frobenius algebras of stanley-reisner ringsmatfxe/gsu-usc/atlanta2015santiagozarzuela.pdf ·...

Frobenius algebras Stanley-Reisner rings Examples Applications

Frobenius algebras of Stanley-Reisner rings

Santiago Zarzuela

University of Barcelona

Recent developments in positive characteristictechniques in commutative algebra:

Frobenius Operators and Cartier Algebras

March 13-15, 2015

Atlanta, GA

Santiago Zarzuela University of Barcelona



Based on joint work with

Josep Alvarez Montaner

and

Alberto F. Boix




1 The algebra of Frobenius Operators of the injective hull

2 The case of Stanley-Reisner rings

3 Examples

4 Applications: Cartier algebras and F -jumping numbers




The algebra of Frobenius Operators of the injective hull

• R a commutative ring of characteristic p > 0.

• F e : R −→ R (r 7→ rpe) the e-th iterated Frobenius map.

• M an R-module, F e∗M the R-module obtained by restriction of

scalars from F e.

This provides a new structure as R-module on M given by

r ·m := rpem

so that we have the Frobenius functor: F e∗ from the category of

(left) R-modules onto itself.




DefinitionA pe-linear map ϕe : M → M is an additive map that satisfies

ϕe(rm) = rpeϕe(m)

for all r ∈ R, m ∈ M.

So it is just an element of the abelian group

Fe(M) := HomR(M,F e∗M)

Note that:

· Composing a pe-linear map and a pe′-linear map we get ap(e+e′)-linear map.

· Each Fe(M) is a left module over F0(M) = EndR(M).




Definition (G. Lyubeznik, K. Smith, 2001)

The ring of Frobenius operators on M is the graded,associative, not necessarily commutative ring

F(M) :=⊕e≥0

Fe(M)

We also call this ring the Frobenius algebra of M.

Question: Is F(M) finitely generated as an F0(M)-algebra?




The following is an alternative description of Fe(M):

• Let R[θ; F e] the e-th Frobenius skew polynomial ring:

R[θ; F e] is the left R-module freely generated by {θi}i≥0 withthe multiplication subject to the rule θr = rpe

θ for all r ∈ R.

Fe(M) may be identified with the set of all (left) R[θ; F e]-modulestructures on M.




Proposition (G. Lyubeznik, K. Smith, 2001)

(1) F(R) ∼= R[θ; F ].

(2) F(Hnm(R)) ∼= R[θ; F ], where Hn

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.




Frobenius algebras of injective hulls have the following explicitdescription:

• Let S = k [[x1, . . . , xn]], where k is a field of characteristicp > 0, and set E the injective hull of k .

• Let I ⊂ S be any ideal, R = S/I and ER the injective hull ofthe residue field of R.

Then,ER ' HomS(S/I,E) ' (0 :E I) ⊂ E




There is a natural Frobenius map F e : E → E for each e ≥ 0.

• Let I[pe] = 〈 rpe | r ∈ I 〉.

Now, the Frobenius map F e can be restricted to ER aftermultiplication by any element g ∈ (I[pe] :S I).

In this way we have pe-linear maps of the form

(gF e : ER → ER) ∈ Fe(ER)




Proposition (M. Blickle, 2001; (R. Fedder, 1983...))

Any pe-linear map ϕe : ER → ER is of the form gF e, whereg ∈ (I[pe] :S I)/I[pe], and there exists an isomorphism ofR-modules

Fe(ER) ∼= ((I[pe] :S I)/I[pe])F e

This isomorphism can be extended in a natural way to anisomorphism of R-algebras

F(ER) ∼=⊕e≥0

((I[pe] :S I)/I[pe])F e




The case of Stanley-Reisner rings

Let S = k [[x1, . . . , xn]], where k is a field of characteristic p > 0,I ⊂ S a squarefree monomial ideal and R = S/I.

Our main goal is to describe in a simple way the ideals (I[q] :S I)for any q = pe, e ≥ 0.

For that we will consider the minimal primary decomposition ofI given in terms of face ideals

I = Iα1 ∩ · · · ∩ Iαs ,

where we denote

Iα := 〈xi | αi 6= 0〉, α ∈ {0,1}n.




Denote a squarefree monomial by

xα := xα11 · · · x

αnn , α ∈ {0,1}n

Then we have the following:

Proposition

(I[q] :S I) = (I[q]α1

:S Iα1) ∩ · · · ∩ (I[q]αs :S Iαs )

= (I[q]α1

+ (xα1)q−1) ∩ · · · ∩ (I[q]αs + (xαs )q−1)

(The first equality was shown in general for the unmixed caseby R. Fedder, 1983.)




Assume for simplicity that Iα1 + · · ·+ Iαs = (x1, . . . , xn).

From the previous formula we get that

(I[q] : I) = I[q] + Jq + (x1)q−1

where the generators xγ = xγ11 · · · x

γnn of Jq satisfy

γi ∈ {0,q − 1,q}

It may happen that Jq ⊂ I[q] + (x1)q−1.

But this will be not the case if and only if there exists agenerator xγ of Jq having γi = q, γj = q − 1, γk = 0 for some1 ≤ i , j , k ≤ n ≥ 3.




Example (A, M. Katzman’s counterexample)

Let I = (xy , xz) = (x) ∩ (y , z). Then:

(I[q] : I) = ((xq) + (xq−1)) ∩ ((yq, zq) + (yz)q−1)

= ((xy)q, (yz)q) + (xq−1yq, xq−1zq) + (xyz)q−1

= I[q] + Jq + (x1)q−1

where Jq = (xq−1yq, xq−1zq) * I[q] + (x1)q−1




Example (B)

Let I = (xy , xz, yz) = (x , y) ∩ (x , z) ∩ (y , z).

Then, (I[q] : I)

= ((xq, yq)+(xy)q−1)∩((xq, zq)+(xz)q−1)∩((yq, zq)+(yz)q−1)

= ((xy)q, (xz)q, (yz)q) + (xyz)q−1 = I[q] + (x1)q−1




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,




Proposition (M. Katzman, 2010)

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 .




In fact, it is enough to show that

xγ /∈ Gβ1G[pβ1 ]β2

G[pβ1+β2 ]β3

· · ·G[pβ1+···+βs−1 ]βs

where Gβj := (xpβj

1 xc22 · · · x

cnn ), with ci ∈ {0,q − 1,q}.

Now, observe that the exponent of x1 in the generator of theproduct is

pβ1+(β1+β2)+···(β1+···+βs) > pβ1+···+βs = pe

since s > 1 and βi ≥ 1 for all i .




On the other side, these kind of generators are somehow”universal”: their shape is the same for any e ≥ 1.

Also, they do not depend on coefficients and so areindependent of the characteristic p.

In particular, it suffices to compute them for p = 2 and e = 1,that is, to compute (I[2] :S I).

So if in F1(ER) there are µ of such kind of special generators,the same will happen in Fe(ER) for each e ≥ 1.

Therefore, we may conclude:




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.




In particular, we have that the complexity sequence of F1(ER)(as defined by F. Enescu and Y. Yao, 2014) is given by

{1,1 + µ, µ, . . . }

and that Frobenius complexity of the ring R is equal to 0.




For our previous two examples we have:

Example (A)

Let I = (xy , xz) = (x) ∩ (y , z). Then: µ = 2 and so F(ER) is notfinitely generated.

Example (B)

Let I = (xy , xz, yz) = (x , y) ∩ (x , z) ∩ (y , z). The µ = 0 andF(ER) is principally generated.




Primarily we can distinguish four cases depending on theminimal primary decomposition of I.

Let I = Iα1 ∩ · · · ∩ Iαs be the minimal primary decomposition of I.

(i) ht (Iαi ) > 1 for all i = 1, . . . , s.In this case both situations may happen so we have:

(a) (I[q] : I) = I[q] + (x1)q−1.(b) (I[q] : I) = I[q] + Jq + (x1)q−1, Jq ( I[q] + (x1)q−1.

(ii) ht (I) = 1 and there is i ∈ {1, . . . , s} such that ht (Iαi ) > 1.In this case (I[q] : I) = Jq + (x1)q−1 with Jq ( (x1)q−1.

(iii) ht (Iαi ) = 1 for all i = 1, . . . , s.Then (I[q] : I) = (x1)q−1.




Proposition

With the previous assumptions:

· F(ER) ∼= R[(x1)(p−1)θ; F ] is principally generated in cases(i .a) and (iii).

· F(ER) is infinitely generated in cases (i .b) and (ii).

Remark. Note that Katzman’s running example (A) satisfies (ii).




Examples

Let us see some concrete examples (most of computationshave been done with CoCoA).

Example 1 (running example (B)).I = (x , y) ∩ (x , z) ∩ (y , z) = (xy , xz, yz).

It is a Cohen-Macaulay, non-Gorenstein ideal with principallygenerated Frobenius algebra.




This table describes the behaviour of the Frobenius algebras inthree variables in the unmixed case:

n = 3 p.g. Gor i.g.ht I = 1 1 1 -ht I = 2 2 1 -ht I = 3 1 1 -




Example 2. I = (x , y) ∩ (x ,w) ∩ (y , z) = (xy , xz, yw)

It is Cohen-Macaulay ideal with infinitely generated Frobeniusalgebra.

We have in this case that (I[q] :R I) =

(xqyq, xqzq, yqwq, xqyq−1zq−1, xq−1yqwq−1, (xyzw)q−1)

and so µ = 2.




In four variables we have the following table for the unmixedcase:

n = 4 p.g. Gor i.g.ht I = 1 1 1 -ht I = 2 4 2 3ht I = 3 3 1 -ht I = 4 1 1 -

The other two cases with infinitely generated Frobenius algebraare:

· I = (x , y) ∩ (z,w) = (xz, xw , yz, yw), µ = 3.· I = (x , y) ∩ (x ,w) ∩ (y ,w) ∩ (z,w) = (xyz, xw , yw), µ = 1.




Example 3. I = (x ,u) ∩ (x ,w) ∩ (x , z) ∩ (x , y) ∩ (y , z,w ,u)

It is a non Cohen-Macaulay, non unmixed ideal with principallygenerated Frobenius algebra.

In five variables we have the following table for the unmixedcase:

n = 5 p.g. Gor i.g.ht I = 1 1 1 -ht I = 2 6 2 13ht I = 3 12 2 10ht I = 4 4 1 -ht I = 5 1 1 -




Example 4. I = (x , y) ∩ (z,w) = (xz, xw , yz, yw)

It is an ideal with infinitely generated Frobenius algebra whoseAlexander dual has a principally generated algebra:

I∨ = (x , z) ∩ (x ,w) ∩ (y , z) ∩ (y ,w) = (xy , zw)




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).




The number µ of new generators in the non principallygenerated case can be arbitrary large.

For instance, letI = Iα1 ∩ · · · ∩ Iαs

be an ideal with disjoint variables such that |αi | > 1 for all i .Then, one can check that:

µ =s∏

i=1

(|αi |+ 1)−s∏

i=1

|αi | − 1




There is a nice combinatorial criteria to decide when theFrobenius algebra F(ER) is principally generated.

Let I ⊂ S be a square free monomial ideal and set ∆ thesimplicial complex with vertex set [n] := {1, . . . ,n} whoseStanley-Reisner ideal I∆ = I.

DefinitionA face F ∈ ∆ is called free if F ∪ {i} is a facet for some i /∈ Fand F ∪ {i} is the unique facet containing F .

Then:




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).




Applications: Cartier algebras and F -jumping numbers

The ”dual” notion of Frobenius algebras are Cartier algebras,as defined by K. Schwede, 2011 and M. Blickle, 2013.

DefinitionA p−e-linear map ψe : M−→M is an additive map that satisfies

ψe(rpem) = rψe(m)

for all r ∈ R, m ∈ M.

We identify the set of p−e-linear maps with the abelian group

Ce(M) := HomR(F e∗M,M)




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

C(M) :=⊕e≥0

Ce(M)




There is also an alternative description of C(M):

• Let R[ε; F e] = R[θ; F e]op the opposite e-th Frobenius skewpolynomial ring:

R[ε; F e] is the right R-module freely generated by {εi}i≥0, withthe multiplication subject to the rule rε = εrpe

for all r ∈ R.

In this way, Ce(M) may be identified with the set of all (left)R[ε; F e]-module structures on M.




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.




This equivalence is compatible with the correspondingidentifications between Frobenius and Cartier operators andring structures.

So by taking R and ER it follows that there is an isomorphism ofalgebras:

C(R) ∼= F(ER)op

For instance, if R is Gorenstein then C(R) is principallygenerated. And so on.




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.




• Let R be a Noetherian ring of characteristic p > 0.

Denote by R0 the set of elements in R that are not in anyminimal prime of R.

• Let a ⊂ R be an ideal such that a ∩ R0 6= ∅.

Definition (N. Hara, K.-I. Yoshida, 2003)

Let I ⊂ R. For any t ∈ R≥0, the at -tight closure I∗at

of I isdefined as the ideal consisting of all elements z ∈ R for whichthere exists c ∈ R0 such that

czqadtqe ⊂ I[q]

for all q = pe � 0.




Definition (N. Hara, K.-I. Yoshida, 2003)

Let (R,m) be an F -finite reduced local ring of characteristicp > 0. For any t ∈ R≥0, we define the (generalized) test idealτ(R, at ) as

τ(R, at ) =⋂I⊂R

(I : I∗at)

where I runs through all ideals of R.

Remark. Under the above hypothesis τ(R, at ) ∩ R0 6= ∅.




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

φ(F e∗ (adt(pe−1)eJ)) ⊂ J

for all e > 0 and φ ∈ HomR(F e∗R,R).




The following property of test ideals is analogous to the one formultiplier ideals:

Proposition

If s ≥ t then τ(R, as) ⊂ τ(R, at ). Furthermore, there exists ε > 0such that τ(R, as) = τ(R, at ) for all s ∈ [t , t + ε).

DefinitionA positive real number ξ is called an F -jumping number of theideal a if τ(R, aξ) 6= τ(R, aξ−ε) for all ε > 0.




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.

Let us see the meaning of that in our situation.




• let S = k [x1, . . . , xn] be the polynomial ring in n variables overa perfect field k .

For any given α = (α1, . . . , αn) ∈ Nn0 we consider the maximun

norm||α|| := max

jαj

This norm induces an increasing filtration

{Sd}d∈N0∪{−∞}

of k -subspaces, where Sd is the k -subspace of S generated bymonomials xα such that ||α|| ≤ d and S−∞ := 0.




• Let M be an S-module finitely generated by m1, . . . ,mk .

The filtration on S induces an increasing filtration on M definedas:

Md :=

{0, if n = −∞,Sd · 〈m1, . . . ,mk 〉, if d ∈ N0.

In this way, one sets

δM : M // N0 ∪ {−∞}

m 7−→

{−∞, if m = 0,d , if m ∈ Md \Md−1.

We say that δ = δM is a gauge for M.




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)

pei + Kp−1 , r ∈ R.

We then say that Kp−1 is the bound of the gauge.




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.




As a consequence, we get that:

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.




Assume that I is a squarefree monomial ideal:

• From the duality between the Frobenius algebra of ER and theCartier algebra of R, and

• from the description of the generators of the ideals (I[pe] :S I),

it is clear that for all e ≥ 0 we can find generators

g1, . . . ,gνe such that δR(gi) ≤ pe for all i

and so the Cartier algebra C(R) is gauge bounded.




Thank you very much!



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