On the notion of sequentially Cohen-Macaulay modules

In this survey paper we first present the main properties of sequentially Cohen-Macaulay modules. Some basic examples are provided to help the reader with quickly getting acquainted with this topic. We then discuss two generalizations of the notion of sequential Cohen-Macaulayness which are inspired by a theorem of J\"urgen Herzog and the third author.


Introduction
This survey note, which is dedicated to the work of Jürgen Herzog on the topic, cannot possibly be complete: the notion of sequentially Cohen-Macaulay module has been central in many papers in the literature from the late 90's, starting maybe with [20]. On the other hand, in the late 90's Herzog's research activity was feverish as he counted very many visitors and collaborators. At the same time the distribution of preprints in the form of postscript files over the internet expedited the dissemination of mathematical ideas. We thus must apologize in advance that our reference list is doomed to be incomplete.
It is in 1997 that Herzog, together with who would become his top co-author, Takayuki Hibi, published a paper on simplicial complexes [26], immediately followed by another series of papers of the two authors together with Annetta Aramova [5,6,7], where numerical problems about simplicial complexes were addressed through the study of Hilbert functions, Gröbner bases techniques and generic initial ideals, see also [3,4,19]. In 1999, another influential paper authored by Herzog and Hibi, is published, [27]: they introduce and study a new class of ideals, called componentwise linear. Componentwise linear Stanley-Reisner ideals I ∆ are characterized as those for which the pure i-th skeleton of the Alexander dual of ∆ is Cohen-Macaulay for every i. Thanks to [20], this means that I ∆ is componentwise linear if and only its Alexander dual is sequentially Cohen-Macaulay, see also [30]. In this way, the authors were also able to generalize a well-known result of Eagon and Reiner, which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay. All the ideas behind the proofs of these facts led also to another fundamental result, which is the main theorem of [31] and provides a somewhat unexpected characterization of graded sequentially Cohen-Macaulay modules over a polynomial ring R in terms of the Hilbert function of the local cohomology modules: Date: April 14, 2023. The first author was partially supported by a grant from the Simons Foundation (41000748, G.C.). The second author was partially supported by the PRIN 2020 project 2020355B8Y "Squarefree Gröbner degenerations, special varieties and related topics". The third author was partially supported by the PRIN project 2017SSNZAW004 "Moduli Theory and Birational Classification" of Italian MIUR. Theorem 1. Let M be a finitely generated graded R-module, and let M ∼ = F/U a free graded presentation of M . Then, F/U is sequentially Cohen-Macaulay if and only if Hilb(H i m (F/U )) = Hilb(H i m (F/ Gin(U ))) for all i 0. Here Gin(U ) denotes the generic initial module of U with respect to the degree reverse lexicographic order.
This paper is organized as follows. In the first section we introduce the definition of sequentially Cohen-Macaulay modules according to Stanley [37], and discuss some basic properties and examples. In Section 2 we present some of the main characterizations, or equivalent definitions, of sequential Cohen-Macaulayness, by recalling Schenzel's results on the dimension filtration of a module, cf. Propositions 2.2, 2.6 and Theorem 2.7, and Peskine's characterization in terms of deficiency modules, cf. Theorem 2.13. In Section 3, we recollect the definition of partial sequential Cohen-Macaulayness introduced by the third and the fourth author in [35], by requiring that only the queue of the dimension filtration is a Cohen-Macaulay filtration, see Definition 3.1 and some basic properties in the graded case in Lemmata 3.4 and 3.5. The next two results, Lemma 3.8 and Proposition 3.10, are dedicated to filling a gap in the proof of a fundamental lemma in [35], and we present the first of our generalizations of Theorem 1 in Theorem 3.11. We start Section 4 by recalling the definition of E-depth, as introduced by the first and second author in [12], see Definition 4.1. E-depth measures how much depth the deficiency modules of a finitely generated standard graded module M have altogether, and sequential Cohen-Macaulayness can be detected by E-depth as observed in Remark 4.2. Some interesting homological properties of E-depth, especially in connection with strictly filter regular elements, are shown in Proposition 4.4. In Definition 4.9 we introduce the other crucial notion for the following, what we might call partial generic initial ideal, making use of a special partial revlex order. Finally, by means of Proposition 4.10, we prove in Theorem 4.11 the main result of this section, which can be seen as another generalization of Theorem 1.
To Jürgen Herzog, a bright example of mathematician and teacher.

Sequentially Cohen-Macaulay modules
Throughout the paper, let (R, m, k) denote either a Noetherian local ring with maximal ideal m and residue field k = R/m, or a standard graded k-algebra R = i 0 R i , with R 0 = k and m = i>0 R i . In the second case, every R-module we consider will be a graded R-module, and homomorphisms will be graded of degree zero.
One of the features that makes the Cohen-Macaulay property significant is its characterization in terms of the vanishing and non-vanishing of local cohomology: for a d-dimensional finitely generated module M with t = depth M , it holds that H i m (M ) = 0 for all i < t and i > d; also, H t m (M ) = 0 and H d m (M ) = 0. These results are originally due to Grothendieck, cf. [10], Theorem 3.5.8, Corollary 3.5.9, Corollary 3.5.11.a) and b). As a consequence, M is a Cohen-Macaulay module if and only if H i m (M ) = 0 for all i = d. Let dim(R) = n. When R is Cohen-Macaulay and has a canonical module ω R , by duality, the above conditions can be checked on the Matlis dual of the local cohomology modules, i.e., on the modules Ext n−i R (M, ω R ). With the above notation, we then have that Ext n−t R (M, ω R ) and Ext n−d R (M, ω R ) are non-zero, and furthermore that Ext i R (M, ω R ) = 0 for all i < n−d and all i > n − t. Moreover, M is Cohen-Macaulay if and only if Ext n−i R (M, ω R ) = 0 for all i = d. Often in the literature the modules Ext n−i R (M, ω R ), i = 0, . . . , n, are called the deficiency modules of M , since they also measure how far a module is from being Cohen-Macaulay.
We now introduce the class of sequentially Cohen-Macaulay modules, following [37], recalling their main properties, and in the next section we discuss some of its equivalent definitions. Our main references here are [31] and [36].
In this case, we say that the above filtration is a sCM filtration for M .
The notion of sequentially Cohen-Macaulay modules appears for the first time in the literature in [37] in the graded setting, in connection with the theory of Stanley-Reisner rings and simplicial complexes. Later on and independently, in [36], the notion of Cohen-Macaulay filtered modules has been introduced in the local case; in the same paper, it is proven that the two notions coincide. Since then, sequentially Cohen-Macaulay modules have been extensively studied especially in connection with shellability and graph theory, see for instance [40] [25] [2], [22], [1], and the definition has been extended in other directions, see [14], [33]. Section 3 and 4 are devoted to two such extensions, due to the third and fourth author and the first two authors respectively. (1) Let N be another sequentially Cohen-Macaulay; then M ⊕ N is sequentially Cohen-Macaulay. To see this, let 0 = N 0 N 1 . . . N s = N be a sCM filtration of N and let δ i = dim(N i /N i−1 ). For convenience, also let d 0 = δ 0 = −1. Then, a filtration whose terms are of the form . . , r} and j < i, the modules M i and M i /M j are sequentially Cohen-Macaulay.
Proof. For Part (1), using the short exact sequences 0 The induced long exact sequences on local cohomology Part (2) . On the contrary, every 0 = x ∈ R is regular, R/(x) is a 1-dimensional R-module and, therefore, always sequentially Cohen-Macaulay by Example 1.2 (2).
In the rest of this section, let S = k[x 1 , . . . , x n ], with the standard grading. Given a monomial ideal I ⊂ S, we let G(I) denote the monomial minimal system of generators of I and m(I) = max{i | x i divides u for some u ∈ G(I)}.
An important class of sequentially Cohen-Macaulay modules is given by rings defined by weakly stable ideals. Definition 1.7. A monomial ideal I ⊆ S is said to be weakly stable if for all monomials u ∈ I and all integers i, j with 1 j < i n, there exists t ∈ N such that x t j u/x i ∈ I, where is the largest integer such that x i divides u.
Observe that the condition of the previous definition is verified if and only if it is verified for all u ∈ G(I). In the literature weakly stable ideals are also called ideals of Borel type, quasi-stable ideals, or ideals of nested type, see [11], [28], [8]. It is easy too see from the definition that stable, strongly stable, and p-Borel ideals are weakly stable. In particular, no matter what the characteristic of the field is, Borel fixed ideals are weakly stable, see also [28,Theorem 4.2.10] and, thus, generic initial ideals are always weakly stable.
Observe that the saturation I sat of a weakly stable ideal I is I : m ∞ = I : x ∞ n , see [28,Proposition 4.2.9]; thus, I sat is again weakly stable and x n does not divide any u ∈ G(I sat ). Proof. We prove the statement by induction on m(I). If m(I) = 1, then I = (x r 1 ) for some positive integer r and S/I is Cohen-Macaulay. Assume now that S/J is sequentially Cohen-Macaulay for every weakly stable ideal J for which m(J) < m(I).
Let S = k[x 1 , . . . , x m(I) ] and I = I ∩ S . Since S/I ∼ = S /I ⊗ k k[x m(I)+1 , . . . , x n ] and a sCM filtration of S /I is easily extended into one of S/I, it is sufficient to prove that S /I is sequentially Cohen-Macaulay. Therefore, without loss of generality, we may assume that m(I) = n, and that S/I sat is sequentially Cohen-Macaulay. Now, I sat /I is a nontrivial Artinian module, and the first non-zero module of a sCM filtration of S/I sat has positive dimension, see Proposition 1.5. We may thus conclude that S/I is sequentially Cohen-Macaulay, cf. Example 1.2 (4). Remark 1.9. By [8,Proposition 3.2], see also [28,Proposition 4.2.9] and [11, Chapter 4]) a monomial ideal I is weakly stable if and only if all its associated primes are generated by initial segments of variables, i.e. are of type (x 1 , . . . , x i ) for some i. Since being sequentially Cohen-Macaulay is independent of coordinates changes on S, the above proposition shows that whenever the associated primes of a monomial ideal I ⊆ S are totally ordered by inclusion, then S/I is sequentially Cohen-Macaulay.
It is easy to verify that I is a weakly stable ideal which is not strongly stable, by checking the definition or by computing its associated primes, by using the previous remark; thus, S/I is sequentially Cohen-Macaulay by Proposition 1.8. We can construct an explicit sCM filtration proceeding as in its proof. We have m(I) = 4, and I sat = I : . This means that the first non-zero module of our filtration will be (x 4 1 , x 2 1 x 2 , x 3 1 x 3 )/I. Now we consider I = (x 4 1 , x 2 1 x 2 , x 3 1 x 3 ) as an ideal of k[x 1 , x 2 , x 3 ] and compute its saturation I = I : x ∞ 3 = (x 3 1 , x 2 1 x 2 ). Hence, the second non-zero module of the filtration is (x 3 1 , x 2 1 x 2 )/I. Proceeding in this way we obtain the filtration 0 ⊆ (x 4 1 , x 2 1 x 2 , x 3 1 x 3 )/I ⊆ (x 3 1 , x 2 1 x 2 )/I ⊆ (x 2 1 )/I ⊆ (1)/I = S/I, and it is easily seen that it is a sCM filtration of S/I. Example 1.11. For the case M = R = S/I, when I = I ∆ is the Stanley-Reisner ideal of a simplicial complex ∆, there is a beautiful characterization of sequential Cohen-Macaulayness due to Duval, [20,Theorem 3.3]. Given a simplicial complex ∆, let ∆(i) be the pure i-th skeleton of ∆, i.e. the pure subcomplex of ∆ whose facets are the faces of ∆ of dimension i.

Characterizations of sequentially Cohen-Macaulay modules
The goal of this section is to present two characterizations of sequentially Cohen-Macaulay modules, due to Schenzel, see Theorem 2.7 and Peskine, see Theorem 2.13. As in the previous section, we let (R, m, k) be either a Noetherian local ring or a standard graded k-algebra with maximal homogeneous ideal m; we let dim(R) = n. In the second case, modules will be graded and homomorphisms homogeneous of degree 0.

Schenzel's characterization.
Our main reference here is [36]. By convention, the dimension of the zero module is set to be −1. Let M be a finitely generated (graded) Rmodule of dimension d. Given a set X of prime ideals of R, we denote by X i = {p ∈ X : dim R/p = i}. Similarly, we define X i and X >i .
otherwise. Now let us consider an irredundant primary decomposition 0 = m j=1 N j inside M , where each M/N j is a p j -primary module.
. Let M be a finitely generated R-module of dimension d. With the above notation, for all i = 0, . . . , d we have that where we let the intersection over the empty set be equal to M .
Proof. We start with the first equality and assume that δ i (M ) = 0. By Remark 2.1 we have that Ass(M ) i = ∅ and in this case . Now assume that δ i (M ) = 0 and observe that, every associated prime of δ i (M ) has dimension i and, therefore, Ass(δ i (M )) ⊆ Ass(M ) i . Thus, there is a power of a i which annihilates δ i (M ), and we get that forcing equality everywhere. Next, assume that p j / ∈ Ass(M ) i . Then, p ⊆ p j for every p ∈ Ass(M ) i since, otherwise, we would have dim(R/p j ) dim(R/p) i and, thus, p j ∈ Ass(M ) i . In this case, by choosing Note that this is consistent with our convention that the intersection over the empty set is equal to M ; in fact, for i = d  (1), and the second equality follows.
Finally, consider the short exact sequence 0 ). Thus, by (1) and (2) we have necessarily that Ass( As a corollary of Propositions 2.2 and 2.3, one can obtain another characterization of the dimension filtration, cf. [29, Proposition 1.1]. Notice that from the definition it immediately follows that if δ i (M )/δ i−1 (M ) = 0, then it has dimension i.
A Cohen-Macaulay filtered module M is sequentially Cohen-Macaulay. In fact, we may let i 1 denote the smallest integer such that δ i 1 (M ) = 0 and set To see that the notions of sequentially Cohen-Macaulay and Cohen-Macaulay filtered modules coincide, we need the following result, which shows that if a CM filtration of a module exists, then it is unique and it is equal to its dimension filtration.
Proof. We proceed by induction on d. If d = 0, then C 0 = M = δ 0 (M ) and this case is complete. Assume henceforth that d > 0, so that by induction we have that Observe that Ass(C 0 ) ⊆ Ass(M ) 0 and, for all p ∈ Ass( With this information, we can inductively show that dim C i i and, accordingly p; then, by prime avoidance we can find p ∈ Ass(M/C d−1 ) such that a ⊆ p and, therefore, p contains a prime p ∈ Ass(M ) d−1 . This is a contradiction, Proof. We only need to show that the "only if" part. Let . In this way we have constructed a filtration C : which is a CM filtration of M . By Proposition 2.6 we may conclude that C i = δ i (M ), and that C is the unique dimension filtration of M . Thus, M is Cohen-Macaulay filtered.
Observe that one can reconstruct M from C; it follows that the modules M i only depend on M as well, and M is also unique. Proof. Observe that, given any i ∈ {0, . . . , d}, we have that δ j (M ) = δ j (δ i (M )) for every

2.2.
Peskine's characterization. We will always assume that R has a canonical module ω R . In our setup, this assumption is not too restrictive: it is always the case when R is standard graded, and it is true for instance if R is complete local, cf. Remark 1.4.
Proof. In the graded case, (1) and (2) (1) and (2) for the completion M of M as an R-module, see Example 1.3 (2). Since for any finitely generated R-module N , we conclude by faithful flatness of R that (1) and (2) hold also in the local case.
Finally, by [10, Theorem 3.3.10 (c) (iii)] we have that N ∼ = Ext n−d R (Ext n−d R (N, ω R ), ω R ) for any Cohen-Macaulay module N of dimension d and, thus, the last statement follows immediately from (1).
Remark 2.11. Let M be as in the above proposition.
(1) By Parts (2) and (3) of the previous proposition, we have an isomorphism , where t = depth M and M 1 is t-dimensional and Cohen-Macaulay. We will show an extension of this fact in Lemma 2.12.
(2) Notice that if M is sequentially Cohen-Macaulay with depth(M ) = 0, then in particular In fact, using [10, Theorem 3.3.10 (c) (iii)] and the short exact sequence . We recall now the following crucial lemma, see [31, Lemma 1.5].
Lemma 2.12. Let R be a Cohen-Macaulay n-dimensional ring with canonical module ω R and M be a finitely generated R-module.
Let also depth(M ) = t and assume that Ext n−t R (M, ω R ) is Cohen-Macaulay of dimension t. Then, there is a natural monomorphism α : is an isomorphism.
Proof. We only give a proof in the graded case; the local one is handled similarly. Without loss of generality we may assume that R is a polynomial ring, as we show next. We write R = S/I, where S is a standard graded polynomial ring of dimension m over a field k and I ⊆ S is homogeneous. Let m and n denote the graded maximal ideals of R and S respectively. By graded Local Duality [10, Theorem 3.
). In particular, we have shown that Ext n−t R (M, ω R ) ∼ = Ext m−t S (M, ω S ) and that we may assume that R is a polynomial ring.
be the minimal graded free resolution of M ; applying the functor Hom R (−, ω R ) we obtain the dual complex Observe that F * • is a complex of free modules, since ω R ∼ = R(−n). If we also let G • denote the minimal graded free resolution of Ext n−t R (M, ω R ), then there is a map of complexes φ • : F * • −→ G • which lifts the identity map of Ext n−t R (M, ω R ). Since the last one is Cohen-Macaulay of dimension t, the length of G • is the same as that of F * • , namely n − t. Applying the functor Hom R (−, ω R ) to φ • , we obtain a map of complexes φ * • : G * • −→ F * * • which, in turn, gives a map on the zero-th cohomology: , which coincides with the map Ext n−t (α). The canonical isomorphism ψ : G * * • −→G • , together with the fact that Ext n−t R (M, ω R ) is Cohen-Macaulay of dimension t, gives an isomorphism

and one can verify that
. Note that H 0 (ψ) is the inverse of the isomorphism of [10, Theorem 3.3.10 (c) (iii)], and this implies that H 0 (φ * * • ) = Ext n−t (α) is the natural isomorphism of the same theorem. To conclude the proof, it remains to be shown that α is a monomorphism. Let N = Ext n−t R (Ext n−t R (M, ω R ), ω R ). We show that α is injective once we localize at every associated prime of N and then we are done, since, if ker(α) = 0, its associated primes would be contained in those of N . Let p ∈ Ass(N ); since N is Cohen-Macaulay of dimension t, we have that dim(R/p) = t and dim(R p ) = n − t. By replacing R with R p , M with M p , and ω R with (ω R ) p ∼ = ω Rp , the proof will be complete once we show that α is injective in the case t = 0. To this end observe that, The equivalence between the first two conditions in the next theorem was announced in [37] without a proof, but citing a spectral sequence argument due to Peskine. Here we give another proof of this fact, see also [36,Theorem 5.5] where the equivalence with the third condition is proved.
By Lemma 2.12 we know that the map Ext n−t (α) is an isomorphism, and therefore β is an isomorphism as well. It follows that Ext j For the converse, it suffices to observe that if the direct sum of two modules is zero or Cohen-Macaulay of a given dimension, then so is each of its summand.
Remark 2.15. We remark that sequential Cohen-Macaulayness behaves well with respect to localization; see for instance [16,Proposition 4.7] If R is Cohen-Macaulay, the fact that M p is sequentially Cohen-Macaulay for all p ∈ Supp(M ) is also a consequence of Theorem 2.13. See [39] for other results about localization and sequentially Cohen-Macaulay modules.
We now recall some definitions needed to state the next result, and that we will use frequently in the next sections. Given an R-module N , we let Ass • (N ) = Ass(N ) {m}.  We conclude this section with the following result, which clarifies how sequential Cohen-Macaulayness behaves with respect to quotients, cf. Example 1.6; it will also be useful later on.
By Theorem 2.13 we have that each non-zero Ext n−i R (M, ω R ) is Cohen-Macaulay of dimension i, and in this case x is Ext n−i R (M, ω R )-regular when i > 0. For i > 1 we then have short exact sequences and it follows that Ext is Cohen-Macaulay of dimension i − 1 for all i > 1. We conclude that M/xM is sequentially Cohen-Macaulay using the implication (3) ⇒ (1) of Theorem 2.13.
For the converse, the fact that x is regular for all non-zero Ext n−i R (M, ω R ) with i > 0 shows that the above long exact sequence of Ext modules breaks into short exact sequences Moreover, in [15] it is investigated how the sequential Cohen-Macaulay property behaves in relation to taking associated graded rings and Rees algebras, see also [38] for more results of this type.

Partially sequentially Cohen-Macaulay modules
We are going to study next the notion of partially sequentially Cohen-Macaulay module, as introduced in [35], which naturally generalizes that of sequentially Cohen-Macaulay modules. Thanks to Schenzel's Theorem 2.7, the definition can be given in terms of the dimension filtration of the module. Throughout this section we let R = k[x 1 , . . . , x n ] be a standard graded polynomial ring over an infinite field k with homogeneous maximal ideal m = (x 1 , . . . , x n ). Recall that, in this case, R has a graded canonical module ω R ∼ = R(−n). We consider finitely generated graded R-modules M ; when M = 0, we set depth(M ) = +∞ and dim(M ) = −1, as usual. We let d = dim(M ).
With the help of Proposition 2.2, we can construct the dimension filtration 0 = δ −1 = δ 0 = δ 1 ⊆ δ 2 = ((x 1 ) ∩ (x 2 , x 3 ))/I ⊆ δ 3 = (x 1 )/I ⊆ δ 4 = R/I of M . Then, δ 4 /δ 3 and δ 3 /δ 2 are Cohen-Macaulay of dimension 4 and 3 respectively, but 0 = δ 2 /δ 1 is not Cohen-Macaulay. Hence, M is an example of a 3-sCM which is not 2-sCM. Given a graded free presentation of M ∼ = F/U , we denote by {e 1 , . . . , e r } a graded basis of F . We consider R together with the pure reverse lexicographic ordering > such that x 1 > . . . > x n ; recall that > is not a monomial order on R, but by definition it agrees with the reverse lexicographic order that refines it on monomials of the same degree.
We extend > to F in the following way: given monomials ue i and ve j of F , set We shall consider this order until the end of the section, and denote by Gin(U ) the generic initial module of U with respect to >. Since the action of GL n (k) on R as change of coordinates can be extended in an obvious way to F , Gin(U ) simply results to be the initial submodule in > (gU ) where g is a general change of coordinates.
We prove next some preliminary facts which are needed later on. Given a graded submodule V ⊆ F with dim(F/V ) = d, for all j ∈ {−1, . . . , d} we denote by V j the R-module such that V j /V = δ j (F/V ). Several results contained in the next two lemmata can be found in [22], where they are proved in the ideal case.
Lemma 3.4. With the above notation, Proof. (1) Notice that Gin(U j )/ Gin(U ) and U j /U have the same Hilbert series, hence the same dimension, which is less than or equal to j. Since Gin(U ) j / Gin(U ) = δ j (F/ Gin(U )), we have the desired inclusion.
(2) Since U ⊆ U j , one inclusion is clear. Now consider the short exact sequence Since the dimensions of U j /U and (U j ) j /U j are less than or equal to j, it follows that also dim((U j ) j /U ) j and, hence, (4) Since U ⊆ U j , it immediately follows from (1) that Gin(U ) ⊆ Gin(U j ) ⊆ Gin(U ) j and, accordingly, Gin(U ) j ⊆ Gin(U j ) j . On the other hand, by Parts (1) and (2), the latter is contained in (Gin(U ) j ) j = Gin(U ) j .
Finally, to prove (4) we show that Hilb(U j /U j−1 ) = Hilb(Gin(U ) j /(Gin(U ) j−1 )) holds for all j i. Actually, we prove more, i.e. that Hilb(U j ) = Hilb(Gin(U ) j ) for all j i; since Gin(U j ) ⊆ Gin(U ) j by Lemma 3.4 (1) for all j, the last equality is equivalent to proving that Gin(U j ) = Gin(U ) j for all j i, and this is what we do.
Consider now, for all j, the short exact sequences 0 → U j /U j−1 → F/U j−1 → F/U j → 0; we see inductively that depth(F/U j ) j + 1 for all j i. For j = d and if U j /U j−1 = 0 this is obvious; otherwise, since M is i-sCM, U j /U j−1 is jdimensional Cohen-Macaulay for all j i and by [10, Proposition 1.2.9] we get that depth F/U j−1 min{j, j + 1} = j. For all graded submodules V ⊆ F it is well-known that depth(F/ Gin(V )) = depth(F/V ) and that F/ Gin(V ) is sequentially Cohen-Macaulay. Thus, j + 1 depth(F/ Gin(U j )) by what we proved above, and Proposition 1.5 together with Theorem 2.7 imply that the latter is also equal to the smallest integer t such that Gin(U j ) Gin(U j ) t . Therefore, we have shown that Gin(U j ) = Gin(U j ) j for all j i. Now the conclusion follows from Lemma 3.4 (4). In the following remark we collect two known facts which are useful in the following.
Remark 3.7. Recall that in our setting R ∼ = ω R (n). It is a well-known fact that for a d-dimensional graded R-module M it holds that dim Ext n−i R (M, ω R ) i for all i.  )) and, thus, dim(Ext n−j R (M, R)) j, contradiction. The converse is analogous, observing that if Ext n−j R (M, R) has dimension at least j and, hence, necessarily equal to j, then it must have a prime of height n − j in its support.
The following lemma can be regarded as an enhanced graded version of [17,Proposition 4.16].
Lemma 3.8. Let M be a finitely generated graded R-module. For a sufficiently general homogeneous element x ∈ m and for all j 0 there is a short exact sequence 0 → δ j (δ(M )/xδ(M )) → δ j+1 (M/xM ) → L j → 0, where L j is a module of finite length.
Proof. Let d = dim(M ). The case d 1 is trivial, therefore we will assume that d 2 and proceed by induction on j 0. Now suppose that the statement of the lemma is proved for j − 1, so that we have a short It is clear from the definition of δ that there is an exact sequence 0 −→ δ j (δ(M )/xδ(M )) ϕ j −→ U −→ L j −→ 0, where we let U = δ j+1 (M/xM ) and L j = coker(ϕ j ). If U has finite length we are done, so let us assume that dim(U ) > 0. In this case we necessarily have that h = dim(δ j (M/xM )) > 0, and since L j−1 has finite length we conclude that dim(δ(M )/xδ(M )) = h. Again because L j−1 has finite length, we can find p 0 such that m p U ⊆ δ j−1 (δ(M )/xδ(M )). Moreover, dim(m p U ) dim(U ) < h, and therefore m p U is contained in δ j (δ(M )/xδ(M )). This shows that m p L j = 0, and thus L j has finite length.
In [35] it is claimed that, if x ∈ R is M -regular, it is possible to prove that M is i-sCM if and only if M/xM is (i − 1)-sCM following the same lines of the proof [36,Theorem 4.7], which is not utterly correct, as we already pointed out in Example 1.6. The claim is indeed false: if x is M -regular and M/xM is (i − 1)-sCM, then M is not necessarily i-sCM, as the following example shows. Example 3.9. Let R be a 2-dimensional domain which is not Cohen-Macaulay, cf. Example 1.6. For all 0 = x ∈ R we have that R/(x) is 0-sCM, but R is not even 2-sCM.
In the following proposition we show that the claimed result holds true under some additional assumption. where L is a module of finite length. Now, either dim(δ j (M/xM )) = dim(δ j (M )/xδ j (M )) = dim(δ j (M )) − 1, or dim(δ j (M/xM )) 0, and in both cases we have that dim(δ j (M/xM )) i − 2. By minimality of j, we have that dim(δ j−1 (M )) > i − 1 and applying iteratively Lemma 3.8 as we did above, we also obtain dim(δ j−1 (M/xM )) = dim(δ j−1 (M )) − 1 > i − 2; thus, we may conclude that Since L has finite length, the associated long exact sequence of Ext-modules yields that We may assume that x, which is N -regular, is also Ext n− R (N, R)-regular for all i − 1, for we proved above that all these modules have positive depth. For  (N, R), which implies that Ext n−1 R (N, R) is Cohen-Macaulay of dimension one by Remark 3.7. Applying Peskine's Theorem 2.9, we have thus showed that N is sequentially Cohen-Macaulay, that is, M is i-sCM.
The next theorem provides a characterization of partially sequentially Cohen-Macaulay modules. It was proved for the first time in [35,Theorem 3.5] in the ideal case. Here, we generalize the result to finitely generated modules and fix the gap in the original proof thanks to Proposition 3.10. We let R [n−1] = k[x 1 , . . . , x n−1 ] ∼ = R/x n R and denote by N [n−1] the R [n−1] -module N/x n N N ⊗ R R/x n R by restriction of scalars. We let x ∈ R be a general linear form which, without loss of generality, we may write as l = a 1 x 1 + · · · + a n−1 x n−1 − x n and consider the map g n : R → R [n−1] , defined by x i → x i for i = 1, . . . , n − 1 and x n → a 1 x 1 + · · · + a n−1 x n−1 . Then, the surjective homomorphism F/U → F [n−1] /g n (U ) has kernel (U + xF )/U and induces the isomorphism (1) F/U is i-sCM; (2) h j (F/U ) = h j (F/ Gin(U )) for all i j d. Proof.
(1) ⇒ (2) is a direct consequence of Lemma 3.5 (3) and (4), since also F/ Gin(U ) is i-sCM. Consider now a sufficiently general linear form x ∈ R. For all j, there are exact sequences and C (j) , and these imply that h j−1 (F/(U +xF )) = (z−1)h j (F/U )+ Hilb(B (j) ) + Hilb(C (j) ) for all j. By (3.1) and (3.2), we obtain Thus, from our hypothesis it follows that the above inequalities are equalities for all j i, that means that Hilb(B (j) ) = Hilb(C (j) ) = 0, i.e. B (j) = C (j) = 0 for j i. Moreover, since C (i−1) = B (i) = 0, it follows that x is regular for all non-zero Ext n−j R (F/U, ω R ) with j i − 1, which thus have positive depth. From the above equalities we also get that h j (F [n−1] /g n (U )) = h j (F [n−1] / Gin(g n (U ))) for all j i − 1; by induction, this implies that F/(U + xF ) ∼ = F [n−1] /g n (U ) is (i − 1)-sCM. The conclusion follows now by a straightforward application of Proposition 3.10.
As a corollary, we immediately obtain Theorem 1. One can wonder whether the equality h i (F/U ) = h i (F/ Gin(U )) is enough to imply that F/U is i-sCM; however, this is not the case. . If we write M 1 ∼ = F 1 /U 1 and M 2 ∼ = F 2 /U 2 , where F 1 and F 2 are graded free R-modules and U 1 , U 2 are graded submodules, then M 1 ⊕ M 2 ∼ = F/U where F = F 1 ⊕ F 2 and U = U 1 ⊕ U 2 , and it follows that Gin(U ) = Gin(U 1 )⊕Gin(U 2 ). Since M 1 is Cohen-Macaulay, hence sequentially Cohen-Macaulay, and depth(F 2 / Gin(U 2 )) = depth(M 2 ) > i we therefore conclude by Theorem 3.12 that h i (F/U ) = h i (F 1 /U 1 ) = h i (F 1 / Gin(U 1 )) = h i (F/ Gin(U )).
(2) The following is another explicit example of such instance in the ideal case.
On the other hand, if I lex is the lexicographic ideal associated with I, then the equality h i (R/I) = h i (R/I lex ) ensures the i-partial sequential Cohen-Macaulayness of R/I. Notice that this is a stronger condition, though, since h i (R/I) h i (R/ Gin(I)) h i (R/I lex ), coefficientwise, see [34,Theorems 2.4 and 5.4]. Actually, in [35,Theorem 4.4] the following result is proved.
Theorem 3.14. Let i be a positive integer and I a homogeneous ideal of R; then, the following conditions are equivalent: If any of the above holds, then I is i-sCM.
We conclude this section by observing that the conditions in the previous theorem are still equivalent if we replace Gin(I) with Gin 0 (I), the zero-generic initial ideal of I introduced in [13]. We also remark that the equivalence between conditions (2) and (3) is not true if we replace the Hilbert series of local cohomology modules with graded Betti numbers, see [32, Theorem 3.1].

E-depth
As in the previous section, k will denote an infinite field, R = k[x 1 , . . . , x n ] a standard graded polynomial ring and m = (x 1 , . . . , x n ) its homogeneous maximal ideal. As before, when M = 0 we let dim(M ) = −1 and depth(M ) = ∞. We start with the main definition of this section.   Our assumption that E-depth(M ) > 0 guarantees that depth(Ext i R (M, R)) > 0 for all i < n. Since is strictly filter regular for M , the multiplication by is injective on Ext i R (M, R) for all i < n, and the long exact sequence breaks into short exact sequences, as claimed, and the graded short exact sequences of local cohomology modules are obtained by graded Local Duality. We now introduce a special grading on a polynomial ring which refines the standard grading, and that can be further refined to the monomial Z n -grading.
Definition 4.5. Let r be a positive integer, A be a Z-graded ring and S = A[y 1 , . . . , y r ] a polynomial ring over A. For i ∈ {0, . . . , r} let η i ∈ Z r+1 be the vector whose (i + 1)-st entry is 1, and all other entries equal 0. We consider S as a Z × Z r -graded ring by letting deg S (a) = deg A (a) · η 0 , for all a ∈ A and deg S (y i ) = η i , for i ∈ {1, . . . , r}.
By means of the previous definition, we may consider R = k[x 1 , . . . , x n ] as a Z × Z r -graded ring for any 0 r n − 1 by letting deg R (x i ) = η 0 for all 1 i n − r and deg R (x i ) = η i for all n − r + 1 i n. Observe that an element f ∈ R is graded with respect to such grading if and only if f can be written as f = f · u, where f ∈ k[x 1 , . . . , x n−r ] is homogeneous with respect to the standard grading, and u is a monomial in k[x n−r+1 , . . . , x n ]. In particular, when r = 0 this is just the standard grading on R, while for r = n − 1 it coincides with the monomial Z n -grading.
Remark 4.6. We can extend in a natural way such a grading to free R-modules F with a basis by assigning degrees to the elements of the given basis. Accordingly, any R-module M is Z × Z r -graded if and only if M ∼ = F/U , where F is a free Z × Z r -graded R-module and U is a Z × Z r -graded submodule of F . We see next that the grading just introduced is very relevant to the purpose of estimating the E-depth of a module; the following can be seen as a refinement of Proposition 1.8.
Proposition 4.7. Let R = k[x 1 , . . . , x n ], and M be a finitely generated Z × Z r -graded Rmodule such that x n , . . . , x n−r+1 is a filter regular sequence for M . Then, x n , . . . , x n−r+1 is a strictly filter regular sequence for M , and E-depth(M ) r.
Proof. Write M as M = F/U , where F is a finitely generated Z × Z r -graded free R-module, and U is a Z×Z r -graded submodule of F . Since U is Z×Z r -graded, x n is Z×Z r -homogeneous and filter regular, U sat = U : F x ∞ n is also Z×Z r graded and, accordingly M/H 0 m (M ) ∼ = F/U sat is too. If F/U sat = 0, then M has dimension zero and, thus, is sequentially Cohen-Macaulay; then E-depth(M ) = n t, and every sequence of non-zero elements of (x 1 , . . . , x n ) is a strictly filter regular sequence for M . In particular, x n , . . . , x n−r+1 is a strictly filter regular sequence for M . Now suppose that F/U sat = 0. By assumption, x n is filter regular for M , and thus regular for F/U sat . Since the latter is Z × Z r -graded, we have that F/U sat ∼ = F /U ⊗ k k[x n ] for some Z × Z r−1 -graded R = k[x 1 , . . . , x n−1 ]-module F , and some Z × Z r−1 -graded submodule U of F such that F /U can be identified with the hyperplane section F/U sat ⊗ R R/(x n ). Thus, for i < n we have . It follows that x n is a non-zero divisor on Ext i R (M, R) for all i < n, and thus E-depth(M ) > 0. Since Ext n R (F/U, R) has finite length it also follows that x n is a strictly filter regular element for M . Now we can consider F /U , and an iteration of this argument will imply the desired conclusion.
We now introduce a weight order on R. Given integers 0 r n, consider the following r × n matrix , and let ω i be its i-th row; then, this induces a "partial" revlex order rev r on R by declaring that a monomial x a = x a 1 1 · · · x an n is greater than a monomial x b = x b 1 1 · · · x bn n if and only if there exists 1 j r such that ω i · a = ω i · b for all i j and ω j+1 · a > ω j+1 · b.
Remark 4.8. Observe that rev n coincides with the usual revlex order on R = k[x 1 , . . . , x n ].
Give a graded free R-module F with basis {e 1 , . . . , e s }, we can write any f ∈ F uniquely as a finite sum f = u j e i j , where u j are monomials and we assume the sum has minimal support. We let the initial form in revr (f ) of f written as above be the sum of those u j e i j for which u j is maximal with respect to the order rev r introduced above. Definition 4.9. Let F be a finitely generated graded free R-module, and U be a graded submodule; let also M = F/U . We say that the r-partial general initial submodule of U satisfies a given property P if there exists a non-empty Zariski open set L of r-uples of linear forms such that F/in revr (g (U )) satisfies P for any = ( n−r+1 , . . . , n ) ∈ L, where g is the automorphism on F induced by the change of coordinates of R which sends i → x i and fixes the other variables. With some abuse of notation, we denote any partial initial submodule in revr (g (U )) which satisfies P the r-partial general submodule of U , and denote it by gin r (U ).
Proposition 4.10. Let F be a finitely generated graded free R-module, and U ⊆ F be a graded submodule. For every 0 r n we have that E-depth(F/ gin r (U )) r.
Proof. For a sufficiently general change of coordinates we have that x n , . . . , x n−r+1 form a filter regular sequence for F/(g (U )). Since in revr (U ) : F x n = in revr (U : F x n ), see [21, 15.7], we have that x n , . . . , x n−r+1 also form a filter regular sequence for F/in revr (g (U )), and hence for F/ gin r (U ). By construction, the module F/ gin r (U ) is Z × Z r -graded, and the claim now follows from Proposition 4.7.
Theorem 4.11. Let F be a finitely generated graded free R-module, and U ⊆ F be a graded submodule. For every 0 r n we have that E-depth(F/U ) r if and only if h i (F/U ) = h i (F/ gin r (U )) for all i ∈ N.
Proof. After performing a sufficiently general change of coordinates, we may assume that V = in revr (U ) has the same properties as gin revr (U ), and that x n , . . . , x n−r+1 form a strictly filter regular sequence for F/U . By using again that in revr (U : F x n ) = in revr (U ) : F x n , and because strictly filter regular sequences are filter regular by Remark 2.17, we have that x n , . . . , x n−r+1 form a filter regular sequence for F/V . Since V is Z × Z r -graded, it follows from Proposition 4.7 that x n , . . . , x n−r+1 is a strictly filter regular sequence for F/V . By [21, 15.7] we also have that V sat = in revr (U ) : F x ∞ n = in revr (U : F x ∞ n ) = in revr (U sat ), and V sat +x n F = in revr (U sat ) + x n F = in revr (U sat +x n F ). Viewing F/(U sat +x n F ) as a quotient of a free S = k[x 1 , . . . , x n−1 ]-module F by a graded submodule U , we see that V sat +x n F can be identified with a submodule V ⊆ F , with V = in rev r−1 (U ) and V has the same properties of gin rev t−1 (U ), see [12,Lemma 3.4] and the proof of [12, Theorem 3.6] for more details.