Frobenius algebras Stanley-Reisner rings Examples Applications Frobenius algebras of Stanley-Reisner rings Santiago Zarzuela University of Barcelona Recent developments in positive characteristic techniques in commutative algebra: Frobenius Operators and Cartier Algebras March 13-15, 2015 Atlanta, GA Santiago Zarzuela University of Barcelona Frobenius algebras of Stanley-Reisner rings
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Frobenius algebras of Stanley-Reisner ringsmatfxe/gsu-usc/Atlanta2015SantiagoZarzuela.pdf · Frobenius algebrasStanley-Reisner ringsExamplesApplications Frobenius algebras of Stanley-Reisner
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m(R) is the top localcohomology module of a complete S2 local ring (R,m) ofdimension n.
In particular, If (R,m) is a complete local Gorenstein ring, andER is the injective hull of the residue field R/m then we havethat F(ER) is principally generated as F0(ER) = R-algebra.The converse is also true if R is F -finite and normal, by M.Blickle, 2013.
Frobenius algebras are not always finitely generated
Example (M. Katzman, 2010)
Let R = k [[x , y , z]]/(xy , xz), where k is any field ofcharacteristic p > 0. Then, F(ER) is not a finitely generatedR-algebra.
We shall use some of the ideas involved in this counterexampleto study in general the generation of the algebra F(ER) for anycomplete Stanley-Reisner ring R.
Now, this condition on Jq determines completely the finitegeneration of the Frobenius algebra of ER.
To see this, we shall use an idea based on M. Katzman’s work.
For any e ≥ 1 denote Ke := (I[pe] :S I) and let (with L1 = 0)
Le :=∑
1 ≤ β1, . . . , βs < eβ1 + · · · + βs = e
Kβ1K [pβ1 ]β2
K [pβ1+β2 ]β3
· · ·K [pβ1+···+βs−1 ]βs
This ideal looks rather complicated but it is just the kind ofcoefficients you get for an homogeneous element of degree eas a product of homogeneous elements of degree < e inF(ER). In other words,
For any e ≥ 1, let F<e be the R-subalgebra of F(ER)generated by F0(ER), . . . ,Fe−1(ER). Then
F<e ∩ Fe(ER) = LeF e
(In fact, this result holds for any regular complete local ring S ofcharacteristic p > 0).
For his example, that is I = (xy , xz), he checked that for alle ≥ 1, the element xq−1yq ∈ Ke does not belong to Le.Therefore Fe(ER) is not contained in F<e and F(ER) is notfinitely generated.
The argument of M. Katzman can be applied more in general.
PropositionAssume that there exists a generator xγ of Jq havingγi = q, γj = q − 1, γk = 0 for some 1 ≤ i , j , k ≤ n. Thenxγ /∈ F<e for any e ≥ 1.
Proof: We may assume e ≥ 2 hence s ≥ 2. We have that Le isa sum of monomial ideals, so xγ ∈ Le if and only if xγ is in oneof the summands.We may assume that
xγ = xq1 xc2
2 · · · xcnn
with ci ∈ {0,q − 1,q} and cj = 0, ck = q − 1 for some j , k .
Proposition (J. Alvarez Montaner, A. F. Boix, S. Z., 2012)
Assume that the minimal number of generators of F1(ER) isequal to µ+ 1, µ ≥ 0.Then F(ER) is finitely generated if and only if µ = 0.In this case, F(ER) is principally generated isomorphic to theskew polynomial ring R[(x1)(p−1)θ; F ] ⊂ R[θ,F ].If µ 6= 0, then the minimal number of new homogeneousgenerators of degree e for any e ≥ 1 is equal to µ.
We also have the following two families of examples withprincipally generated Frobenius algebra:
Proposition
(1) Let I ⊆ k [[x1, ..., xn]] be a squarefree monomial of pureheight n − 1. Then, the Frobenius algebra F(ER) isprincipally generated.
(2) Let Ik ,n ⊆ k [[x1, ..., xn]] be the squarefree monomial idealobtained as intersection of all the face ideals of height k(squarefree Veronese of type (n, k)). Then, the Frobeniusalgebra F(ER) is principally generated.
(Our running example (B) belongs to both families).
Proposition (J. Alvarez Montaner, K. Yanagawa, 2014)
Let R = S/I. The following are equivalent:
(a) The Frobenius algebra F(ER) is principally generated.(b) ∆ does not have a free face.
It is completely obvious now that for our running example (A),∆ has free faces (the vertices y , z) and that for example (B), ∆has not free faces (it just consists on three vertices x , y , z).
Analogously to the case of Frobenius algebras we have thefollowing facts:
· Composing a p−e-linear map and a p−e′-linear map in theobvious way as additive maps we get a p−(e+e′)-linearmap.· Each Ce(M) is a right module over C0(M) = EndR(M).
Thus we may define:
DefinitionThe ring of Cartier operators on M is the graded, associative,not necessarily commutative ring
Assume that (R,m) is complete, local and F -finite.
By work of M. Blickle and G. Bockle, 2011, and R. Y. Sharp andY. Yoshino, 2011, we have that Matlis duality induces anequivalence of categories between:
Left R[θ; F e]-modules which are co-finite as R-modules.
Right R[θ,F e]-modules which are finitely generated asR-modules.
Equivalently, R-finitely generated left R[ε,F e]-modules.
The ring of Cartier operators (or the more general notion ofR-Cartier algebras) allows to extend to a non reduced settingthe generalized test ideals.
Generalized test ideals were defined, among others, by N. Haraand K.-I. Yoshida, 2003, as characteristic p > 0 analogs ofmultiplier ideals in characteristic 0.
Their original definition was in terms of (generalized) tightclosure.
There is an alternative characterization for (generalized) testideals in terms of Cartier operators, given first by M. Blickle, M.Mustata and K. Smith, 2008, in the regular case, and then by K.Schwede, 2011, in the reduced case:
• Let a ⊂ R a non-zero ideal and t ∈ R≥0.
The test ideal τ(R, at ) is the unique smallest non-zero idealJ ⊂ R such that
A major interest in F -jumping numbers revolves around provingthat the set of F -jumping numbers form a discrete set ofrational numbers, as it is the case for multiplier ideals incharacteristic zero.
M. Blickle, 2013, has considered this problem by looking to thespecial property of being gauge bounded.
S has a gauge δS induced by the generator 1: it is the degreegiven by grading the monomials with the maximum norm.
If R = S/I is a quotient ring, then the generator 1R of R inducesa gauge δR which we shall call standard gauge.
DefinitionThe R-Cartier algebra C(R) is gauge bounded if for each/somegauge δ on R there exists a set {ψi | ψi ∈ Cei (R)}i∈I whichgenerates C+(R) as a right R-module, and a constant K suchthat for each ψi , one has δ(ψi(r)) ≤ δ(r)
The following is the result relating the behavior of Cartieralgebras with F -jumping numbers:
Proposition (M. Blickle, 2013)
(1) If C(R) is gauge bounded and a ⊂ R is an ideal, theF-jumping numbers corresponding to the generalized testideals τ(R, at ) for t ∈ R≥0 form a discrete set.
(2) If C(R) is finitely generated then it is gauge bounded.
We have seen that if R is Stanley-Reisner ring the Cartieralgebra C(R) is not always finitely generated.
Nevertheless, by using the explicit description of its generators,obtained by duality from the generators of the Frobeniusalgebra F(ER), we are able to prove the following:
Proposition (J. Alvarez Montaner, A. F. Boix, S. Z., 2012)
The R-Cartier algebra C(R) associated to a Stanley-Reisnerring R over a perfect field is gauge bounded.
Corollary (J. Alvarez Montaner, A. Fernandez Boix, S. Z., 2012)
Let a ⊆ R be an ideal of a Stanley-Reisner ring R over a perfectfield k. Then the F-jumping numbers of the generalized testideals τ(R, at ) are a discrete set of R≥0.
We proved our result on the gauge boundedness of the Cartieralgebra of Stanley-Reisner rings by a direct computation overthe generators of the Cartier-Algebra. Instead, one could usethe following recent result by M. Katzman and W. Zhang:
Proposition (M. Katzman, W. Zhang, 2014)
Let I ⊂ S be an ideal such that we can find a constant K and,for all e ≥ 0, a set of generators g1, . . . ,gνe of (I[pe] :S I) suchthat δR(gi) ≤ Kpe for all 1 ≤ i ≤ νe. Then, the R-Cartier algebraC(R) associated to R/I is gauge bounded.