Decomposition of local cohomology tables of modules with large E-depth
aa r X i v : . [ m a t h . A C ] O c t DECOMPOSITION OF LOCAL COHOMOLOGY TABLES OF MODULESWITH LARGE E-DEPTH
GIULIO CAVIGLIA AND ALESSANDRO DE STEFANI
Dedicated to Professor Bernd Ulrich on the occasion of his 65th birthday
Abstract.
We introduce the notion of E-depth of graded modules over polynomial ringsto measure the depth of certain Ext modules. First, we characterize graded modules overpolynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partialgeneral initial submodules preserve the Hilbert function of local cohomology modules supportedat the irrelevant maximal ideal, extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules. Second, we describe the cone of local cohomology tables of modules withsufficiently high E-depth, building on previous work of the second author and Smirnov. Finally,we obtain a non-Artinian version of a socle-lemma proved by Kustin and Ulrich. Introduction
Let S = k [ x , . . . , x n ] be a standard graded polynomial ring over an infinite field k . A finitelygenerated Z -graded S -module M is called sequentially Cohen-Macaulay if, for each integer i ,the module Ext iS ( M, S ) is either zero, or Cohen-Macaulay of maximal possible dimension n − i .Sequentially Cohen-Macaulay modules were introduced by Stanley [Sta83] from a different pointof view (see Definition 2.1), and later reinterpreted by Peskine as above (for instance, see [HS02,Theorem 1.4]).This definition suggests to consider modules for which each Ext iS ( M, S ), if not Cohen-Macaulay of maximal dimension, at least has “sufficiently large” depth. To better quantifythis, we introduce a numerical invariant of a graded module, which we call E-depth (see Defi-nition 2.3). Sequentially Cohen-Macaulay S -modules can be characterized as those which havemaximal E-depth, equal to n (see Proposition 2.11). On the other hand, modules with largeE-depth still satisfy desirable properties. For instance, if M is a module of positive depth andpositive E-depth, and ℓ ∈ S is a sufficiently general linear form, then the Hilbert function ofthe modules H i m ( M ) can be read from that of H i − m ( M/ℓM ) for all i >
0. Here, H i m ( − ) denotesthe i -th graded local cohomology functor, with support in the irrelevant maximal ideal m of S .Section 2 is devoted to study how the E-depth behaves under some basic operations (seeProposition 2.12), and to provide a key example of modules with positive E-depth, which iscrucially used in the following sections (see Example 2.15).The starting point of Section 3 is an important characterization of sequentially Cohen-Macaulay modules in terms of generic initial modules, due to Herzog and Sbarra [HS02], thatwe now recall. Let HF( − ) denote the Hilbert function of a Z -graded S -module. Let M bea finitely generated Z -graded S -module, that we write as a quotient of a graded free module Mathematics Subject Classification.
Key words and phrases.
Local cohomology tables, general initial modules, revlex-orders, sequentially Cohen-Macaulay modules. F by a graded submodule U . Then M ∼ = F/U is sequentially Cohen-Macaulay if and only ifHF( H i m ( F/U )) = HF( H i m ( F/ gin revlex ( U ))).In order to extend this, for any given integer t ∈ { , . . . , n } we introduce a weight-order,denoted by rev t , and we consider general initial modules gin rev t .We characterize modules with sufficiently large E-depth: Theorem A (see Theorem 3.6) . Let S = k [ x , . . . , x n ], with the standard grading, and M bea finitely generated graded S -module. Write M = F/U , where F is a graded free S -module,and U is a graded submodule of F . For a given integer t >
0, we have that E-depth( M ) > t ifand only if HF( H i m ( F/U )) = HF( H i m ( F/ gin rev t ( U ))) for all i ∈ Z .Keeping in mind that M is sequentially Cohen-Macaulay if and only if E-depth( M ) = n ,and that an initial ideal with respect to rev n coincides with the initial ideal with respect to theusual revlex order, Theorem A can be viewed as an extension of [HS02, Theorem 3.1].In Section 4, we focus on studying the cone generated by local cohomology tables of S -modules M with sufficiently large E-depth.Let M be a finitely generated Z -graded S -module. We let [ H • m ( M )] ∈ Mat n +1 , Z ( Q ) be thematrix whose ( i + 1 , j )-th entry records dim k ( H i m ( M ) j ), and we consider the cone Q > · { [ H • m ( M )] | M is a finitely generated Z -graded S -module } . The study of this object was initiated in [DSS20] by Smirnov and the second author. Its interestis motivated by the well-known Boij-S¨oderberg theory for the cone of Betti diagrams [BS08].Eisenbud and Schreyer proved the conjectures for the cone of Betti diagrams of Cohen-Macaulaymodules by exhibiting a subtle duality with the cone of cohomology tables of vector bundles onprojective space [ES16]. Later, Boij and S¨oderberg extended the techniques employed in [ES16]to all coherent sheaves, obtaining a description of the full cone of Betti diagrams of finitelygenerated graded S -modules [BS12]. Motivated by the original Boij-S¨oderberg theory, andgiven that local cohomology and sheaf cohomology are very much connected, Daniel Ermanasked whether one could describe the cone of local cohomology tables of finitely generatedgraded S -modules. More specifically, this means whether one can identify the extremal rays ofsuch cone, and the equations of its supporting hyperplanes.The main results of [DSS20] contain a complete description of the extremal rays of the coneof local cohomology tables of modules of dimension at most 2, as well as the equations of thesupporting hyperplanes. We improve this result by determing the extremal rays of the conespanned by modules with sufficienty large E-depth. Theorem B (see Theorem 4.7) . Let S = k [ x , . . . , x n ], with the standard grading, and M bea Z -graded S -module with E-depth( M ) > n −
2. For i = 0 , . . . , n let S i = k [ x , . . . , x i ], and let J = ( x , x ) S . We have a decomposition[ H • m ( M )] = n X i =0 X j ∈ Z r i,j [ H • m ( S i ( − j ))] + X m> X j ∈ Z r ′ m,j [ H • m ( J m ( − j ))] , where r i,j ∈ Z > , r ′ m,j ∈ Q > , and all but finitely of them are equal to zero. Moreover, the setΛ = { [ H • m ( k [ x , . . . , x s ]( − j )] , [ H • m ( J m ( − j ))] | s n, j ∈ Z , m > } is minimal, and it describes the extremal rays of the cone spanned by local cohomology tablesof modules of E-depth at least n − ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 3 If M is an S -module satisfying E-depth( M ) > dim( M ) −
2, then one can still apply TheoremB (see Remark 4.6). In particular, since modules M of dimension at most two automaticallysatisfy E-depth( M ) > dim( M ) −
2, Theorem B is indeed an extension of [DSS20, Theorem 4.6].Using Theorem B and a description of the facets of the cone of local cohomology tables indimension two [DSS20, Theorem 6.2], we provide equations for the supporting hyperplanes ofthe cone of local cohomology tables of modules M with E-depth( M ) > n − M be the Q -vector space of ( n + 1) × Z matrices with finite support. Given thelocal cohomology table [ H • m ( M )] of a finitely generated Z -graded S -module, we can produce anew table ∆[ H • m ( M )] which belongs to M (see Section 4, or [DSS20, Section 6] for more detailsabout this construction). Consider the cone C seq = Q > { ∆[ H • m ( M )] | M is a sequentially Cohen-Macaulay Z -graded S -module } . Theorem C (see Proposition 4.10 and Theorem 4.13) . Let A = ( a i,j ) ∈ M . Then A ∈ C seq ifand only if a i,j > i and j .Finally, in Section 5 we extend a “socle-lemma” due to Kustin and Ulrich to the non-Artiniancase. The original version states that, if I ⊆ J are two m -primary homogeneous ideals, andHF(soc( S/I )) HF(soc(
S/J )), then I = J . To extend this result to arbitrary dimension, weneed to assume that our modules have sufficiently large E-depth, and the Hilbert functions ofthe socles of certain local cohomology modules satisfy an analogous inequality. For simplicity,here we only state our result in the sequentially Cohen-Macaulay case: Theorem D (see Theorem 5.2 and Corollary 5.3) . Let S = k [ x , . . . , x n ], and F be a gradedfree S -module. Let A ⊆ B be graded submodules of F such that F/A and
F/B are sequentiallyCohen-Macaulay. If HF(soc( H i m ( F/A ))) HF(soc( H i m ( F/B ))) for all i ∈ Z , then A = B . Acknowledgments.
We thank the anonymous referees for pointing out some inaccuraciescontained in a previous version of this article, and for several very useful comments.2.
E-depth: definitions and basic properties
Let S = k [ x , . . . , x n ], where k is a field and each variable is given degree equal to one. Wewill also assume that k is infinite, since reducing to this case via a faithfully flat extension doesnot affect our considerations. Given a Z -graded S -module M = L i ∈ Z M i , and j ∈ Z , we denoteby M ( j ) its shift by j , that is, the Z -graded S -module whose i -th graded component is M i + j .Throughout, m will always denote the maximal homogeneous ideal of S , and M will denotea finitely generated Z -graded S -module. For convenience, we let depth(0) = + ∞ . Given afinitely generated Z -graded S -module M , we denote by H i m ( M ) the i -th graded local cohomologymodule of M , with support in m . By definition, this is the i -th cohomology of ˇC • ⊗ S M , whereˇC • is the ˇCech complex on x , . . . , x n .We start by recalling the notion of sequentially Cohen-Macaulay module. Definition 2.1. An S -module M is said to be sequentially Cohen-Macaulay if there exists afiltration 0 = M ⊆ M ⊆ . . . ⊆ M r = M such that each quotient M i +1 /M i is Cohen-Macaulay with dim( M i +1 /M i ) > dim( M i /M i − ) forall i = 1 , . . . , r − GIULIO CAVIGLIA AND ALESSANDRO DE STEFANI
Sequentially Cohen-Macaulay modules were introduced by Stanley [Sta83]. An equivalentformulation, due to Peskine, is the following: M is sequentially Cohen-Macaulay if and only if,for every i ∈ Z , the module Ext iS ( M, S ) is either zero, or Cohen-Macaulay of dimension n − i . Example 2.2.
Cohen-Macaulay modules are sequentially Cohen-Macaulay. One dimensionalmodules are also sequentially Cohen-Macaulay, since either M is Cohen-Macaulay, or the fil-tration 0 ⊆ H m ( M ) ⊆ M has Cohen-Macaulay subquotients, and dim( M/H m ( M )) = 1 > H m ( M )).We observe that, if M is a sequentially Cohen-Macaulay S -module and ℓ is a linear non-zerodivisor on M , as well as on Ext iS ( M, S ) and Ext i +1 S ( M, S ), thenExt i +1 S ( M/ℓM, S ) ∼ = Ext iS ( M ( − , S ) /ℓ Ext iS ( M, S ) . In fact, we will see that ℓ only needs to be a non-zero divisor on M/H m ( M ) and on the twoExt modules for this to be true, not necessarily on M . This simple observation often allows toreduce the dimension of a sequentially Cohen-Macaulay module, yet controlling features suchas depth and regularity. In this sense, the case when M is sequentially Cohen-Macaulay is thebest possible, since all Ext-modules have maximal depth. We introduce the notion of E-depthof a module M to measure the number of times that the above procedure can be re-iterated,without altering the cohomological features of M . Definition 2.3.
Let S = k [ x , . . . , x n ], and M be a finitely generated graded S -module. Foran integer t ∈ Z > , we say that M satisfies condition ( E t ) if depth(Ext iS ( M, S )) > min { t, n − i } for all i . We defineE-depth( M ) = min (cid:26) n, sup { t ∈ Z > | M satisfies ( E t ) } (cid:27) . Remark 2.4.
The definition of E-depth is here given in the standard graded setting, because itsuits the level of generality that we consider in this article. The same definitions, and completelyanalogous considerations, can be made for finitely generated modules over local rings.We now study some basic properties of the E-depth of a module. We start by noticing thatthere is no general relation between E-depth( M ) and depth( M ), even when M is a k -algebra. Example 2.5.
Let S = k [ x, y, z, w ], and let R = S/ p , where p is the kernel of the map ϕ : S → k [ s, t ] defined as follows: ϕ ( x ) = s , ϕ ( y ) = s t, ϕ ( z ) = st , ϕ ( w ) = t . The only two non-zero Ext modules are Ext S ( R, S ) and Ext S ( R, S ). It can be checked thatExt S ( R, S ) has finite length, and therefore E-depth( R ) = 0 is forced. On the other hand,depth( R ) = 1. Example 2.6.
Let S = k [ x, y ] and R = S/I , with I = ( x , xy ). It is clear that, depth( R ) = 0.On the other hand, the only two non-zero Ext modules Ext S ( R, S ) and Ext S ( R, S ) are bothCohen-Macaulay of dimension one and zero, respectively. So R is sequentially Cohen-Macaulay,and thus E-depth( R ) = 2.We recall the definition of filter and strictly filter regular sequence. Definition 2.7.
Let S = k [ x , . . . , x n ], and M be a Z -graded S -module. A homogeneouselement ℓ ∈ m is called a filter regular element for M if ℓ / ∈ S p ∈ Ass ◦ ( M ) p , where Ass ◦ ( M ) = ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 5
Ass( M ) r m . A sequence ℓ , . . . , ℓ t is called a filter regular sequence for M if ℓ i is filter regularfor M/ ( ℓ , . . . , ℓ i − ) M for all i .Equivalently, ℓ is filter regular for M if 0 : M ℓ has finite length, and in this case one has H m ( M ) = 0 : M m ∞ ⊆ M ℓ ∞ ⊆ H m ( M ) , hence forcing equality everywhere. A related notion is that of strictly filter regular element. Definition 2.8.
Let S = k [ x , . . . , x n ], and M be a Z -graded S -module. For all i ∈ Z , let X i = Ass ◦ (Ext iS ( M, S )). A homogeneous element ℓ ∈ m is called a strictly filter regular elementfor M if ℓ / ∈ S i ∈ Z S p ∈ X i p . A sequence ℓ , . . . , ℓ t is called a strictly filter regular sequence for M if ℓ i is strictly filter regular for M/ ( ℓ , . . . , ℓ i − ) M for all i .We will simply say that ℓ , . . . , ℓ t is a filter (resp. strictly filter) regular sequence whenever themodule M is clear from the context. It follows from [BS98, 11.3.9] that Ass ◦ ( M ) ⊆ S i S p ∈ X i p ,therefore a strictly filter regular sequence is automatically a filter regular sequence. Lemma 2.9.
Let S = k [ x , . . . , x n ], and M be a finitely generated Z -graded S -module. Let N = M/H m ( M ), and ℓ be a strictly filter regular element for M of degree δ >
0. We have thatE-depth( M ) > / / Ext n − iS ( M, S ) · ℓ / / Ext n − iS ( M ( − δ ) , S ) / / Ext n − i +1 S ( N/ℓN, S ) / / / / H i − m ( N/ℓN ) / / H i m ( M )( − δ ) · ℓ / / H i m ( M ) / / → N ( − δ ) · ℓ −→ N → N/ℓN → i > Proof.
By previous observations, we have that ℓ is also filter regular for M , hence it is regularfor N . Assume that E-depth( M ) >
0, and consider the graded short exact sequence 0 → N ( − δ ) · ℓ −→ N → N/ℓN →
0. This gives a long exact sequence · · · / / Ext n − iS ( N, S ) · ℓ / / Ext n − iS ( N ( − δ ) , S ) / / Ext n − i +1 S ( N/ℓN, S ) / / · · · Observe thatb Ext n − iS ( N, S ) = 0 for i
0. Moreover, it follows from the short exact sequence0 → H m ( M ) → M → N → nS ( M, S ) = Ext nS ( H m ( M ) , S ), while Ext n − iS ( M, S ) ∼ =Ext n − iS ( N, S ) for all i >
0. As E-depth( M ) >
0, and ℓ is strictly filter regular, we have that ℓ is regular on Ext n − iS ( M, S ) ∼ = Ext n − iS ( N, S ) for all i >
0. In particular, multiplication by ℓ on Ext iS ( M, S ) in the long exact sequence above is injective for all i >
0, and the longexact sequence breaks into short exact sequences. The statement for local cohomology modulesfollows at once from graded local duality [BS98, 13.4.3].Conversely, assume that the sequences above are exact. From the Ext-sequence we deducethat either Ext n − iS ( M, S ) = 0, or ℓ is a non-zero divisor for it. In particular, we have thatdepth(Ext n − iS ( M, S )) > i >
0. Since Ext nS ( M, S ) has finite length, and Ext n − iS ( M, S ) =0 for i <
0, it follows that E-depth( M ) > (cid:3) Our next goal is to provide a more explicit relation between E-depth and sequentially Cohen-Macaulay modules. We first need a lemma.
Lemma 2.10.
Let S = k [ x , . . . , x n ], with the standard grading, and M be a finitely generated Z -graded S -module of dimension d such that depth(Ext n − dS ( M, S )) > d −
1. If d >
1, furtherassume that depth(Ext n − ( d − S ( M, S )) >
0. Then depth(Ext n − dS ( M, S )) = d . GIULIO CAVIGLIA AND ALESSANDRO DE STEFANI
Proof.
We proceed by induction on d >
0. If d = 0 there is nothing to show. If d = 1, thenlet ℓ be a strictly filter regular element for M of degree δ >
0, and N = M/H m ( M ). From theshort exact sequence 0 → N ( − δ ) · ℓ −→ N → N/ℓN → n − S ( N/ℓN, S ) / / Ext n − S ( N, S ) · ℓ / / Ext n − S ( N ( − δ ) , S ) / / Ext nS ( N/ℓN, S ) . However, since dim(
N/ℓN ) = 0, we have Ext n − S ( N/ℓN, S ) = 0, that is, ℓ is a non-zero divisoron Ext n − S ( N, S ). Since Ext n − S ( M, S ) ∼ = Ext n − S ( N, S ), we have that depth(Ext n − S ( M, S )) = 1.If d >
1, then let ℓ be a strictly filter regular element for M , and consider the same short exactsequence as above, which gives an exact sequence0 / / Ext n − dS ( N, S ) · ℓ / / Ext n − dS ( N ( − δ ) , S ) / / Ext n − ( d − S ( N/ℓN, S ) / / ann Ext n − ( d − S ( N,S ) ( ℓ ) , where the zero on the left follows again from the fact that Ext n − dS ( N/ℓN, S ) = 0, sincedim(
N/ℓN ) < d . As above, we also have Ext n − dS ( M, S ) ∼ = Ext n − dS ( N, S ) and Ext n − ( d − S ( M, S ) ∼ =Ext n − ( d − S ( N, S ). Since depth(Ext n − ( d − S ( M, S )) = depth(Ext n − ( d − S ( N, S )) >
0, and because ℓ is a strictly filter regular element, we have that ann Ext n − ( d − S ( N,S ) ( ℓ ) = 0. In particular, weobtain that depth(Ext n − ( d − S ( N/ℓN, S )) = depth(Ext n − dS ( M, S )) − > d − . Applying the inductive hypothesis to the module
N/ℓN , which has dimension d −
1, gives thatdepth(Ext n − ( d − S ( N/ℓN, S )) = d −
1, and thus depth(Ext n − dS ( M, S )) = d , as claimed. (cid:3) Proposition 2.11.
Let S = k [ x , . . . , x n ], with the standard grading, and M be a finitelygenerated Z -graded S -module. The following are equivalent:(a) M is sequentially Cohen-Macaulay.(b) E-depth( M ) = n .(c) M satisfies condition ( E t ) for some t > dim( M ) − Proof.
The implications (a) ⇒ (b) ⇒ (c) are clear from the definitions. Let d = dim( M ), andassume that M satisfies ( E t ) for some t > d −
1. Let i be such that Ext iS ( M, S ) = 0. Sinceht(ann S ( M )) = n − d , we must have i > n − d . For i > n − d we have that t > d − > n − i ,and thus n − i = min { t, n − i } depth(Ext iS ( M, S )) dim(Ext iS ( M, S )) n − i. In particular, Ext iS ( M, S ) is Cohen-Macaulay of dimension n − i . If i = n − d , by assumptionwe have that depth(Ext n − dS ( M, S )) > min { t, d } > d −
1. Since dim(Ext n − dS ( M, S )) = d , andbecause depth(Ext n − ( d − S ( M, S )) = d − > d >
1, we conclude by Lemma 2.10that Ext n − dS ( M, S ) is Cohen-Macaulay of dimension d , and therefore M is sequentially Cohen-Macaulay. (cid:3) Proposition 2.12.
Let S = k [ x , . . . , x n ], with the standard grading, and M , M ′ be twofinitely generated graded S -modules. We have:(1) E-depth( M ⊕ M ′ ) = min { E-depth( M ) , E-depth( M ′ ) } .(2) E-depth( M ) = E-depth( M/H m ( M )).(3) Let N = M/H m ( M ). If E-depth( M ) > ℓ is a homogeneous strictly filter reg-ular element, then either E-depth( N/ℓN ) = E-depth( M ) = n , or E-depth( N/ℓN ) =E-depth( M ) − ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 7
Proof.
The proof of (1) follows immediately from the definitions. Let N = M/H m ( M ). For i = n , we have that Ext iS ( M, S ) ∼ = Ext iS ( N, S ). Since Ext nS ( N, S ) = 0, while Ext nS ( M, S ) hasfinite length, it is clear that M satisfies condition ( E t ) for some t if and only if N does, and part(2) follows. We now prove (3). Let N = M/H m ( M ). If E-depth( M ) = n , then M is sequentiallyCohen-Macaulay by Proposition 2.11. By (2) it follows that N is sequentially Cohen-Macaulay,and so is N/ℓN by [HS02, Corollary 1.9]. In particular, as a module over S , we have thatE-depth( N/ℓN ) = n , again by Proposition 2.11. Now assume that E-depth( M ) = t < n . Bypart (2) we have that E-depth( N ) = t , which is positive by assumption. By Lemma 2.9, if welet δ be the degree of ℓ , we have graded short exact sequences0 / / Ext iS ( M, S ) · ℓ / / Ext iS ( M ( − δ ) , S ) / / Ext i +1 S ( N/ℓN, S ) / / i < n . Let i be such that Ext iS ( M, S ) = 0. The short exact sequences above show thatdepth(Ext i +1 S ( N/ℓN, S )) = depth(Ext iS ( M, S )) − > min { t − , n − ( i + 1) } As this holds for all i + 1 n , we have that E-depth( N/ℓN ) > t −
1. On the other hand,since E-depth( M ) = t < n , there must exist i such that depth(Ext iS ( M, S )) = t < n − i . Thendepth(Ext i +1 S ( N/ℓN, S )) = t − < n − ( i + 1), which shows that E-depth( N/ℓN ) = t − (cid:3) Remark 2.13.
Observe that if E-depth( M ) is not assumed to be positive in Proposition 2.12(3), then E-depth( N/ℓN ) can even increase. Indeed, Example 2.5 exhibits an integral k -algebra R such that E-depth( R ) = 0, but E-depth( R/ℓR ) = 4 for any non-zero linear form ℓ , since R/ℓR is one-dimensional, hence sequentially Cohen-Macaulay over S = k [ x, y, z, w ].We conclude the section by providing examples of classes of modules with a given E-depth.The relevance of the following construction will become clearer in the upcoming sections.Let t >
0, and R = A [ y , . . . , y t ] be a polynomial ring over a Z -graded ring A . We put a Z × Z t -grading on R as follows. Let η i ∈ Z t +1 be the vector with 1 in position i and 0 everywhereelse. We set deg R ( a ) = deg A ( a ) · η for all a ∈ A , and deg R ( y i ) = η i +1 . Example 2.14.
Let A = k [ x , x ], with the standard grading, and R = A [ x , x ]. Then R is Z × Z -graded. For instance, we have deg R ( x ) = deg R ( x ) = (1 , , R ( x ) = (0 , , R ( x ) = (0 , , S as A [ y , . . . , y t ], where A = k [ x , . . . , x n − t ] and y i = x n − t + i , we see that S is a Z × Z t -graded ring. Observe that a non-zero polynomial of S is homogeneous with respect tothis grading if and only if it is a monomial in the last t variables, and is homogeneous withrespect to the standard grading in the first n − t variables. Throughout, whenever we claimthat a module is Z × Z t graded for some t >
0, we mean that it is graded with respect tothis grading. When t = 0, this simply means that the module is Z -graded with respect to thestandard grading on S .Similar considerations can be done in the subrings S j = k [ x , . . . , x j ] of S . That is, if j > n − t ,we can view S j as A j [ y , . . . , y j − ( n − t ) ], where A j = k [ x , . . . , x n − t ] and y i = x n − t + i . In this way, S j is a Z × Z j − ( n − t ) -graded ring.Observe that, if M is a Z × Z t -graded S -module, with t >
0, we have that
M/x n M is still Z × Z t -graded, and it can be identified with a Z × Z t − -graded module over S n − . Example 2.15 (Key Example) . Let S = k [ x , . . . , x n ], and M be a finitely generated Z × Z t -graded S -module such that x n , . . . , x n − t +1 is a filter regular sequence for M . Then E-depth( M ) > t , and x n , . . . , x n − t +1 is a strictly filter regular sequence for M . GIULIO CAVIGLIA AND ALESSANDRO DE STEFANI
In fact, we can write M = F/U , where F is a free S -module, and U is a Z × Z t -gradedsubmodule of F . Consider the saturation U sat = U : x ∞ n = { α ∈ F | x rn α ∈ U for some r ≫ } .Since x n is a Z × Z t -homogeneous element, we have that U sat is Z × Z t -graded itself, and sois F/U sat . Since x n is assumed to be filter regular, we actually have F/U sat ∼ = M/H m ( M ). If F/U sat = 0, then M is zero dimensional, hence sequentially Cohen-Macaulay. In particular,E-depth( M ) = n > t , and any filter regular sequence is automatically strictly filter regular.Assume that F/U sat = 0, so that x n is a non-zero divisor on F/U sat . Since
F/U sat is Z × Z t -graded, we can write it as F/U sat = F /U ⊗ k k [ x n ], where F is a free graded S n − -module, and U is a Z × Z t − -graded submodule of F such that F /U can be identified with
F/U sat ⊗ S S/x n S .In particular, for all i < n we haveExt iS ( M, S ) ∼ = Ext iS ( F/U sat , S ) ∼ = Ext iS n − ( F /U , S n − ) ⊗ k k [ x n ] . Hence x n is a non-zero divisor on Ext iS ( M, S ) for all i < n , and it is then a strictly filter regularelement for M . Moreover, we have that E-depth( M ) >
0. Iterating this argument t -times givesthe desired claim.We will make a more systematic use of the methods of Example 2.15 in the next sections.3. Partial general initial modules and E-depth
Given integers 0 t n , we consider the following t × n matrix:Ω t,n = . . . . . . −
10 0 . . . . . . − . . . − . . . . . . − . . . . . . − . . . If we let S = k [ x , . . . , x n ], then Ω t,n induces a “partial revlex” term order on S . Given a finitelygenerated Z -graded S -module M , we can present it as M = F/U , where F is a finitely generated Z -graded free S -module, with graded basis { e , . . . , e r } . Notice that an element f ∈ F can bewritten uniquely as a finite sum of monomials of F , that is, we can write f = P j u j e i j wherethe elements u j are monomials in S and the sum has minimal support. Then, the initial formin rev t ( f ) of f with respect to the grading induced by Ω t,n will be the sum of elements of the form u j e i j from f for which u j is maximal with respect to the order induced by Ω t,n on S . Observethat, in general, in rev t ( f ) may not be of the form f ′ e i for some i = 1 , . . . , r . In other words, itmay not live in one single free summand of F . And even if it is of that form, the coefficient f ′ may not be a monomial of S .Given that the order in rev t can be extended to F , it makes sense to consider the initialsubmodule in rev t ( U ) of U in F . Remark 3.1.
One can check that the one defined is a partial reverse lexicographic order (see[Eis95, 15.7] for more details). In particular, we have(i) in rev t ( U : F x sn ) = in rev t ( U ) : F x sn for all s > rev t ( U + x n F ) = in rev t ( U ) + x n F . ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 9
More generally, one could take the partial orders induced by the following ( t + 1) × n matrix:Ω ′ t,n = . . . . . . −
10 0 . . . . . . − . . . − . . . . . . − . . . . . . − . . . . . . . . . , which takes also the degree of a monomial into account, or the one induced by Ω ′ t,n , and thatsuccessively defines that ue i > ve j if i < j . In both these cases, properties (i) and (ii) listedabove are still satisfied. Similarly, we would like to point out that one can also take into accountthe degrees of a graded basis of F . However, in order to define a revlex order (according to[Eis95, 15.7]) satisfying properties (i) and (ii) above, such degrees should be considered onlyafter all rows of Ω t,n have been evaluated.For the rest of this section we assume that k is infinite. The goal is to define a “partialgeneral initial submodule” of a given submodule U of a free S -module F . Definition 3.2.
Let S = k [ x , . . . , x n ], F be a Z -graded free S -module, and U be a graded S -submodule of F . We say that the partial general initial submodule of U satisfies a givenproperty (P) if there exists a non-empty Zariski open set L of t -uples of linear forms suchthat for every point ℓ = ( ℓ n − t +1 , . . . , ℓ n ) ∈ L the module F/ in rev t ( g ℓ ( U )) satisfies property (P),where g ℓ is the change of coordinates sending ℓ i x i and that fixes the other variables.For instance, we will consider properties (P) such as having a specific Hilbert function, or aspecific value for regularity, Betti numbers, etc.In fact, it is easy to see that such invariants and the corresponding non-empty Zariski openset where the property is constantly true or constantly false can be computed in the followingway: let e k = k ( α ij | n − t + 1 n t, j n ) be a purely transcendental field extensionof k , and let e ℓ i = P nj =1 α ij x j . Consider the change of coordinates e g ℓ sending e ℓ i x i and thatfixes the other variables, and compute in rev t ( e g ℓ ( U )) and any of the invariants mentioned above.The algorithm for such calculations is based on repeated Gr¨obner bases computations. Collectall non-zero coefficients in e k which appear in the calculations. Observe that they are finitelymany rational functions in k [ α ij | n − t + 1 i n, j n ]. We set L to be the Zariskiopen set of points where such functions are defined, and do not vanish. Since k is infinite, theintersection is not empty.By abusing notation, we will call any such submodule a general partial initial submodule of U ,and denote it by gin rev t ( U ). Thanks to the discussion above, we will therefore consider featuressuch as the Hilbert function, the
Betti numbers, and the
Hilbert function of local cohomologymodules of gin rev t ( U ).Let L be a Zariski open set consisting of t -uples of linear forms, that we can view as a Zariskiopen set in a projective space P = P ( n − × t . To each point ℓ ∈ L is associated a linear changeof coordinates g ℓ defined as above. Vice versa, to each g ℓ we can associate a point ℓ ∈ P . Byabusing notation, we will henceforth refer to a Zariski open set of transformations of the form g ℓ to mean the above scenario.Now consider the closed subspace P UP of P consisting of “upper triangular” t -uples of linearforms, that is, elements of the form ( ℓ n − t +1 , . . . , ℓ n ) where ℓ i is a linear form supported on the variables x , . . . , x i . Observe that, for ℓ ∈ L ∩ P UP , the corresponding change of coordinates g ℓ can be represented by an upper triangular matrix. Remark 3.3.
In order to test whether a property (P) of a general initial submodule holds,it is sufficient to produce a non-empty Zariski open set in P UP where (P) holds. In fact, let L be a Zariski open set of t -uples of linear forms. Associated to ℓ ∈ L we have a change ofcoordinates g ℓ as in Definition 3.2, which can be represented as a matrix of the form(3.1) g ℓ = (cid:20) I n − t ⋆ ⋆ (cid:21) , where I n − t is the identity matrix of size n − t , and (cid:20) ⋆⋆ (cid:21) has size n × t . By possibly shrinkingthe open set L , we can factor such a matrix in the product of a lower triangular matrix withall entries equal to one on the main diagonal, and an upper triangular matrix: g ℓ = g LOW ℓ g UP ℓ .Moreover,(3.2) in rev t ( g ℓ ( U )) = in rev t ( g LOW ℓ g UP ℓ ( U )) = in rev t ( g UP ℓ ( U )) , where the last equality follows from standard properties of revlex-type orders. Thus, startingfrom a non-empty Zariski open set L ⊆ P where property (P) holds, one can produce a non-empty Zariski open set L UP ⊆ P UP where (P) still holds. Vice versa, assume that we are given anon-empty Zariski open set L UP inside P UP , so that the change of coordinates g ℓ correspondingto points in L UP are upper triangular. By acting on the set of such transformations with thefollowing group of n × n matrices (cid:26) A = (cid:20) I n − t ⋆ (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) A = { a ij } is lower triangular, and a i,i = 1 for all i = 1 , . . . , n (cid:27) , one obtains a non-empty Zariski open set of matrices g ℓ of the form (3.1) on which property(P) still holds by (3.2). In other words, this gives a non-empty Zariski open subset L ⊆ P where (P) still holds.The following is an extension of [Gre10, Proposition 2.14] to our setting, which will be used inthe proof of the main result of this section. Even if the argument is similar, it is more technical.Thus, we provide a proof for sake of completeness. Lemma 3.4.
Let S = k [ x , . . . , x n ] with the standard grading, N be a non-negative integerand t be a positive integer. Let F be a free S -module, and U be a graded submodule of F . Forany sufficiently general linear form h = P ni =1 α i x i , we can identify (( U : h N ) + hF ) /hF witha submodule V h of a free S n − = k [ x , . . . , x n − ]-module F by setting x n = − α − n ( P n − i =1 α i x i ).Consider a property (P). There exists a non-empty Zariski open set of linear forms H suchthat, for all h ∈ H , the module ((gin rev t ( U ) : x Nn ) + x n F ) /x n F satisfies (P) if and only ifgin rev t-1 ( V h ) satisfies (P). Here, gin rev t-1 denotes a general partial initial submodule computedin S n − . Proof.
First of all, observe that to compute in rev t we can first compute the initial submodulein rev with respect to the first row of the matrix Ω t,n introduced above, and then compute theinitial submodule with respect to the remaining t − rev t-1 .By Remark 3.3, we can reduce to considering upper triangular changes of coordinates. Inparticular, we can find a non-empty Zariski open set L UP ⊆ (cid:0) P ( n − × t (cid:1) UP such that the linearchange of coordinates g ℓ = g UP ℓ introduced in Definition 3.2 is upper triangular and the given ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 11 property (P) holds for ((gin rev t ( U ) : x Nn ) + x n F ) /x n F if and only if it holds for ((in rev t ( g UP ℓ ( U )) : x Nn ) + x n F ) /x n F for all ℓ ∈ L UP .For ℓ = ( λ n − t +1 , . . . , λ n − , λ n ) ∈ L UP , set λ = ( λ n − t +1 , . . . , λ n − ). We can factor g UP ℓ asthe composition g UP λ ◦ k λ n , defined as follows: k λ n is the change of coordinates that fixes x i for i = n , and sends λ n x n , while g UP λ is the change of coordinates such that x i x i for1 i n − t and i = n , and such that λ i x i for n − t + 1 i n − U ′ = k λ n ( U ). Proceeding in a similar manner as in [Gre10, Section 6], where Greenconstructs partial elimination ideals for the lex order, we can write U ′ as a disjoint union ofsets U ′ = U ′ [0] ⊔ U ′ [1] x n ⊔ U ′ [2] x n ⊔ · · · , where U ′ [ i ] x in consists of the elements of U ′ that are divisible by x in , and not by x i +1 n . It caneasily be checked that U ′ [ i ] ⊆ U ′ [ j ] if i j . Given a polynomial f = f ( x , . . . , x n ) ∈ S , weset f = f ( x , . . . , x n − , ∈ S n − . Now, if ( f , . . . , f r ) ∈ F = S ⊕ r , we define ( f , . . . , f r ) =( f , . . . , f r ) ∈ F , where F is a free S n − -module, which can be identified with F/x n F . Since U ′ [ i ] is a subset of F , it makes sense to define U ′ [ i ] = { u | u ∈ U [ i ] } and g UP λ ( U ′ [ i ] ) = { u | u ∈ g UP λ ( U ′ [ i ] ) } .Since the change of coordinates g UP λ fixes x n , and each other linear form λ i involved in suchtransformation does not have x n in its support, one can check thatin rev ( g UP λ ( U ′ )) = g UP λ ( U ′ [0] ) ⊔ g UP λ ( U ′ [1] ) x n ⊔ g UP λ ( U ′ [2] ) x n ⊔ · · · = g UP λ (cid:16) U ′ [0] ⊔ U ′ [1] x n ⊔ U ′ [2] x n ⊔ · · · (cid:17) = g UP λ (in rev ( U ′ )) . Therefore(in rev t ( g UP ℓ ( U )) : x Nn ) + x n F = (in rev t ( g UP λ ( U ′ )) : x Nn ) + x n F = (in rev t-1 ( g UP λ (in rev ( U ′ ))) : x Nn ) + x n F = (in rev t-1 ( g UP λ ( U ′ [0] ⊔ U ′ [1] x n ⊔ U ′ [2] x n ⊔ · · · )) : x Nn ) + x n F = (cid:16) in rev t-1 (cid:16) g UP λ ( U ′ [0] ) ⊔ g UP λ ( U ′ [1] ) x n ⊔ · · · (cid:17) : x Nn (cid:17) + x n F. Because of how in rev t-1 is defined, we have that in rev t-1 (cid:16) g UP λ ( U ′ [ i ] ) x in (cid:17) = in rev t-1 (cid:16) g UP λ ( U ′ [ i ] ) (cid:17) x in .Therefore the last formula is equal to (cid:16)(cid:16) in rev t-1 (cid:16) g UP λ ( U ′ [0] ) (cid:17) ⊔ in rev t-1 (cid:16) g UP λ ( U ′ [1] ) (cid:17) x n ⊔ · · · (cid:17) : x Nn (cid:17) + x n F. One can check that, as a set, in rev t-1 (cid:16) g UP λ ( U ′ [ i ] ) x in (cid:17) : x Nn equals in rev t-1 (cid:16) g UP λ ( U ′ [ i ] ) x i − Nn (cid:17) if i > N ,and it equals in rev t-1 (cid:16) g UP λ ( U ′ [ i ] ) (cid:17) if i < N . Since U ′ [ i ] ⊆ U ′ [ j ] if i j , the above expression isequal to (cid:16) in rev t-1 (cid:16) g UP λ ( U ′ [ N ] ) (cid:17) ⊔ in rev t-1 (cid:16) g UP λ ( U ′ [ N +1] ) (cid:17) x n ⊔ · · · (cid:17) + x n F = in rev t-1 (cid:16) g UP λ ( U ′ [ N ] ) (cid:17) + x n F = in rev t-1 (cid:16) g UP λ ( U ′ [ N ] ) (cid:17) + x n F. Let π : (cid:0) P ( n − × t (cid:1) UP → P n − be the map which sends ℓ = ( λ n − t +1 , . . . , λ n ) to λ n , and let H = π ( L UP ), which is a non-empty Zariski open set. Moreover, let L UP h = { λ = ( λ n − t +1 , . . . , λ n − ) | ( λ n − t +1 , . . . , λ n − , h ) ∈ L UP } . We have shown that, in order to decide whether ((gin rev t ( U ) : x Nn ) + x n F ) /x n F satisfies the given property (P), one can just check whether for h ∈ H and λ = ( λ n − t +1 , . . . , λ n − ) ∈ L UP h , after the change of coordinates k λ n this is true for thesubmodule in rev t-1 ( g UP λ ( U ′ [ N ] )) of the S n − -module F .For any h ∈ H , we have that k h (( U : h N ) + hF ) = ( U ′ : x Nn ) + x n F = U ′ [ N ] + x n F . Thus,after the change of coordinates k h , we may identify V h with U ′ [ N ] . Note that, for λ ∈ L UP h , thetransformation g UP λ can be viewed as an upper triangular change of coordinates in S n − .By possibly shrinking the open set L UP h , we may assume that the module gin rev t-1 ( V h ) satisfies(P) if and only if in rev t-1 ( g UP λ ( U ′ [ N ] )) does for every λ ∈ L UP h . Finally, observe that a partialinitial submodule in rev t-1 computed over F is the same as a partial initial submodule with respectto the matrix Ω t − ,n − . Putting all these facts together, by Definition 3.2 we can then say thatfor h ∈ H the module gin rev t-1 ( V h ) satisfies (P) if and only if ((gin rev t ( U ) : x Nn ) + x n F ) /x n F does, and this concludes the proof. (cid:3) We exhibit a first relation between gin rev t of a module and the notion of E-depth. Proposition 3.5.
Let S = k [ x , . . . , x n ], and M be a Z -graded S -module of dimension n .Write M = F/U , where F is a graded free S -module. If t is an integer with 0 t n , thenE-depth( F/ gin rev t ( U )) > t . Proof.
Observe that, by construction, F/ gin rev t ( U ) is Z × Z t -graded. In the notation of Defi-nition 3.2, we have that x n , . . . , x n − t +1 forms a filter regular sequence for F/ ( g ℓ ( U )). It followsfrom 3.1 (i) that they also form a filter regular sequence for F/ in rev t ( g ℓ ( U )) = F/ gin rev t ( U ).The claim now follows from Example 2.15. (cid:3) Recall that, given a graded S -module M , we denote by HF( M ) its Hilbert function, that is,the row vector whose entry in position j equals dim k ( M j ).The following is the main theorem of this section. As it will be pointed out later, this can beviewed as extension of the main result of Herzog and Sbarra in [HS02]. We follow closely thesteps of their proof. Theorem 3.6.
Let S = k [ x , . . . , x n ], and M be a finitely generated Z -graded S -module, thatwe can write as a quotient M = F/U , where F is a graded free S -module. For a given integer0 t n , we have that E-depth( M ) > t if and only if HF( H i m ( F/U )) = HF( H i m ( F/ gin rev t ( U )))for all i ∈ Z . Proof.
After a general change of coordinates, we may assume that V = in rev t ( U ) has the sameproperties as gin rev t ( U ). We may also assume that x n , . . . , x n − t +1 forms a strictly filter regularsequence for F/U . Since strictly filter regular sequences are filter regular sequences, it followsfrom Remark 3.1 (i) that x n , . . . , x n − t +1 forms a filter regular sequence for F/V . Since V is Z × Z t -graded, it then follows from Example 2.15 that x n , . . . , x n − t +1 forms a strictly filterregular sequence also for F/V . Furthermore, by Remark 3.1 we have V sat = in rev t ( U ) : F x ∞ n = in rev t ( U : F x ∞ n ) = in rev t ( U sat ) , and also(3.3) V sat + x n F = in rev t ( U sat ) + x n F = in rev t ( U sat + x n F ) . We first prove that if E-depth( M ) > t , then there is equality for the Hilbert functions oflocal cohomology modules. We proceed by induction on t >
0, the case t = 0 being trivial (notethat in rev ( U ) = U ). By Proposition 2.12 (2) we have that E-depth( F/U sat ) > t >
0, where
ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 13 U sat = U : F x ∞ n . It follows from Proposition 2.12 (3) that E-depth( F/ ( U sat + x n F )) > t − H i m ( F/ ( U sat + x n F ))) = HF( H i m ( F/ ( V sat + x n F ))) for all i ∈ Z .Since E-depth( F/U sat ) > t >
0, by Lemma 2.9 we have short exact sequences0 / / H i m ( F/ ( U sat + x n F )) / / H i +1 m ( F/U )( − · x n / / H i +1 m ( F/U ) / / i >
0. By Proposition 3.5 and Lemma 2.9, we have analogous short exact sequences for
F/V : 0 / / H i m ( F/ ( V sat + x n F )) / / H i +1 m ( F/V )( − · x n / / H i +1 m ( F/V ) / / . Let HS( M ) = P j ∈ Z dim k ( M j ) z j be the Hilbert series of M . For i >
0, we haveHS( H i +1 m ( F/U ))( z −
1) = HS( H i m ( F/ ( U sat + x n F )))= HS( H i m ( F/ ( V sat + x n F ))) = HS( H i +1 m ( F/V ))( z − , and thus HF( H i m ( F/U )) = HF( H i m ( F/V )) for all i >
0. Finally, we have HF(
F/U ) = HF(
F/V ),and also HF(
F/U sat ) = HF( F/ in rev t ( U sat )) = HF( F/V sat ), by Remark 3.1 (i). Therefore weconclude that HF( H m ( F/U )) = HF( U sat /U ) = HF( V sat /V ) = HF( H m ( F/V )) . To prove the converse, assume that the local cohomology modules of
F/U and
F/V have thesame Hilbert function. We want to prove by induction on t > F/U ) > t . If t = 0there is nothing to show. By Proposition 2.12 (2), it suffices to show that E-depth( F/U sat ) > t .We claim that the set S = { j ∈ Z > | H j m ( F/U sat ) · x n −→ H j m ( F/U sat ) is not surjective } is empty.If not, let i = min S , and observe that i >
0, since H m ( F/U sat ) = 0. Then we have an exactsequence 0 / / H i − m ( F/ ( U sat + x n F )) / / H i m ( F/U )( − · x n / / H i m ( F/U ) . Since the rightmost map is not surjective by choice of i , there exists j ∈ Z such thatdim k ( H i − m ( F/ ( U sat + x n F )) j ) > dim k ( H i m ( F/U ) j − ) − dim k ( H i m ( F/U ) j )= dim k ( H i m ( F/V ) j − ) − dim k ( H i m ( F/V ) j )= dim k ( H i − m ( F/ ( V sat + x n F )) j ) . Here, we used that dim k ( H i m ( F/U ) j ) = dim k ( H i m ( F/V ) j ) for all j ∈ Z by assumption, and thatthe sequence0 / / H i − m ( F/ ( V sat + x n F )) / / H i m ( F/V )( − · x n / / H i m ( F/V ) / / H i − m ( F/ ( U sat + x n F ))) HF( H i − m ( F/ in rev t ( U sat + x n F ))) = HF( H i − m ( F/ ( V sat + x n F ))) . This contradicts the inequality obtained above, and hence the set S is empty. This implies thatE-depth( F/U ) > H i m ( F/ ( U sat + x n F ))) = HF( H i m ( F/ ( V sat + x n F ))) for all i ∈ Z . Viewing F/ ( U sat + x n F ) as a quotient F /U of a finitely generated free S n − -module F , one still has equality of Hilbert functions for the local cohomology modules of F /U and
F /V , where V = in rev t − ( U ). Moreover, by Lemma 3.4, V has the same propertiesas gin rev t-1 ( U ). By induction, we then have E-depth( F/ ( U sat + x n F )) > t − over S n − . Since E-depth( F/U sat ) = E-depth(
F/U ) > F/U ) = E-depth(
F/U sat + x n F ) = n > t , orE-depth( F/U ) < n and E-depth( F/U ) = E-depth( F/ ( U sat + x n F )) + 1 > t . In both cases, thedesired inequality is obtained. (cid:3) Observe that the proof of Theorem 3.6 can be adapted to the general initial submodule ofany revlex order which satisfies conditions (i), (ii) of Remark 3.1, and Lemma 3.4. In particular,we recover [HS02, Theorem 3.1].
Corollary 3.7.
Let S = k [ x , . . . , x n ], and M be a finitely generated Z -graded S -module. Write M = F/U , where F is a free S -module and U is a graded submodule. Then M is sequentiallyCohen-Macaulay if and only if HF( H i m ( F/U )) = HF( H i m ( F/ gin revlex ( U ))) for all i ∈ Z .4. Decomposition of local cohomology tables
For convenience of the reader, we recall the grading introduced in Section 2. Let t >
0, and R = A [ y , . . . , y t ] be a polynomial ring over a Z -graded ring A . We put a Z × Z t -grading on R as follows. Let η i ∈ Z t +1 be the vector with 1 in position i and 0 everywhere else. We setdeg R ( a ) = deg A ( a ) · η for all a ∈ A , and deg R ( y i ) = η i +1 . We recall the notation S j for thesubring k [ x , . . . , x j ] of S , and we let m j = ( x , . . . , x j ) S j . If j > n − t >
0, we recall that S j isa Z × Z j − ( n − t ) -graded ring. Lemma 4.1.
Let S = k [ x , . . . , x n ], and M be a Z × Z t -graded module. Assume that x n , x n − , . . . , x n − t +1 forms a filter regular sequence for M . There exists a family {N j } nj = n − t of modules that satisfies the following conditions: • N n = M . • Each N j is either zero, or is a finitely generated Z × Z j − ( n − t ) -graded module over S j ofKrull dimension dim( N j ) = dim( M ) − ( n − j ). • x j , . . . , x n − t +1 is a strictly filter regular sequence on N j for every j = n − t + 1 , . . . , n . • For all j we have N j /H m j ( N j ) ∼ = N j − ⊗ k k [ x j ]. Proof.
We construct such modules inductively, starting from N n = M . Assume that N j hasbeen constructed for some n − t + 1 j n . Since N j is Z × Z j − ( n − t ) -graded over S j ,and x j is filter regular over N j , we have that N j /H m j ( N j ) = N j − ⊗ k k [ x j ] for some finitelygenerated Z × Z ( j − − ( n − t ) -graded S j − -module, that we call N j − . From this, we see that N j /H m j ( N j ) ⊗ S j S j /x j S j can be identified with N j − . If dim( N j ) = 0, then N j /H m j ( N j ) = 0,and thus N j − = 0. Otherwise N j − has dimension dim( N j ) − M ) − ( n − j + 1).Moreover, for i < j we haveExt iS j ( N j , S j ) ∼ = Ext iS j ( N j /H m j ( N j ) , S j ) ∼ = Ext iS j − ( N j − , S j − ) ⊗ k k [ x j ] . This shows that x j is a non-zero divisor on Ext iS j ( N j , S j ) for all i < j , and therefore it is astrictly filter regular element for N j . By construction, we see that it is also strictly filter regularfor N s , for all s > j , and the proof is complete. (cid:3) Notation 4.2.
Let S = k [ x , . . . , x n ], and M be a finitely generated Z -graded S -module suchthat x n , . . . , x n − j +1 is a filter regular sequence for M . We consider the following chain ofsubmodules of M : Q = 0 : M x ∞ n ⊆ Q = ( x n ) M : M x ∞ n − ⊆ . . . ⊆ Q j = ( x n − j +1 , . . . , x n ) M : M x ∞ n − j . ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 15
We set M = Q , and for 1 i j we let M i = Q i Q i − + x n − i +1 M .
Observe that ( x n − i +1 , . . . , x n ) M i = 0, hence we will view M i as a finitely generated Z -graded S n − i -module. Because of its definition, it is also a finite dimensional graded k -vector space. Remark 4.3. If {N j } nj = n − t is the family of modules constructed in Lemma 4.1 for M , then onecan readily check that H m n − j ( N n − j ) ∼ = M j for all j as graded k -vector spaces. Proposition 4.4.
Let S = k [ x , . . . , x n ], and M be a finitely generated S -module. Assumethat M is Z × Z t -graded for some t >
1, and that x n , x n − , . . . , x n − t +1 form a filter regularsequence on M . Let {N j } nj = n − t be the family of modules constructed in Lemma 4.1. We havegraded k -vector space isomorphisms: H j m ( M ) ∼ = ( M j ⊗ k H j m j ( S j ) if 0 j t − H j − t m n − t ( N n − t ) ⊗ k H t m t ( S t ) if t j n Proof.
First, observe that H m ( M ) = M ∼ = M ⊗ k k = M ⊗ k H m ( S ) , so the statement is trivially true for j = 0. By the K¨unneth formula for local cohomology, andby Lemma 4.1, for j > s min { j, t } we have graded k -vector space isomorphisms: H j m ( M ) ∼ = H j m ( N n − s ⊗ k k [ x n − s +1 , . . . , x n ]) ∼ = M i (cid:16) H j − i m n − i ( N n − s ) ⊗ k H i ( x n − s +1 ,...,x n ) ( k [ x n − s +1 , . . . , x n ]) (cid:17) ∼ = H j − s m n − s ( N n − s ) ⊗ k H s m s ( S s ) , where we used that k [ x n − s +1 , . . . , x n ] ∼ = S s and the only non-vanishing cohomology of S s occursfor i = s . In particular, for j > t we have H j m ( M ) ∼ = H j − t m n − t ( N n − t ) ⊗ k H t m t ( S t ), while for j t − H j m ( M ) ∼ = H m n − j ( N n − j ) ⊗ k H j m j ( S j ), which by Remark 4.3 is isomorphic to M j ⊗ k H j m j ( S j ). (cid:3) Remark 4.5.
In what follows, we will view S j = k [ x , . . . , x j ] as an S -module, identifying itwith S/ ( x j +1 , . . . , x n ). In this way, the two local cohomology modules H j m ( S j ) and H j m j ( S j ) areisomorphic, and we will switch from one to the other without further justifications. Remark 4.6.
Let M be a finitely generated S -module of dimension d . Consider a gradedNoether normalization R of S/ ann S ( M ), and observe that M is a finitely generated graded R -module. Since R is isomorphic to a polynomial ring in d many variables, and becauselocal cohomology does not change when viewing M as an R -module, we may view the localcohomology table of M both as an S and an R -module, with no distinctions. Moreover, we mayalways assume that k is infinite without affecting considerations on the cone of local cohomologytables (see [DSS20, Lemma 2.2]). Finally, if M is not sequentially Cohen-Macaulay one canshow that E-depth( M ) is unaffected by viewing it as an R -module instead of an S -module. Onthe other hand, if M is sequentially Cohen-Macaulay as an S -module, it is also sequentiallyCohen-Macaulay as an R -module, with E-depth( M ) = d . We now turn our attention to local cohomology tables. Let S = k [ x , . . . , x n ], with thestandard grading. Let M be a finitely generated graded Z -module. We consider the localcohomology table of M : [ H • m ( M )] = ( h ij ) ∈ Mat n +1 , Z ( Z )where h ij := dim k ( H i m ( M ) j ). We can then consider the cone spanned by the local cohomologytables, with non-negative rational coefficients: Q > · { [ H • m ( M )] | M is a finitely generated Z -graded S -module } . A description of the extremal rays and supporting hyperplanes of this cone has been given in[DSS20] when restricting to local cohomology tables of modules of dimension at most two.Given a finitely generated Z -graded module M , we will consider its (local cohomology) Hilbertseries: HS( H • m ( M )) = n X i =0 X j ∈ Z h ij u i z j ∈ Z J z ± K [ u ] . If N is a finitely generated Z -graded S n − j -module, then by the proof of Proposition 4.4 wehave(4.1) HS( H • m ( N ⊗ k k [ x n − j +1 , . . . , x n ])) = u j ( z (1 − z − )) j HS( H • m ( N )) = u j ( z − j HS( H • m ( N )) . We start by proving a decomposition theorem for S -modules satisfying E-depth( M ) > dim( S ) −
2. By Remark 4.6, if E-depth( M ) > dim( M ) −
2, we can still reduce to this case.In particular, since modules of dimension at most two automatically satisfy this condition, thefollowing is an extension of [DSS20, Theorem 4.6].
Theorem 4.7.
Let S = k [ x , . . . , x n ] be a standard graded polynomial ring, and M be a Z -graded S -module. Let t = E-depth( M ), and assume that t > n −
2. Let J = ( x , x ) S . Wehave a decomposition[ H • m ( M )] = n X i =0 X j ∈ Z r i,j [ H • m ( S i ( − j ))] + X m> X j ∈ Z r ′ m,j [ H • m ( J m ( − j ))] , where r i,j ∈ Z > , r ′ m,j ∈ Q > , and all but finitely many of them are equal to zero. Moreover,the set Λ = { [ H • m ( S i ( − j ))] , [ H • m ( J m ( − j ))] | i n, j ∈ Z , m > } is minimal, that is, none of the tables from Λ can be written as a non-negative rational linearcombination of the other tables from the same set. Proof.
We may assume that n >
2, otherwise the result follows from [DSS20, Theorem 4.6].Since t > n −
2, by Theorem 3.6 we can replace M by a partial general initial module F/ gin rev n-2 ( U ) which is Z × Z n − -graded, and is such that x n , . . . , x forms a filter regularsequence. Let h ij = dim k ( H i m ( M ) j ). By Proposition 4.4, for all 0 i n − k -vector space isomorphisms H i m ( M ) ∼ = M i ⊗ k H i m ( S i ) ∼ = L j ∈ Z H i m ( S i ( − j )) ⊕ r i,j , where r i,j = dim k { m ∈ M i | deg( m ) = j } . Thus(4.2) n − X i =0 X j ∈ Z h ij u i z j = n − X i =0 X j ∈ Z r i,j HS( H • m ( S i ( − j ))) . ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 17
On the other hand, for n − i n we have H i m ( M ) ∼ = H i − ( n − m ( N ) ⊗ k H n − m n − ( S n − ). Thereforeby (4.1) we have n X i = n − X j ∈ Z h ij u i z j = u n − ( z − n − HS( H • m ( N )) . If N = 0, then the proof is complete, since (4.2) gives the desired decomposition. If N = 0,then it is a finitely generated Z -graded S -module. It follows from [DSS20, Theorem 4.6] thatHS( H • m ( N )) = X i =0 X j ∈ Z s i,j HS( H • m ( S i ( − j ))) + X m> X j ∈ Z s ′ m,j HS( H • m ( m m ( − j ))) , where s i,j ∈ Z > and s ′ m,j ∈ Q > are all but finitely many equal to zero.Applying (4.1) to the module J m ∼ = m m ⊗ k k [ x , . . . , x n ], we obtain the following relation:HS( H • m ( J m )) = u n − ( z − n − HS( H • m ( m m )) . Similarly, for i = 0 , , H • m ( S i +( n − )) = u n − ( z − n − HS( H • m ( S i )).Putting these relations together we see that:(4.3) n X i = n − X j ∈ Z h ij u i z j = n X i = n − X j ∈ Z r i,j HS( H • m ( S i ( − j ))) + X m> X j ∈ Z r ′ m,j HS( H • m ( J m ( − j ))) , where r i,j = s i − ( n − ,j and r ′ m,j = s ′ m,j . Finally, summing (4.2) and (4.3) up and passing to localcohomology tables gives the desired decomposition:[ H • m ( M )] = n X i =0 X j ∈ Z r i,j [ H • m ( S i ( − j ))] + X m> X j ∈ Z r ′ m,j [ H • m ( J m ( − j ))] . For minimality, the strategy of the proof is completely analogous to that of Theorem [DSS20,Theorem 4.6], combined with the use of (4.1) as above. (cid:3)
As a corollary of the proof, we obtain a very special decomposition for local cohomologytables in the sequentially Cohen-Macaulay case.
Corollary 4.8.
Let S = k [ x , . . . , x n ] with the standard grading, and M be a finitely generated Z -graded S -module of dimension d . Assume that M is sequentially Cohen-Macaulay. Then[ H • m ( M )] = d X i =0 X j ∈ Z r i,j [ H • m ( S i ( − j ))] , where r i,j ∈ Z > , and all but finitely many of them are zero. Proof.
By Remark 4.6, we may assume that n = d . By Corollary 3.7 (or [HS02, Theorem3.1]), we can replace M by its general initial module, and assume that M is Z × Z d − -graded,and that x d , x d − , . . . , x forms a filter regular sequence on M . By Proposition 4.4, for all i = 0 , . . . , d we have graded k -vector spaces M i and graded isomorphisms H i m ( M ) ∼ = M i ⊗ k H i m ( S i ) ∼ = L j ∈ Z H i m ( S i ( − j )) ⊕ r i,j , where r i,j = dim k { m ∈ M i | deg( m ) = j } , and the proof iscomplete. (cid:3) Supporting hyperplanes.
The goal is to provide a description of the supporting hyper-planes of the cone of local cohomology tables of modules with sufficiently large E-depth. Asdone in [DSS20], to do so we must reduce to “finitely supported” local cohomology tables. Wewill now explain this process.We start by recalling a notation introduced in [DSS20, Notation 4.1].
Notation 4.9.
Let N = L j ∈ Z N j be a Z -graded k -vector space that satisfies dim k ( N j ) < ∞ for all j ∈ Z . For t > t ( N ) inductively as follows. If t = 0 thenwe set ∆ ( N ) = HF( N ). If t >
0, we define ∆ t ( N ) to be the vector whose j -th entry is∆ t ( N ) j = ∆ t − ( N ) j − ∆ t − ( N ) j +1 .Let S = k [ x , . . . , x n ], with the standard grading. Recall that the local cohomology table ofa finitely generated Z -graded S -module is a point in the Q -vector space of ( n + 1) × Z -matricesMat n +1 , Z ( Q ). We now define an operator ∆( − ), which takes a local cohomology table [ H • m ( M )]and transforms it into a new table ∆[ H • m ( M )] as follows:[ H • m ( M )] = HF( H m ( M ))HF( H m ( M ))...HF( H n m ( M )) ∆[ H • m ( M )] = ∆ ( H m ( M ))∆ ( H m ( M ))...∆ n ( H n m ( M )) . It is important to observe that, even though a local cohomology table typically has infinitelymany non-zero entries, the table ∆[ H • m ( M )] only has finitely many non-zero entries. This followsfrom the fact that the functions HF( H i m ( M )) : j dim k ( H i m ( M ) j ) = dim k (Ext n − iS ( M, S ( − n )) − j )coincide with polynomials of degree at most i − j ≪ H • m ( M )) = n X i =0 X j ∈ Z h ij u i z j ∆HS( H • m ( M )) = n X i =0 X j ∈ Z h ij u i z j (1 − z − ) i , and from the fact that ∆[ H • m ( M )] has finite support we deduce that ∆HS( H • m ( M )) is a Laurentpolynomial in Z [ z ± , u ]. Since local cohomology modules are zero in sufficiently high degrees,it can be checked that, given two modules M and N , HS( H • m ( M )) = HS( H • m ( N )) if and only if∆HS( H • m ( M )) = ∆HS( H • m ( N )). In the rest of the section, we will often switch from the pointof view of tables to that of Hilbert series, and vice versa.For convenience, we will adopt the convention that the first row of a matrix A = ( a i,j ) ∈ Mat n +1 , Z ( Q ) corresponds to the index i = 0, the second row to the index i = 1, and so on. Thismakes the association between the rows of A and invariants of local cohomology modules witha given cohomological index more natural.Let M be the Q -vector subspace of Mat n +1 , Z ( Q ) generated by tables with finitely manynon-zero entries. Then M can be filtered as M = S a b } . Let C [ a,b ] be the cone spanned by tables ∆[ H • m ( M )] that are contained in M [ a,b ] , where M is a Z -graded S -module such that E-depth( M ) > n −
2. Let C seq[ a,b ] be the cone spanned by tables∆[ H • m ( M )] that are contained in M [ a,b ] , where M is a sequentially Cohen-Macaulay Z -graded S -module. Clearly, C seq[ a,b ] ⊆ C [ a,b ] . We start with a description of the smaller cone. ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 19
Proposition 4.10.
For integers a < b , consider the cone defined by D seq[ a,b ] = { A ∈ M [ a,b ] | a i,j > i n and j ∈ Z } . Then C seq[ a,b ] = D seq[ a,b ] . Proof.
Let H ∈ C seq[ a,b ] be a table, that we may assume being equal to ∆[ H • m ( M )] for somesequentially Cohen-Macaulay graded S -module M . By Corollary 4.8 we have that [ H • m ( M )]can be written as a sum, with positive coefficients and shifts, of tables of the form [ H • m ( S i )]. Itis easy to see that, for all i , the table ∆[ H • m ( S i )] has non negative entries, and thus H ∈ D seq[ a,b ] .Conversely, let H ∈ D seq[ a,b ] . We can represent H as a Laurent polynomial, namely P ( H ) = P ni =0 P j ∈ Z r i,j u i z j , where r i,j denotes the entry of H in position ( i, j ). Observe that r i,j > H • m ( S i ( − i ))) = z i HS( H • m ( S i )) = ( uz ) i ( z − i HS( H • m ( k )) = u i (1 − z − ) i , and thus ∆HS( H • m ( S i ( − i ))) = u i . Therefore P ( H ) = n X i =0 X j ∈ Z r i,j u i z j = n X i =0 X j ∈ Z r i,j z j ∆HS( H • m ( S i ( − i ))) = n X i =0 X j ∈ Z r i,j ∆HS( H • m ( S i ( − j − i ))) . In the language of tables, this means that H = P ni =0 P j ∈ Z r i,j ∆[ H • m ( S i ( − j − i ))]. Since eachmodule S i ( − j − i ) is sequentially Cohen-Macaulay, it follows that ∆[ H • m ( S i ( − j − i ))] ∈ C seq[ a,b ] whenever r i,j = 0. Thus, H ∈ C seq[ a,b ] . (cid:3) To describe the cone C [ a,b ] we need to introduce more functionals, which come from [DSS20]. Definition 4.11.
Let A = ( a i,j ) ∈ M . For j ∈ Z we set τ j ( A ) = a n − ,j + X s j − a n,s . Given an integer m > j ∈ Z , we set π m,j ( A ) = X s>j + m a n − ,s + ( m + 1) a n − ,j + m + m − X s =0 ( s + 1) a n,j + s . Finally, given an integer 0 i n and j ∈ Z , we let µ ( i ) j = a i,j .Let a < b be integers, and consider the following list of functionals on M : H [ a,b ] = µ ( i ) j for a j b, < i n − i = 0 , n,τ j for a j < b,π ,j for a + 1 j b,π m,j for a + 1 m b − , a + 1 j < b − m . Remark 4.12.
Observe that the functionals τ j , π ,j , π m,j are precisely those appearing in[DSS20, Theorem 6.2], with the only difference that the variables a ,j are here replaced by a n − ,j and the variables a ,j are replaced by a n,j . Roughly speaking, if a matrix A ∈ M [ a,b ] sat-isfies τ j ( A ) > , π ,j ( A ) > , π m,j ( A ) > µ ( i ) j ( A ) > i = n − , n , then the submatrixcorresponding to the last three rows of A satisfies the inequalities given by the functionals from[DSS20, Theorem 6.2]. Theorem 4.13.
For integers a < b , consider the cone D [ a,b ] = { A ∈ M [ a,b ] | φ ( A ) > φ ∈ H [ a,b ] } . We have D [ a,b ] = C [ a,b ] . Proof. If n
1, then every finitely generated S -module is sequentially Cohen-Macaulay, andthe result trivially follows from Proposition 4.10. If n = 2, the result follows from [DSS20,Theorem 6.2]. Henceforth, we will assume that n > H ∈ C [ a,b ] be a table. As in Proposition 4.10, we may assume that H = ∆[ H • m ( M )], where M is a graded S -module of Krull dimension d n , and E-depth( M ) > n − d < n , then E-depth( M ) > n − > d −
1, and thus M is sequentially Cohen-Macaulay by Proposition 2.11. It follows from Proposition 4.10 that H satisfies µ ( i ) j ( H ) > a j b and 0 i n . From this, it can easily be checked that φ ( H ) > φ ∈ H [ a,b ] , that is, H ∈ D [ a,b ] .Now assume that d = n . Apply the operator ∆( − ) to a decomposition of [ H • m ( M )] obtainedfrom Theorem 4.7:∆[ H • m ( M )] = n X i =0 X j ∈ Z r i,j ∆[ H • m ( S i ( − j ))] + X m> X j ∈ Z r ′ j,m ∆[ H • m ( J m ( − j ))] , where J = ( x , x ) S , the rational numbers r i,j , r ′ j,m are non-negative, and only finitely many ofthem are not zero. Thus, it suffices to show that every single table appearing on the right-hand side of the equation satisfies the inequalities defining D [ a,b ] . For tables of the form∆[ H • m ( S i ( − j ))], the argument is the same as the one used above when M is sequentially Cohen-Macaulay. For ∆[ H • m ( J m ( − j ))], we pass through its Hilbert series ∆HS( H • m ( J m ( − j ))). Using(4.1), straightforward calculations give that∆HS( H • m ( J m ( − j ))) = ( uz − ) n − ∆HS( H • m ( m m ( − j ))) = u n − ∆HS( H • m ( m m ( − j + n − . We conclude that the table ∆[ H • m ( J m ( − j ))] satisfies the desired inequalities because the table∆[ H • m ( m m ( − j + n − S -module m m ( − j + n −
2) satisfies analogous functionals indimension two, by [DSS20, Theorem 6.2] (see Remark 4.12).Conversely, let H ∈ D [ a,b ] . We show that H belongs to C [ a,b ] by constructing the first n − r i,j be the ( i, j )-th entry of H ,and consider the associated Laurent polynomial P ( H ) = P ni =0 P j ∈ Z r i,j u i z j . Since µ ( i ) j ( H ) > i n −
3, as in Proposition 4.10 we have that(4.4) n − X i =0 X j ∈ Z r i,j u i z j = n − X i =0 X j ∈ Z r i,j ∆HS( H • m ( S i ( − j − i ))) , with r i,j >
0. As already observed in Proposition 4.10, we have that ∆[ H • m ( S i ( − j − i ))] ∈C seq[ a,b ] ⊆ C [ a,b ] whenever r i,j = 0. We now construct the last three rows of H . By Remark 4.12,the submatrix H ′ ∈ Mat , Z ( Q ) consisting of the last three rows of H satisfies the inequali-ties given by the functionals from [DSS20, Theorem 6.2]. As a consequence of such theorem,there exist finitely generated Z -graded S -modules N , . . . , N p and positive rational numbers t , . . . , t p such that H ′ = P ps =1 t s ∆[ H • m ( N s )]. Let P ( H ′ ) = P i =0 r i + n − ,j u i z j denote the Lau-rent polynomial associated to H ′ . Passing to Hilbert series, the equation above gives that P ( H ′ ) = P ps =1 t s ∆HS( H • m ( N s )). For s = 1 , . . . p , we let N ′ s = ( N s ⊗ k k [ x , . . . , x n ]) (2 − n ), so ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 21 that by (4.1) and with a straightforward calculation we get(4.5) p X s =1 t s ∆HS( H • m ( N ′ s )) = u n − p X s =1 t s ∆HS( H • m ( N s )) = u n − P ( H ′ ) = n X i = n − X j ∈ Z r i,j u i z j . Observe that E-depth( N ′ s ) > n −
2, so that ∆[ H • m ( N ′ s )] ∈ C [ a,b ] for all s . Putting (4.4) and (4.5)together, from the point of view of tables we finally get H = n − X i =0 X j ∈ Z r i,j ∆[ H • m ( S i ( − j − i ))] + p X s =1 t s ∆[ H • m ( N ′ s )] ∈ C [ a,b ] . (cid:3) If the socle of local cohomology modules fits
Let S = k [ x , . . . , x n ] with the standard grading, and m = ( x , . . . , x n ). Let M be a Z -graded S -module. We denote by soc( M ) = ann M ( m ) the socle of M . In [KU92], Kustin and Ulrichprove that if I ⊆ J are two homogeneous ideals such that S/I is Artinian, and that satisfythe pointwise inequality HF(soc(
S/I )) HF(soc(
S/J )), then I = J . The proof can easily beadapted to submodules A ⊆ B of a fixed S -module C such that C/A is Artinian, and we includeit here for the sake of completeness.
Lemma 5.1.
Let S = k [ x , . . . , x n ], and C be a finitely generated graded S -module. Let A ⊆ B be two graded submodules of C such that C/A is Artinian. If HF(soc(
C/A )) HF(soc(
C/B )),then A = B . Proof.
Assume that the natural map soc(
C/A ) → soc( C/B ) is not injective. Let j be the largestdegree in which it is not injective. Then, because of our assumptions, the map soc( C/A ) j → soc( C/B ) j is not surjective either. This means that there exists a non-zero c ∈ soc( C/B ) j suchthat its lift c ∈ C satisfies m c A . Choose an element f ∈ S of largest positive degree withthe property that f c / ∈ A . Then f c ∈ L i>j soc( C/A ) i by construction, and such an elementmaps to zero in soc( C/B ). This contradicts our choice of j , and concludes the proof that themap soc( C/A ) → soc( C/B ) is injective. To conclude the proof, observe that there is an exactsequence 0 → soc( B/A ) → soc( C/A ) → soc( C/B ). As we proved that the rightmost map isinjective, we have that soc(
B/A ) = 0. However,
B/A has finite length, so its socle cannot bezero unless A = B . (cid:3) The following is a generalization to Kustin and Ulrich’s socle lemma to the non-Artiniancase.
Theorem 5.2.
Let S = k [ x , . . . , x n ], with the standard grading, and F be a graded free S -module. Let A ⊆ B be two graded submodules of F . Let ℓ , . . . , ℓ t be a filter regular sequencefor both F/A and
F/B consisting of linear forms such that HF(( A +( ℓ , . . . , ℓ t ) F ) sat ) = HF(( B +( ℓ , . . . , ℓ t ) F ) sat ). Furthermore, assume that HF(soc( H i m ( F/A ))) HF(soc( H i m ( F/B ))) for all0 i t . If min { E-depth(
F/A ) , E-depth(
F/B ) } > t −
1, then A = B . Proof.
We prove the result by induction on t >
0, treating the cases t = 0 and t = 1 separately.If t = 0 we have A sat = B sat by assumption. Call such module C . Moreover, observe that H m ( F/A ) = A sat /A = C/A and H m ( F/B ) = B sat /B = C/B . By assumption, we have thatHF(soc(
C/A )) HF(soc(
C/B )), hence A = B follows from Lemma 5.1. Now assume that t = 1, and set ℓ = ℓ . We observe that H i m ( F/A ) ∼ = H i m ( F/A sat ) for all i >
0. We have a graded short exact sequence0 / / F/A sat ( − · ℓ / / F/A sat / / F/ ( A sat + ℓF ) / / , which induces a graded long exact sequence on local cohomology0 / / H m ( F/ ( A sat + ℓF )) / / H m ( F/A )( − · ℓ / / H m ( F/A ) / / · · ·· · · / / H i − m ( F/ ( A sat + ℓF )) / / H i m ( F/A )( − · ℓ / / H i m ( F/A ) / / · · · Taking Hom S ( k , − ) in the first part of the sequence gives0 / / soc( H m ( F/ ( A sat + ℓF ))) / / soc( H m ( F/A ))( − · ℓ / / soc( H m ( F/A )) . Multiplication by ℓ is zero on socles, hence soc( H m ( F/ ( A sat + ℓF ))) ∼ = soc( H m ( F/A ))( − F/B . Hence we still have an inequality HF(soc( H m ( F/ ( A sat + ℓF )))) HF(soc( H m ( F/ ( B sat + ℓF )))). Let C = ( A + ℓF ) sat , which is also equal to ( B + ℓF ) sat by assumption. Finally, observe that H m ( F/ ( A sat + ℓF )) = ( A sat + ℓF ) sat / ( A sat + ℓF ) = C/ ( A sat + ℓF ). Similarly, H m ( F/ ( B sat + ℓF )) = C/ ( B sat + ℓF ). Since we have containments( A sat + ℓF ) ⊆ ( B sat + ℓF ) ⊆ C , with an appropriate inequality on socles, it follows from Lemma5.1 that A sat + ℓF = B sat + ℓF . Since we always have a containment A sat ⊆ B sat , and ℓ is anon-zero divisor on F/A sat and
F/B sat , we must have A sat = B sat . To conclude, we now argueas in the case t = 0.We now assume that t >
2. Set ℓ = ℓ . Since min { E-depth(
F/A ) , E-depth(
F/B ) } > t − > / / H i m ( F/ ( A sat + ℓF )) / / H i +1 m ( F/A )( − · ℓ / / H i +1 m ( F/A ) / / i >
0. In particular, taking Hom S ( k , − ) and using again that multiplication by ℓ iszero on socles, we get isomorphisms soc( H i m ( F/ ( A sat + ℓF ))) ∼ = soc( H i +1 m ( F/A )( − i >
0. A similar argument holds for F/ ( B sat + ℓF ), so that HF(soc( H i m ( F/ ( A sat + ℓF )))) HF(soc( H i m ( F/ ( B sat + ℓF )))) for all 0 i t − ℓ , . . . , ℓ t are still a filter regular sequence for both F/ ( A sat + ℓF ) and F/ ( B sat + ℓF ). Moreover,( A sat + ℓF + ( ℓ , . . . , ℓ t ) F ) sat = ( A + ( ℓ , . . . , ℓ t ) F ) sat = ( B + ( ℓ , . . . , ℓ t ) F ) sat = ( B sat + ℓF + ( ℓ , . . . , ℓ t ) F ) sat , and thus HF(( A sat + ℓF + ( ℓ , . . . , ℓ t ) F ) sat ) = HF(( B sat + ℓF + ( ℓ , . . . , ℓ t ) F ) sat ).Since min { E-depth( F/ ( A sat + ℓF )) , E-depth( F/ ( B sat + ℓF )) } > t − A sat + ℓF = B sat + ℓF . To conclude the proof wenow proceed as in the case t = 1. (cid:3) If both
F/A and
F/B are sequentially Cohen-Macaulay, then we only need to check thecondition on the Hilbert functions of the socles of (all) local cohomology modules.
Corollary 5.3.
Let S = k [ x , . . . , x n ], with the standard grading, and F be a graded free S -module. Let A ⊆ B be two graded submodules of F that satisfy HF(soc( H i m ( F/A ))) ECOMPOSITION OF LOCAL COHOMOLOGY TABLES AND E-DEPTH 23
HF(soc( H i m ( F/B ))) for all i ∈ Z . If F/A and
F/B are sequentially Cohen-Macaulay, then A = B . Proof.
Observe that the inequality between socles forces
F/A and
F/B to have the same Krulldimension d . Let ℓ , . . . , ℓ d be a filter regular sequence for both F/A and
F/B . Since ℓ , . . . , ℓ d is a full system of parameters for F/A , we have that ( A + ( ℓ , . . . , ℓ d ) F ) sat = F , because F/ ( A + ( ℓ , . . . , ℓ d ) F ) has finite length. A similar statement holds for B as well. The Corollarynow follows from Theorem 5.2, since the assumptions that HF(( A +( ℓ , . . . , ℓ d ) F ) sat ) = HF(( B +( ℓ , . . . , ℓ d ) F ) sat ) and min { E-depth(
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