OPEN LOCI OF GRADED MODULES

CHRISTEL ROTTHAUS AND LIANA M. ŞEGA

Abstract. Let A = ⊕_{i∈ℕ}A_{i} be an excellent homogeneous Noetherian graded ring and let M = ⊕_{n∈ℤ}M_{n} be a finitely generated graded A-module. We consider M as a module over A_{0} and show that the (S_{k})-loci of M are open in Spec(A_{0}). In particular, the Cohen-Macaulay locus U_{CM}^{0} = {𝔭 ∈ Spec(A_{0}) | M_{𝔭} is Cohen-Macaulay} is an open subset of Spec(A_{0}). We also show that the (S_{k})-loci on the homogeneous parts M_{n} of M are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module M over an excellent ring A and for an ideal I ⊆ A which is not contained in any minimal prime of M the (S_{k})-loci for the modules M/I^{n}M are eventually stable.

introduction

A well-known theorem of Grothendieck states that if M is a finitely generated module over an excellent Noetherian ring A then for all k ∈ ℕ the (S_{k})-locus of M:

U_{S_{k}}(M) = {𝔭 ∈ Spec(A) | M_{𝔭} satisfies (S_{k})}

is an open subset of Spec(A). As usual, (S_{k}) denotes the Serre condition, that is, M_{𝔭} satisfies (S_{k}) if for all 𝔮 ∈ Spec(A) with 𝔮 ⊆ 𝔭 it holds that:

depth_{A_{𝔮}}(M_{𝔮}) ≥ min(k, dim(M_{𝔮})).

It also follows that for such modules M the Cohen-Macaulay locus:

U_{CM}(M) = {𝔭 ∈ Spec(A) | M_{𝔭} is Cohen-Macaulay}

is an open subset of Spec(A).

Let A = ⨁_{n≥0} A_{n} be Noetherian graded excellent homogeneous ring and M = ⨁_{i∈ℤ} M_{i} a finitely generated graded A-module. Considered as a module over the base ring A_{0}, M is a direct sum of finitely generated A_{0}-modules. Moreover, if the base ring A_{0} is local the standard notion of depth is meaningful for the A_{0}-module M and we may consider its (S_{k})-loci:

U_{S_{k}}^{0}(M) = {𝔭 ∈ Spec(A_{0}) | M_{𝔭} satisfies S_{k}}

where M_{𝔭} denotes the localization of M at the multiplicative set A_{0} ∖ 𝔭. In this paper we prove that under these assumptions the (S_{k})-loci of the A_{0}-module M are open subsets of Spec(A_{0}). In particular, the Cohen-Macaulay locus of M (as an A_{0}-module):

U_{CM}^{0}(M) = {𝔭 ∈ Spec(A_{0}) | M_{𝔭} is Cohen-Macaulay}

is an open subset of Spec(A_{0}).

The proof follows the main ideas of Grothendieck’s proof. It is, however, not merely a copy of the proof in EGA and requires a number of modifications. For the benefit of the reader we have included complete proofs of the results. Our proof is based on the following two observations: First, if A is a polynomial ring over the base ring A_{0}, then every graded resolution of M by finitely generated graded free Amodules provides a free resolution of the A_{0}-module M which is finitely generated on the homogeneous parts. The second is a result by Hochster and Roberts which states for the A-module M that there is an element a ∈ A_{0} ∖ (0) so that M_{a} is a free (A_{0})_{a}-module provided that the ring A_{0} is a domain.

Date: September 28, 2004.

The paper is organized as follows:

The first section contains basic facts about graded rings and modules which are relevant for the rest of the paper. As a main result we obtain that the Auslander-Buchsbaum formula holds for the A_{0}-module M.

The second section shows that the codepth-loci of M are open in Spec(A_{0}). This is the main step in proving the openness of the (S_{k})-loci which we present in the next section.

In Section 4 we consider the homogeneous parts of the graded module M. We show that the codepth-loci and (S_{k})-loci of the homogeneous parts of M are eventually stable. This is applied in the last section to the case of a finitely generated module M over an excellent Noetherian ring A. If I ⊆ A is an ideal we recover a well-known result by Kodiyalam [7], namely that for k ≥ k_{0}:

depth(M/I^{k}M) = depth(M/I^{k_{0}}M).

We also show that if M is a Cohen-Macaulay module over A and if I ⊆ A is not contained in a minimal prime of M, then the codepth- and (S_{k})-loci of M/I^{n}M are eventually stable.

1. basic facts

In this paper we assume that A = ⨁_{i∈ℕ} A_{i} is a Noetherian homogeneous graded ring and that M = ⨁_{i∈ℤ} M_{i} is a finitely generated A-module. As usual, we let A_{+} denote the irrelevant ideal of A, that is, A_{+} = ⨁_{i≥1} A_{i}.

If 𝔭 ∈ Spec(A_{0}) is a prime ideal of A_{0}, then M_{𝔭} denotes the localization S^{−1}M where S = A_{0} ∖ 𝔭. Note that M_{𝔭} is a graded module over the graded ring A_{𝔭}.

Our goal is to show that if A is excellent then the codepth-loci and the (S_{k})-loci of M, considered as a module over the base ring A_{0}, are open subsets of Spec(A_{0}).

1.1. General remarks. We begin our investigation with some well known facts about graded modules. Since these results are frequently used throughout the paper we include them together with their (short) proofs in this introductory section. 1.1.1. Lemma. There exists an integer t so that ann_{A_{0}}(M_{t}) = ann_{A_{0}}(M_{k}) for all k ≥ t.

Proof. For all k ∈ ℤ set J_{k} = ann_{A_{0}}(M_{k}) . Since A is homogeneous and M is a finitely generated A-module, there exists s_{0} ∈ ℤ such that

A_{1}M_{k} = M_{k+1} for all k ≥ t_{0} .

We conclude J_{k} ⊆ J_{k+1} for all k ≥ t_{0}. Since A_{0} is Noetherian, there exists then s ≥ t_{0} so that J_{k} = J_{t} for all k ≥ s. □

1.1.2. Lemma. The following two functions are well-defined and surjective: (1) The function φ: Supp_{A}(M) → Supp_{A_{0}}(M) defined by φ(P) = P ∩ A_{0}.

(2) The function ψ: Ass_{A}(M) → Ass_{A_{0}}(M) defined by ψ(P) = P ∩ A_{0}.

Proof. (1) If P ∈ Supp_{A}(M), then M_{P} ≠ 0 and in particular M_{𝔭} ≠ 0, where 𝔭 = P ∩ A_{0}. This shows that φ is well defined. Let 𝔭 ∈ Supp_{A_{0}}(M), then

M_{𝔭} = ⨁_{i∈ℤ}(M_{i})_{𝔭} ≠ 0

and we may consider M_{𝔭} as a graded module over the graded ring A_{𝔭}. Note that A_{𝔭} is a ^{∗}local ring with unique graded maximal ideal 𝔪 = 𝔭(A_{0})_{𝔭} ⨁(A_{+})_{𝔭}. Since all minimal primes of Supp_{A_{𝔭}}(M_{𝔭}) are graded, 𝔪 ∈ Supp_{A_{𝔭}}(M_{𝔭}). Thus there is a prime P ∈ Supp_{A}(M) with P ∩ A_{0} = 𝔭.

(2) If P ∈ Ass_{A}(M), then there exists y ∈ M so that ann_{A}(y) = P. Thus ann_{A_{0}}(y) = P ∩A_{0} = 𝔭 and 𝔭 ∈ Ass_{A_{0}}(M). Conversely, let 𝔭 ∈ Ass_{A_{0}}(M). Consider again the graded A_{𝔭}-module M_{𝔭}. There exists z ∈ M_{𝔭} so that ann_{(A_{0})_{𝔭}}(z) = 𝔭(A_{0})_{𝔭}, and therefore

𝔭(A_{0})_{𝔭} ⊆ ⋃_{Q∈Ass_{A𝔭} (M_{𝔭})} Q

Since M_{𝔭} is a finitely generated A_{𝔭}-module, there exists Q ∈ Ass_{A_{𝔭}}(M_{𝔭}) with 𝔭(A_{0})_{𝔭} ⊆ Q. Since A_{𝔭} is ^{∗}local with unique graded maximal ideal 𝔭(A_{0})_{𝔭} ⊕ (A_{+})_{𝔭} we obtain Q ∩ (A_{0})_{𝔭} = 𝔭(A_{0})_{𝔭} and a preimage P ∈ Spec(A) of Q is an associated prime of the A-module M, with P ∩ A_{0} = 𝔭. □

Lemma 1.1.2 shows in particular that M as an A_{0}-module has a finite set of associated primes. 1.1.3. Lemma. Let A and M be as above and set I = ann_{A_{0}}(M). For any 𝔭 ∈ Spec(A_{0}) the following hold:

(1) If M_{𝔭} = 0, then there is an element a ∈ A_{0} ∖ 𝔭 with M_{a} = 0.

(2) ann_{(A_{0})_{𝔭}}(M_{𝔭}) = I(A_{0})_{𝔭}

Proof. (1) This is a basic fact about Noetherian modules using that M is a finitely generated module over A and A_{0} ∖ 𝔭 is a multiplicative subset of A.

(2) Obviously, I(A_{0})_{𝔭} ⊆ ann_{(A_{0})_{𝔭}}(M_{𝔭}). Let x ∈ ann_{(A_{0})_{𝔭}}(M_{𝔭}) with x = \frac{b}{s} where b ∈ A_{0} and s ∈ A_{0} ∖ 𝔭. Assume that m_{1}, …, m_{r} is a system of generators of the A-module M. Since x\frac{m_{i}}{1} = 0 for all 1 ≤ i ≤ r there is an element t ∈ A_{0} ∖ 𝔭 with tbm_{i} = 0 for all 1 ≤ i ≤ r. We have that tb ∈ I and hence x = \frac{b}{s} ∈ I(A_{0})_{𝔭}. □

1.2. The Auslander-Buchsbaum formula. Let A = ⨁_{i≥0} A_{i} be a graded Noetherian homogeneous ring with (A_{0}, 𝔪_{0}) local and let M = ⨁_{i∈ℤ} M_{i} be a finitely generated A-module. Since M is (in general) not finitely generated as an A_{0}- module, we need to verify that the classical definition of A_{0}-depth works in the case of a finitely generated graded module. First note that an element z ∈ 𝔪_{0} is regular on M if and only if z is regular on M_{i} for all i ∈ ℤ with M_{i} ≠ 0. Let x_{1}, …, x_{s} ∈ 𝔪_{0} and y_{1}, …, y_{t} ∈ 𝔪_{0} be two maximal regular M-sequences (as an A_{0}-module). Then for all i ∈ ℤ with M_{i} ≠ 0 the two sequences are regular on the A_{0}-module M_{i} and the sets

Ass_{A_{0}}(M/(x_{1}, …, x_{s})M) = ⋃_{i∈ℤ} Ass_{A_{0}}(M_{i}/(x_{1}, …, x_{s})M_{i})

Ass_{A_{0}}(M/(y_{1}, …, y_{t})M) = ⋃_{i∈ℤ} Ass_{A_{0}}(M_{i}/(y_{1}, …, y_{t})M_{i})

are finite by Lemma 1.1.2. The maximality of the first sequences yields that there is an i ∈ ℤ with M_{i} ≠ 0 and 𝔪_{0} ∈ Ass_{A_{0}}(M_{i}/(x_{1}, …, x_{s})M_{i}). Since the second sequence is also regular on M_{i} we have that t ≤ s. A similar argument shows that s ≤ t and we obtain that two maximal regular sequences on M have the same length. Therefore the classical definition of depth is efficient and we put: 1.2.1. Definition. Let A and M be as above with (A_{0}, 𝔪_{0}) local. We define the depth of M as an A_{0}-module to be the number:

depth_{A_{0}}(M) := sup{n ∈ ℕ | ∃ an M-sequence of lenght n} .

In general, for a (not necessarily finitely generated) module M over a Noetherian local ring A, the depth of M is defined by means of Koszul homology (see [2, Definition 9.1.1]). In our setting, the definition above coincides with the one in [2].

The aim of this section is to prove the Auslander-Buchsbaum theorem for finitely generated graded modules M over ^{∗}local graded Noetherian rings A when M is considered a module over the base ring A_{0}. There is a generalized version of the Auslander-Buchsbaum theorem which applies to our case (see [3, (12.2)] or [6, Theorem (2.1)]). For the convenience of the reader we include a proof of this theorem in the graded case, which only makes use of the classical definition of depth as given above. 1.2.2. Lemma. Let A and M be as above and assume that (A_{0}, 𝔪_{0}) is local. Then:

(1) dim_{A_{0}}(M) = sup{dim_{A_{0}}(M_{i}) | i ∈ ℤ}

(2) depth_{A_{0}}(M) = inf{depth_{A_{0}}(M_{i}) | i ∈ ℤ with M_{i} ≠ 0}

(3) projdim_{A_{0}}(M) = sup{projdim_{A_{0}}(M_{i}) | i ∈ ℤ}

Proof. (1) By Lemma 1.1.1 there is an integer s ∈ ℤ so that ann_{A_{0}}(M_{k}) = ann_{A_{0}}(M_{s}) for all k ≥ s. In particular, for all k ≥ s: dim_{A_{0}}(M_{k}) = dim_{A_{0}}(M_{s}) and

dim_{A_{0}}(M) = dim_{A_{0}}(M_{r} ⊕ M_{r−1} ⊕ … ⊕ M_{s−1} ⊕ M_{s})

where r ∈ ℤ is the smallest integer j with M_{j} ≠ 0. The dimension of a finite direct sum of A_{0}-modules is the maximum of the dimensions of its summands.

(2) If r_{1}, …, r_{s} ∈ A_{0} is a regular sequence on M, then r_{1}, …, r_{s} is a regular sequence on M_{i} for all i ∈ ℤ with M_{i} ≠ 0. Thus depth_{A_{0}}(M) ≤ depth_{A_{0}}(M_{i}) for all i ∈ ℤ with M_{i} ≠ 0 and hence

depth_{A_{0}}(M) ≤ inf{depth_{A_{0}}(M_{i}) | i ∈ ℤ with M_{i} ≠ 0} .

In order to show the other inequality we proceed by induction on t = depth_{A_{0}}(M). Note that by Lemma 1.1.3, Ass_{A_{0}}(M) is a finite set. If t = 0 then 𝔪_{0} ∈ Ass_{A_{0}}(M) and there is an i ∈ ℤ so that 𝔪_{0} ∈ Ass_{A_{0}}(M_{i}).

Thus:

inf{depth_{A_{0}}(M_{i}) | i ∈ ℤ with M_{i} ≠ 0} = 0 .

Now assume that t = depth_{A_{0}}(M) > 0. This implies that

⋃_{𝔭∈Ass_{A_{0}} (M)} 𝔭 ≠ 𝔪_{0} .

Consider an element

r ∈ 𝔪_{0} ∖ ⋃_{𝔭∈Ass_{A_{0}} (M)} 𝔭

Since r is regular on M, and therefore is regular on M_{i} for all i ∈ ℤ with M_{i} ≠ 0, we obtain

depth_{A_{0}}(M/rM) = depth_{A_{0}}(M) − 1

and for all i ∈ ℤ with M_{i} ≠ 0:

depth_{A_{0}}(M_{i}/rM_{i}) = depth_{A_{0}}(M_{i}) − 1 .

By induction hypothesis:

depth_{A_{0}}(M/rM) = inf{depth_{A_{0}}(M_{i}/rM_{i}) | i ∈ ℤ and M_{i}/rM_{i} ≠ 0} .

The assertion follows.

(3) For all i ∈ ℤ let F_{•}^{(i)} be a finite free resolution of M_{i}. Then

F_{•} = ⨁_{i∈ℤ} F_{•}^{(i)}

is a free resolution of the A_{0}-module M yielding:

projdim_{A_{0}}(M) ≤ sup{projdim_{A_{0}}(M_{i}) | i ∈ ℤ} .

In order to show the other inequality, assume that projdim_{A_{0}}(M) = r and consider for all i ∈ ℤ the rth-syzygy T_{r}^{(i)} of M_{i} and the exact sequence:

0 ⟶ T_{r}^{(i)} ⟶ F_{r−1}^{(i)} ⟶ … ⟶ F_{0}^{(i)} ⟶ M_{i} ⟶ 0 .

By taking direct sums we see that

⨁_{i∈ℤ} T_{r}^{(i)}

is an rth-syzygy of M and thus projective. Therefore every T_{r}^{(i)} is a projective finitely generated A_{0}-module. Since A_{0} is a local Noetherian ring every, T_{r}^{(i)} is a free A_{0}-module and thus for all i ∈ ℤ:

projdim_{A_{0}}(M_{i}) ≤ r .

This shows (3). □

1.2.3. Proposition. Let A and M be as above with (A_{0}, 𝔪_{0}) a local ring. Then the Auslander-Buchsbaum formula holds for M as an A_{0}-module. That is, if projdim_{A_{0}}(M) is finite, then:

depth_{A_{0}}(M) + projdim_{A_{0}}(M) = depth(A_{0}) .

Proof. Let projdim_{A_{0}}(M) = r < ∞, then by Lemma 1.2.2(2) there is an i ∈ ℤ with projdim_{A_{0}}(M) = projdim_{A_{0}}(M_{i}) and for all j ∈ ℤ:

projdim_{A_{0}}(M_{j}) ≤ r.

The Auslander-Buchsbaum formula holds for finitely generated A_{0}-modules:

depth_{A_{0}}(M_{j}) + projdim_{A_{0}}(M_{j}) = depth_{A_{0}}(A_{0}) for all j ∈ ℤ

and therefore:

depth_{A_{0}}(M_{j}) ≥ depth_{A_{0}}(M_{i}) for all j ∈ ℤ

Using Lemma 1.2.2(1), we conclude depth_{A_{0}}(M) = depth_{A_{0}}(M_{i}). The Auslander-Buchsbaum formula for M_{i} gives then the desired formula. □

2. Openness of the codepth locus

Throughout this section we assume that A = ⨁_{i∈ℕ_{0}} A_{i} is a graded Noetherian homogeneous ring and that M = ⨁_{i∈ℤ} M_{i} is a finitely generated A-module. Our aim is to generalize and/or modify existing theorems for finitely generated modules over Noetherian rings to the graded case where the module M is considered a module over the base ring A_{0}. We begin with a result on the flat locus of the A_{0}-module M.

2.1. The flat locus of M. Our first result is a modification of [8, Theorem 24.3]. The proof follows the proof in Matsumura’s book. A key observation is that for a finitely generated graded module M the localizations M_{𝔭} are I-adically separated for every ideal I ⊆ (A_{0})_{𝔭}. Proposition. Let A and M be as above. The flat locus of M as an A_{0}-module:

U^{0}(M) = {𝔭 ∈ Spec(A_{0}) | M_{𝔭} is flat over A_{0}}

is open in Spec(A_{0}).

Proof. According to Nagata’s criterion on the openness of loci [8, Theorem 24.2] we have to show:

(a) If 𝔭, 𝔮 ∈ Spec(A_{0}) with 𝔭 ∈ U^{0}(M) and 𝔮 ⊆ 𝔭 then 𝔮 ∈ U^{0}(M).

(b) If 𝔭 ∈ U^{0}(M) then U^{0}(M) contains a nonempty open subset of V^{0}(𝔭) = {𝔫 ∈ Spec(A_{0}) | 𝔭 ⊆ 𝔫}.

(a) is trivial. Let 𝔭 ∈ U^{0}(M), that is, assume that M_{𝔭} is flat over A_{0}. Set Ā_{0} = A_{0}/𝔭. By [8, Theorem 22.3] for every 𝔮 ∈ V^{0}(𝔭) the module M_{𝔮} is flat over A_{0} if and only if (M/𝔭M)_{𝔮} is flat over Ā_{0} and Tor_{1}^{A_{0}}(M_{𝔮}, Ā_{0}) = 0. A similar argument as in the proof of [8, Theorem 23.2] shows that Tor_{1}^{A_{0}}(M, Ā_{0}) is a finitely generated module over A. Therefore there is an element a ∈ A_{0}∖𝔭 so that (Tor_{1}^{A_{0}}(M, Ā_{0}))_{a} = 0. By applying [8, Theorem 24.1] to the Ā_{0}-module M/𝔭M we obtain an element b ∈ A_{0}∖𝔭 so that (M/𝔭M)_{b} is a free (Ā_{0})_{b}-module. Set D_{ab}^{0} = {𝔮 ∈ Spec(A_{0}) | ab ∉ 𝔮}, then for all 𝔮 ∈ V^{0}(𝔭)∩D_{ab}^{0} we have that Tor_{1}^{A_{0}}(M_{𝔮}, Ā_{0}) = 0 and that (M/𝔭M)_{𝔮} is flat over (Ā_{0})_{𝔮}. Thus by [8, Theorem 22.3] the module M_{𝔮} is flat over (A_{0})_{𝔮} and M_{𝔮} is flat over A_{0}. □

2.2. A proposition by Auslander. As before let A be a Noetherian graded homogeneous ring and let M be a finitely generated A-module. The following Proposition is an extension of a proposition in EGA [4, (6.11.1) and (6.11.2)] to the (not finitely generated) A_{0}-module M. Proposition. The function γ : Spec(A_{0}) ⟶ ℕ defined by

γ(𝔭) = projdim_{(A_{0})_{𝔭}}(M_{𝔭}) for all 𝔭 ∈ Spec(A_{0})

is upper semicontinuous. That is, for all n ∈ ℕ the set

U_{n}^{0}(M) = {𝔭 ∈ Spec(A_{0}) | projdim_{(A_{0})_{𝔭}}(M_{𝔭}) ≤ n}

is open in Spec(A_{0}).

Proof. Note that the ring A is the homomorphic image of the polynomial ring B = A_{0}[x_{1}, …, x_{t}], and that, with the standard grading on the polynomial ring B, the graded B-module M is finitely generated. We may replace A by B and assume that A is a graded polynomial ring over A_{0}. Let 𝔭 ∈ Spec(A_{0}) with projdim_{(A_{0})_{𝔭}}(M_{𝔭}) ≤ n.

Consider a graded finitely generated free resolution of the A-module M:

F_{n} ⟶^{φ_{n}} F_{n−1} ⟶^{φ_{n−1}} ⋯ ⟶^{φ_{1}} F_{1} ⟶^{φ_{0}} M → 0

where the F_{i} are finitely generated graded free A-modules and the φ_{i} are homogeneous A-linear maps. Let T be the nth syzygy of M, yielding an exact sequence of graded A-modules:

(^{∗}) 0 → T ⟶^{δ} F_{n−1} ⟶^{φ_{n−1}} … ⟶^{φ_{1}} F_{1} ⟶^{φ_{0}} M → 0 .

Since all the homogeneous parts of F_{i} are free A_{0}-modules and since T is a graded A-module we obtain for all k ∈ ℤ an exact sequence of A_{0}-modules:

0 → T_{k} ⟶^{(δ)_{k}} (F_{n−1})_{k} ⟶^{(φ_{n−1})_{k}} … ⟶^{(φ_{1})_{k}} (F_{1})_{k} ⟶^{(φ_{0})_{k}} M_{k} → 0

with (F_{i})_{k} a finitely generated free A_{0}-module. Therefore by considering (^{∗}) as an exact sequence of A_{0}-modules we obtain that every module F_{i} is free over A_{0} and T is an nth syzygy of the A_{0}-module M. Localization at 𝔭 yields exact sequences:

0 → T_{𝔭} ⟶^{δ_{𝔭}} (F_{n−1})_{𝔭} ⟶^{(φ_{n−1})_{𝔭}} … ⟶^{(φ_{1})_{𝔭}} (F_{1})_{𝔭} ⟶^{(φ_{0})_{𝔭}} M_{𝔭} → 0 .

Since projdim_{(A_{0})p}(M_{𝔭}) ≤ n it follows that T_{𝔭} is a projective (A_{0})_{𝔭}-module. Therefore T_{𝔭} is a free (A_{0})_{𝔭}-module. Since T is a finitely generated graded A-module it follows from Proposition 2.1 that the set

U^{0}(T) = {𝔮 ∈ Spec(A_{0}) | T_{𝔮} is a flat over (A_{0})_{𝔮}}

is an open subset of Spec(A_{0}). Since T is a finitely generated graded A-module:

T = ⨁_{i∈ℤ} T_{i}

we have for 𝔮 ∈ Spec(A_{0})

T_{𝔮} = ⨁_{i∈ℤ}(T_{i})_{𝔮} .

If T_{𝔮} is flat over (A_{0})_{𝔮} then, by [1, chapter 1, §2.3, Proposition 2], for all i ∈ ℤ, (T_{i})_{𝔮} is flat over (A_{0})_{𝔮}. Since every (T_{i})_{𝔮} is a finitely generated (A_{0})_{𝔮}-module, each (T_{i})_{𝔮} is a free (A_{0})_{𝔮}-module and

U^{0}(T) = {𝔮 ∈ Spec(A_{0}) | T_{𝔮} is a free over (A_{0})_{𝔮}} .

This shows that 𝔭 ∈ U^{0}(T) and

U^{0}(T) ⊆ {𝔮 ∈ Spec(A_{0}) | projdim_{(A_{0})_{𝔮}}(M_{𝔮}) ≤ n} .

The set {𝔮 ∈ Spec(A_{0}) | projdim_{(A_{0})_{𝔮}}(M_{𝔮}) ≤ n} is thus open in Spec(A_{0}). □

2.3. A dimension formula.

Proposition. Let A and M be as above. Assume that A_{0} is catenary and let 𝔭 be a prime ideal in A_{0} with 𝔭 ∈ Supp_{A_{0}}(M). Then there is an open subset U in Spec(A_{0}) such that 𝔭 ∈ U and for all 𝔮 ∈ U ∩ V^{0}(𝔭) we have:

dim(M_{𝔮}) = dim(M_{𝔭}) + dim((A_{0}/𝔭)_{𝔮}) .

Proof. Set S = A_{0}/ ann_{A_{0}}(M) and choose an element a ∈ S∖𝔭 so that the following equality on the set of minimal primes holds:

Min(S_{𝔭}) = Min(S_{a}) .

Assume that dim(M_{𝔭}) = ht(𝔭S) = t and choose elements y_{1}, y_{2}, …, y_{t} ∈ S so that:

y_{1} not in a minimal prime of S_{𝔭}

y_{2} not in a minimal prime of y_{1}S_{𝔭}

…
y_{t} not in a minimal prime of (y_{1}, …, y_{t−1})S_{𝔭}.

Then there is an element b ∈ S ∖ 𝔭 so that:

y_{1} not in a minimal prime of S_{b}

y_{2} not in a minimal prime of y_{1}S_{b}

…
y_{t} not in a minimal prime of (y_{1}, …, y_{t−1})S_{b} .

Let a, b also denote preimages of a and b in A_{0} and put U = D_{ab} = {𝔮 ∈ Spec(A_{0}) | ab ∉ 𝔮}. Then for every 𝔮 ∈ U ∩V^{0}(𝔭) the elements y_{1}, …, y_{t} extend to a system of parameters of S_{𝔮}. Since S_{𝔭} and S_{𝔮} have the same set of minimal primes and since S is catenary we obtain that:

dim(S_{𝔮}) = dim(S_{𝔭}) + dim((S/𝔭)_{𝔮}) .

This is the same as:

dim(M_{𝔮}) = dim(M_{𝔭}) + dim((A_{0}/𝔭)_{𝔮}) .  □

2.4. The special case of A_{0} regular. Let (R, 𝔪) be a local Noetherian ring and M an R-module. Then we define:

codepth_{R}(M) := dim_{R}(M) − depth_{R}(M) .

As usual the depth of the zero module is defined to be ∞, and the dimension of the zero module is −∞, implying that the codepth of the zero module is −∞.

The following proposition extends a result by Auslander [4, (6.11.2)] to the graded case. Proposition. Let A and M be as above and assume that A_{0} is a homomorphic image of a regular ring. The function φ: Spec(A_{0}) ⟶ ℕ defined by

φ(𝔭) = codepth_{(A_{0})_{𝔭}}(M_{𝔭}) for all 𝔭 ∈ Spec(A_{0})

is upper semicontinuous, that is, for all n ∈ ℕ, the set

U_{C_{n}}^{0}(M) = {𝔭 ∈ Spec(A_{0}) | codepth_{(A_{0})_{𝔭}}(M_{𝔭}) ≤ n}

is open in Spec(A_{0}).

Proof. If A_{0} is a homomorphic image of a regular ring R_{0}, then the dimension and the depth of the R_{0}-module M are identical to the dimension and depth of M considered as an R_{0}-module. If we show that the set

Ũ_{C_{n}}^{0}(M) = {𝔮 ∈ Spec(R_{0}) | codepth_{(R_{0})_{𝔮}}(M_{𝔮}) ≤ n}

is open in Spec(R_{0}) (where M is considered a R_{0}-module), then the corresponding set for the A_{0}-module M is given by

U_{C_{n}}^{0}(M) = Ũ_{C_{n}}^{0}(M) ∩ V (J)

where A_{0} = R_{0}/J. Thus we may assume that A_{0} is a regular ring. We may also assume that A is a polynomial ring over A_{0} equipped with the standard grading.

Let 𝔭 ∈ Spec(A_{0}). By Proposition 1.2.3, the Auslander-Buchsbaum formula holds:

depth_{(A_{0})_{𝔭}}(M_{𝔭}) = depth((A_{0})_{𝔭}) − projdim_{(A_{0})_{𝔭}}(M_{𝔭}) .

Let I = ann_{A_{0}}(M). By Lemma 1.1.3, I_{𝔭} = ann_{(A_{0})_{𝔭}}(M_{𝔭}) and we have that:

dim_{(A_{0})_{𝔭}}(M_{𝔭}) = dim((A_{0})_{𝔭}) − ht(I(A_{0})_{𝔭}) .

Suppose that 𝔭 ∈ Spec(A_{0}) is such that

codepth_{(A_{0})_{𝔭}}(M_{𝔭}) ≤ n .

If M_{𝔭} = 0 then 𝔭 ⊉ I. Take an element a ∈ I ∩ (A_{0} ∖ 𝔭). Then for all

𝔮 ∈ D_{a} = {𝔴 ∈ Spec(A_{0}) | a ∉ 𝔴}

we have that M_{𝔮} = 0 and codepth_{(A_{0})_{𝔮}}(M_{𝔮}) = −∞ ≤ n.

If M_{𝔭} ≠ 0 pick an element a_{1} ∈ A_{0} ∖ 𝔭 so that (A_{0})_{𝔭} and (A_{0})_{a_{1}} have the same minimal primes and put U_{1} = D_{a_{1}} = {𝔴 ∈ Spec(A_{0}) | a_{1} ∉ 𝔴}. Then for all 𝔮 ∈ U_{1} ∩ V^{0}(I):

ht(I(A_{0})_{𝔮}) ≥ ht(I(A_{0})_{𝔭}) .

Let projdim_{(A_{0})_{𝔭}}(M_{𝔭}) = t, then by Proposition 2.2 there is an open subset U_{2} in Spec(A_{0}) so that

projdim_{(A_{0})_{𝔮}}(M_{𝔮}) ≤ t for all 𝔮 ∈ U_{2} .

Using the Auslander-Buchsbaum formula and the fact that A_{0} is regular we obtain for all 𝔮 ∈ U_{2} ∩ U_{1} ∩ V^{0}(I):

codepth_{(A_{0})_{𝔮}}(M_{𝔮}) = dim_{(A_{0})_{𝔮}}(M_{𝔮}) − depth_{(A_{0})_{𝔮}}(M_{𝔮})

= dim((A_{0})_{𝔮}) − ht(I(A_{0})_{𝔮}) − dim((A_{0})_{𝔮}) + projdim_{(A_{0})_{𝔮}}(M_{𝔮})

= projdim_{(A_{0})_{𝔮}}(M_{𝔮}) − ht(I(A_{0})_{𝔮}).

This implies that for all 𝔮 ∈ U = U_{1} ∩ U_{2}:

codepth_{(A_{0})_{𝔮}}(M_{𝔮}) ≤ codepth_{(A_{0})_{𝔭}}(M_{𝔭})

and it follows that U_{C_{n}}^{0}(M) is an open subset of Spec(A_{0}). □

2.5. A local formula. Using the fact that a complete local Noetherian ring is the homomorphic image of a regular local ring, we obtain a result similar to [4, (6.11.5)]: Lemma. Let A be a Noetherian graded homogeneous ring and let M be a finitely generated graded A-module. Then for all prime ideals 𝔭, 𝔮 ∈ Spec(A_{0}) with 𝔭 ⊆ 𝔮 we have that:

codepth_{(A_{0})_{𝔮}}(M_{𝔮}) ≥ codepth_{(A_{0})_{𝔭}}(M_{𝔭}) .

Proof. By replacing A_{0} by (A_{0})_{𝔮} (and A by A_{𝔮}) we may assume that (A_{0}, 𝔪_{0}) is a local ring. Then we have to show:

codepth_{A_{0}}(M) ≥ codepth_{(A_{0})_{𝔭}}(M_{𝔭}) .

Let 𝔭̂ ∈ Spec(Â_{0}) be a minimal prime ideal over 𝔭Â_{0}. Then 𝔭̂ ∩ A_{0} = 𝔭 and (Â_{0})𝔭̂ is flat over (A_{0})_{𝔭} with trivial special fiber. Moreover:

M_{𝔭} ⊗_{(A_{0})_{𝔭}} (Â_{0})𝔭̂ = (⨁_{i∈ℤ}(M_{i})_{𝔭}) ⊗_{(A_{0})_{𝔭}} (Â_{0})𝔭̂
= ⨁_{i∈ℤ}((M_{i})_{𝔭} ⊗_{(A_{0})_{𝔭}} (Â_{0})𝔭̂)
≅ ⨁_{i∈ℤ}(M̂_{i})𝔭̂

where M̂_{i} ≅ M_{i} ⊗_{A_{0}} Â_{0}. We have that:

depth_{A_{0}}(M) = inf{depth_{A_{0}}(M_{i}) | M_{i} ≠ 0}
dim_{A_{0}}(M) = sup{dim_{A_{0}}(M_{i}) | i ∈ ℤ}

By [8, Theorem 23.3], for all i ∈ ℤ:

depth_{(Â_{0})𝔭̂}((M̂_{i})𝔭̂) = depth_{(A_{0})_{𝔭}}((M_{i})_{𝔭}) + depth((Â_{0})𝔭̂/𝔭(Â_{0})𝔭̂)
= depth_{(A_{0})_{𝔭}}((M_{i})_{𝔭})

and by [8, Theorem 15.1]:

dim_{(Â_{0})𝔭̂}((M̂_{i})𝔭̂) = dim_{(A_{0})_{𝔭}}((M_{i})_{𝔭}) + dim((Â_{0})𝔭̂/𝔭(Â_{0})𝔭̂)
= dim_{(A_{0})_{𝔭}}((M_{i})_{𝔭})

Let

M̃ := ⨁_{i∈ℤ}M̂_{i} ≅ M ⊗_{A_{0}} Â_{0}

and note that M̃ is a finitely generated graded module over the Noetherian homogeneous graded ring

Ã:= A ⊗_{A_{0}} Â_{0} .

The computation above shows that

codepth_{(Â_{0})𝔭̂}(M̃ 𝔭̂) = codepth_{(A_{0})_{𝔭}}(M_{𝔭}) =: n .

Since Â_{0} is a homomorphic image of a regular local ring, by Proposition 2.3 the set U_{C_{n−1}}^{0}(M̃) is open in Spec(Â_{0}). This implies that

codepthÂ_{0}(M̃) ≥ codepth_{(Â_{0})𝔭̂}(M̃ 𝔭̂) .

The same argument as above shows that

codepthÂ_{0}(M̃) = codepth_{A_{0}}(M)

which proves the claim:

codepth_{A_{0}}(M) ≥ codepth_{(A_{0})_{𝔭}}(M_{𝔭}) .  □

2.6. Formulas for depth and codepth. In this section we make the same assumption as at the beginning, namely, A is a positively graded Noetherian homogeneous ring and M is a finitely generated graded A-module. The following proposition is the graded version of [4, (6.10.6)]: 2.6.1. Proposition. Let A and M be as above and assume that A is excellent. Then for every 𝔭 ∈ Spec(A_{0}) there is an open subset U^{0} ⊆ Spec(A_{0}) with 𝔭 ∈ U^{0} so that for all 𝔮 ∈ U^{0} ∩ V^{0}(𝔭):

depth_{(A_{0})_{𝔮}}(M_{𝔮}) = depth_{(A_{0})_{𝔭}}(M_{𝔭}) + depth((A_{0})_{𝔮}/𝔭(A_{0})_{𝔮}) .

Proof. Let 𝔭 ∈ Spec(A_{0}), then by Lemma 2.5 for all 𝔮 ∈ V^{0}(𝔭):

codepth_{(A_{0})_{𝔮}}(M_{𝔮}) ≥ codepth_{(A_{0})_{𝔭}}(M_{𝔭}),

or equivalently:

(^{∗}) dim_{(A_{0})_{𝔮}}(M_{𝔮}) − depth_{(A_{0})_{𝔮}}(M_{𝔮}) ≥ dim_{(A_{0})_{𝔭}}(M_{𝔭}) − depth_{(A_{0})_{𝔭}}(M_{𝔭}) .

According to Proposition 2.3 there is an open subset U_{1} ⊆ Spec(A_{0}) with 𝔭 ∈ U_{1} so that for all 𝔮 ∈ U_{1} ∩ V^{0}(𝔭):

dim_{(A_{0})_{𝔮}}(M_{𝔮}) = dim_{(A_{0})_{𝔭}}(M_{𝔭}) + dim((A_{0}/𝔭)_{𝔮}) .

Since A_{0} is excellent, there is an open subset U_{2} ⊆ Spec(A_{0}) so that 𝔭 ∈ U_{2} and for all 𝔮 ∈ U_{2} ∩ V^{0}(𝔭) the local ring:

(A_{0}/𝔭)_{𝔮} is Cohen-Macaulay.

There is also an open subset U_{3} ⊆ Spec(A_{0}) so that 𝔭 ∈ U_{3} and for all 𝔮 ∈ U_{3} ∩ V^{0}(𝔭) we have equality on the set of minimal primes:

Min_{(A_{0})_{𝔮}}(I(A_{0})_{𝔮}) = Min_{(A_{0})_{𝔭}}(I(A_{0})_{𝔭})

where I := ann_{A_{0}}(M) denotes the A_{0}-annihilator of M. In particular, for all 𝔮 ∈ U_{3} ∩ V^{0}(𝔭):

ht(I(A_{0})_{𝔮}) = ht(I(A_{0})_{𝔭}) .

Put Ũ_{1} = U_{1} ∩ U_{2} ∩ U_{3}, then for all 𝔮 ∈ Ũ_{1} ∩ V^{0}(𝔭):

dim_{(A_{0})_{𝔮}}(M_{𝔮}) = dim((A_{0}/I)_{𝔮}) and dim_{(A_{0})_{𝔭}}(M_{𝔭}) = dim((A_{0}/I)_{𝔭}).

Since A is excellent, the ring A_{0} is universally catenary and for all 𝔮 ∈ Ũ_{1}∩V^{0}(𝔭):

dim((A_{0}/I)_{𝔮}) − dim((A_{0}/I)_{𝔭}) = dim((A_{0}/𝔭)_{𝔮}) = depth((A_{0}/𝔭)_{𝔮}) .

From (^{∗}) we obtain:

depth_{(A_{0})_{𝔮}}(M_{𝔮}) − depth_{(A_{0})_{𝔭}}(M_{𝔭}) ≤ depth((A_{0}/𝔭)_{𝔮})

for all 𝔮 ∈ Ũ_{1} ∩ V^{0}(𝔭).

In order to prove the other inequality:

depth_{(A_{0})_{𝔮}}(M_{𝔮}) − depth_{(A_{0})_{𝔭}}(M_{𝔭}) ≥ depth((A_{0}/𝔭)_{𝔮})

assume that depth_{(A_{0})_{𝔭}}(M_{𝔭}) = t and let f_{1}, …, f_{t} ∈ 𝔭 be such that f_{1}, …, f_{t} is a regular sequence on M_{𝔭}. A prime avoidance argument shows that there is an element a ∈ A_{0} ∖ 𝔭 so that f_{1}, …, f_{t} is a regular sequence on M_{a}. (The argument makes again use of the fact that the sets Ass_{A_{0}}(M) and Ass_{A_{0}}(M/(f_{1}, …, f_{i})M) for all 1 ≤ i ≤ t are finite.)

Put

\overline{M} := M/(f_{1}, …, f_{t})M

and consider the associated graded module:

gr_{𝔭}(\overline{M}) = ⨁_{i∈ℕ} 𝔭^{i}\overline{M}/𝔭^{i+1}\overline{M}.

The module \overline{M} is finitely generated over A and gr_{𝔭}(\overline{M}) is a finitely generated gr_{𝔭}(A)-module. Also note that gr_{𝔭}(A) is a finitely generated algebra over A/𝔭A and that A/𝔭A is a finitely generated algebra over A_{0}/𝔭. Thus gr_{𝔭}(A) is a finitely generated A_{0}/𝔭-algebra. By [8, Theorem 24.1] there is an element b ∈ A_{0} ∖ 𝔭 so that the (A_{0}/𝔭)_{b}-module:

gr_{𝔭}(\overline{M})_{b} = ⨁_{i∈ℕ}(𝔭^{i}\overline{M}/𝔭^{i+1}\overline{M})_{b}

is free. Set Ũ_{2} = D_{b} = {𝔮 ∈ Spec(A_{0}) | b ∉ 𝔮} and fix a prime ideal 𝔮 ∈ Ũ_{2} ∩ V^{0}(𝔭). Assume that

depth((A_{0}/𝔭))_{𝔮} = s

and let g_{1}, …, g_{s} ∈ 𝔮 be such that g_{1}, …, g_{s} is a regular sequence on (A_{0}/𝔭)_{𝔮}.

Claim 1. g_{1} is a regular element on \overline{M}_{𝔮}.

Claim 2. Set N_{1} := \overline{M}_{𝔮}/g_{1}\overline{M}_{𝔮}, then gr_{𝔭}(N_{1}) ≅ gr_{𝔭}(\overline{M}_{𝔮})/g_{1} gr_{𝔭}(\overline{M}_{𝔮}).

Assuming the claims, we finish the proof. From the second claim it follows that gr_{𝔭}(N_{1}) is a free (A_{0}/(g_{1}, 𝔭)A_{0})_{𝔮}-module. Since g_{2} is a regular element on (A_{0}/(g_{1}, 𝔭)A_{0})_{𝔮}, we may apply claims 1 and 2 to N_{1}. Note that N_{1} is also a finitely generated graded A_{𝔮}-module. This yields that g_{2} is a regular element on N_{1} and that with N_{2} = N_{1}/g_{2}N_{1}:

gr_{𝔭}(N_{2}) ≅ gr_{𝔭}(N_{1})/g_{2} gr_{𝔭}(N_{1}) .

An induction argument yields that g_{1}, …, g_{s} is a regular sequence on \overline{M}_{𝔮} and we have that:

depth_{(A_{0})_{𝔮}}(M_{𝔮}) ≥ depth_{(A_{0})_{𝔭}}(M_{𝔭}) + depth((A_{0}/𝔭)_{𝔮} .

This inequality holds for all 𝔮 ∈ Ũ_{2} ∩ V^{0}(𝔭). Assuming the claims the proposition is now proved with U^{0} = Ũ_{1} ∩ Ũ_{2}.

In order to prove the claims, set g = g_{1} and N = N_{1}. Proof of Claim 1. Let z ∈ \overline{M}_{𝔮} with gz = 0. Consider the image z̄ of z in \overline{M}_{𝔮}/𝔭\overline{M}_{𝔮}. Since \overline{M}_{𝔮}/𝔭\overline{M}_{𝔮} is a free module over (A_{0}/𝔭)_{𝔮} and since g is regular on (A_{0}/𝔭)_{𝔮} we obtain that z̄ = 0 and z ∈ 𝔭\overline{M}_{𝔮}. Now consider the image of z in 𝔭\overline{M}_{𝔮}/𝔭^{2}\overline{M}_{𝔮} and repeat the argument. This yields

z ∈ ⋂_{j=0}^{∞} 𝔭^{j}\overline{M}_{𝔮} .

Note that

\overline{M}_{𝔮} = ⨁_{i∈ℤ}(\overline{M}_{i})_{𝔮} with (\overline{M}_{i})_{𝔮} = (M_{i})_{𝔮}/(f_{1}, …, f_{t})(M_{i})_{𝔮} .

In particular,

𝔭^{j}\overline{M}_{𝔮} = ⨁_{i∈ℤ} 𝔭^{j}(\overline{M}_{i})_{𝔮}

and every (\overline{M}_{i})_{𝔮} is a finitely generated (A_{0})_{𝔮}-module. This shows that z = 0. Proof of Claim 2. By assumption, we have that gr_{𝔭}(\overline{M}_{𝔮}) is a free (A_{0}/𝔭)_{𝔮}- module and 𝔭^{j}\overline{M}_{𝔮}/𝔭^{j+1}\overline{M}_{𝔮} is a direct summand of gr_{𝔭}(\overline{M}_{𝔮}). Thus 𝔭^{j}\overline{M}_{𝔮}/𝔭^{j+1}\overline{M}_{𝔮} is a free (A_{0}/𝔭)_{𝔮}-module and g is regular on (A_{0}/𝔭)_{𝔮}. Therefore:

(^{∗∗}) 𝔭^{j}\overline{M}_{𝔮} ∩ g\overline{M}_{𝔮} = g𝔭^{j}\overline{M}_{𝔮}

and thus:

\begin{aligned}𝔭^{j}\overline{M}_{𝔮}/g𝔭^{j}\overline{M}_{𝔮} & ≅ & 𝔭^{j}\overline{M}_{𝔮}/(𝔭^{j}\overline{M}_{𝔮} ∩ g\overline{M}_{𝔮}) \\ & ≅ & 𝔭^{j}(\overline{M}_{𝔮}/g\overline{M}_{𝔮}) \end{aligned}

From the commutative diagram:

[figure: 0; 𝔭^{j+1}N; 𝔭^{j}N; 𝔭^{j}N/𝔭^{j+1}N; 0; 0; 𝔭^{j+1}(\overline{M}_{𝔮}/g\overline{M}_{𝔮}); ≅; 𝔭^{j}(\overline{M}_{𝔮}/g\overline{M}_{𝔮}); ≅; 𝔭^{j}(\overline{M}_{𝔮}/g\overline{M}_{𝔮})/𝔭^{j+1}(\overline{M}_{𝔮}/g\overline{M}_{𝔮}); 0; 0; 𝔭^{j+1}\overline{M}_{𝔮}/g𝔭^{j+1}\overline{M}_{𝔮}; 𝔭^{j}\overline{M}_{𝔮}/g𝔭^{j}\overline{M}_{𝔮}; 𝔭^{j}\overline{M}_{𝔮}/(g𝔭^{j}\overline{M}_{𝔮} + 𝔭^{j+1}\overline{M}_{𝔮}); 0]

we obtain that:

gr_{𝔭}(N) = ⨁_{j∈ℕ} 𝔭^{j}N/𝔭^{j+1}N
≅ ⨁_{j∈ℕ} 𝔭^{j}\overline{M}_{𝔮}/(g𝔭^{j}\overline{M}_{𝔮} + 𝔭^{j+1}\overline{M}_{𝔮})
≅ ⨁_{j∈ℕ}(𝔭^{j}\overline{M}_{𝔮}/𝔭^{j+1}\overline{M}_{𝔮})/g(𝔭^{j}\overline{M}_{𝔮}/𝔭^{j+1}\overline{M}_{𝔮})
≅ gr_{𝔭}(\overline{M}_{𝔮})/g(gr(\overline{M}_{𝔮}).

This proves the claim, and finishes the proof. □

Similarly to [4, (6.11.8.1)] we have in the graded case: 2.6.2. Corollary. Let A and M be as above and assume that A is excellent. Then for every 𝔭 ∈ Spec(A_{0}) there is an open subset U^{0} ⊆ Spec(A_{0}) with 𝔭 ∈ U^{0} so that for all 𝔮 ∈ U^{0} ∩ V^{0}(𝔭):

codepth_{(A_{0})_{𝔮}}(M_{𝔮}) = codepth_{(A_{0})_{𝔭}}(M_{𝔭}) + codepth((A_{0})_{𝔮}/𝔭(A_{0})_{𝔮}) .

Proof. Let 𝔭 ∈ Spec(A_{0}) and U_{1}^{0} be as in Proposition 2.6.1, so that 𝔭 ∈ U_{1}^{0} and for all 𝔮 ∈ U_{1}^{0} ∩ V^{0}(𝔭):

depth_{(A_{0})_{𝔮}}(M_{𝔮}) = depth_{(A_{0})_{𝔭}}(M_{𝔭}) + depth((A_{0})_{𝔮}/𝔭(A_{0})_{𝔮}) .

By Proposition 2.3 there is an open subset U_{2}^{0} in Spec(A_{0}) so that 𝔭 ∈ U_{2}^{0} and for all 𝔮 ∈ U_{2}^{0} ∩ V^{0}(𝔭):

dim_{(A_{0})_{𝔮}}(M_{𝔮}) = dim_{(A_{0})_{𝔭}}(M_{𝔭}) + dim((A_{0}/𝔭)_{𝔮}) .

Thus with U^{0} = U_{1}^{0} ∩ U_{2}^{0} we have that 𝔭 ∈ U^{0} and for all 𝔮 ∈ U^{0} ∩ V^{0}(𝔭):

codepth_{(A_{0})_{𝔮}}(M_{𝔮}) = codepth_{(A_{0})_{𝔭}}(M_{𝔭}) + codepth((A_{0})_{𝔮}/𝔭(A_{0})_{𝔮}) .  □

We are now ready to prove the graded version of [4, (6.11.2)(a)]: 2.6.3. Theorem. Let A = ⨁_{i∈ℕ} A_{i} be an excellent graded homogeneous ring and M = ⨁_{i∈ℤ} M_{i} be a finitely generated graded A-module. Then for all n ∈ ℕ the set

U_{C_{n}}^{0}(M) = {𝔭 ∈ Spec(A_{0}) | codepth_{(A_{0})_{𝔭}}(M_{𝔭}) ≤ n}

is open in Spec(A_{0}).

Proof. According to Nagata’s criterion on openness of loci (see [8, Theorem 24.2]) we need to show:

(a) If 𝔭, 𝔮 ∈ Spec(A_{0}) with 𝔮 ⊆ 𝔭 and 𝔭 ∈ U_{C_{n}}^{0}(M) then 𝔮 ∈ U_{C_{n}}^{0}(M).

(b) If 𝔭 ∈ U_{C_{n}}^{0}(M) then U_{C_{n}}^{0}(M) contains a nonempty open subset of V (𝔭).

(a) Let 𝔭, 𝔮 ∈ Spec(A_{0}) with 𝔮 ⊆ 𝔭. By Lemma 2.5:

codepth_{(A_{0})_{𝔭}}(M_{𝔭}) ≥ codepth_{(A_{0})_{𝔮}}(M_{𝔮})

and thus 𝔭 ∈ U_{C_{n}}^{0}(M) implies that 𝔮 ∈ U_{C_{n}}^{0}(M).

(b) Let 𝔭 ∈ U_{C_{n}}^{0}(M). By Corollary 2.6.2 there is an open subset U_{1}^{0} in Spec(A_{0}) so that 𝔭 ∈ U_{1}^{0} and for all 𝔮 ∈ U_{1}^{0} ∩ V^{0}(𝔭):

codepth_{(A_{0})_{𝔮}}(M_{𝔮}) = codepth_{(A_{0})_{𝔭}}(M_{𝔭}) + codepth((A_{0})_{𝔮}/𝔭(A_{0})_{𝔮}) .

Since A and A_{0} are excellent, there is an open subset U_{2}^{0} in Spec(A_{0}) so that 𝔭 ∈ U_{2}^{0} and for all 𝔮 ∈ U_{2}^{0} ∩ V^{0}(𝔭) the ring (A_{0}/𝔭)_{𝔮} is Cohen-Macaulay. Therefore with U^{0} = U_{1}^{0} ∩ U_{2}^{0} we have that 𝔭 ∈ U^{0} and for all 𝔮 ∈ U^{0} ∩ V^{0}(𝔭):

codepth_{(A_{0})_{𝔮}}(M_{𝔮}) = codepth_{(A_{0})_{𝔭}}(M_{𝔭}) .

This implies that U^{0} ∩ V^{0}(𝔭) ⊆ U_{C_{n}}^{0}(M) and the Theorem is proved. □

2.6.4. Corollary. Let A and M be as in Theorem 2.6.3. Then the Cohen-Macaulay locus of the A_{0}-module M:

U_{CM}^{0}(M) = U_{C_{0}}^{0}(M) = {𝔭 ∈ Spec(A_{0}) | M_{𝔭} is a CM module over (A_{0})_{𝔭}}

is open in Spec(A_{0}). □

3. Openness of the (S_{n})-locus

Throughout this section we assume that R = A_{0} is the base ring of a graded Noetherian homogeneous ring A = ⨁_{i≥0} A_{i} and M is a finitely generated graded Amodule. This includes the case of a finitely generated module M over a Noetherian ring R. For those modules we prove that the openness of the C_{n}-loci of M implies the openness of the (S_{k})-loci of M. The argument is due to Grothendieck [4, (5.7.2) and (6.11.2)(b)] but we include it here for the convenience of the reader. The proof also shows that the (S_{k})-loci of M only depend on the C_{n}-loci of M and on the annihilator of M, so that two R-modules M and N with the same annihilators and C_{n}-loci have identical (S_{k})-loci.

Let M be an R-module and suppose that for all n ∈ ℕ_{0} the set:

U_{C_{n}}(M) = {𝔭 ∈ Spec(R) | codepth_{R_{𝔭}}(M_{𝔭}) ≤ n}

is open in Spec(R). Define:

Z_{n} = V (𝔟_{n}) = Spec(R) ∖ U_{C_{n}}(M)

where 𝔟_{n} ⊆ R is a reduced ideal. Obviously, for all n ∈ ℕ:

U_{C_{n}}(M) ⊆ U_{C_{n+1}}(M)

and therefore:

Z_{n+1} ⊆ Z_{n} and 𝔟_{n} ⊆ 𝔟_{n+1} .

Since R is Noetherian there is an m ∈ ℕ so that for all t ∈ ℕ:

𝔟_{m} = 𝔟_{m+t} and Z_{m} = Z_{m+t} .

3.1. Lemma. Let m ∈ ℕ be as above. Then Z_{m} = ∅.

Proof. If 𝔭 ∈ Z_{m} then 𝔭 ∈ Z_{m+t} for all t ∈ ℕ. By definition of Z_{n+t}:

codepth_{(R)_{𝔭}}(M_{𝔭}) ≥ m + t for all t ∈ ℕ .

But codepth_{R_{𝔭}}(M_{𝔭}) ≤ dim((R)_{𝔭}) ≤ ∞ and therefore Z_{m} = ∅. □

Recall that the R-module M satisfies Serre’s condition (S_{k}) if for all 𝔭 ∈ Spec(R):

(^{∗}) depth_{R_{𝔭}}(M_{𝔭}) ≥ min(dim(M_{𝔭}), k).

From now on let m denote the minimal m ∈ ℕ with Z_{m} = ∅.

3.2. Lemma. With the assumptions as above put \overline{R} = R/ ann_{R}(M) and let k ∈ ℕ. Then the R-module M satisfies (S_{k}) if and only if for all 0 ≤ n < m:

ht(𝔟_{n}\overline{R}) > n + k .

Proof. Suppose that M satisfies (S_{k}) and fix an integer n with 0 ≤ n < m. Let 𝔭 ∈ Spec(R) with 𝔟_{n} ⊆ 𝔭. Then 𝔭 ∈ Z_{n} and therefore:

codepth_{R_{𝔭}}(M_{𝔭}) > n

or equivalently:

dim_{R_{𝔭}}(M_{𝔭}) − depth_{R_{𝔭}}(M_{𝔭}) > n .

Since M satisfies (S_{k}) we obtain that whenever

dim_{R_{𝔭}}(M_{𝔭}) − depth_{R_{𝔭}}(M_{𝔭}) ≠ 0

then

depth_{R_{𝔭}}(M_{𝔭}) ≥ k .

Thus, if 𝔭 ∈ Z_{n} then

dim_{R_{𝔭}}(M_{𝔭}) ≥ n + k

which implies that ht(𝔟_{n}\overline{R}) ≥ n + k.

Conversely, fix an integer k and assume that for all 0 ≤ n < m:

ht(𝔟_{n}\overline{R}) > n + k .

Let 𝔭 ∈ Spec(R).

If M_{𝔭} = 0 then depth_{R_{𝔭}}(M_{𝔭}) = ∞ and condition (^{∗}) is satisfied.

Now assume M_{𝔭} ≠ 0. If M_{𝔭} is a Cohen-Macaulay R-module, then condition (^{∗}) is satisfied. Now assume that:

codepth_{R_{𝔭}}(M_{𝔭}) > 0

and let n ∈ ℕ_{0} with

codepth_{R_{𝔭}}(M_{𝔭}) = n + 1 .

Thus 𝔭 ∈ Z_{n} and 𝔟_{n} ⊆ 𝔭. By assumption

ht(𝔟_{n}\overline{R}) > n + k ⇒ ht(𝔟_{n}\overline{R}_{𝔭}) > n + k ⇒ dim(\overline{R}_{𝔭}) > n + k .

This implies that

codepth_{R_{𝔭}}(M_{𝔭}) = n + 1
= dim(\overline{R}_{𝔭}) − depth_{R_{𝔭}}(M_{𝔭})
≥ n + 1 + k − depth_{R_{𝔭}}(M_{𝔭})

and therefore

depth_{R_{𝔭}}(M_{𝔭}) ≥ k .

Thus M_{𝔭} satisfies condition (^{∗}) and the R-module M satisfies Serre’s condition (S_{k}). □

For all 0 ≤ n < m consider the closed subset of Spec(R):

Y_{n,k} = \{𝔮 ∈ V (𝔟_{n}) | ht(𝔟_{n}\overline{R}_{𝔮}) ≤ n + k\} ,

and its complement

V_{n,k} = Spec(R) − Y_{n,k} ,

an open subset of Spec(R). By Lemma 3.2:

U_{S_{k}}(M) = ⋂_{0≤n<m} V_{n,k}

is an open subset of Spec(R). We have shown: 3.3. Theorem. Let M be an R-module as above. If for all n ∈ ℕ_{0} the C_{n}-locus U_{C_{n}}(M) is open in Spec(R) then for all k ∈ ℕ the (S_{k})-locus:

U_{S_{k}}(M) = {𝔭 ∈ Spec(R) | M_{𝔭} satisfies (S_{k})}

is open in Spec(R).

In the graded case the theorem states: 3.4. Corollary. Let A = ⨁_{i∈ℕ} A_{i} be an excellent graded homogeneous ring and let M = ⨁_{i∈ℤ} M_{i} be a finitely generated graded A-module. Then for all k ∈ ℕ the set

U_{S_{k}}^{0}(M) = {𝔭 ∈ Spec(A_{0}) | the (A_{0})_{𝔭}-module M_{𝔭} satisfies (S_{k})}

is open in Spec(A_{0}).

The proof of the theorem also yields the following corollary: 3.5. Corollary. Suppose that M and N are R-modules as above. Assume that ann_{R}(M) = ann_{R}(N) and that for all n ∈ ℕ_{0} the sets U_{C_{n}}(M) = U_{C_{n}}(N) are open in Spec(R). Then for all k ∈ ℕ:

U_{S_{k}}(M) = U_{S_{k}}(N)

and the (S_{k})-loci are open subsets of Spec(R).

4. Stability on the homogeneous parts

Let A = ⨁_{i∈ℕ} A_{i} be an excellent graded homogeneous Noetherian ring and let M = ⨁_{i∈ℤ} M_{i} be a finitely generated graded A-module. In this section we prove that there is a k ∈ ℕ so that for all n ∈ ℕ and all i ≥ k:

U_{C_{n}}^{0}(M_{i}) = U_{C_{n}}^{0}(M_{k}) and U_{S_{n}}^{0}(M_{i}) = U_{S_{n}}^{0}(M_{k}),

that is, the codepth and (S_{n})-loci of the homogeneous parts of M are eventually stable (considered as an A_{0}-module). As before we define for all t ∈ ℤ:

N_{t} = ⨁_{i≥t} M_{i},

and observe the following simple facts: Let k_{1} ∈ ℕ be an integer so that for all t ≥ k_{1}: ann_{A_{0}}(M_{t}) = ann_{A_{0}}(M_{k_{1}}). Then for all t ≥ k_{1}:

U_{C_{n}}^{0}(N_{t}) ⊇ U_{C_{n}}^{0}(N_{k_{1}}) and U_{S_{n}}^{0}(N_{t}) ⊇ U_{S_{n}}^{0}(N_{k_{1}}) .

Since A_{0} is Noetherian there is an integer k_{2} ∈ ℤ so that k_{2} ≥ k_{1} and

U_{C_{n}}^{0}(N_{t}) = U_{C_{n}}^{0}(N_{k_{2}}) and U_{S_{n}}^{0}(N_{t}) = U_{S_{n}}^{0}(N_{k_{2}}) .

We may also assume for large enough k_{2} that:

N_{k_{2}} = AM_{k_{2}},

which implies that for all t ≥ k_{2}:

N_{t} = AM_{t}.

4.1. Lemma. With the assumptions as above assume additionally that (A_{0}, 𝔪_{0}) is a local ring. Then there is a k_{3} ∈ ℤ so that for all t ≥ k_{3}:

depth_{A_{0}}(M_{t}) = depth_{A_{0}}(M_{k_{3}}) = depth_{A_{0}}(N_{k_{3}}) .

Proof. Let k_{1} and k_{2} be as above and take an integer k with k > k_{2}. Then codepth_{A_{0}}(N_{k}) = n for some n ∈ ℕ and therefore:

𝔪_{0} ∈ U_{C_{n}}^{0}(N_{k}) and 𝔪_{0} ∉ U_{C_{n−1}}^{0}(N_{k}) .

Since k ≥ k_{2} we have for all t ≥ k:

codepth_{A_{0}}(N_{k}) = n = codepth_{A_{0}}(N_{t}) .

For all t ≥ k_{1} we also have that ann_{A_{0}}(N_{t}) = ann_{A_{0}}(N_{k}), and therefore for all t ≥ k:

depth_{A_{0}}(N_{t}) = s = depth_{A_{0}}(N_{k}) .

Let r_{1}, …, r_{s} be a maximal regular sequence on N_{k} and put

\overline{N}_{k} = N_{k}/(r_{1}, …, r_{s})N_{k} with homogeneous parts \overline{M}_{i} = M_{i}/(r_{1}, …, r_{s})M_{i}

for i ≥ k. Note that the torsion submodule Γ_{A_{+}}(\overline{N}_{k}) is a finitely generated Asubmodule of \overline{N}_{k}. This implies that there is an integer k_{3} ≥ k so that Γ_{A_{+}}(\overline{N}_{k}) ∩ N_{k_{3}} = 0 = Γ_{A_{+}}(\overline{N}_{k_{3}}). Thus for k_{3} large enough the A-module \overline{N}_{k_{3}} is A_{+}-torsion free. Since by assumption depth_{A_{0}}(N_{k}) = s = depth_{A_{0}}(N_{k_{3}}) there is an integer i ≥ k_{3} and an element x̄ ∈ \overline{M}_{i} so that x̄ ≠ 0 and 𝔪_{0}x̄ = 0. Since \overline{N}_{k_{3}} is A_{+}-torsion free we obtain:

(A_{+})^{l}x̄ ≠ 0 for all l ∈ ℕ .

Thus for k_{4} = i > k_{3} we have that depth_{A_{0}}(\overline{M}_{k_{4}+l}) = 0 for all l ∈ ℕ_{0} and therefore for all t ≥ k_{4}:

depth_{A_{0}}(M_{t}) = depth_{A_{0}}(M_{k_{4}}) = s  □

Choose an integer k_{0} ∈ ℤ so that the following conditions are satisfied: (a) N_{k_{0}} = AM_{k_{0}}, that is, N_{k_{0}} is generated in the lowest nonvanishing degree.

(b) For all t ≥ k_{0}: ann(M_{k_{0}}) = ann(M_{t}).

(c) For all n ∈ ℕ_{0} and all t ≥ k_{0}:

U_{C_{n}}^{0}(N_{t}) = U_{C_{n}}^{0}(N_{k_{0}}) and U_{S_{n}}^{0}(N_{t}) = U_{S_{n}}^{0}(N_{k_{0}}) .

As before put:

Z_{n} = Spec(A_{0}) ∖ U_{C_{n}}^{0}(N_{k_{0}}) = V (𝔟_{n})

where 𝔟_{n} ⊆ A_{0} is a reduced ideal. Then 𝔟_{n} ⊆ 𝔟_{n+1} yielding an increasing sequence of ideals:

𝔟_{0} ⊆ 𝔟_{1} ⊆ … ⊆ 𝔟_{m−1} ⊆ …

We have seen before that the sequence stops with some 𝔟_{m} = A_{0} and let m be minimal with this property, that is, let 𝔟_{m} = A_{0} and 𝔟_{m−1} ≠ A_{0}. For all 0 ≤ j ≤ m − 1 we consider the set of minimal prime divisors of 𝔟_{j}:

Min(A_{0}/𝔟_{j}) = {𝔭_{j1}, …, 𝔭_{jr_{j}}} .

By Lemma 4.1 for all 0 ≤ j ≤ m − 1 and all r_{j} ≥ h ≥ 1 there is an integer k_{jh} ∈ ℕ with k_{jh} ≥ k_{0} so that for all i ≥ k_{jh}:

depth_{(A_{0})_{𝔭_{jh}}} ((M_{i})_{𝔭_{jh}} = depth_{(A_{0})_{𝔭_{jh}}} ((M_{k_{jh}})_{𝔭_{jh}} = constant .

Let k = max{k_{jh} | 0 ≤ j ≤ m − 1; 1 ≤ h ≤ r_{j}}, then for all i ≥ k:

depth_{(A_{0})_{𝔭_{jh}}} ((M_{i})_{𝔭_{jh}}) = depth_{(A_{0})_{𝔭_{jh}}} ((M_{k})_{𝔭_{jh}}) = depth_{(A_{0})_{𝔭_{jh}}} ((N_{k})_{𝔭_{jh}}) .

By assumption on the annihilators we also have for all i ≥ k:

dim_{(A_{0})_{𝔭_{jh}}} ((M_{i})_{𝔭_{jh}}) = dim_{(A_{0})_{𝔭_{jh}}} ((M_{k})_{𝔭_{jh}}) = dim_{(A_{0})_{𝔭_{jh}}} ((N_{k})_{𝔭_{jh}})

which implies that for all i ≥ k and all primes 𝔭_{jh}:

codepth_{(A_{0})_{𝔭_{jh}}} ((M_{i})_{𝔭_{jh}}) = codepth_{(A_{0})_{𝔭_{jh}}} ((M_{k})_{𝔭_{jh}}) = codepth_{(A_{0})_{𝔭_{jh}}} ((N_{k})_{𝔭_{jh}}) .

We are now ready to prove: 4.2. Theorem. Let k be as above. Then for all i ≥ k and all 𝔭 ∈ Spec(A_{0}):

codepth_{(A_{0})_{𝔭}}((M_{i})_{𝔭}) = codepth_{(A_{0})_{𝔭}}((M_{k})_{𝔭}) .

Proof. Let 𝔭 ∈ Spec(A_{0}). If 𝔟_{0} ⊈ 𝔭 then (N_{k})_{𝔭} is a Cohen-Macaulay module over(A_{0})_{𝔭} it follows that (M_{i})_{𝔭} is Cohen-Macaulay for all i ≥ k.

Assume that 𝔟_{0} ⊆ 𝔭 and let g be minimal so that 𝔟_{g} ⊆ 𝔭 and 𝔟_{g+1} ⊈ 𝔭. In this case codepth_{(A_{0})_{𝔭}}((N_{k})_{𝔭}) = g +1 and there is an integer 1 ≤ j ≤ r_{j} so that 𝔭_{gj} ⊆ 𝔭. By [4, (6.11.5)], the nongraded version of Lemma 2.5, for all i ≥ k:

codepth_{(A_{0})_{𝔭}}((M_{i})_{𝔭}) ≥ codepth_{(A_{0})_{p_{gj}}} ((M_{i})_{p_{gj}}) = codepth_{(A_{0})_{𝔭_{gj}}} ((N_{k})_{𝔭_{gj}}) > g .

In order to verify the other inequality consider

codepth_{(A_{0})_{𝔭}}((N_{k})_{𝔭}) = g + 1 = dim((N_{k})_{𝔭}) − depth_{(A_{0})_{𝔭}}((N_{k})_{𝔭})

and assume that depth_{(A_{0})_{𝔭}}((N_{k})_{𝔭}) = s. Let x_{1}, …, x_{s} be a regular sequence on (N_{k})_{𝔭}, then x_{1}, …, x_{s} is a regular sequence on (M_{i})_{𝔭} for all i ≥ k. Since N_{k} and M_{i} have the same annihilators we obtain that:

codepth_{(A_{0})_{𝔭}}((N_{k})_{𝔭}) = g + 1 ≥ codepth_{(A_{0})_{𝔭}}((M_{i})_{𝔭})

for all i ≥ k. This shows that for all i ≥ k:

codepth_{(A_{0})_{𝔭}}((M_{i})_{𝔭}) = g + 1 .  □

4.3. Corollary. There is an integer k ∈ ℕ so that for all i ≥ k and all n ∈ ℕ:

U_{C_{n}}^{0}(M_{i}) = U_{C_{n}}^{0}(M_{k}) = U_{C_{n}}^{0}(N_{k}) .  □

4.4. Corollary. There is an integer k ∈ ℕ so that for all i ≥ k and all n ∈ ℕ:

U_{S_{n}}^{0}(M_{i}) = U_{S_{n}}^{0}(M_{k}) = U_{S_{n}}^{0}(N_{k}) .

Proof. The second corollary follows from the first by using Corollary 3.5. □

5. Applications

Let A be an excellent ring, M a finitely generated A-module, and let I ⊆ A be an ideal of A. By applying the results of the previous section to the Rees algebra/module and to the associated graded ring/module, respectively, we see that there is an integer k ∈ ℕ so that for all i ≥ k and all n ∈ ℕ:

U_{C_{n}}(I^{i}M) = U_{C_{n}}(I^{k}M)  and U_{C_{n}}(I^{i}M/I^{i+1}M) = U_{C_{n}}(I^{k}M/I^{k+1}M)
U_{S_{n}}(I^{i}M) = U_{S_{n}}(I^{k}M)  and U_{S_{n}}(I^{i}M/I^{i+1}M) = U_{S_{n}}(I^{k}M/I^{k+1}M)

In the following we want to apply these results to the (S_{n})- and codepth-loci of the modules M/I^{k}M. We want to show that these loci are again eventually stable provided that M is a Cohen-Macaulay module over A.

5.1. Lemma. Let A be any Noetherian ring, I ⊆ A an ideal, and M a finitely generated A-module. Then for all k ∈ ℕ:

Supp(M/I^{k}M) = Supp(M/IM) .

Proof. It suffices to show that for all k ∈ ℕ:

Supp(M/I^{k}M) = Supp(M/I^{k+1}M) .

Since M/I^{k}M is a homomorphic image of M/I^{k+1}M we have Supp(M/I^{k}M) ⊆ Supp(M/I^{k+1}M). Consider the exact sequence:

0 → I^{k}M/I^{k+1}M → M/I^{k+1}M → M/I^{k}M → 0

and let 𝔭 ∈ Spec(A) with I ⊆ 𝔭. The sequence stays exact after localization:

0 → (I^{k}M/I^{k+1}M)_{𝔭} → (M/I^{k+1}M)_{𝔭} → (M/I^{k}M)_{𝔭} → 0 .

If (M/I^{k}M)_{𝔭} = 0 with (M/I^{k+1}M)_{𝔭} ≠ 0, then

(I^{k}M/I^{k+1}M)_{𝔭} = (M/I^{k+1}M)_{𝔭},

which implies by Nakayama that (M/I^{k+1}M)_{𝔭} = 0, a contradiction. □

A more general version of the next result was proved, using different methods, by Kodiyalam [7, Corollary 9]. 5.2. Theorem. Suppose that (A, 𝔪) is a local Noetherian ring, I ⊆ A an ideal of A, and let M be a finitely generated A-module. Then there is a k ∈ ℕ so that for all i ≥ k:

depth_{A}(M/I^{i}M) = depth_{A}(M/I^{k}M) .

Proof. Let Â be the 𝔪-adic completion of A. Then for any finitely generated A-module T:

depth_{A}(T) = depthÂ(T ⊗_{A} Â)

and we may replace A by Â and M by M ⊗_{A} Â and assume that A is excellent. By Lemma 4.1 there is a k_{1} ∈ ℕ so that for all t ≥ k_{1}:

depth_{A}(I^{t}M/I^{t+1}M) = depth_{A}(I^{k_{1}}M/I^{k_{1}+1}M) = g.

For all t ≥ k_{1} consider the exact sequence:

0 → I^{t}M/I^{t+1}M → M/I^{t+1}M → M/I^{t}M → 0

which leads to an exact sequence on the cohomology modules:

⋯ → H_{𝔪}^{i}(M/I^{t+1}M) → H_{𝔪}^{i}(M/I^{t}M) → 0 → ⋯ → 0 → ⋯
⋯ → H_{𝔪}^{g−1}(M/I^{t+1}M) → H_{𝔪}^{g−1}(M/I^{t}M) → H_{𝔪}^{g}(I^{t}M/I^{t+1}M) →
→ H_{𝔪}^{g}(M/I^{t+1}M) → H_{𝔪}^{g}(M/I^{t}M) → ⋯

where g is minimal with H_{𝔪}^{g}(I^{t}M/I^{t+1}M) ≠ 0.

case 1: There is an i ≤ g − 1 and a t_{0} ≥ k_{1} so that H_{𝔪}^{i}(M/I^{t_{0}}M) ≠ 0. Then for all t ≥ t_{0} H_{𝔪}^{i}(M/I^{t}M) ≠ 0. Let h ≤ g − 1 be the minimal i with this property, then

depth_{A}(M/I^{t}M) = h for all t ≥ t_{0} .

case 2: For all i ≤ g − 1 and all t ≥ k_{1}:

H_{𝔪}^{i}(M/I^{t}M) = 0 .

This implies that depth_{A}(M/I^{t}M) ≥ g − 1 for all t ≥ k_{1}. case 2.1: There are infinitely many t ≥ k_{1} so that

H_{𝔪}^{g−1}(M/I^{t}M) ≠ 0 .

From the long exact sequence we observe that H_{𝔪}^{g−1}(M/I^{t}M) ≠ 0 implies that H_{𝔪}^{g−1}(M/I^{t−1}M) ≠ 0 whenever t − 1 ≥ k_{1}. Thus in this case there is a t_{1} ≥ k_{1} so that for all t ≥ t_{1}:

H_{𝔪}^{g−1}(M/I^{t}M) ≠ 0 ,

and therefore for all t ≥ t_{1}: depth_{A}(M/I^{t}M) = g − 1.

case 2.2: There is a t_{2} ≥ k_{1} so that for all t ≥ t_{2}: H_{𝔪}^{g−1}(M/I^{t}M) = 0. Then for all t ≥ t_{2}:

depth_{A}(M/I^{t}M) = g .  □

5.3. Theorem. Let A be an excellent ring and M a finitely generated Cohen-Macaulay A-module. Let I ⊆ A be an ideal of A which is not contained in any minimal prime ideal of M. Then there is an integer k ∈ ℕ so that for all t ≥ k and all n ∈ ℕ_{0}:

(1) U_{C_{n}}(M/I^{t}M) = U_{C_{n}}(M/I^{k_{0}}M).

(2) U_{S_{n}}(M/I^{t}M) = U_{S_{n}}(M/I^{k_{0}}M).

Proof. (1) Fix n ∈ ℕ and let k ∈ ℕ so that for all t ≥ k:

U_{C_{n}}(I^{t}M) = U_{C_{n}}(I^{k}M) .

We claim that for all i ≥ k and all 𝔭 ∈ V (I):

depth_{A_{𝔭}}((M/I^{i}M)_{𝔭}) = depth_{A_{𝔭}}((M/I^{k}M)_{𝔭}) .

Obviously, for all i ≥ k: dim((I^{i}M)_{𝔭}) = dim((I^{k}M)_{𝔭}) and thus because of the stability of the codepth-loci we have for all 𝔭 ∈ V (I) and all i ≥ k that:

depth_{A_{𝔭}}((I^{i}M)_{𝔭}) = depth_{A_{𝔭}}((I^{k}M)_{𝔭}) .

Fix an integer i ≥ k and a prime ideal 𝔭 ∈ V (I) and consider the exact sequence:

0 → (I^{i}M)_{𝔭} → M_{𝔭} → (M/I^{i}M)_{𝔭} → 0 .

With d = dim_{A_{𝔭}}(M_{𝔭}) = depth_{A_{𝔭}}(M_{𝔭}) we obtain a long exact sequence of the local cohomology modules:

⋯ → 0 → H_{𝔭}^{i−1}((M/I^{i}M)_{𝔭}) → H_{𝔭}^{i}((I^{i}M)_{𝔭}) → 0 → ⋯ → 0 →
→ H_{𝔭}^{d−1}((M/I^{i}M)_{𝔭}) → H_{𝔭}^{d}((I^{i}M)_{𝔭}) → H_{𝔭}^{d}(M_{𝔭}) → 0 = H_{𝔭}^{d}((M/I^{i}M)_{𝔭})

where H_{𝔭}^{d}((M/I^{i}M)_{𝔭}) = 0 since dim_{A_{𝔭}}((M/I^{i}M)_{𝔭}) ≤ d − 1. This shows that

depth_{A_{𝔭}}((M/I^{i}M)_{𝔭}) = depth_{A_{𝔭}}((I^{i}M)_{𝔭}) − 1 = depth_{A_{𝔭}}((I^{k}M)_{𝔭}) − 1

and the claim is proven. For all i ≥ k and all 𝔭 ∈ V (I) we have:

depth_{A_{𝔭}}((M/I^{i}M)_{𝔭}) = depth_{A_{𝔭}}((M/I^{k}M)_{𝔭})
dim((M/I^{i}M)_{𝔭}) = dim((M/I^{k}M)_{𝔭}) .

The last equation is obtained from Lemma 5.1. This yields that for all n ∈ ℕ and for all i ≥ k:

U_{C_{n}}(M/I^{i}M) = U_{C_{n}}(M/I^{k}M) .

The second assumption follows with Corollary 3.5. □

5.4. Corollary. Let A, M, and I be as in the theorem and assume that IM ≠ M. Then there is an element a ∈ A so that for all k ∈ ℕ:

(1) (M/I^{k}M)_{a} ≠ 0.

(2) (M/I^{k}M)_{a} is a Cohen-Macaulay module. □ 5.5. Corollary. Let A be an excellent ring and M a finitely generated A-module. Suppose that the ideal I ⊆ A satisfies the following conditions:

(i) I is not contained in a minimal prime of M. (ii) If 𝔞 ⊆ A is the defining ideal of the non-Cohen-Macaulay locus of M then

𝔞 ⊈ \sqrt{(IM : M)}. Then there is an element a ∈ A so that for all k ∈ ℕ:

(1) (M/I^{k}M)_{a} ≠ 0.

(2) (M/I^{k}M)_{a} is a Cohen-Macaulay module.

Proof. Choose an element b ∈ 𝔞 ∖ \sqrt{(IM : M)}. In order to prove the assertion apply the previous corollary to the Cohen-Macaulay A_{b}-module M_{b}. □

References

[1] N. Bourbaki, Commutative algebra, chapters 1-7, Springer Verlag, New York, 1989

[2] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge studies in advanced mathematics, Vol. 39, revised edition , Cambridge, 1998

[3] H. B. Foxby, Hyperhomological algebra and commutative rings, in preparation

[4] A. Grothendieck, Éléments de géométrie algébrique IV, Inst. Hautes Études Sci. Publ. Math 24, (1965)

[5] M. Hochster, J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. Math. 13 (1974), 115–175.

[6] S. Iyengar, Depth for complexes, and intersection theorems, Math. Z. 230 (1999), 545–569 [7] V. Kodiyalam, Homological invariants of powers of an ideal, Proc. Amer. Math. Soc. 118

(1993), 757–764.

[8] H. Matsumura, Commutative ring theory, Cambridge studies in advanced mathematics, Vol. 8, Cambridge, 1986.

Department of Mathematics, Michigan State University, East Lansing, MI 48824

  E-mail address: rotthaus@math.msu.edu

Department of Mathematics, Michigan State University, East Lansing, MI 48824

  E-mail address: lsega@math.msu.edu