Serre depth and local cohomology

A Cohen-Macaulay ring is a Noetherian local ring R in which every system of parameters forms a regular sequence. Denote by depth(R) the depth of R. Then depth(R) is always less than or equal to dim(R), and equality holds if and only if R is Cohen-Macaulay. Thus, the depth of R measures how far the ring is from being Cohen-Macaulay. Similarly, one can define Cohen-Macaulay modules.

There is a natural refinement of the Cohen-Macaulay property. We say that a R-module satisfies Serre’s condition (S r ) if depth R P (M P ) ≥ min{r, dim R P (M P )}, for all primes P belonging to the support Supp R (M ).

It is easily seen that M satisfies (S 1 ) if and only if it does not have embedded associated primes. Furthermore, M is Cohen-Macaulay if and only if it satisfies (S r ) for all r ≥ 1. Serre’s conditions (S r ) stratify Noetherian local rings, provide a flexible generalization of the Cohen-Macaulay property, and play a crucial role in the study of singularities and homological behaviour of finitely generated modules.

Recently, inspired by results of Schenzel (see [35,36,37]), Muta and Terai introduced in [30] (see also [24,31,32]) the notion of Serre depth of a finitely generated graded module M in the polynomial ring setting as a measure of the deviation of M from satisfying Serre’s condition (S r ). The authors extensively studied the Serre depth of squarefree monomial ideals and their symbolic powers.

In this paper, our goal is to extend the notion of Serre depth in the setting of arbitrary Noetherian local rings or standard graded algebras over a field.

Let (R, m) be a Noetherian local ring with residue class field R/m ∼ = K. Let r ≥ 1 be a positive integer, and let M be a R-module. Let E R (K) be an injective hull of K, and for a R-module N we denote by D R (N ) = Hom R (N, E R (K)) the Matlis dual of N . Then, the r-th Serre depth of M is defined as S r -depth R (M ) = min{j : dim D R (H j m (M )) ≥ j -r + 1} if M ̸ = 0; otherwise we set S r -depth R (0) = -∞. Similar definitions can be made in the graded setting. In Proposition 1.4 we prove that:

(a) dim(M )

If M is finitely generated, then S r -depth(M ) = depth(M ), for all positive integers r ≥ dim(M ). Statement (b) indicates that the Serre depth plays a role analogous to the usual depth. By (a), S r -depth(M ) ≤ dim(M ). In Theorem 3.6, we prove that if equality holds, then M is unmixed, hence equidimensional, and satisfies (S r ).

Our main result is:

Theorem A. Let (R, m) be a Noetherian local ring or a standard graded K-algebra.

Let M be a finitely generated R-module, which we assume is homogeneous if R is a K-algebra. Consider the following statements.

(a) M is equidimensional and satisfies Serre’s condition (S r ). (b) S r -depth(M ) = dim(M ).

Then (b) ⇒ (a), and the converse implication holds if R is a homomorphic image of a Gorenstein ring. Furthermore, in (a) the condition that M is equidimensional can be dropped if r ≥ 2, R is a homomorphic image of a Gorenstein ring, and the module M is either indecomposable or a quotient of R.

This result highlights the role of Serre depths as a natural substitute of the usual depth, in measuring deviation from Serre’s conditions.

Theorem A substantially improves several scattered results in the literature. An earlier version of this result appeared in [35,Lemma (2.1)], where Schenzel proved that (a) and (b) are equivalent only when M = R = S/I is a homomorphic image of a Gorenstein ring S. This version also appears (without proof) in the book of Vasconcelos [43,Proposition 3.51] under the extra unnecessary assumption that R is equidimensional, and was recently proved explicitly by Dao, Ma and Varbaro in [9,Proposition 2.11], but only in the polynomial ring case.

Subsequently, Schenzel extended this result to modules (see [36,Lemma 3.2.1] and [37,Lemma 1.9]), again under the assumption that R = S/I, and the extra unnecessary assumption that M is equidimensional.

We emphasize that implication (b) ⇒ (a) was previously unknown for general rings. Unfortunately, in general the converse is false, as we show in Example 3.7. On the other hand, the class of rings that are homomorphic images of a Gorenstein ring is quite large. It includes all complete Noetherian local rings (by Cohen’s structure theorem [29,Theorem 29.4(ii)]), all Cohen-Macaulay rings having a canonical module (see [4,Theorem 3.3.6]) and all standard graded K-algebras. Thus, for all these rings, statements (a) and (b) in Theorem A are equivalent.

We now present a geometric reformulation of Theorem A.

Let X be a Noetherian affine local scheme. That is, X = Spec(R) is the spectrum of a Noetherian local ring R. Let F be a coherent sheaf on X. Recall that for each x ∈ X (which corresponds to a prime ideal P ∈ Spec(R)