Contemporary Mathematics

Depth and amplitude for unbounded complexes

Hans–Bjørn Foxby and Srikanth Iyengar

Abstract. We prove that over a commutative noetherian ring the three approaches to introducing depth for complexes: via Koszul homology, via Ext modules, and via local cohomology, all yield the same invariant. Using this result, we establish a far reaching generalization of the classical Auslander-Buchsbaum formula for the depth of finitely generated modules of finite projective dimension. We extend also Iversen’s amplitude inequality to unbounded complexes. As a corollary we deduce: Given a local homomorphism Q → R, if there is a non-zero finitely generated R-module that has finite flat dimension both over Q and over R, then the flat dimension of R over Q is finite. This last result yields a module theoretic extension of a characterization of regular local rings in characteristic p due to Kunz and Rodicio

Introduction

This paper concerns the theory of depth for complexes over commutative noetherian rings. Let 𝔞 be an ideal in such a ring R and K the Koszul complex on a finite set of n generators for 𝔞. It is well known that the following numbers are equal when M is a finitely generated R-module:

• n − sup{ ℓ ∈ ℤ | H_{ℓ}(K ⊗_{R} M) ≠ 0 };

• inf{ ℓ ∈ ℤ | Ext_{R}^{ℓ} (R/𝔞, M) ≠ 0 };

• inf{ ℓ ∈ ℤ | H_{𝔞}^{ℓ} (M) ≠ 0 }, where H_{𝔞}^{∗} (M) is the 𝔞-local cohomology of M. If R is local, then this common value coincides with the length of the longest

M-regular sequence in 𝔞, that is to say, the 𝔞-depth of M. Since each of the quantities displayed above is meaningful even when M is a complex of R-modules, they can, and have been, taken as a starting point for building a theory of depth for complexes. These have proved to be of immense utility even in dealing with problems concerning modules alone.

The natural question arises: Do these three approaches yield the same invariant? Results of Foxby and Iyengar settle this question in the affirmative for about a third of the complexes. Namely, those whose homology is bounded above, noteworthy among these being complexes with bounded homology and that includes also all modules, finitely generated or not.

\begin{matrix}c \\ ◯ \end{matrix}2000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 13C15, 13C25. Secondary 18G15, 13D45.

H.–B. F. was partially supported by the Danish Natural Science Research Council.

S. I. was supported by a grant from the E.P.S.R.C.

One of our main results answers the question with an unqualified: Yes.

Theorem I. For any complex of R-modules M the numbers obtained from the three formulas above coincide.

Thus one may now speak of the depth of a complex without having to specify which of the preceding formulas is being used to define it. This flexibility in computing depth allows one to extend many familiar results on depth for modules to identical statements for arbitrary complexes.

For instance, calculating depth via the vanishing of Ext leads to the following theorem concerning the depth (with respect to the maximal ideal) of complexes over local rings. It is a vast generalization of the classical Auslander-Buchsbaum equality: depth R = depth_{R} P +pd_{R} P for any finitely generated R-module P whose projective dimension pd_{R} P is finite.

Theorem II. Let R be a local ring and let M and P be complexes of R-modules. If pd_{R} P is finite and H(P) is nonzero and finitely generated, then

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

Here M ⊗_{R}^{L} P stands for M ⊗_{R} F, where F is any finite free resolution of P. Our proof of the next result makes critical use of the characterization of depth

via Koszul homology and also of the preceding theorem. Recall that the amplitude of M is the number

amp M = sup{ℓ | H_{ℓ}(M) ≠ 0} − inf{ℓ | H_{ℓ}(M) ≠ 0} .

We write fd_{R} F for the flat dimension of F over R.

Theorem III. Let R → S be a local homomorphism and F a complex of Smodules with H(F) nonzero and finitely generated. If fd_{R} F is finite, then for any complex of R-modules M with H(M) degreewise finite one has

amp M ≤ amp (M ⊗_{R}^{L} F ) .

The first step in our proof of this theorem is a reduction to the case where S = R; this is readily accomplished by using the theory of Cohen factorizations of Avramov, Foxby, and Herzog [3]. Now, when H(M) is bounded the result we seek is precisely the amplitude inequality, proved by Iversen [14]; it is an avatar of Paul Roberts’ remarkable New Intersection Theorem [21]. Thus, the new information provided by our theorem concerns unbounded complexes; the issues that come into play in proving it are of a different nature and not as involved. Nevertheless, as the following corollary demonstrates, it too has its uses.

Theorem IV. Let Q → R → S be local homomorphisms and F a nonzero finitely generated S-module. Assume that fd_{R} F < ∞. Then

fd_{Q} R ≤ fd_{Q} F ≤ fd_{Q} R + fd_{R} F .

In particular, fd_{Q} R and fd_{Q} F are finite simultaneously.

The inequality on the right is classical. Since regularity descends along maps of finite flat dimension, the one on the left implies that if R is regular and fd_{Q} F is finite, then Q is regular. The special case where R = S of this latter result had been discovered by Apassov [5].

Theorem IV yields also the following characterization of regular rings of characteristic p that extends those of Kunz [18] and Rodicio [22]. In its statement, ^{φ^{n}}F denotes the R module structure on F induced by restriction of scalars along φ^{n}.

Theorem V. Let R be a local ring of characteristic p, and let φ: R → R be the Frobenius endomorphism of R. The following conditions are equivalent.

(a) R is regular;

(b) φ^{n} is flat for each integer n ≥ 0;

(c) there exists a positive integer n and a nonzero finitely generated R-module F such that both fd_{R} F and fd_{R}(^{φ^{n}}F) are finite.

The results in this paper have further applications to the study of the Frobenius endomorphism. These will be dealt with in [4] and [17].

Theorem IV has implications also for flat dimensions of homomorphisms. In the sequel, the flat dimension of any homomorphism ψ: R → S is the flat dimension of S over R; it is denoted fd ψ.

Consider a diagram of local homomorphisms Q ⟶^{φ} R ⟶^{ψ} S. As is well known, if both φ and ψ have finite flat dimension, then so does the composition ψ ∘ φ. The ‘factorization problem’ for maps of finite flat dimension reads: Suppose that fd(ψ ∘ φ) is finite. What can be said about fd φ and fd ψ? We are now in a position to give a complete answer to this question.

Indeed, if fd ψ is finite, then fd φ is also finite; this is by Theorem IV.

On the other hand, it could happen that fd φ is finite, but fd ψ is not: Let (Q, 𝔪, k) be a regular local ring and f a nonzero element in 𝔪^{2}. Then the diagram Q → Q/(f) → k furnishes the requisite counterexample.

Theorems I and II are proved in Section 2. That section establishes also the basic properties of depth for complexes. Most of these generalize results of Iversen, Foxby, and Iyengar. The proofs of Theorems III, IV, and V are contained in Section 3. Section 4 concerns certain results on width for complexes analogous to those on depth described in Section 2.

Most of our calculations occur in the derived category of modules for they are most transparent there. For this reason, in Section 1, we collect a few useful facts concerning the same, as well as results on local cohomology and derived completions required in this work.

We stumbled upon the main results described in this paper at the conference in Grenoble.^{1} It is a pleasure to thank the organizers for inviting us, and also for giving us an opportunity to present our results in this forum.

1. Homological algebra

Let R be a ring; the standing assumption in this paper is that rings are commutative and noetherian. We are concerned with complexes of the form

⋯ → M_{ℓ+1} ⟶^{∂_{ℓ+1}} M_{ℓ} ⟶^{∂_{ℓ}} M_{ℓ−1} → ⋯

Given a complex M of R-modules set

sup M = sup{ℓ | H_{ℓ}(M) ≠ 0} and inf M = inf{ℓ | H_{ℓ}(M) ≠ 0} ,

with the understanding that sup ∅ = −∞ and inf ∅ = ∞. When sup M < ∞ (respectively, inf M > ∞), one says that H(M) is bounded above (respectively, bounded below). Naturally, H(M) is bounded if it is bounded both above and below. We label H(M) degreewise finite if the R-module H_{i}(M) is finitely generated for each integer i; it is labelled finite if, in addition, it is bounded.

1Between the morning and afternoon sessions. Not during.

Associated to any complex M and integers i, s are the truncated complexes

τ_{⩾ i} (M) : ⋯ → M_{i+2} → M_{i+1} → Ker(∂_{i}) → 0 ,

τ_{⩽ s} (M) : 0 → \frac{M_{s}}{∂(M_{s+1})} → M_{s−1} → M_{s−2} → · · · .

The canonical inclusion τ_{⩾ i} (M) → M induces an isomorphism in homology in degrees ≥ i, whilst the map in homology induced by the surjection M → τ_{⩽ s} (M) is an isomorphism in degrees ≤ s. In particular, if i ≤ inf M (respectively, s ≥ sup M), then the morphism τ_{⩾ i} (M) → M (respectively, M → τ_{⩽ s} (M)) is a quasiisomorphism, that is to say, a homology isomorphism.

The nth suspension of M is the complex Σ^{n}M with (Σ^{n}M)_{i} = M_{i−n} and differential given by ∂(m) = (−1)^{|n|}∂_{i−n}(m) for m ∈ (Σ^{n}M)_{i}.

Koszul homology. For an element x in the ring R denote K(x) the complex 0 → R ⟶^{x} R → 0 concentrated in degrees 0 and 1. The Koszul complex on a sequence x = { x_{1}, …, x_{n} } is the complex

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

In the sequel we take recourse to the following properties of Koszul complexes. 1.1. Let K be the Koszul complex on an element x in R and let M be a complex.

There is a long exact sequence

⋯ → H_{i}(M) ⟶^{x} H_{i}(M) → H_{i}(K ⊗_{R} M) → H_{i−1}(M) ⟶^{x} · · · .

Indeed, the canonical inclusion R = K_{0} ↪ K extends to a short exact sequence of complexes of free modules 0 → R → K → ΣR → 0. When tensored with M, this induces the exact sequence of complexes

0 → M → K ⊗_{R} M → ΣM → 0 .

The corresponding homology long exact sequence is the one we seek.

1.2. Let K be the Koszul complex on a set of elements { x_{1}, …, x_{n} } in R, and set 𝔞 = (x_{1}, …, x_{n}). For any complex of R-modules M one has

𝔞 H(K ⊗_{R} M) = 0 and 𝔞 H(Hom_{R} (K, M) ) = 0 .

This follows from the fact that multiplication by x_{i} is homotopic to 0 on K(x_{i}), and so on K ⊗_{R} M and Hom_{R} (K, M) . Thus, x_{i} annihilates both H(K ⊗_{R} M) = 0 and H(Hom_{R} (K, M) ), so the same holds for any element in 𝔞.

1.3. Let (R, 𝔪, k) be a local ring, M a complex of R-modules, and let K be the Koszul complex on a set of n elements in 𝔪. If H(M) is degreewise finite, then

(1) sup M ≤ sup (K ⊗_{R} M) ≤ sup M + n;

inf (K ⊗_{R} M) = inf M . (2)

For any element x in R, it follows from (1.1) that since H(M) is degreewise finite, so is H(K(x) ⊗ M). So it suffices to verify the claim for K = K(x). Then, invoking (1.1) once again, one deduces that sup(K ⊗_{R} M) ≤ sup M + 1 and inf (K ⊗_{R} M) ≥ inf M. By Nakayama’s lemma, since H(M) is degreewise finite, H_{n}(K ⊗_{R} M) = 0 entails H_{n}(M) = 0. This yields the remaining inequalities.

The derived category. As stated in the introduction, our arguments usually take place in D(R), the derived category of R-modules. Recall that the objects of D(R) are complexes of R-modules. We use the symbol ≃ to signal an isomorphism in D(R). A morphism of complexes (in the category of complexes of R-modules) represents an isomorphism in D(R) precisely when it is a quasiisomorphism.

The left derived functor of the tensor product functor of complexes of Rmodules is denoted − ⊗_{R}^{L} −, and the right derived functor of the homomorphisms functor is denoted RHom_{R} (−, −) . Given complexes M and N, the complexes M ⊗_{R}^{L} N and RHom_{R} (M, N) are defined uniquely (up to canonical isomorphism) in D(R). Set

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

When M and N are modules, these coincide with the classical notions. The reader can refer to Hartshorne [13], or Gelfand and Manin [11], for a thorough discussion of the construction and basic properties of the derived category and derived functors.

Given our context, it is crucial that the derived functors are defined without any boundedness restrictions imposed on the homology of the complexes involved. This is feasible thanks to the work of Spaltenstein [24], wherein he constructs appropriate projective and injective resolutions. For the purpose on hand, we do not need to be concerned with the precise structure of such resolutions. All that is required are the following notions.

Let M be a complex of R-modules such that H(M) is bounded below. Then M admits a bounded below flat resolution, that is, a complex F isomorphic to M (in D(R)) with each module F_{ℓ} flat, and F_{ℓ} = 0 for ℓ ≪ 0. The flat dimension of M is the number

fd_{R} M = inf{ sup{ ℓ | F_{ℓ} ≠ 0 } | F a flat resolution of M } .

As is well known, when M is a finitely generated module over a local ring (R, 𝔪, k), one has that fd_{R} M = sup(k ⊗_{R}^{L} M); this is a direct consequence of M having a minimal free resolution. We require the following extension, obtained by Avramov and Foxby [2, (5.5.F)], of this result.

1.4. Let φ: (R, 𝔪, k) → S be a local homomorphism and M a complex of Smodules such that H(M) is finite. Then

fd_{R} M = sup(k ⊗_{R}^{L} M) .

An analogous notion is that of a projective resolution of M, and also of the projective dimension of M, which we denote fd_{R} M.

1.5. Accounting principles. Let M and N be complexes of R-modules. The following assertions may be verified without much ado; alternatively, look into [9].

(1)  inf(M ⊗_{R}^{L} N) ≥ inf M + inf N ;

equality holds if i = inf M and j = inf N are finite, and H_{i}(M) ⊗_{R} H_{j}(N) ≠ 0.

(2)  sup(M ⊗_{R}^{L} N) ≤ sup M + fd_{R} N ;
(3)  sup RHom_{R} (M, N) ≤ sup N − inf M ;

equality holds i = inf M and s = sup N are finite and Hom_{R} (H_{i}(M), H_{s}(N)) ≠ 0.

(4) If H(M) is bounded, then

sup RHom_{R} (M, N) ≤ sup{ sup RHom_{R} (H_{ℓ}(M), N) − ℓ | ℓ ∈ ℤ } .

For ease of reference, we now recall a few facts concerning local cohomology and derived completions. Some of these are well known for modules, and perhaps also for bounded complexes. In the generality we desire, they are to be found in the paper of Alonso, Jeremías, and Lipman [1]. Having said this, in what follows we refer to Lipman’s article [19] whenever possible.

Local cohomology. Let 𝔞 be an ideal in R and M an R-module. The 𝔞torsion submodule of M is the R-module

Γ_{𝔞}(M) = { m ∈ M | 𝔞^{n}m = 0 for some integer n }

The association M ↦ Γ_{𝔞}(M) extends to define an additive, left exact functor on the category of complexes of R-modules; its right derived functor is denoted RΓ_{𝔞} (−). More precisely, given a complex of R-modules M, let I be an appropriate injective resolution of M and set RΓ_{𝔞} (M) = Γ_{𝔞}(I). There is a natural morphism

RΓ_{𝔞} (M) → M in D(R) .

It is traditional to set

H_{𝔞}^{i} (M) = H_{−i}(RΓ_{𝔞} (M)) for each i ∈ ℤ .

This is the ith local cohomology M with support in 𝔞. We need the following properties of local cohomology.

1.6. If 𝔞 H(M) = 0 then the morphism RΓ_{𝔞} (M) → M is an isomorphism. This is a special case of [19, (3.2.1)]. The argument runs as follows: One has

a spectral sequence that lies in a rectangular strip in the left half-plane, with

2E_{p,q} = H_{𝔞}^{−p} (H_{q}(M)) and ^{r}d_{p,q}: ^{r}E_{p,q} → ^{r}E_{p−r,q+r−1} ,

and converges to H_{𝔞}^{−p−q} (M). For each q, since 𝔞 H_{q}(M) = 0, one can construct an injective resolution I of H_{q}(M) with Γ_{𝔞}(I) = I. This yields H_{𝔞}^{0} (H_{q}(M)) = H_{q}(M) and H_{𝔞}^{−p} (H_{q}(M)) = 0 for p ≠ 0. Thus, the spectral sequence collapses and the edge homomorphisms H_{𝔞}^{−q} (M) → H_{q}(M) are isomorphisms.

1.7. For any complex of R-modules N there is a canonical morphism

RΓ_{𝔞} (RHom_{R} (M, N) ) → RHom_{R} (M, RΓ_{𝔞} (N)) ;

it is an isomorphism when H(M) is finite and pd_{R} M < ∞.

Indeed, one has an inclusion Γ_{𝔞} Hom_{R} (M, N) ⊆ Hom_{R} (M, Γ_{𝔞}N) , as is immediate from the definition of the torsion submodule. Its derived version is the morphism given above. It is evidently an isomorphism when M = R and hence also when M is a finitely generated projective module. When both H(M) and pd_{R} M are finite, M can be replaced by a bounded complex of such modules; this explains the assertion about the isomorphism.

The displayed morphism is an isomorphism also when H(M) is degreewise finite and bounded below and H(N) is bounded above, and, in this generality, it contains Grothendieck’s local duality theorem; see [8, (6.1), (6.3)]. A sheaf theoretic analogue of the isomorphism above is given by [1, (5.2.1)]. With no restrictions on M or on N, one cannot expect the morphism in (1.7) to be an isomorphism.

We provide two examples to substantiate our claim; both involve a complete local ring R, with maximal ideal 𝔞, and the injective hull E of its residue field. For the first, we let M = E and N = E, so that RΓ_{𝔞} (N) = N. Thus, the complexes RHom_{R} (M, N) and RHom_{R} (M, RΓ_{𝔞} (N)) are both isomorphic to R, and the morphism in (1.7) is the canonical morphism RΓ_{𝔞} (R) → R. This last morphism is not an isomorphism unless R happens to be zero dimensional. This example shows that one cannot do away with the degreewise finiteness of H(M).

Our second example illustrates the role of the boundedness hypothesis on H(N) when H(M) is not bounded. Let M and N be complexes with trivial differentials, with M_{n} = R and N_{n} = E for each non-negative integer n and 0 otherwise. Now, RΓ_{𝔞} (N) = N and M is a bounded below complex of projectives, so RHom_{R} (M, N) = Hom_{R} (M, N) . This last complex has trivial differentials, so it is easy to compute its homology:

H_{n}(Hom_{R} (M, N) ) = Hom_{R} (M, N)_{n} = ∏_{i ⩾ n} E for each integer n.

In particular, the modules H_{n}(Hom_{R} (M, N) ) are not 𝔞-torsion, unless R happens to be artinian. However the homology modules of RΓ_{𝔞} (RHom_{R} (M, N) ) are all 𝔞-torsion; this is a particular case of the following phenomenon: for any complex X, each local cohomology module H_{𝔞}^{i} (X) is 𝔞-torsion. This result is contained in the work of Dwyer and Greenlees; cf. [7, (5.3)].

Derived completions. Let 𝔞 be an ideal in R and M an R-module. The 𝔞-adic completion of M is the R-module

Λ_{𝔞}(M) = \underleftarrow{lim}_{n}(M ⊗_{R} R/𝔞^{n}) .

The mapping M ↦ Λ_{𝔞}(M) extends to an additive functor on the category of complexes of R-modules. This functor admits a left derived functor that we denote LΛ_{𝔞} (−) following [19]. This is defined as follows: given a complex of R-modules M, let F be an appropriate flat resolution of M, and set LΛ_{𝔞} (M) = Λ_{𝔞}(F).

It is not entirely obvious that this construction yields a well defined functor: given two flat resolutions F and G of M, one has to prove that Λ_{𝔞}(F) ≃ Λ_{𝔞}(G). This entails proving that for any complex of R-module X, if H(X) = 0, then H(Λ_{𝔞}(X)) = 0 as well. The latter assertion may be verified, for example, by arguments akin to those used in the last part of the proof of Theorem 4.1. There is another option: one way to define the left derived functor of Λ_{𝔞}, and indeed, any additive functor on the category of complexes, is via projective resolutions. Then, since any two such are homotopy equivalent, it is evident that the functors thus obtained are well defined. However, for most applications it is crucial that one be able to compute the left derived functor of Λ_{𝔞}(−) via flat resolutions.

For each complex of R-modules M, there is a natural morphism

M → LΛ_{𝔞} (M) in D(R) .

For each integer i, the ith derived completion of M with respect to 𝔞 is the R-module

H_{i}^{𝔞} (M) = H_{i}(LΛ_{𝔞} (M)) .

We require the following facts concerning derived completions.

1.8. If 𝔞 H(M) = 0 then the morphism M → LΛ_{𝔞} (M) is an isomorphism. Here is one justification of this assertion: By [1, (0.3)], see also [19, (4.1)],

there is a natural isomorphism RHom_{R} (RΓ_{𝔞} (R) , M) → LΛ_{𝔞} (M). The projective dimension of RΓ_{𝔞} (R) is finite; cf. [8, (6.5)]. Thus, a spectral sequence analogous to the one in (1.6) allows one to reduce the problem to the case where M is an R-module with 𝔞M = 0. Then the canonical map M → Λ_{𝔞}M is an isomorphism so the same holds for the morphism M → LΛ_{𝔞} (M), by [12, (4.1)].

Theorem 1.8 is valid under the far weaker assumption that for each integer i the R-module H_{i}(M) is 𝔞-adically complete. Similarly, Theorem 1.6 holds, more generally, whenever each H_{i}(M) is 𝔞-torsion. A good way to understand these phenomenon is via the theory of Bousfield colocalizations and localizations; confer, for example, the article of Dwyer and Greenlees [7].

1.9. For any integer i, if 𝔞 H_{i}^{𝔞} (M) = H_{i}^{𝔞} (M), then H_{i}^{𝔞} (M) = 0.

Indeed, this follows from [23, (1.4)] as in [10], that is, by replacing M by an appropriate resolution P and noting that then H_{i}^{𝔞} (M) equals H_{i}(Λ_{𝔞}(P)).

1.10. For any complex of R-modules N, there is a canonical morphism

LΛ_{𝔞} (M) ⊗_{R}^{L} N → LΛ_{𝔞} (M ⊗_{R}^{L} N) ;

it is an isomorphism, if H(N) is finite and pd_{R} N < ∞.

As noted in (1.8) above, the complexes RHom_{R} (RΓ_{𝔞} (R) , M) and LΛ_{𝔞} (M) are canonically isomorphic. The desired morphism results from the natural morphism

RHom_{R} (RΓ_{𝔞} (R) , M) ⊗_{R}^{L} N → RHom_{R} (RΓ_{𝔞} (R) , M ⊗_{R}^{L} N) .

As to the bit about the isomorphism: It is immediate when N = R, and hence also when N is a finitely generated projective module. This settles it, since N is isomorphic in D(R) to a bounded complex consisting of such modules.

2. Depth

The following is one of our main results. It contains Theorem I discussed in the introduction, and extends results of Iyengar [16, §6] who operates under the additional hypothesis that H(M) is bounded above.

2.1. Theorem. Let R be a noetherian ring and M a complex of R-modules. Let 𝔞 be an ideal in R and let K be the Koszul complex on a sequence of n generators for 𝔞. In this case, one has that

sup(K ⊗_{R} M) − n = sup Hom_{R} (K, M)
= sup RHom_{R} (R/𝔞, M)
= sup RΓ_{𝔞} (M) .

The proof of this theorem uses the proposition below.

In fact, this latter result can be extended so that it holds, more generally, for any X such that RΓ_{𝔞} (X) ≃ X, but justifying this last claim requires the use of sophisticated tools. The same comment applies also to (4.2). What is more, these results, and others in their vein, have implications that go beyond the present application; we plan to turn to these matters in future work.

For now, we state and prove only the weaker version below for it suffices for the present purpose and its proof is elementary.

2.2. Proposition. Let M and X be complexes of R-modules. If H(X) is bounded and 𝔞^{d} H(X) = 0 for some positive integer d, then

sup RHom_{R} (X, M) ≤ sup RHom_{R} (R/𝔞, M) − inf X .

Proof. Since H(X) is bounded, (1.5.4) yields the estimate

sup RHom_{R} (X, M) ≤ sup{ sup RHom_{R} (H_{ℓ}(X), M) − ℓ | ℓ ∈ ℤ } .

This reduces the problem to the case where X is concentrated in degree 0. By hypothesis 𝔞^{d}X = 0, so X is in fact an R/𝔞^{d}-module. For any R/𝔞^{d}-module T, one has the isomorphism

RHom_{R} (T, M) ≃ RHom_{R/𝔞^{d}} (T, RHom_{R} (R/𝔞^{d}, M) ) .

By (1.5.3), this implies that

(†)  sup RHom_{R} (T, M) ≤ sup RHom_{R} (R/𝔞^{d}, M) .

Therefore, it suffices to prove the result in the case where X = R/𝔞^{d}. To do this, we resort to an induction on the integer d, the base case d = 1 being tautological. Suppose that the desired estimate holds for X = R/𝔞^{d} for some integer d ≥ 1 . Applying Hom_{R} (−, M) to the short exact sequence

0 → 𝔞^{d}/𝔞^{d+1} → R/𝔞^{d+1} → R/𝔞^{d} → 0 ,

yields the long exact sequence

⋯ → Ext_{R}^{m} (R/𝔞^{d}, M) → Ext_{R}^{m} (R/𝔞^{d+1}, M) → Ext_{R}^{m} (𝔞^{d}/𝔞^{d+1}, M) → · · · .

Since 𝔞^{d}/𝔞^{d+1} is an R/𝔞^{d}-module, (†) yields that sup RHom_{R} (𝔞^{d}/𝔞^{d+1}, M) ≤ sup RHom_{R} (R/𝔞^{d}, M). This estimate, along with the induction hypothesis and the long exact sequence above allows us to complete the induction step, and hence the proof of the proposition. □

Proof of Theorem 2.1. Since K is a bounded complex of finitely generated free R-modules, the complexes K ⊗_{R}^{L} M and K ⊗_{R} M are isomorphic. For this reason, in the ensuing discussion, we identity these two complexes; ditto for the complexes RHom_{R} (K, M) and Hom_{R} (K, M) .

Proof of sup(K ⊗_{R} M) − n = sup Hom_{R} (K, M) . Koszul complexes are self dual: There is an isomorphism K ≅ Σ^{n}Hom_{R} (K, R) of complexes of R-modules and this induces the isomorphism K ⊗_{R} M ≅ Σ^{n}Hom_{R} (K, M) . In particular,

sup (K ⊗_{R} M) = sup Hom_{R} (K, M) + n .

Proof of sup Hom_{R} (K, M) = sup RHom_{R} (R/𝔞, M) . Since H(K) is bounded with 𝔞 H(K) = 0, the proposition above yields the inequality

sup RHom_{R} (K, M) ≤ sup RHom_{R} (R/𝔞, M) .

As to the opposite inequality, note that

sup RHom_{R} (R/𝔞, RHom_{R} (K, M) ) = sup RHom_{R} ((R/𝔞) ⊗_{R}^{L} K, M)

= sup RHom_{R} (∐_{i ⩾ 0} Σ^{i}(R/𝔞)^{\binom{n}{i}}, M)

= sup RHom_{R} (R/𝔞, M) .

The desired result follows from the calculation above and the fact that, by (1.5.3), sup RHom_{R} (R/𝔞, RHom_{R} (K, M) ) ≤ sup RHom_{R} (K, M) .

Proof of sup Hom_{R} (K, M) = sup RΓ_{𝔞} (M). In the chain below, the isomorphism on the left is by (1.6), since 𝔞 H(Hom_{R} (K, M) ) = 0, while the one on the right is given by (1.7).

Hom_{R} (K, M) ≃ RΓ_{𝔞} (Hom_{R} (K, M) ) ≃ RHom_{R} (K, RΓ_{𝔞} (M)) .

This yields the equality below, while the inequality follows from (1.5.3).

sup Hom_{R} (K, M) = sup RHom_{R} (K, RΓ_{𝔞} (M)) ≤ sup RΓ_{𝔞} (M) .

On the other hand, for each integer n ≥ 0, one has that

sup RHom_{R} (R/𝔞^{n}, M) ≤ sup RHom_{R} (R/𝔞, M) = sup Hom_{R} (K, M) ,

where the equality has been established already, and the inequality is provided by (2.2). Since H_{𝔞}^{i} (M) = \underrightarrow{lim}_{n} Ext_{R}^{i} (R/𝔞^{n}, M) for each integer i, the inequalities above imply that sup RΓ_{𝔞} (M) ≤ sup Hom_{R} (K, M) .

This completes the proof of the desired equality and of the theorem. □

A compendium. As has been explained in the introduction, Theorem 2.1 may be interpreted as stating that the various ways of introducing depth for complexes: via Koszul homology and cohomology, via the Ext functor, and via local cohomology, all lead to the same invariant. Thus, there is no ambiguity in speaking of the depth of a complex. Let us record this fact.

2.3. Definition. Let R be a noetherian ring, 𝔞 an ideal in R and K the Koszul complex on a sequence of n generators for 𝔞. For a complex of R-modules M, the 𝔞-depth of M over R is defined by one of the following equivalent formulas:

depth_{R}(𝔞, M) = \begin{cases}n − sup(K ⊗_{R} M) ; \\ inf\{ ℓ ∈ ℤ | H_{−ℓ}(Hom_{R} (K, M) ) ≠ 0 \}; \\ inf\{ ℓ ∈ ℤ | Ext_{R}^{ℓ} (R/𝔞, M) ≠ 0 \}; \\ inf\{ ℓ ∈ ℤ | H_{𝔞}^{ℓ} (M) ≠ 0 \} . \end{cases}

We write depth (𝔞, M) for the 𝔞-depth of M when the ring R is clear from the context. By the by, the equality above implies that in computing depth via Koszul (co)homology, one may choose any finite generating sequence for 𝔞.

For a complex M over a local ring (R, 𝔪, k), the depth of M is the number

depth_{R} M = depth_{R}(𝔪, M) .

This is abbreviated to depth M, if omitting R does not lead to much confusion.

In the remainder of this section we state, and prove, the fundamental properties that depth enjoys without imposing unnecessary boundedness conditions. In this process, it becomes clear that in dealing with depth no one definition can be singled out as being best suited to every purpose.

The first result subsumes Theorem II from the introduction. When H(M) is bounded above, it is precisely [16, (2.1)]; in turn that extends results in [9], [15].

2.4. Theorem. Let (R, 𝔪, k) be a local ring and M a complex of R-modules. Let P be bounded complex of R-modules with H(P) ≠ 0 and such that fd_{R} P < ∞.

If either H(M) is bounded above or H(P) is degreewise finite, then

depth_{R}(M ⊗_{R}^{L} P) = depth_{R} M − sup(k ⊗_{R}^{L} P) .

Proof. As noted above, the case where H(M) is bounded above is settled by [16, (2.1)]; confer also [8, (1.8)]. That proof works with the Koszul homology characterization of depth. We give an argument that handles both parts simultaneously.

The complex of R-modules RHom_{R} (k, M) is isomorphic to a graded k-vector space; for example, see [9]. This engenders the isomorphism in the diagram below, whereas the morphism θ is the canonical one.

RHom_{R} (k, M) ⊗_{k}^{L} (k ⊗_{R}^{L} P) ≃ RHom_{R} (k, M) ⊗_{R}^{L} P ⟶^{θ} RHom_{R} (k, M ⊗_{R}^{L} P ) .

Under either hypothesis, θ is an isomorphism: When H(M) is bounded above, this is [2, (4.4.F)]; a similar argument also goes through when H(P) is degreewise finite. If H(M) is bounded above, then sup RHom_{R} (k, M) < ∞, by (1.5.3). When H(P) is degreewise finite, H(k ⊗_{R}^{L} P) is non-zero, by (1.5.1), so (1.5.2) yields that sup(k ⊗_{R}^{L} P) is finite. Thus, the isomorphisms above implies an equality

sup RHom_{R} (k, M ⊗_{R}^{L} P ) = sup RHom_{R} (k, M) + sup(k ⊗_{R}^{L} P) .

This is the result we seek. □

At this point, it is expedient to record the following remarks which are handy for many of the subsequent arguments.

2.5. Let R be a local ring, 𝔞 a proper ideal in R, and let M be a complex of R-modules with H(M) degreewise finite. Then

depth_{R}(𝔞, M) = −∞ ⟺ sup M = ∞;
depth_{R}(𝔞, M) = ∞ ⟺ sup M = −∞ .

These are immediate from (1.3.1), once we compute depth via Koszul homology. It is crucial that R be local, as the following example illustrates. 2.6. Example. Let R be a noetherian ring containing a set of non-trivial ideals

{ 𝔞_{n} }_{n ⩾ 0} with the property that 𝔞_{i} + 𝔞_{j} = R for i ≠ j, and set

M = ∐_{n ⩾ 0} Σ^{n}(R/𝔞_{n}) .

Then, H(M) is degreewise finite with sup M = ∞, while for any non-negative integer n, one finds that depth_{R}(𝔞_{n}, M) = −n (compute via Koszul complexes).

The following lower bound for depth is well known; see [9] or [16, (2.3)].

2.7. Let (R, 𝔪, k) be a local ring and M a complex such that sup M = s is finite. Then depth M ≥ −s and equality holds if and only if 𝔪 ∈ Ass H_{s}(M).

Here is what we have to say about upper bounds for depth.

2.8. Let 𝔞 be an ideal in R and set a = max{ i | H_{𝔞}^{j} (R) ≠ 0 }. Let M a complex of R-modules. From the local cohomology spectral sequence encountered in (1.6) it follows that if there are integers s and d such that

(a) H_{𝔞}^{d} (H_{s}(M)) ≠ 0, whilst

(b) H_{𝔞}^{d−1−j} (H_{s−i}(M)) = 0 for 1 ≤ j ≤ d − 1, and

(c) H_{𝔞}^{d+1+j} (H_{s+j}(M)) = 0 for 1 ≤ j ≤ a − d − 1

then ^{2}E_{−d,s} survives to ^{∞}E_{−d,s} so H_{𝔞}^{d−s} (M) ≠ 0. Thus, depth_{R}(𝔞, M) ≤ d − s. Conditions (a)–(c) may seem contrived, but they hold in the following case. Lemma. Let (R, 𝔪, k) be a local ring and M be a complex of R-modules. If

there is an integer s such that (a) 𝔪 ∈ Ass H_{s}(M) and

(b) H_{j}(M) is 𝔪-torsion for each integer s + 1 ≤ j ≤ s + dim R − 1 , then depth_{R} M ≤ −s. In particular, if sup M = ∞ and H_{j}(M) is 𝔪-torsion for all j ≫ 0, then depth_{R} M = −∞. □

Next we improve on [16, (5.3)]. The proof of loc. cit. uses the Auslander-Buchsbaum equality [16, (2.1)], which is why its validity was restricted to complexes with bounded above homology. Now, thanks to (2.4), the same argument establishes this result without any boundedness hypothesis on the complex M.

2.9. Proposition. Let 𝔞 be an ideal in a noetherian ring R, and let K be the Koszul complex on a finite sequence x of generators for 𝔞. Let M be a complex of Rmodules such that its 𝔞-depth is finite. Set d = depth_{R}(𝔞, M) and s = sup(K⊗_{R}M).

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

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

The proposition below describes the local nature of depth. It builds on the proof of [16, (5.4)] which is good enough to handle the case when depth_{R}(𝔞, M) > −∞. Thus the only situation that remains to be tackled is when depth_{R}(𝔞, M) = −∞. This is more involved than one might suspect.

2.10. Proposition. Let 𝔞 be an ideal in a noetherian ring R and let M be a complex of R-modules. Then

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

Remark. If 𝔞 = 0, then depth_{R}(𝔞, M) = − sup M, and the proposition reads

− sup M = inf{ depth_{R_{𝔭}} M_{𝔭} | 𝔭 ∈ Spec R } .

This formula in conjunction with the preceding proposition explains (2.7).

Caveat. Evidently, when depth_{R}(𝔞, M) is finite, the infimum is achieved at some prime 𝔭 ∈ V(𝔞). This need not be the case if depth_{R}(𝔞, M) = −∞, as is illustrated by the following example.

Let R and M be as in (2.6). Then depth ((0), M) = sup M = −∞, whilst H(M_{𝔭}) is bounded above for any prime ideal 𝔭 ∈ Spec R, so depth_{R_{𝔭}} M_{𝔭} > −∞. Note that if 𝔭_{i} is a minimal prime in V(𝔞_{i}), then depth_{R_{𝔭_{i}}} M_{𝔭_{i}} = −i, so the infimum here is −∞, as predicted by the proposition.

It is not too hard to cook up another such example wherein the ring R is local.

Proof of Proposition 2.10. As mentioned before, [16, (5.4)] resolves the case when depth_{R}(𝔞, M) > −∞. The argument is short, so it bears repeating: Utilizing the characterization of depth via either local cohomology or Koszul homology it is easy to establish the following inequalities; cf. [16, (5.2)].

depth_{R}(𝔞, M) ≤ depth_{R_{𝔭}}(𝔞_{𝔭}, M_{𝔭}) ≤ depth_{R_{𝔭}} M_{𝔭} .

Now we may assume that depth_{R}(𝔞, M) is finite, in which case Proposition 2.9 provides us with a prime 𝔭 at which the inequalities above become equalities.

For the remainder of the proof depth_{R}(𝔞, M) = −∞; in particular, sup M = ∞. We have to establish that depth_{R_{𝔭}} M_{𝔭} has no lower bound as 𝔭 varies in V(𝔞). To this end, it is convenient to engineer ourselves into a situation where M is supported in V(𝔞), by the following device: Let K be the Koszul complex on a sequence of n generators for 𝔞. Then

depth (𝔞, K ⊗_{R} M) = n − sup(K ⊗_{R} (K ⊗_{R} M))
= n − (2n − depth (𝔞, M))
= depth (𝔞, M) − n ,

where the first, respectively, the second, equality is due to the fact that K, respectively, K ⊗_{R} K, detects depth with respect to 𝔞. Thus, depth (𝔞, K ⊗_{R} M) = −∞ as well. Moreover, for any prime 𝔭 ∈ V(𝔞), the complex K_{𝔭} is minimal, so that sup (k(𝔭) ⊗_{R_{𝔭}}^{L} K_{𝔭}) = n, where k(𝔭) denotes the residue field of R_{𝔭}. In particular, Theorem 2.4 yields

depth_{R_{𝔭}}(K_{𝔭} ⊗_{R_{𝔭}} M_{𝔭}) = depth_{R_{𝔭}} M_{𝔭} − n .

Therefore, by passing to K ⊗_{R} M we may assume in addition that 𝔞 H(M) = 0.

Define V_{n} = { 𝔮 ∈ V(𝔞) | n ≤ sup M_{𝔮} < ∞ }, for each integer n. These are subsets of Spec R with ⋯ ⊇ V_{n} ⊇ V_{n+1} ⊇ · · ·. There are two possibilities.

Suppose that V_{n} ≠ ∅ for each integer n. Fix an integer d ≥ 0 and choose an element 𝔮 ∈ V_{d}, so that sup M_{𝔮} = s ≥ d and finite. If 𝔭 is a prime associated to H_{s}(M_{𝔮}), then 𝔞 ⊆ 𝔭 because 𝔞 H(M) = 0. Moreover, we deduce from (2.7) that depth_{R_{𝔭}} M_{𝔭} = −s ≤ −d, which yields the desired conclusion, since d was arbitrary.

Suppose that V_{n} = ∅ for some integer n. Then, since 𝔞 H(M) = 0 and sup M = ∞, this implies that the set U = { 𝔮 ∈ V(𝔞) | sup M_{𝔮} = ∞ } is non-empty. Pick an prime ideal 𝔭 which is minimal in U, that is to say, 𝔮 ⊄ 𝔭 for any element 𝔮 ∈ U. By choice of 𝔭, one has sup M_{𝔭} = ∞, whilst for i ≥ n the R_{𝔭}-module H_{i}(M_{𝔭}) is supported only at the maximal ideal 𝔭R_{𝔭}. It remains to invoke Lemma 2.8 to conclude that depth_{R_{𝔭}} M_{𝔭} = −∞.

This completes the proof of the proposition. □

The preceding proposition allows us to extend [16, (5.5)] to complexes which are not necessarily bounded above.

2.11. Proposition. Let 𝔞 and 𝔟 be ideals in a noetherian ring R, and let M be a complex of R-modules.

(1) depth (𝔞𝔟, M) = depth (𝔞 ∩ 𝔟, M) = min{ depth (𝔞, M) , depth (𝔟, M) }.

(2) depth (𝔟, M) = depth (𝔞, M) if rad(𝔟) = rad(𝔞).

If R is local and H(M) is degreewise finite, then

(3) depth M ≤ depth (𝔞, M) + dim R/𝔞 .

Proof. (1) This first equality is a corollary of (2.10) since V(𝔞𝔟) = V(𝔞 ∩ 𝔟). As to the second, consider the equalities

depth (𝔞 ∩ 𝔟, M) = inf_{𝔭∈V(𝔞∩𝔟)} depth_{R_{𝔭}} M_{𝔭}
= min{ inf_{𝔭∈V(𝔞)} depth_{R_{𝔭}} M_{𝔭} , inf_{𝔭∈V(𝔟)} depth_{R_{𝔭}} M_{𝔭} }
= min{ depth (𝔞, M) , depth (𝔟, M) } ,

where the first and the third are by (2.10), and the middle one reflects the identity: V(𝔞 ∩ 𝔟) = V(𝔞) ∪ V(𝔟).

(2) This too follows from (2.10) since V(𝔞) = V(𝔟).

(3) Thanks to (2.5), it suffices to consider the case when sup M is finite. At this point, we may refer to [16, (5.5.4)], but for completeness we give (a slight variant of) the argument: Since R is local, one can find elements x_{1}, …, x_{d} in R whose image under the canonical surjection R → R/𝔞 forms a system of parameters for R/𝔞. In particular, rad(x, 𝔞) is the maximal ideal 𝔪 of R, and d = dim R/𝔞. Thus, RΓ_{𝔪} (M) ≃ RΓ_{(x)} (RΓ_{𝔞} (M)); for example, see [19]. Since sup M < ∞, one has sup RΓ_{𝔞} (M) < ∞, so sup RΓ_{(x)} (RΓ_{𝔞} (M)) ≥ sup RΓ_{𝔞} (M) − d, which is the inequality we seek. □

3. Amplitude inequality

This section is dedicated to the proof of the following extension of Theorem III from the introduction.

3.1. Theorem. Let R → S be a local homomorphism and let F be a complex of S-modules with H(F) non-trivial and finite. If the flat dimension of F over R is finite, then for any complex of R-modules M with H(M) degreewise finite, one has

sup M + inf F ≤ sup (M ⊗_{R}^{L} F ) ;
inf M + inf F = inf (M ⊗_{R}^{L} F ) .

In particular, amp M ≤ amp (M ⊗_{R}^{L} F ).

This theorem has the following surprising (to us) corollary. It contain Theorem IV stated in the introduction; as has been explained there, this latter result is a significant generalization of [5, Theorem R].

3.2. Theorem. Let Q → R → S be local homomorphisms and let F be a complex of S-modules such that H(F) is non-trivial and finite. Assume that fd_{R} F is finite. One has

fd_{Q} R + inf F ≤ fd_{Q} F ≤ fd_{Q} R + fd_{R} F .

In particular, fd_{Q} R and fd_{Q} F are finite simultaneously.

Proof. Let k be the residue field of Q, and let M = k ⊗_{Q}^{L} R. Observe that the complexes M ⊗_{R}^{L} F and k ⊗_{Q}^{L} F are isomorphic. In particular, by (1.4), one has fd_{Q} R = sup M and fd_{Q} F = sup(M ⊗_{R}^{L} F), so the inequality on the left is a consequence of the preceding theorem. The one on the right is given by (1.5.2). □

From the preceding result one can deduce (a complex extension of) Theorem V from the introduction. The equivalence of the first two conditions is contained in a result of Kunz [18, (2.1)]; that of the first and the third was discovered by Rodicio [22, 2] in the special case where F = R. The reader may consult the survey article of C. Miller [20] in these proceedings for other developments that are inspired by the work of Kunz and Rodicio.

3.3. Theorem. Let R be a local ring of characteristic p, and let φ: R → R be the Frobenius endomorphism of R. The following conditions are equivalent.

(a) R is regular;

(b) φ^{n} is flat for each integer n ≥ 0;

(c) there exists a positive integer n and a complex of R-modules F with H(F) finitely generated such that both fd_{R} F and fd_{R}(^{φ^{n}}F) are finite.

Proof. Let 𝔪 be the maximal ideal of R. For each natural number n, one has depth_{R}(^{φ^{n}}R) = depth_{R}(φ^{n}(𝔪), R) = depth R. Indeed, the first equality is immediate from the Koszul complex characterization of depth whilst the second is given by (2.11), since rad(φ^{n}(𝔪)) = rad(𝔪). These equalities, coupled with equality sup(k ⊗_{R}^{L} (^{φ^{n}}R)) = fd_{R}(^{φ^{n}}R), provided by (1.4), and Theorem 2.4 yield

(†)  fd(φ^{n}) < ∞ ⟹ fd(φ^{n}) = 0 .

Now for the proof of the desired equivalences.

(a) ⟹ (b): Since R is regular, fd φ^{n} is finite, and hence fd φ^{n} = 0, by (†).

(b) ⟹ (c): Pick a positive integer n and set F = R.

(c) ⟹ (a): Applied to the diagram R ⟶^{φ} R ⟶^{φ^{n−1}} R, Theorem 3.2 entails fd(φ) < ∞. From this inequality, (†) allows us to draw the stronger conclusion that fd φ = 0. It remains to invoke [18, (2.1)]. □

Proof of Theorem 3.1. If H(M) = 0, then H(M ⊗_{R}^{L}F) = 0, and the desired (in)equalities are immediate. For the rest of the proof it is assumed that H(M) ≠ 0.

Let 𝔪 be the maximal ideal of R and set k = R/𝔪. Let φ denote the homomorphism R → S. The first step is to reduce to the case where φ is surjective; then H(F) would be finite over R itself.

Let Ŝ denote the completion of S at its maximal ideal, and set F̂ = F ⊗_{S} Ŝ. Since the S-module Ŝ is faithfully flat, H(F̂) ≅ H(F)⊗_{S} Ŝ; hence H(F̂) is degreewise finite over Ŝ. For the same reason, for each complex of R-modules X, one has

sup (X ⊗_{R}^{L} F̂) = sup (X ⊗_{R}^{L} F ) and inf (X ⊗_{R}^{L} F̂) = inf (X ⊗_{R}^{L} F ) .

Thus, it suffices to establish the (in)equalities we seek with F̂ in place of F. Moreover, the special case X = k of the equality above concerning the suprema, in conjunction with (1.4), yields: fd_{R} F̂ = fd_{R} F. Hence the flat dimension of F̂ over R is finite. So, passing to Ŝ and F̂, one can assume that S is complete.

By [3, (1.1)], the homomorphism φ has a factorization R ⟶^{φ̇} R′ ⟶^{φ′} S such that the R-module R′ is flat, the ring R′/𝔪R′ is regular, and the map φ′ is surjective. Set M′ = M ⊗_{R} R′. Then, M′⊗_{R′}^{L} F ≃ M ⊗_{R}^{L} F, and, since φ̇ is faithfully flat, sup M′ = sup M and inf M′ = inf M. Furthermore, it follows from arguments analogous to [3, (3.2)] that the flat dimension of fd_{R′} F is finite. Thus, replacing R and M by R′ and M′ respectively, one can assume that φ is a surjective homomorphism.

From this point onwards the ring S plays no role in the picture.

The homomorphism φ is surjective, so H(F) is finite over R. Moreover, fd_{R} F is finite. Thus, F is isomorphic to a complex F′ of finitely generated, free modules with F′_{i} = 0 for i ≥ fd_{R} F + 1. Replacing F by F′, one can assume henceforth that F is of this form. In particular, M ⊗_{R}^{L} F ≃ M ⊗_{R} F.

Proof that sup M +inf F ≤ sup (M ⊗_{R} F). If sup M = ∞, then depth M = −∞ by (2.5), so depth(M ⊗_{R} F) = −∞, by the Auslander-Buchsbaum formula (2.4); another application of (2.5) yields sup(M ⊗_{R} F) = ∞. Now we may assume that sup M is finite. The next step is to reduce to case where inf M is also finite, so that H(M) is bounded.

Set s = sup(M ⊗_{R} F). We claim that −∞ < s < ∞.

Indeed, since sup M < ∞ and F is finite free, s < ∞; see (1.5.2). By (2.5), the inequality sup M > −∞ implies depth M < ∞, and hence, by (2.4), that depth (M ⊗_{R} F) < ∞. Another appeal to (2.5) yields that s > −∞.

Set M′ = τ_{⩾ u} (M), where u = s − pd_{R} F − 2. The following equalities hold.

(1) sup M′ = sup M;

(2) sup(M′ ⊗_{R} F) = sup(M ⊗_{R} F). To see this, note that the canonical inclusion M′ → M induces the morphism of complexes M′⊗_{R} F → M ⊗_{R} F. This map is the identity in degrees s−1 and higher so that H_{n}(M′ ⊗_{R} F) = H_{n}(M ⊗_{R} F) for n ≥ s; this contains (2). In particular, H(M′) ≠ 0 and hence u ≤ sup M; thus sup M′ = sup M.

At this point, we substitute M′ for M and assume that H(M) is bounded. In view of [21], the result of Iversen [14, (3.2)] gives amp(M) ≤ amp(M ⊗_{R} F). Unravelling this inequality, keeping in mind that inf M + inf F = inf(M ⊗_{R} F), by (1.5.1), yields the desired inequality.

This completes the justification of the inequality concerning the suprema. Proof that inf M +inf F = inf (M ⊗_{R} F). When inf M is finite, (1.5.1) provides

the desired equality. Suppose that inf M = −∞.

Let K be the Koszul complex on a finite generating set for 𝔪. For any complex of R-modules X with H(X) degreewise finite, inf (K ⊗_{R} X) = inf X; this follows from (1.1). In particular, inf (K ⊗_{R} M) = −∞; moreover,

inf (M ⊗_{R} F) = −∞ ⟺ inf ((K ⊗_{R} M) ⊗_{R}^{L} F ) = −∞ .

So, substituting K ⊗_{R} M for M, one can assume that the R-module H_{i}(M) has finite length; see (1.2). Let E be the injective hull of k - the residue field of R - and set M^{∨} = Hom_{R} (M, E) . Then H_{i}(M^{∨}) has finite length, and in particular, finitely generated, and sup M^{∨} = − inf M = ∞. This explains the equality in the calculation below; the inequality is by the already established part of the theorem, since Hom_{R} (F, R) is finite free.

sup (M^{∨} ⊗_{R} Hom_{R} (F, R) ) ≥ sup M^{∨} + inf Hom_{R} (F, R) = ∞ .

The complexes M^{∨} ⊗_{R} Hom_{R} (F, R) and Hom_{R} (F ⊗_{R} M, E) are isomorphic, since F is finite free. So sup Hom_{R} (F ⊗_{R} M, E) = ∞; this entails inf (M ⊗_{R} F) = −∞.

This completes the proof of the theorem. □

4. Width

In this section we prove the following result; its statement parallels that of Theorem 2.1. It extends the result of Frankild [10] that treats the case of complexes with bounded below homology.

4.1. Theorem. Let R be a noetherian ring and M a complex of R-modules. Let 𝔞 be an ideal in R and K the Koszul complex on a finite generating sequence for 𝔞. Then

inf(K ⊗_{R} M) = inf((R/𝔞) ⊗_{R}^{L} M) = inf LΛ_{𝔞} (M) .

The proof becomes more transparent with the following result on hand.

4.2. Proposition. Let M and X be complexes of R-modules. If H(X) is bounded and 𝔞^{d} H(X) = 0 for some positive integer d, then

inf (X ⊗_{R}^{L} M) ≥ inf (R/𝔞 ⊗_{R}^{L} M) + inf X .

Proof. Let E be a faithfully injective R-module. Then from Proposition 2.2, applied to the complexes of R-modules RHom_{R} (M, E) and X, and adjointness one obtains that

sup RHom_{R} (X ⊗_{R}^{L} M, E) ≤ sup RHom_{R} (R/𝔞 ⊗_{R}^{L} M, E) − inf X .

This gives the desired result as sup RHom_{R} (L, E) = − inf L for any complex L. □

Proof of Theorem 4.1. As before, we identify K ⊗_{R}^{L} M and K ⊗_{R} M. Proof of inf (K ⊗_{R}^{L} M) = inf ((R/𝔞) ⊗_{R}^{L} M). Let E be a faithfully injective

R-module. Then inf L = − sup RHom_{R} (L, E) for any complex L; thus, by adjointness, the desired equality is equivalent to

− sup RHom_{R} (K, RHom_{R} (M, E) ) = − sup RHom_{R} (R/𝔞, RHom_{R} (M, E) ) .

This is a special case of (2.1).

Proof of inf(K ⊗_{R} M) = inf LΛ_{𝔞} (M). We may assume that one of the two quantities in consideration is not −∞. Suppose that inf LΛ_{𝔞} (M) > −∞. By (1.8), since 𝔞 H(K ⊗_{R} M) = 0, one has the the first of the following isomorphisms, whilst the second is a special case of (1.10), since K is finite and free.

K ⊗_{R} M ≃ LΛ_{𝔞} (K ⊗_{R} M) ≃ K ⊗_{R} LΛ_{𝔞} (M) .

This entails inf(K ⊗_{R} M) = inf(LΛ_{𝔞} (M)), by (1.9) and (1.5.1).

Suppose now that inf (K ⊗_{R} M) = i > −∞. We know now that this is equivalent to inf (R/𝔞 ⊗_{R}^{L} M) = i. By (4.2), this implies that

(†)  inf (R/𝔞^{n} ⊗_{R}^{L} M) ≥ i  for each integer n ≥ 1 .

By definition, LΛ_{𝔞} (M) ≃ Λ_{𝔞}(F), where F is an appropriate flat resolution of M. The complex Λ_{𝔞}(F) is defined by the exactness of the sequence of complexes

0 → Λ_{𝔞}(F) → ∏((R/𝔞^{n}) ⊗_{R} F) ⟶^{θ} ∏((R/𝔞^{n}) ⊗_{R} F) −−→ 0 ,

where the degree ℓ th component of ∏((R/𝔞^{n}) ⊗_{R} F) is ∏((R/𝔞^{n}) ⊗_{R} F_{ℓ}), and θ(f_{n}) = (f_{n}−π^{n+1}(f_{n+1})), with π^{n+1} : (R/𝔞^{n+1})⊗_{R}F → (R/𝔞^{n})⊗_{R}F the canonical surjection. The inverse system { (R/𝔞^{n})⊗_{R}F }_{n≥0} being surjective, the map θ in the diagram above is surjective. For each integer n, since (R/𝔞^{n}) ⊗_{R} F ≃ R/𝔞^{n} ⊗_{R}^{L} M, inequality (†) ensures that inf((R/𝔞^{n}) ⊗_{R} F) is at least i. Therefore, from the homology long exact sequence arising from the short exact sequence of complexes above we deduce that inf Λ_{𝔞}(F) ≥ i − 1, that is to say, inf LΛ_{𝔞} (M) ≥ i − 1. In particular, inf LΛ_{𝔞} (M) > −∞, which brings us back to preceding case.

This completes the proof of the theorem. □

4.3. Definition. Let 𝔞 be an ideal in R and K the Koszul complex on 𝔞. For a complex of R-modules M, the 𝔞-width of M over R is defined by one of the following equivalent formulas:

width_{R}(𝔞, M) = \begin{cases}inf(K ⊗_{R} M) ; \\ inf\{ ℓ ∈ ℤ | Tor_{ℓ}^{R} (R/𝔞, M) ≠ 0 \}; \\ inf\{ ℓ ∈ ℤ | H_{ℓ}^{𝔞} (M) ≠ 0 \} . \end{cases}

For a complex M over a local ring (R, 𝔪, k), the width of M is the number

width_{R} M = width_{R}(𝔪, M) .

For complexes whose homology is bounded below, width was introduced by Yassemi in [25] over local rings. In full generality, width for complexes was studied by Christensen, Foxby and Frankild, who defined it via Kozsul homology [6].

Next we establish a tight relation between 𝔞-depth and 𝔞-width.

4.4. Proposition. Let 𝔞 be an ideal in R and M a complex of R-modules. Let E a faithful injective R-module and set M^{∨} = RHom_{R} (M, E) . Then

depth_{R}(𝔞, M) = width_{R}(𝔞, M^{∨}) and width_{R}(𝔞, M) = depth_{R}(𝔞, M^{∨}) .

Proof. Let K be the Koszul complex on a finite sequence of generators for 𝔞. Since K is bounded and degreewise finite and free, K ⊗_{R} M^{∨} and Hom_{R} (K, M)^{∨} are isomorphic (see, for example, the proof of [2, (4.4)] ). This gives the second equality below, whereas the first is due to the equality sup L = − inf(L^{∨}), which holds for any complex L.

− sup Hom_{R} (K, M) = inf Hom_{R} (K, M)^{∨} = inf (K ⊗_{R} M^{∨}) .

So depth_{R}(𝔞, M) = width_{R}(𝔞, M^{∨}); the proof of the other equality is analogous. □

The preceding proposition allows us to obtain many a results concerning width from those on depth. For instance, the following observation is a immediate consequence of (2.5).

4.5. Let 𝔞 be an ideal in R and let M be a complex of R-modules with H(M) degreewise finite. Then

width_{R}(𝔞, M) = ∞ ⟺ inf M = ∞;
width_{R}(𝔞, M) = −∞ ⟺ inf M = −∞ .

One can translate also (2.4) - the Auslander-Buchsbaum formula - into a statement concerning width. We end this section with the formula for the width of the derived tensor product, and the derived homomorphisms, of a pair of complexes.

4.6. Proposition. Let (R, 𝔪, k) be a noetherian local ring, and let M and N be complexes of R-modules.

width_{R}(M ⊗_{R}^{L} N) = width_{R} M + width_{R} N ;
depth_{R}(RHom_{R} (M, N) ) = width_{R} M + depth_{R} N .

Proof. The associativity formula (M ⊗_{R}^{L} N) ⊗_{R}^{L} k ≃ (M ⊗_{R}^{L} k) ⊗_{k} (k ⊗_{R}^{L} N) leads to the first equality below, while the second holds since k is a field.

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

This establishes the first formula; the argument for the second one is similar. □

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Matematisk Afdeling, Københavns Universitet, Universitetsparken 5, DK–2100 København Ø, Denmark

    E-mail address: foxby@math.ku.dk

Mathematics Department, University of Missouri, Columbia, MO 65211, USA

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