

dLrml~1Il £ ~ (4) - -

'Ll(~)d)l)e) tQ.)b)t)c\)~J !~1)::

t ( Q.) bc~) . ct (d ~ e ,) ~ ) ) ~ { 1)) . .

~~J~ : f6 t M) -= ~ ~ t>! reQ/\t\IOJ\, 4 'Ut,1:

(w\\~.) r.1, N-cAt\tlt eCi'b~c..\ -'\) ~- ) ('t\., ~ :unt""t ~~kM.t

.) SO("P1A. f1\M.CW8btA o! . .. - :un~~t i.tfAQ.ttd. crU'

.

11818;,

Page 6: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

. b o~ttO()\' @.) ~:'= t ~~~)~ ~~& ~~ o~c\,} ~ t.Prtr'fJ

0\ (VL~tWl.at ofs. .qt.o<'tru.t!IA:<:alld 'Mtvc 1fv\Stit,{iOM rn.o~ ~1..Q.! \;.~1.

.) I d-}\o~ 1JhdM.u.~ S bd '

,~ ~ t ~ ) ~ ) --) ~ at 'U ( ! ) ~I ~ -! ij (c~ 5 ) d ) -,) t 'U l ~ ) ~ ~) . - . )so ~~~ fi = t ~lM) IS \ .\.~ Q. d.o - ()pfl\.a.a ,l .

. 'F' , J ~";:: - ~~ v!.::;.:..J :

~~/ll.Q.d 21(h~~al OpfA,Q,tl:OlkS ~ +t. tI1~ w~ p~ .

. .-H*\ fb~ ~~MQ..~a1J~ b~ {-\~O<J..~f\;A)

~ ~ ~t-'\I1.1t opttJ,Q.ct t>! (n.~~Ql, OptJl,Q.~O~l f)~ +!. tok('rr'tO~'1.

1_. ?~b~: W~~ i~ t~ ~('M-okp~ ~~t... bd ~~ {~(v\,-O\.bt~ ~~-) ~bt~. bJ VF(/\..~cl~?

~~ PQhh.C,w..lQltt ~ "k)~t 1,~ }l '-t g) ~ -

"

W~ ~po~~b ~ U(h~ttQM&~.i . ~11~;~~~:i~~~~~~~i~ ~ ~ '" quat - .

~ QAt, II p1..ooj~" {~., ~,~-1Q(1nQ/L~Mo\ b~t ~ ~ mot~ h@ II!f.:IAbYA,. -' ': - . ¥""" ~~

Ob3fAAeL ~~~.:.l.;::j~~~~~!- ~r.~Cht., GO~~ ~~ ~tA ~,t,

b£t ~lw~f4, .' t\

~~~~~~~~j ='> I' ~ 9lKAA rt\Oa.~oJ. °\JIA.Ct.UolM." ~~~ U & C - , - j .

II1II'~,

"";I!,I!~~tj._.;ft~;

Page 7: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

-~ '

qt~~~~1~f4~. @6' ... ~ IAQ.dI14~ k .1 J I , t4.a.c:l,

~ OS~ o~ (CO~) ~) a,s J 'PoiuJ -.. ) I A -. fjJ -at t.bt.Q.

e, (A, A) .. - tt. "&LftA_-'- _I 1 -j.-. ., 3~!!~~ ~s~~~ dL ,,- . r.ii4:~q+I; ( A; A) , - -. , r a

I Ex: ~"' ~ ~f -. lilt , I ~. -A~ ~.~) ""CofW\ ': "ttQAl1.) H ~ ~ CE) -~. /0 ('rIL ~ dL ~ tit(, (~::1~~~~~;~ iO -= ( 8

(~ ) t ) :IP- n. u II /0' '>0 ~ . -..'1_,-"tftl. ": ass ~l, t (\t - - d (1~c,~~

~~~ t>l ~Q.tfAf1.al t>pUAf,LOt'\t.1 ()(M, c~ (A;f-')- -

~~~=~I~~ ." ~ ~ ~(.. =. '> '

~f -= C~E:) ..w 1,1 .... ) .a\f '" )Cf-:(Lj L ~ ~ (/\ L, L

& ~~L ~.rIM1Q..\ to Bass b~ g~lt1~~~ "iJk~tta.cl o! -

!;~~~, t.'UL!..., {\I'IIlv'J~t b)/VvL (YYW1...(..., (J i,fJ':,CJ.AJ.h ~t"Q4\1. I), ~ ,\ (.I&,~. ~

rtW\8_~/1'_~jJ.AJ (J \9, It I fl(). ~.:.-r~--~~ ~ { ~ O(k~ ~iL. b1.Q.C~

3 ~ w.d.. .l +I 0 { ~ lA:.c-) a ~L .11~i~~~~~ 1 b fA. t eo ~ b j c.. P ~ e..- cc., f.io (\\ t .~ bt, af.~~>Ql6~~~~fi;J :-tJ. t , ~~~!~~Q4.. VlWV\, l TJ- \ V~I ': V'_-=--' .:::'~-- ('W')fl ~,

-==-

~ ~

~~, ~Co~.]GC~t lot. Qt~to.e 6>.

~. q~1/{J\1, fJ, fi)llcM,~ & ~,t. r>! BOJ. ~ t,Wnt,~ bJ f.

~ ftJ(A,ti~: d..Ll~bl. R.t. DI ~~ Q.l1.d1 al3(...b~ Bf(i\)h."..-l,..O "aI~\..."I\... nnt"4,'nt, 1.1"4( ~.t'(O)'

..I"

Page 8: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

~

~~I~~~~~~. ()1- ~~DtJ,. Q: k ko ~ ~ \Ai = '> ~ 6> tV B 6' ! .t~~aJ~ co.~t.A~(\4,l. ~ l~ & 1 II. -- t.O""~~~

.#

8, (0) (\.c.p\L&~~S frI.().~o.I." tJLrrt~t" ~ c, (A)~)~ ~!:!* (8,(01) .- ('nQ,bMQt c,trAs8u t(Jk -H~ (A)"A).

.'§~~~~}~.: ((!>a.llto)6as~) "'~(A,~ '

~ ,& ~ ~Ik ~ Ik [~",J -'>- f«.):,~ 1 --.') 1< (~\t 1. -')0 --

rwRJAL £5' ~s o.Jl r ~~ t " u.~~\4J~ It ~~ ~ ~t\al ~ '

~ ~w~ +\ . ( 6 atS ( 0)) &o.t~) 'C o.- -

-. S~~~:"e~: t ~~(. ( 0) } ~ ~t) Q.t 1A.a.Q.~.It) p 0 -1l ~~ -4 ~ -+ ~ '--~ l -"\.. ,-

. "*-tAw! O'\L -H ~ e f\c., l 0) ) &~ ) '": O.

~ -

. ~ c.aX1 +t ~ t B 6 { 0 ) ~ tf) -1&.,~~, ~ {Vu, ~~.JX~.~~)~ 1 .

{~~~~i~;~~(~'> S<X.Le ~ ~ - ~O1., "J1.(.a.W~~L\

oJ,~(,~! ~1 ~1A,1rial- I CoS- ~fei~ J°tfnlf,) \(,o~~f("Y\.(,t /.

1-J~~~~~~.. I~A( ~Q~S tQg,s l 0)) ~~u *Q.~c. ) ~ lk J '. .

.,,', -

~~ &~~ tW f)p f'dOt~ (Mt ('nLU-&~td ~~~ ~,tl

..:1~:J==I~~OP~~~ ~o~fJ ~1II.~k ~~,~ ~n~~.;. h/~1lL 0 pl.{,l>-d ~ ~ QJu ~ ~u ~ {Ctc.,b

~ 6> 1\.1 f4~ut 'kottwf

Page 9: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

I ~~~~~ K6 O~'M~e 2.0~.(, Atlus tla.tV ~ ,Ko.1\uf tf7 ) 8(/ '1,1 d°l.~ O/t\cl,

. . ~ f tV Qf r

~ Q ~ ~L(D.k ~M Ico &It.l.l ,

H9t(,'tsQ.tiD",,1: fJa.&l .'. ..,,' qVtll- f3 ~ .," ~ ~-" ~'.t.

& ... ~. . , .COt.." , .~- .'

~ ~~O'"' g~~ U ~ Roft 1WA.8 IW'L O~ .

~~~c- W~t Qd.~w~al ~top".,t1 fYM.Q.Jc.c.~ f\,OpL

!Wt~ UQ,~,(,3 Jp,:.,~~~a£ (),~okt at~)~(, a.t.cI &~ ~

IIiI

Page 10: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

~~~0fM3 b~ ~\&~f.! 6~:) '::' ~L , (\

" . .) ~j oj (1) tok.ts about ~S ~Q.~l.1 .~ . . ) (1) ~~ ~l"'~~\lial. 11141 pfA:cc.. .uOnc.t .

.I I@ () '~~ , ..,;J ~~ (f¥") - -+ e ~f,.(~)

i

I' t cL I ,J ~

. ~1..L-~t (X",) -.J Xc.-.) ~ fJ1L-Z:t (x~) ) ~~) tJ)

d, ~L.\'(~ ~ ~NM.rfL rk~~l !o~A.eC(,.

'. G)<=-> ~ ~ ~ ~ tx~I-.~)X~)

! (1) ~ '> Q, CAOI1,Q-ct ~ \ ~ . () ~ t1 ~ ~ to oiL~JW;l1 :w,

m - ~ oR Lb~.. ~ ~ b! c9LQJ~cb~\G.~ ,,! ~ e.l.-Lf'fflI,,~t t i1.i. 1't v )

PI 1/rIo 'tVtSYttt, ~ ~ ~ ~! ~ ~ 1. ~ 0 '~ t;I)IO ~ ~ d ~ UA1. ~ fr\..o 4\;0 k .

(1)\ 'tK~t cl..:M - OpUQ.tt>,.1 ('\c>D'lk ~ p'LoQtLt.,sg) ~~.~~I ~ SH. i . . 0

F; 1~ OL4,t p...b~ \! r.Ik~~r..Q,ttAi\a..t ~ij~ ~

~~ f)p~t 6f,t~(.tJ~;,:, 1U(.c~ bWlJ1L1 (W.dA..tt.L~) L1. ~IM'J

.) ~~~~~QMl ~O1, ~Rt6~. ) -1YnI.t. r..od. ~ 'I) ~ J f. t ~ krc. tiP 114. ~ CI '.

to co.£CAA. ~k~~ ~ ~ ~ ~) J "t, "'frill- ~- C<>/'/VL I~ !CJt ~~~ ~l {.o vc.. "1 -=~~~t~:

Page 11: Cohomology operations and the Deligne conjecture Martin ... · Cohomology operations and the Deligne conjecture Martin Markl The aim of the talk, which raises more questions than

@'i=YflMllb~: X "E- ~ ( H) 0~~~ ) ,. } "f'Y\

~~~~b~~\ ~ ~.DA~ t ~ £, at f ~ t tMat 0 r rAQ;t OI,.! tJa. Q. t

t.o1IL X*,) ..#)X~ ~ t\ttCto\' r'ttd~ ~ r.-t>;l.t q,tlW

I ~- -~~;~~-"'~-;:~~"d:[~~XJ ~~ .' ~~t ,:~~,~ 0

~\ ~\ ¥iI.oW\4., ~J;. ~ -= ~ I ~ [ X~ [. ')(.\ ~111 4 ptltfAl\., 0 '

~~ :---; ~ J--

~t -t14,~ Okt i~t ~\, "",,{Q.tu!. V {~S ~~b~~ ~S

(~d~) \!'...~~.=~=~. (!) 1hn 0(3 oR!~' . f/1,Q.t~~-- . ~ aM, ,ilfA.Q.itd. C-} -] .

~ ~ 'La.{'r'k. : fV\O.vM at '

~ t\4J ~ b oa& ~ ~-I.I."" di 1 i. () p "- ~ tr;t, 1

foi ~3~\.tU) ~kI., JOt}(, Q.pp~ ~i~ oc. ~~ p5 0 ~

O~ ~~ ~~a.th"e.o.t ,tA'ttrm.l.I\1,~ .fI.o.~ ~ PC'kt1 ~~~tMaJ.

;u\1.\/QMCIJ'II.t~-~~~~~t~ 'QII-L . -- ~. TtX]t)::. '\l)('i - V,X - (~)'(] .I

. ~ ~~) l: =-~V cx,tll--:i?~ ) ~,1 (i) ..\

~~ttA h~ .tf.A, ~~~u ~~~ h-l.t . qt. fl.Lfatt.~ IJtCf-rltOf*P~~~~~~~~~ .

~t.o ~~