Math. Z. 230, 545–567 (1999) Corrected version

DEPTH FOR COMPLEXES, AND INTERSECTION THEOREMS

S. IYENGAR

Abstract. This paper introduces a new notion of depth for complexes; it agrees with the classical definition for modules, and coincides with earlier extensions to complexes, whenever those are defined. Techniques are developed leading to a quick proof of an extension of the Improved New Intersection Theorem (this uses Hochster’s big Cohen-Macaulay modules), and also a generalization of the “depth formula” for tensor product of modules. Properties of depth for complexes are established, extending the usual properties of depth for modules.

Introduction

This paper studies homological invariants of modules and complexes over noetherian local rings. We introduce a new notion of depth for complexes; it agrees with the classical definition for modules, and coincides with earlier extensions to complexes, whenever those are defined. Our interest in this approach arises from its efficacy in dealing with a class of problems centered around “intersection theorems” in commutative algebra. On the one hand, it leads to a quick proof of an extension of the Improved New Intersection Theorem; this uses Hochster’s big Cohen-Macaulay modules, and hence is restricted to rings containing fields. On the other hand, it yields a generalization of the “depth formula” for tensor product of modules.

For these and other applications, we extend the classical Auslander-Buchsbaum Equality: if a finitely generated module M, over a noetherian local ring R, has finite projective dimension, then depth R = depth M + pd_{R} M. Generalizing this to complexes, in Section 2 we prove that for a complex M of finite projective dimension

depth N = depth(M⊗_{R}^{L}N) + pd_{R} M ,

where N is any complex with H_{i}(N) = 0 for i ≫ 0, and M⊗_{R}^{L}N stands for F ⊗_{R} N, with F any free resolution of M. As a complement to this result we prove that in certain cases the depth of M⊗_{R}^{L}N can be computed in terms of the depth of its homology modules Tor_{i}^{R} (M, N) . The basic technique employed in this paper is to compare the depth of M⊗_{R}^{L}N obtained by applying the two theorems.

As direct corollaries of these results, we strengthen a theorem of Auslander, and extend a result of Foxby on the behavior of depth under flat base change.

In Section 3, we use the same tools to prove, over a ring R containing a field, that if F is finite free complex with length H_{i}(F) finite for i > 0, then

length(F) ⩾ dim R − dim(R/I)

for each ideal I annihilating a minimal generator of H_{0}(F). This theorem is an avatar of a result of Bruns and Herzog, and reduces to the Improved New Intersection Theorem of Evans and Griffith, when I is 𝔪-primary.

In Section 4, we show that if M and N are Tor-independent modules, then

depth(M ⊗_{R} N) + depth R = depth M + depth N provided one of them is of finite CI-dimension, in the sense of Avramov, Gasharov, and Peeva. This is both a common generalization and an extension of results by Auslander, and by Huneke and R. Wiegand.

Date: November 3, 1999.

1991 Mathematics Subject Classification. 13C15, 13D25, 18G15.

The results discussed thus far were for complexes over local rings. In fact, we define, for a finitely generated ideal I in a commutative ring R, the I-depth of a complex M to be the number

depth_{R}(I, M) = n − sup{i | H_{i}(K ⊗_{R} M) ≠ 0}

where K is the Koszul complex on a set of n generators for I (the depth of complex over a noetherian local ring is the depth at the maximal ideal.) This definition is motivated by a classical result of Auslander and Buchsbaum: for a finitely generated module over a noetherian ring, this number is equal to the one obtained via regular sequences.

Section 5 is devoted to proving that depth for complexes enjoys the usual properties of depth for modules. Some of these extend results of Foxby, who defines depth for complexes with bounded above homology, using injective resolutions.

In Section 6 we prove that for a large class of complexes,

depth_{R}(I, M) = inf{i | Ext_{R}^{i} (R/I, M) ≠ 0} = inf{i | H_{I}^{i}(M) ≠ 0} .

where H_{I}^{i}(M) are the local cohomology modules of the complex M. In particular, our notion of depth coincides with that defined by Foxby, and Iversen. Moreover, when M is bounded above, any one of a multitude of complexes may be used to compute the depth of M. Noteworthy among these are finite free complexes with non-zero finite length homology.

The arguments in this paper use a modicum of the homological theory of complexes, reviewed in Section 1.

1. Complexes

Let R be a commutative ring. We consider complexes of the form

⋯ ⟶ M_{i+1} ⟶^{∂_{i+1}} M_{i} ⟶^{∂_{i}} M_{i−1} ⟶ · · · .

The k’th suspension of M is the complex M[k] with M[k]_{i} = M_{i−k} and differential given by ∂(m) = (−1)^{k}∂_{i−k}(m) for m ∈ M[k]_{i}. For a complex M, we set

sup M = sup{i | H_{i}(M) ≠ 0} and inf M = inf{i | H_{i}(M) ≠ 0} .

(so sup(0) = −∞ and inf(0) = ∞). A complex with sup M (respectively, inf M) finite is said to be bounded above (respectively, below). A bounded complex is one which is bounded both above and below. We write M^{♮} for the graded module underlying a complex M.

A quasi-isomorphism ϕ: M ⟶ N is a degree zero chain map such that H(ϕ) is an isomorphism. Complexes M and N are quasi-isomorphic (denoted M ≃ N) if they are linked by a chain of quasi-isomorphisms. A complex M with t = inf M finite (respectively, s = sup M finite) is quasi-isomorphic to τ_{⩾ t} (respectively, τ_{⩽ s}), where

τ_{⩾ t}(M): ⋯ ⟶ M_{t−1} ⟶ M_{t} ⟶ ∂(M_{t}) ⟶ 0 ,
τ_{⩽ s}(M): 0 ⟶ ∂(M_{s+1}) ⟶ M_{s} ⟶ M_{s−1} ⟶ · · · .

We use some basic facts concerning resolutions of complexes, for which we refer to [8], [19] or [22].

For a complex F of flat R-modules with F^{♮} bounded below, if G and G′ are quasi-isomorphic, then so are F ⊗_{R} G and F ⊗_{R} G′. Every bounded below complex M has a flat resolution, that is, a complex F quasi-isomorphic to M such that each F_{i} is a flat module, and F^{♮} is bounded below. The symbol M⊗_{R}^{L}N denotes any complex quasi-isomorphic to F ⊗_{R} N, where N is a complex of R-modules. Set

Tor_{i}^{R} (M, N) = H_{i}(M⊗_{R}^{L}N) .

When M and N are R-modules, this gives the classical notion.

If a complex N has a flat resolution G, then

M ⊗_{R} G ≃ F ⊗_{R} G ≃ F ⊗_{R} N .

Thus, any one of the complexes above may be used to compute Tor^{R} (M, N) .

A complex M has finite flat dimension if M ≃ F, where F^{♮} is a bounded complex of flat R-modules.

A projective resolution of M is a complex P quasi-isomorphic to M, with each P_{i} a projective R-module, and P^{♮} bounded below. The projective dimension of M is defined by

pd_{R} M = inf{sup F^{♮} | F a projective resolution of M} .

For an element x ∈ R, we let K(x) denote the complex

0 ⟶ Re_{x} ⟶^{∂} R ⟶ 0

with ∂(e_{x}) = x, and Re_{x} in degree 1. The Koszul complex on a set x = {x_{1}, …, x_{n}} ⊆ R is the complex

K(x) = K(x_{1}) ⊗_{R} ⋯ ⊗_{R} K(x_{n}) .

If M is a complex of R-modules, then we set H(x; M) = H(K(x) ⊗_{R} M). The following properties of Koszul complexes are well known:

1.1. If y = {x_{1}, …, x_{n}, y} and M is a complex, then there is a long exact sequence

⋯ ⟶ H_{i}(x; M) ⟶^{±y} H_{i}(x; M) ⟶ H_{i}(y; M) ⟶ H_{i−1}(x; M) ⟶^{±y} H_{i−1}(x; M) ⟶ · · · .

Indeed, as K(y) = K(x) ⊗_{R} K(y), there is a short exact sequence

0 ⟶ K(x) ⟶ K(x) ⊕ (K(x) ⊗_{R} Re_{y}) ⟶ K(x) ⊗_{R} Re_{y} ⟶ 0 .

This sequence is split exact, and hence remains exact upon tensoring with M. The corresponding homology exact sequence is the long exact sequence above.

1.2. If y ∈ (x_{1}, …, x_{n}) and M is a complex, then y H(x; M) = 0.

It is easy to see that multiplication by x_{i} is homotopic to zero on K(x_{i}), so the same holds for K ⊗_{R} M. Thus, x_{i} H(x; M) = 0 for each x_{i}, and hence for all y ∈ (x_{1}, …, x_{n}).

1.3. If y = {y_{1}, …, y_{m}} with (y_{1}, …, y_{m}) = (x_{1}, …, x_{n}) and M is a complex, then

n − sup H(x; M) = m − sup H(y; M) .

Replacing y by {x_{1}, …, x_{n}, y_{1}, …, y_{m}}, we may assume that x ⊆ y. Inducing on m, we can further restrict our attention to y = {x_{1}, …, x_{n}, y}. Since y ∈ (x_{1}, …, x_{n}), 1.2 yields y H(x; M) = 0, and hence sup H(y; M) = sup H(x; M) + 1, by 1.1.

2. Depth of a complex

Let I be an finitely generated ideal in a commutative ring R, and let K be the Koszul complex on a set of n generators for I. For a complex of R-modules M, we introduce the I-depth of M over R as the number

depth_{R}(I, M) = n − sup(K ⊗_{R} M) .

By 1.3, it is independent of the choice of a generating set for I. When the ring R is clear from the context, we write depth(I, M) for the I-depth of M.

Observation. We catalogue a few elementary properties of depth:

1. If M ≃ N, then depth(I, M) = depth(I, N). Indeed, since K is a complex of free modules with K^{♮} bounded, (K ⊗_{R} M) ≃ (K ⊗_{R} N).

2. depth(I, M) = ∞ if and only if H(K ⊗_{R} M) = 0.

3. depth(I, M) = −∞ if and only if sup(K ⊗_{R} M) = ∞. In particular, depth(I, M) > −∞ when M is bounded above.

4. depth M[k] = depth M − k

5. For a module M over a noetherian ring R, one has depth_{R}(I, M) = 0 if and only if there is a prime ideal 𝔭 ∈ Ass M with I ⊆ 𝔭, cf. [4, 9.1].

For a complex M over a noetherian local ring (R, 𝔪, k), the depth of M is defined to be

depth_{R} M = depth_{R}(𝔪, M) ;

when the ring R is clear from the context, we write depth M.

The following theorem extends a result of Foxby [8, 12.ai], where it is proved under the more restrictive hypothesis that N is bounded.

Theorem 2.1. Let (R, 𝔪, k) be a noetherian local ring, and let N be a bounded above complex of R-modules.

If a complex M has finite flat dimension, then

depth_{R}(M⊗_{R}^{L}N) = \begin{cases}depth_{R} N − sup(M⊗_{R}^{L}k) & when H(M⊗_{R}^{L}k) ≠ 0 \\ +∞ & when H(M⊗_{R}^{L}k) = 0 . \end{cases}

If M is bounded below, and the homology modules of M are finitely generated, then there exists a complex of free R-modules F ≃ M such that ∂(F) ⊆ 𝔪F and sup F^{♮} = pd_{R} M, cf. [19, Chapter 2, 2.4]. In particular, sup(M⊗_{R}^{L}k) = pd_{R} M.

Therefore, the following is a special case of Theorem 2.1.

Corollary 2.2. If M is a complex with finitely generated homology modules, and pd_{R} M < ∞, then

depth_{R} N = depth_{R}(M⊗_{R}^{L}N) + pd_{R} M .  □

Note that, when M is an R-module, and N = R, the corollary reduces to the classical Auslander-Buchsbaum Equality: depth R = depth M + pd_{R} M.

The ‘unknown’ entity in the expressions in 2.1 and 2.2 is depth(M⊗_{R}^{L}N). Typically depth cannot be determined by the depth of the homology modules H_{n}(M⊗_{R}^{L}N) = Tor_{n}^{R} (M, N) . The next result will allow us to do so in certain special cases.

Theorem 2.3. Let L be a complex over a noetherian local ring (R, 𝔪, k). If sup L = s is finite, and depth H_{s}(L) − s ⩽ depth H_{i}(L) − i, for i ⩽ s, then

depth L = depth H_{s}(L) − s.

Before proving theorems 2.1 and 2.3, we present some corollaries.

The basic technique in this paper is to compare the depth of M⊗_{R}^{L}N obtained via 2.2 and 2.3. The proofs of the next two results illustrate this procedure.

The first was proved by Auslander, using different techniques, for finitely generated N, cf. [2, 1.2].

Corollary 2.4. Let M be a finitely generated module over a local ring R such that pd_{R} M < ∞. Let N be an R-module, and set s = sup Tor^{R} (M, N) . If either s = 0, or depth Tor_{s}^{R} (M, N) ⩽ 1, then

depth_{R} N = depth_{R} Tor_{s}^{R} (M, N) − s + pd_{R} M .

Proof. Since H(M⊗_{R}^{L}N) = Tor^{R} (M, N) , the hypotheses implies that M⊗_{R}^{L}N satisfies the conditions of 2.3. Thus

depth(M⊗_{R}^{L}N) = depth Tor_{s}^{R} (M, N) − s ,  by 2.3
depth N = depth(M⊗_{R}^{L}N) + pd_{R} M .  by 2.2

Combining these two equalities gives the desired result. □

An advantage in allowing N to be arbitrary is that one can deduce, using Hochster’s big Cohen-Macaulay modules, the Intersection Theorem for rings containing a field. This follows from the New Intersection Theorem, see remarks after 3.1, but we give a direct proof which offers a slightly different perspective on the result.

2.5. (Intersection Theorem.) Let R be a noetherian local ring containing a field. If M and N are finitely generated R-modules such that length(M ⊗_{R} N) is finite, then dim N ⩽ pd_{R} M.

Proof. The statement is trivial unless pd_{R} M is finite. Let N′ be a big Cohen-Macaulay module over R/ ann N, so that depth N′ = dim(R/ ann N) = dim N, cf. 3.2. Since M ⊗_{R} N has finite length, one has Supp M ∩ Supp N = {𝔪}, and hence Supp M ∩ Supp N′ = {𝔪}. A standard argument using localizations shows that Supp Tor_{s}^{R} (M, N′) = {𝔪}, for s = sup Tor^{R} (M, N′) , and hence that depth Tor_{s}^{R} (M, N′) = 0. Consequently,

depth N′ = −s + pd_{R} M

by 2.4. Therefore, dim N = depth N′ ⩽ pd_{R} M. □

The next corollary of Theorem 2.1 is well known for finitely generated modules, and extends a result in [8, Chapter 22], where it is proved for bounded complexes.

Corollary 2.6. Let ϕ: (R, 𝔪, k) ⟶ (R′, 𝔪′, k′) be a local homomorphism, and let M be a bounded above complex of R-modules. If ϕ is flat, then

depth_{R′}(M ⊗_{R} R′) = depth_{R} M + depth(R′/𝔪R′) .

Proof. Let K′ be the Koszul complex on a generating set for 𝔪′. Note that depth_{R′}(M ⊗_{R} R′) > −∞, since M is bounded above. Thus, the isomorphism of complexes (M ⊗_{R} R′) ⊗_{R′} K′ ≅ M ⊗_{R} K′ implies that sup(M ⊗_{R} K′) < ∞.

Since K′ is a bounded complex of flat R-modules, by 2.1

depth_{R}(M ⊗_{R} K′) = depth M − sup(K′ ⊗_{R} k)  (*)

When M ⊗_{R} K′ ≃ 0, one has depth M = ∞ by (*), and depth_{R′}(M ⊗_{R} R′) = ∞ by definition; hence equality holds.

In remains to consider the case where sup(M ⊗_{R} K′) is finite. By 1.2, for each integer i

𝔪 H_{i}(M ⊗_{R} K′) ⊆ 𝔪′ H_{i}(M ⊗_{R} K′) = 0 .

In particular, depth_{R} H_{i}(M ⊗_{R} K′) = 0 whenever H_{i}(M ⊗_{R} K′) ≠ 0, so by 2.3

depth_{R}(M ⊗_{R} K′) = − sup(M ⊗_{R} K′)
= depth_{R′}(M ⊗_{R} R′) − sup K′^{♮}

Combining this equation for depth_{R}(M ⊗_{R} K′) with the one in (*), and rearranging terms, yields

depth_{R′}(M ⊗_{R} R′) = depth M + sup K′ − sup(K′ ⊗_{R} k) .

But depth(R′/𝔪′R′) = sup K′^{♮} − sup(K′ ⊗_{R} k), as K′ ⊗_{R} k is a Koszul complex on a set of generators of the maximal ideal of R′/𝔪R′. This gives the required result. □

Now we prove theorems 2.1 and 2.3.

Proof of Theorem 2.1. To begin with, since s = sup N is finite, N is quasiisomorphic to τ_{⩽ s}(N). Thus, we may assume that N ≄ 0, and that N^{♮} itself is bounded above.

Let F ≃ M be a flat resolution of M with F^{♮} bounded; thus M⊗_{R}^{L}N ≃ F ⊗_{R} N and M⊗_{R}^{L}k ≃ F ⊗_{R} k. Let K be the Koszul complex on a generating set for 𝔪, and set G = K ⊗_{R} F ⊗_{R} N.

The spectral sequence of the filtration K ⊗_{R} (F_{⩽ p})⊗_{R} N of G converges to H(G). As F is a complex of flat modules, the spectral sequence has

1E_{p,q} = H_{q}(K ⊗_{R} N) ⊗_{R} F_{p} and ^{2}E_{p,q} = H_{p}(H_{q}(K ⊗_{R} N) ⊗_{R} F).

By 1.2, each H_{q}(K ⊗_{R} N) is a k-vector space, and so

2E_{p,q} = H_{q}(K ⊗_{R} N) ⊗_{k} H_{p}(F ⊗_{R} k)

If either H(M⊗_{R}^{L}k) = 0, or H(K ⊗_{R} N) = 0, then ^{2}E_{p,q} = 0 for all p, q. This shows that H(K ⊗_{R} N ⊗_{R} F) = 0, and hence that depth(N ⊗_{R} F) = ∞; this is the desideratum.

When H(M⊗_{R}^{L}k) ≠ 0, the expression above for ^{2}E_{p,q} shows that

2E_{p,q} = 0 for p > sup(F ⊗_{R} k) and q > sup(K ⊗_{R} N) ;
2E_{p,q} ≠ 0 for p = sup(F ⊗_{R} k) and q = sup(K ⊗_{R} N) .

Therefore, by a standard “corner” argument,

sup G = sup(K ⊗_{R} N) + sup(F ⊗_{R} k) ,

which is equivalent to

depth(F ⊗_{R} N) = depth N − sup(F ⊗_{R} k) = depth N − sup(M⊗_{R}^{L}k) .

This is the required equality. □

Proof of Theorem 2.3. Let K be the Koszul complex on a generating set for 𝔪. Since L ≃ L_{⩽ s}, we may assume that L^{♮} is bounded above. The spectral sequence of the filtration K_{⩽ p} ⊗_{R} L of G = K ⊗_{R} L converges to H(G) with

1E_{p,q} = K_{p} ⊗_{R} H_{q}(L) and ^{2}E_{p,q} = H_{p}(K ⊗_{R} H_{q}(L)).

Rewriting the hypotheses in the form

sup(K ⊗_{R} H_{q}(L)) ⩽ sup(K ⊗_{R} H_{s}(L)) + s − q for q ⩽ s

we get the first formula below (the second one is tautological) :

2E_{p,q} = 0 for p + q > sup(K ⊗_{R} H_{s}(L)) + s ;
2E_{p,q} ≠ 0 for p = sup(K ⊗_{R} H_{s}(L)) and q = s .

Furthermore, ^{2}E_{p,q} = 0 for q > s. The standard “corner argument” shows that

sup(K ⊗_{R} L) = sup(K ⊗_{R} H_{s}(L)) + s ,

and this implies that depth L = depth H_{s}(L) − s. □

Information on the depth of L_{i}, or on the depth of the homology modules H_{i}(L), leads to useful lower bounds on the depth of L:

Proposition 2.7. Let L ≄ 0 be a complex over a noetherian local ring (R, 𝔪, k).

(1) depth_{R} L ⩾ inf{ depth_{R} L_{i} − i | L_{i} ≠ 0}.

(2) depth_{R} L ⩾ inf{ depth_{R} H_{i}(L) − i | H_{i}(L) ≠ 0}.

Proof. Let K be the Koszul complex on a generating set for 𝔪, and set G = K ⊗_{R}L. (1) The spectral sequence of the filtration K ⊗_{R} L_{⩽ p} converges to H(G) with

1E_{p,q} = H_{q}(K ⊗_{R} L_{p}) .

If d is the right hand side of the expression in (1) above, then

sup H(K ⊗_{R} L_{p}) ⩽ sup K^{♮} − d − p for each p ,

and this forces ^{2}E_{p,q} = 0 for p+q > sup K^{♮} −d. As the spectral sequence converges, the same holds for ^{∞}E_{p,q}. In particular

sup(K ⊗_{R} L) ⩽ sup K^{♮} − d .

This is the desired result.

(2) Consider the spectral sequence associated to the filtration K_{⩽ p} ⊗_{R} L of G, used in the proof of 2.3, and argue as in the proof of (1) above. □

The next example shows that the inequalities in the proposition may be strict:

Example. Let R = k[[x, y]] be the power series ring over a field k, and consider

L: 0 ⟶ R ⟶^{∂_{2}} R^{3} ⟶^{∂_{1}} R ⟶ 0 with ∂_{2} = (\begin{matrix}0 \\ 0 \\ 1 \end{matrix}) , ∂_{1} = (x, y, 0) .

Clearly, H_{0}(L) = k, H_{1}(L) ≅ R and H_{2}(L) = 0, hence

inf{ depth_{R} L_{i} − i | L_{i} ≠ 0} = −1 ;
inf{ depth_{R} H_{i}(L) − i | H_{i}(L) ≠ 0} = 0

On the other hand, L is quasi-isomorphic to the complex M: 0 ⟶ R^{2} ⟶^{(x,y)} R ⟶ 0. Since pd_{R} M = 1, one has depth L = depth M = 1, by 2.2.

3. Intersection theorems

The purpose of this section is to prove a theorem that contains the equicharacteristic case of a series of “intersection theorems” in commutative algebra. When I is an 𝔪-primary, it specializes to the Improved New Intersection Theorem of Evans and Griffith, see [6, 1.6] and [10, 2.6]; it is not known whether that result holds for all local rings. When H_{0}(F) ≠ 0 has finite length, this is the New Intersection Theorem, proved by Hochster [11, 6.1], Peskine and Szpiro [17], and Paul Roberts [18]; Roberts [20] extended the New Intersection Theorem to all local rings.

Recall that a minimal generator of an R-module C is an element e ∈ C \ 𝔪C.

Theorem 3.1. Let (R, 𝔪, k) be a noetherian local ring containing a field, and let

F: 0 ⟶ F_{n} ⟶ … ⟶ F_{0} ⟶ 0

be a complex of finitely generated free R-modules. If length H_{i}(F) is finite for each i > 0, and I is an ideal which annihilates a minimal generator of H_{0}(F), then

n ⩾ dim R − dim(R/I) .

The result above is a variant of a theorem of Bruns and Herzog, cf. 3.5 below. Their proof uses Hochster’s big Cohen-Macaulay modules and the Buchsbaum-Eisenbud Acyclicity Criterion. Our proof of Theorem 3.1, based on techniques developed in Section 2, uses only big Cohen-Macaulay modules.

3.2. Big Cohen-Macaulay modules. An R-module N is said to be big Cohen-Macaulay if depth N = dim R. Such modules were constructed by Hochster [9] when R contains a field. Hochster [11, 5.4] has shown that such a ring also has a balanced big Cohen-Macaulay module N, that is, one for which every system of parameters of R is N-regular. The reader is referred to [4, Chapter 8] for details on the existence and properties of big Cohen-Macaulay modules. R. Sharp [21, 3.2] has shown that if N is a balanced big Cohen-Macaulay module, then dim R_{𝔭} ⩽ depth N_{𝔭} for each prime ideal 𝔭 ∈ Spec R; in particular, if depth N_{𝔭} is finite, then depth N_{𝔭} = dim R_{𝔭}.

When I is 𝔪-primary, the following lemma is proved in [6, 1.5].

Lemma 3.3. Let I be a ideal in a noetherian local ring R. Let C be an R-module, and and let e ∈ C \ 𝔪C be such that Ie = 0. If N is an R-module with N ≠ 𝔪N, then depth(C ⊗_{R} N) ⩽ dim R/I.

Proof. Set D = C ⊗_{R} N. The natural surjection D ⟶ (C/𝔪C) ⊗_{k} (N/𝔪N) maps e ⊗_{R} N onto a submodule isomorphic to N/𝔪N. In particular, there is an element f ∈ D \ 𝔪D, with If = 0.

We induce on the depth of D: the case where depth D = 0 is trivial.

Suppose that depth D > 0. By going to a faithfully flat extension (with a zero dimensional fiber), of R, we may assume that there is an x ∈ 𝔪, which is D-regular; see [4, 9.1.3]. Set J = ann(f), and note that I ⊆ J. As x is D-regular, x ∉ 𝔭, for each prime ideal 𝔭 ∈ Min(J). Furthermore, (J, x) annihilates the image of e in D/xD. As depth(D/xD) = depth D − 1, the induction step yields

depth D − 1 = depth(D/xD) ⩽ dim R/(J, x) ⩽ dim(R/J) − 1 ⩽ dim(R/I) − 1.

This is the desired inequality. □

Proof of Theorem 3.1. Let N be a balanced big Cohen-Macaulay R-module, cf. 3.2, set C = H_{0}(F) and s = sup H(F ⊗_{R} N).

Take a prime 𝔭 ∈ Ass H_{s}(F ⊗_{R} N), so that depth_{R_{𝔭}} H_{s}(F_{𝔭} ⊗_{R_{𝔭}} N_{𝔭}) = 0. Clearly L = F_{𝔭} ⊗_{R_{𝔭}} N_{𝔭} satisfies the condition in 2.3, so depth_{R_{𝔭}}(F_{𝔭} ⊗_{R_{𝔭}} N_{𝔭}) = −s. This, in conjunction with 2.2, yields

dim R_{𝔭} = depth_{R_{𝔭}} N_{𝔭} = depth_{R_{𝔭}}(F_{𝔭} ⊗_{R_{𝔭}} N_{𝔭}) + pd_{R_{𝔭}} F_{𝔭}  (*)
= −s + pd_{R_{𝔭}} F_{𝔭} .

If s > 0, then we claim that 𝔭 = 𝔪. Indeed, if 𝔭 ≠ 𝔪, then F_{𝔭} ≃ C_{𝔭}, and the expression above, along with the classical Auslander-Buchsbaum Equality yields a contradiction:

dim R_{𝔭} = −s + pd_{R_{𝔭}} C_{𝔭} ⩽ −s + depth R_{𝔭} < depth R_{𝔭} .

Thus, 𝔭 = 𝔪, hence (*) reduces to

dim R = −s + pd F ⩽ −s + n < n ,

which implies the inequality we seek.

Let now s = 0, that is, (F ⊗_{R} N) ≃ (C ⊗_{R} N). By assumption, there is an element e ∈ C \ 𝔪C such that Ie = 0, so depth(C ⊗_{R} N) ⩽ dim(R/I) by 3.3. Hence

dim R = depth_{R} N = depth_{R}(F ⊗_{R} N) + pd_{R} F  by 2.2
= depth_{R}(C ⊗_{R} N) + pd_{R} F
⩽ dim(R/I) + n

which is the desired result. □

In order to discuss the relation between Theorem 3.1 and the result of Bruns and Herzog mentioned above, we recall:

3.4. Buchsbaum-Eisenbud Acyclicity Criterion For a complex of finitely generated free R-modules

F: 0 ⟶ F_{n} ⟶^{ϕ_{n}} … ⟶ F_{1} ⟶^{ϕ_{1}} F_{0} ⟶ 0 ,

and 1 ≤ i ≤ n, set r_{i} = ∑_{j=i}^{n}(−1)^{j−i} rank F_{j} and let I_{r_{i}}(ϕ_{i}) be the ideal of r_{i} × r_{i} minors of ϕ_{i}. The following conditions on an R-module N are equivalent:

(a) F ⊗ N is acyclic;

(b) depth(I_{r_{i}}(ϕ_{i}), N) ≥ i for i = 1, …, n.

For a complex F as above, Bruns and Herzog introduce

codim F = inf{dim R − dim(R/I_{r_{i}}(ϕ_{i})) − i | for 1 ≤ i ≤ n} .

If codim F ≥ 0, and N is a balanced big Cohen-Macaulay module, then I_{r_{i}}(ϕ_{i}) contains a part of a system of parameters x_{1}, …, x_{i}, and so by the Acyclicity Criterion, F ⊗ N is acyclic, (the preceding remark is Lemma [4, 9.1.8]). Thus, the argument in the last paragraph of the proof of Theorem 3.1 yields:

3.5. Theorem [4, 9.4.1]. Let (R, 𝔪, k) be a noetherian local ring containing a field, and let F be a complex of finitely generated free R-modules. If codim F ≥ 0, and I is an ideal which annihilates a minimal generator of H_{0}(F), then

n ⩾ dim R − dim(R/I) .  □

Next, we deduce Theorem 3.1 from the theorem above. It suffices to assume that n ≥ dim R. Let 1 ≤ i ≤ n be an integer, and let 𝔭 ∈ Spec R be a prime ideal with height I_{r_{i}}(ϕ_{i}) = height 𝔭. If height 𝔭 = dim R, then height I_{r_{i}}(ϕ_{i}) ≥ dim R ≥ i. If height 𝔭 < dim R, then

height I_{r_{i}}(ϕ_{i}) ≥ depth(I_{r_{i}}(ϕ_{i}) ⊗_{R} R_{𝔭}, R) = depth(I_{r_{i}}(ϕ_{i} ⊗_{R} R_{𝔭}), R) ≥ i .

where the last inequality is by 3.4, since F_{𝔭} is acyclic. In either case, it follows that

dim R − dim(R/(I_{r_{i}}(ϕ_{i})) ≥ height I_{r_{i}}(ϕ_{i}) ≥ i ,

which shows that codim F ≥ 0. Now 3.5 gives n ⩾ dim R − dim(R/I).

4. Depth of tensor products

In this section, we explore the relation between the depths of M, N, and M ⊗_{R}N. The following consequence of Theorem 2.1 was pointed out to us by Foxby.

Theorem 4.1. Let (R, 𝔪, k) be a noetherian local ring, and let N be a complex this is bounded above.

If a complex M has finite flat dimension, then

depth(M⊗_{R}^{L}N) + depth R = depth M + depth N .

Proof. By 2.1, for any bounded above complex N, if H(M⊗_{R}^{L}k) = 0, then depth(M⊗_{R}^{L}N) = ∞. In particular, with N = R, one has depth M = ∞. Since N is bounded above, depth N > −∞, and the desired equality holds trivially.

Suppose that H(M⊗_{R}^{L}k) ≠ 0. In this case, one has

depth(M⊗_{R}^{L}N) = depth N − sup(M⊗_{R}^{L}k)
depth M = depth R − sup(M⊗_{R}^{L}k) ,

where the first equality is by 2.1, and the second equality is the special case N = R of the first. The desired expression is obtained by subtracting the second equality from the first, and rearranging terms. □

Modules M and N over a ring R are Tor-independent if Tor_{i}^{R} (M, N) = 0 for i > 0. This condition is equivalent to the fact that the canonical augmentation M⊗_{R}^{L}N ⟶ M ⊗_{R} N is a quasi-isomorphism. Thus, Theorem 4.1 specializes to

Corollary 4.2. If M and N are Tor-independent modules over a local ring R, and M has finite flat dimension, then

depth(M ⊗_{R} N) + depth R = depth M + depth N .  □

In the case where M and N are finitely generated, this can also be deduced from a theorem of Auslander [2, 1.2] (see Corollary 2.4 above). Recently, Huneke and Wiegand [12, 2.5] proved that this equation holds for any two finitely generated, Tor-independent modules over a complete intersection ring.

Following [1], we say that a module M ≠ 0 over a noetherian local ring R has finite CI-dimension if there exists a diagram of local homomorphisms R ⟶ R′ ⟵ Q such that R ⟶ R′ is flat, R′ ⟵ Q is surjective with kernel generated by a regular sequence, and the flat dimension of Q(M ⊗_{R} R′) over Q is finite.

Evidently, if pd_{R} M < ∞, then M has finite CI-dimension. Furthermore, it is easy to see that all modules over a complete intersection have finite CI-dimension. Hence, the following theorem extends both results above.

Theorem 4.3. Let M and N be Tor-independent modules over a noetherian local ring R. If M has finite CI-dimension, then

depth(M ⊗_{R} N) + depth R = depth M + depth N .

Remark . The statement of the theorem imposes two homological conditions on the modules M and N: the first is that they are Tor-independent, and the second is that one of them has finite CI-dimension.

When M and N are not Tor-independent, the equation above may fail to hold, even over a regular ring R. Indeed, by the Auslander-Buchsbaum Equality the assertion in 4.3 is equivalent to pd(M ⊗_{R} N) = pd M + pd N.

When dim R > 0, for M = N = k one has

pd(M ⊗_{R} N) = dim R < 2 dim R = pd M + pd N .

On the other hand, since R is a UFD, if an ideal J is two-generated, then pd(R/J) ⩽ 2. Burch [5] constructs an ideal I = (a, b, c) with pd(R/I) = dim R. Thus, for M = R/(a, b) and N = R/c, we get pd(M ⊗_{R} N) > pd M + pd N, whenever dim R > 3.

The proof of Theorem 4.3 (which is new also when R is a complete intersection) starts with the following:

Computation 4.4. Let x = x_{1}, …, x_{c} be a regular sequence in a commutative ring Q. The Koszul complex K(x) is a Q-free resolution of R = Q/(x). Thus, if M is a Q-module annihilated by (x), then

Tor_{q}^{Q} (M, R) = H_{q}(x; M) = ∧^{q}(R^{c}) ⊗_{R} M, for all q ⩾ 0 .

This generalizes to:

Lemma 4.5. Let x = x_{1}, …, x_{c} be a regular sequence in a commutative ring Q. If M and N are Tor-independent modules over R = Q/(x), then

Tor_{n}^{Q} (M, N) = ∧^{n}(R^{c}) ⊗_{R} (M ⊗_{R} N) for all n ⩾ 0 .

Proof. Consider the standard change of rings spectral sequence with

2E_{p,q} = Tor_{p}^{R} (Tor_{q}^{Q} (M, R) , N) ⟹ Tor_{p+q}^{Q} (M, N) .

Using 4.4, we get

2E_{p,q} = ∧^{q}(R^{c}) ⊗_{R} Tor_{p}^{R} (M, N) .

Since M and N are Tor-independent over R, we have ^{2}E_{p,q} = 0 for p > 0, and hence Tor_{n}^{Q} (M, N) = ^{2}E_{0,n}, hence the required equality. □

Proof of Theorem 4.3. Consider a diagram of local homomorphisms R ⟶ R′ ⟵ Q given by the hypothesis that the CI-dimension of M is finite. As R ⟶ R′ is flat,

Tor_{p}^{R′} (M ⊗_{R} R′, N ⊗_{R} R′) ≅ Tor_{p}^{R} (M, N) ⊗_{R} R′ , for all p > 0.

Thus, M′ and N′ are Tor-independent R′-modules, as M and N are Torindependent. Furthermore,

depth_{R′}(L ⊗_{R} R′) = depth_{R} L + depth(R′/𝔪R′)

for any R-module L, cf. 2.6. Thus, it is enough to prove the depth formula for the R′-modules M ⊗_{R} R′ and N ⊗_{R} R′. Replacing R′ by R, we may assume that R = Q/(x), with x = x_{1}, …, x_{c} a Q-regular sequence, and that the flat dimension of M over Q is finite.

Note that depth_{R} L = depth_{Q} L for any R-module L, since Q ⟶ R is surjective. Hence, it is enough to establish that

depth_{Q}(M ⊗_{R} N) + depth_{Q} R = depth_{Q} M + depth_{Q} N

If M ⊗_{R} N = 0, then 4.5 shows that Tor_{n}^{Q} (M, N) = 0, for all n, so M and N are Tor-independent over Q. Therefore, depth_{Q} M + depth_{Q} N = depth_{Q}(M ⊗_{Q} N) = ∞, by 4.2, and the depth formula holds.

Consider the case where M ⊗_{R} N ≠ 0. Lemma 4.5 tells us two things: first, that sup Tor^{Q} (M, N) = c; second that depth_{Q} Tor_{p}^{Q} (M, N) = depth_{Q}(M ⊗_{R} N), for all 0 ⩽ p ⩽ c. Thus, the complex H(M⊗_{Q}^{L}N) satisfies the conditions of 2.3, since H(M⊗_{Q}^{L}N) = Tor^{Q} (M, N) , and hence

depth_{Q}(M⊗_{Q}^{L}N) = depth_{Q} Tor_{c}^{Q} (M, N) − c = depth_{Q}(M ⊗_{R} N) − c .

Furthermore, 4.1 applies, as the flat dimension of M over Q is finite, so

depth_{Q}(M⊗_{Q}^{L}N) = depth_{Q} M + depth_{Q} N − depth Q .

Combining the two equalities, and rearranging terms yields

depth_{Q}(M ⊗_{R} N) + depth Q − c = depth_{Q} M + depth_{Q} N .

As x is a regular sequence, depth_{Q} R = depth Q−c. This is the required result. □

5. Properties of Depth

In this section we show that many of the properties of depth for modules extend to identical statements for complexes.

The first result captures the behavior of depth on short exact sequences.

Proposition 5.1. Let R be a commutative ring. Consider a short exact sequence of complexes of R-modules:

0 ⟶ L ⟶ M ⟶ N ⟶ 0 .

For each finitely generated ideal I of R, there are inequalities

depth(I, M) ⩾ min{depth(I, L), depth(I, N)} ,
depth(I, N) ⩾ min{depth(I, L) − 1, depth(I, M)} ,
depth(I, L) ⩾ min{depth(I, M), depth(I, N) + 1} .

Proof. Let K be the Koszul complex on a system of generators for I. As K is a complex of free modules, the sequence of complexes

0 ⟶ K ⊗_{R} L ⟶ K ⊗_{R} M ⟶ K ⊗_{R} N ⟶ 0 ,

is exact. A routine analysis of the homology long exact sequence yields the required inequalities. □

The next proposition tracks depth under change of rings or ideals.

Proposition 5.2. Let ϕ: R ⟶ R′ be a homomorphism of rings, and let I and I′ be finitely generated ideals in R and R′, respectively.

For a complex of R′-modules N

(1) depth_{R}(I, N) = depth_{R′}(IR′, N)

(2)  depth_{R}(I, N) ⩽ depth_{R′}(I′, N)  if ϕ(I) ⊆ I′.

For a complex of R-modules M

(3) depth_{R}(I, M) ⩽ depth_{R′}(IR′, M ⊗_{R} R′) if ϕ is flat.

(4) depth_{R}(I, M) = depth_{R′}(IR′, M ⊗_{R} R′) if ϕ is faithfully flat.

Proof. (1) If K is the Koszul complex on a set of generators for I, then K′ = K⊗_{R}R′ is the Koszul complex on a set of generators for IR′. The equality that we seek follows from

K ⊗_{R} N = K ⊗_{R} S ⊗_{S} N = K′ ⊗_{S} N .

(2) By (1), it is enough to show that depth_{R′}(IR′, N) ⩽ depth_{R′}(I′, N)

Let x be a finite set of generators for IR′, and take a generating set for I′ of the form x ∪ {y_{1}, …, y_{m}}. By inducing on m, we may restrict ourselves to the case where m = 1. The inequality follows from the long exact sequence in 1.1.

(3) and (4) Let K and K′ be as above. Since

K′ ⊗_{R′} (M ⊗_{R} R′) = (K ⊗_{R} M) ⊗_{R} R′

and R ⟶ R′ is flat, sup(K′ ⊗_{R′} (M ⊗_{R} R′)) ⩽ sup(K ⊗_{R} M), with equality when R ⟶ R′ is faithfully flat. □

For a module over a noetherian ring R, it is well known that I-depth may be computed locally. The next proposition extends this to complexes, and characterizes those primes at which the I-depth is attained. In it, we denote by V(I) the set of prime ideals containing I.

Proposition 5.3. Let R be a noetherian ring, and let M be a bounded above complex of R-modules. Let I be an ideal in R, and let K be the Koszul complex on a finite set x of generators for I, and set s = sup H(K ⊗_{R} M)

(1) The R-module H_{s}(K ⊗_{R} M) is independent up to isomorphism of the choice of x.

(2) If depth_{R}(I, M) = d is finite, then

{𝔭 ∈ V(I) | depth_{R_{𝔭}} M_{𝔭} = d} = Ass_{R} H_{s}(K ⊗_{R} M) .

Proof. (1) Suppose that y = {y_{1}, …, y_{n}} generates I. As in the proof of 1.3, one can reduce to the case where y = x ∪ y. Since y ∈ (x), one has y H(x; M) = 0, by 1.2, and hence H_{s}(x; M) ≅ H_{s+1}(y; M), by 1.1.

(2) Since I ⊂ 𝔭, the complex K_{𝔭} is minimal, and so pd_{R_{𝔭}} K_{𝔭} = sup K^{♮}. In particular

depth_{R_{𝔭}} M_{𝔭} = depth_{R_{𝔭}}(K_{𝔭} ⊗_{R_{𝔭}} M_{𝔭}) + sup K^{♮}  (*)

by 2.2. Note that 𝔭 ∈ Ass H_{s}(K ⊗_{R} M) if and only if depth H_{s}(K_{𝔭} ⊗_{R_{𝔭}} 𝔭) = 0.

If depth_{R_{𝔭}} H_{s}(K_{𝔭} ⊗_{R_{𝔭}} M_{𝔭}) = 0, then L = K_{𝔭} ⊗_{R_{𝔭}} M_{𝔭} satisfies the condition in 2.3. This implies that depth_{R_{𝔭}}(K_{𝔭} ⊗_{R_{𝔭}} M_{𝔭}) = −s, and so equation (*) reduces to

depth_{R_{𝔭}} M_{𝔭} = −s + sup K^{♮} = depth_{R}(I, M) .

On the other hand, if depth_{R_{𝔭}} H_{s}(K_{𝔭} ⊗_{R_{𝔭}} M_{𝔭}) > 0, then

depth_{R_{𝔭}} H_{i}(K_{𝔭} ⊗_{R_{𝔭}} M_{𝔭}) − i > −s for i ⩽ sup(K_{𝔭} ⊗_{R_{𝔭}} M_{𝔭})

since sup(K_{𝔭} ⊗_{R_{𝔭}} M_{𝔭}) ⩽ s. Thus, depth_{R_{𝔭}}(K_{𝔭} ⊗_{R_{𝔭}} M_{𝔭}) > −s, by 2.7.2, and so (*) yields

depth_{R_{𝔭}} M_{𝔭} > −s + sup K^{♮} = depth_{R}(I, M) .  □

Proposition 5.4. Let I be an ideal in a noetherian ring R. If a complex M is bounded above, then

depth_{R}(I, M) = inf{depth_{R_{𝔭}} M_{𝔭} | 𝔭 ∈ V(I)} .

Proof. By 5.2.2, the inclusion IR_{𝔭} ⊆ 𝔭R_{𝔭} yields the first inequality below:

depth M_{𝔭} ⩾ depth(I_{𝔭}, M_{𝔭}) ⩾ depth(I, M) ;

the second inequality holds as the homomorphism R ⟶ R_{𝔭} is flat, cf. 5.2.3.

Now we may assume that depth(I, M) is finite. Thus, there exists a prime ideal 𝔭 ∈ V(I) such that depth(I, M) = depth M_{𝔭}, by 5.3. □

Proposition 5.5. Let I and J be finitely generated ideals in a commutative ring R. For a complex M that is bounded above

(1) depth(IJ, M) = min{depth(I, M), depth(J, M)}.

(2)  depth(IJ, M) = depth(I ∩ J, M)  if I ∩ J is finitely generated.

(3) depth(J, M) = depth(I, M) if rad(J) = rad(I) and R is noetherian.

If, in addition, R is a noetherian local ring, and M has finitely generated homology modules, then

(4) depth M ⩽ depth(I, M) + dim R/I .

Proof. (1) To begin with, depth(IJ, M) ⩽ min{depth(I, M), depth(J, M)}, by 5.2.2. Now, we may assume that depth(IJ, M) is finite. Furthermore, in view of 5.2.1, a standard argument allows us to assume that R is noetherian. By Proposition 5.4, there exists a prime ideal 𝔭 ⊇ IJ with depth M_{𝔭} = depth(IJ, M).

As IJ ⊆ 𝔭, it must be that I ⊆ 𝔭 or J ⊆ 𝔭. Assume, without loss of generality, that I ⊆ 𝔭. By 5.4, depth_{R}(I, M) ⩽ depth M_{𝔭} . Thus depth(I, M) ⩽ depth(IJ, M), and we are done.

(2) When I ∩ J is finitely generated, the required result follows from (1) and the obvious inequalities

depth(IJ, M) ⩽ depth(I ∩ J, M) ⩽ min{depth(I, M), depth(J, M)} .

(3) Clearly, we may assume that J = rad(I). As R is noetherian, we can choose an integer n such that J^{n} ⊆ I. The result follows from

depth(J, M) = depth(J^{n}, M) ⩽ depth(I, M) ⩽ depth(J, M) ,

where the equality comes from (1), and the inequalities from 5.2.2.

(4) Since R is a noetherian local ring, we can pick elements y = {y_{1}, …, y_{n}} ⊂ R, whose image under the canonical surjection R ⟶ R/I forms a system of parameters for R/I. In particular, rad(I, y) = 𝔪 and n = dim R/I.

Since H_{i}(M) is finitely generated for each i, a standard argument using 1.1 shows that depth((I, y_{1}), M) ⩽ depth(I, M) + 1, and hence

depth(𝔪, M) = depth((I, y_{1}, …, y_{n}), M) ⩽ depth(I, M) + n ,

where the equality follows from rad(I, y) = 𝔪 and (3) above. Since depth M = depth(𝔪, M), we are done. □

6. Depth sensitivity

For modules over a noetherian ring, it is well known that the depth with respect to an ideal may be computed via injective resolutions. In this section, we extend this to complexes, under certain conditions on the homology modules. (Refer to 6.5 for a recap on injective resolutions of complexes.)

Theorem 6.1. Let I be an ideal in a noetherian ring R. If M is a bounded above complex of R-modules, then

depth_{R}(I, M) = inf{i | Ext_{R}^{i} (R/I, M) = 0} .

Moreover, when depth_{R}(I, M) = d is finite

{𝔭 ∈ V(I) | depth_{R_{𝔭}} M_{𝔭} = d} = Ass_{R} Ext_{R}^{d} (R/I, M) .

In particular, with I = 𝔪, the theorem shows that our notion of depth for complexes, given in Section 1, extends that of Foxby [8, Chapter 12], and Iversen [14].

The next result relates depth to local cohomology modules, cf. [22].

Theorem 6.2. Let I be an ideal in a noetherian ring R. If M is a bounded above complex of R-modules, then

depth_{R}(I, M) = inf{i | H_{I}^{i}(M) ≠ 0}.

Moreover, when depth_{R}(I, M) = d is finite

{𝔭 ∈ V(I) | depth_{R_{𝔭}} M_{𝔭} = d} = Ass_{R} H_{I}^{d}(M) .

Note . When depth_{R}(I, M) is finite, the theorem above, and Proposition 5.3 yield: Ass_{R} H_{I}^{d}(M) = Ass_{R} H_{s}(K ⊗_{R} M), where K is the Koszul complex on a generating set for I, and s = sup(K ⊗_{R} M). In particular, if the R-module H_{i}(M) is finitely generated for all i, then Ass H_{I}^{d}(M) is finite. This fact is folklore, at least when M is a finitely generated R-module.

We deduce the preceding theorems from the result below. It identifies a large family of complexes which may be used, in lieu of the Koszul complex, to measure depths of complexes.

Lemma 6.3. Let I be an ideal in a noetherian ring R, M a complex of R-modules, and let F be a complex of flat R-modules

0 → F_{n} ⟶^{∂_{n}} F_{n−1} ⟶ ⋯

such that F_{n} is faithfully flat and ∂_{n}(F_{n}) ⊆ IF_{n−1}. If either M^{♮} is bounded above and (F ⊗_{R} M)_{𝔭} is exact for each prime ideal 𝔭 ∈ Spec(R) \ V(I), or M is bounded above, F^{♮} is bounded, and F_{𝔭} is exact for each prime ideal 𝔭 ∈ Spec(R) \ V(I), then

depth_{R}(I, M) = n − sup(F ⊗_{R} M) .

Moreover, if depth_{R_{𝔭}}(I, M) = d is finite, then

{𝔭 ∈ V(I) | depth_{R_{𝔭}} M_{𝔭} = d} = Ass_{R} H_{s}(F ⊗_{R} M)
where s = sup(F ⊗_{R} M) .

Proof. When F^{♮} is bounded, if s = sup M is finite, then F ⊗_{R}M is quasi-isomorphic to F ⊗_{R} τ_{⩽ s}(M). Moreover, if F_{𝔭} exact, then (F ⊗_{R} M)_{𝔭} is exact. Thus, it suffices to consider the case where M^{♮} is bounded above and (F ⊗_{R} M)_{𝔭} is exact for each prime ideal 𝔭 ∈ Spec(R) \ V(I).

Let K be the Koszul complex on a finite set of generators for I, and set G = K ⊗_{R} F ⊗_{R} M. The hypotheses ensure that G^{♮} is bounded above.

The spectral sequence of the filtration K ⊗_{R} (F_{⩽ p}) ⊗_{R} M converges to H(G). Arguing as in the proof of 2.1 (with 𝔪 replaced by I), one sees that

sup G = sup(K ⊗_{R} M) + n .  (*)

A similar argument with the spectral sequence of the filtration K_{⩽ p} ⊗_{R} F ⊗_{R} M, this time using the fact that (F ⊗_{R} M)_{𝔭} is exact for each 𝔭 ∈ Spec(R) \ V(I), yields

sup G = sup(F ⊗_{R} M) + sup K^{♮} .

Thus, we have an equality: sup(F ⊗_{R} M) + sup K^{♮} = sup(K ⊗_{R} M) + n, and this gives the expression for depth M, upon rearrangement.

The proof of the statement regarding the associated primes is along the lines of that of 5.3. Note that (*) is equivalent to depth_{R}(I, M) = depth_{R}(I, F ⊗_{R} M) + n. Thus, for any prime ideal 𝔭 ∈ V(I), since ∂_{n}((F_{n})_{𝔭})) ⊆ 𝔭(F_{n−1})_{𝔭}, one has

depth_{R_{𝔭}} M_{𝔭} = depth_{R_{𝔭}}(F_{𝔭} ⊗_{R_{𝔭}} M_{𝔭}) + n .

For the complex L = F_{𝔭} ⊗_{R_{𝔭}} M_{𝔭}, by 2.3, we get depth L ⩾ −s, with equality if and only if 𝔭 ∈ Ass_{R} H_{s}(F ⊗_{R} M). This, and the equality above, give the required result. □

A special case of the lemma above is given in

Corollary 6.4. Let (R, 𝔪, k) be a noetherian local ring, and let

F: 0 ⟶ F_{n} ⟶ ⋯ ⟶ F_{0} ⟶ 0

be a complex of free R-modules with ∂(F) ⊆ 𝔪F and H_{i}(F) of finite length, for each integer i. If M is a bounded above complex, then

depth M = n − sup(F ⊗_{R} M).  □

Before proving Theorem 6.1, we recall some basic facts concerning injective resolutions of complexes, cf. [22].

6.5. Injective resolutions. When E is a complex of injective modules with E^{♮} bounded above, if P ≃ P′, then Hom_{R} (P, E) ≃ Hom_{R} (P′, E). A bounded above complex M has an injective resolution, that is, a complex E ≃ M, with each E_{i} an injective module, and E^{♮} bounded above. The symbol RHom_{R} (N, M) denotes any complex quasi-isomorphic to Hom_{R} (N, E), for a complex of R-modules N. Set

Ext_{R}^{i} (N, M) = H_{−i}(RHom_{R} (N, M)) .

When M and N are R-modules, this gives the classical notion. If P ≃ N is is a projective resolution of N then,

Hom_{R} (N, E) ≃ Hom_{R} (P, E) ≃ Hom_{R} (P, M) .

Thus, any one of the complexes above may be used to compute Ext_{R} (N, M) .

Proof of Theorem 6.1. This equality holds trivially when M ≃ 0. Consider the case where sup M = s is finite; thus M ≃ M_{⩽ s}. Since depth(I, M) = depth(I, M_{⩽ s}) and Ext_{R} (R/I, M) ≅ Ext_{R} (R/I, M_{⩽ s}) , we may replace M with M_{⩽ s} and assume that M ≄ 0 and M^{♮} is bounded above. Let P be a minimal free resolution of R/I, and set F = Hom_{R} (P, R). Since Ext_{R} (R/I, M)_{𝔭} = 0 for each prime ideal 𝔭 ∈ V(I), the complexes F and M satisfy the conditions of Lemma 6.3. Thus,

depth_{R}(I, M) = sup F^{♮} − sup(F ⊗_{R} M) = − sup(F ⊗_{R} M) .

Note that F ⊗_{R} M ≅ Hom_{R} (P, M), and hence Ext_{R}^{i} (R/I, M) = H_{−i}(F). The assertion on the associated primes is also contained in 6.3. □

Let x_{1}, …, x_{n} be a set of generators for the ideal I. For U = {s_{1}, …, s_{i}}, we set R_{U} = R_{x_{s_{1}} ···x_{s_{i}}}. The complex

C: ⋯ ⟶ ⨁_{\begin{matrix}U⊆\{1,...,n\} \\ card(U)=i \end{matrix}} R_{U} ⟶^{∂} ⨁_{\begin{matrix}V ⊆\{1,...,n\} \\ card(V)=i+1 \end{matrix}} R_{V} ⟶ ⋯

where the component of ∂ on R_{U} ⟶ R_{V} is (−1)^{j} times the canonical localization map R_{U} ⟶ (R_{V})_{x_{s_{j}}} if U = V \ {s_{j}}, and 0 otherwise, is called the Čech complex on I. More often than not, it is defined as a cohomological complex (with upper indices). The notation used here is in line with that of the rest of the paper.

Proof of Theorem 6.2. Let C be the complex described above. It is well known that local cohomology modules may be computed via Čech complexes; for example, see [4, §3.5], or [15, §3]:

H_{I}^{i}(M) ≅ H_{n−i}(C ⊗_{R} M) .

Since C satisfies the hypotheses of Lemma 6.3, one has

depth_{R}(I, M) = n − sup(C ⊗_{R} M) = inf{i | H_{I}^{i}(M) ≠ 0} ,

and also the statement regarding the associated primes. □

Acknowledgement . This paper has evolved over numerous conversations with my thesis advisor Luchezar L. Avramov. I should like to thank him for raising the right questions. I also thank Hans-Bjørn Foxby for valuable suggestions on an earlier version, and the referee for drawing my attention to Theorem 3.5. Anders Frankild pointed out the inaccuracy in the published version of Lemma 6.3; I thank him for the same.

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Department of Mathematics, Purdue University, W. Lafayette, IN 47907, U.S.A.

E-mail address: iyengar@math.purdue.edu