Comm. Algebra (to appear)

ON A DEPTH FORMULA FOR MODULES OVER LOCAL RINGS

SANGKI CHOI* DEPARTMENT OF MATHEMATICS EDUCATION KON-KUK UNIVERSITY, SEOUL, KOREA AND SRIKANTH IYENGAR** 1993 NETTLEBROOK ST. WESTLAKE VILLAGE, CA 91361

Abstract. We prove that for modules M and N over a local ring R, the depth formula: depth_{R} M + depth_{R} N − depth R = depth_{R} Tor_{s}^{R} (M, N) − s, where s = sup{i | Tor_{i}^{R} (M, N) ≠ 0}, holds under certain conditions. This adds to the list cases where the depth formula, which extends the classical Auslander-Buchsbaum equality, is satisfied.

Introduction

The celebrated Auslander-Buchsbaum equality asserts that if a finitely generated module M over a noetherian local ring R has finite projective dimension, then depth_{R} M +pd_{R} M = depth R. There have been a number of generalizations of this formula, one of which is due to Auslander [1, 1.2] himself, who proved that with M as before and N a finitely generated R-module, if for s = sup{n | Tor_{n}^{R} (M, N) ≠ 0} either s = 0 or depth_{R} Tor_{s}^{R} (M, N) ≤ 1, then

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

Observe that with N = k, the residue field of R, we recover the Auslander-Buchsbaum equality. As a matter of fact, Auslander established that depth_{R} N = depth_{R} Tor_{s}^{R} (M, N) +pd M −s, which, in view of the Auslander-Buchsbaum equality, translates to the formula above.

Let us note that all the terms in (∗) are defined and finite for any pair of Rmodules M and N as long as sup{n | Tor_{n}^{R} (M, N) ≠ 0} < ∞. This leads us to the following

Problem. Let M and N be finitely generated modules over a noetherian local ring R such that s = sup{n | Tor_{n}^{R} (M, N) ≠ 0} is finite. Discover conditions under which (*) holds.

When this occurs, we say that the depth formula holds for M and N. Over the last few years it has emerged that if M has finite CI-dimension and either one of the following conditions holds:

(1) s = 0;

(2) depth_{R} Tor_{s}^{R} (M, N) ≤ 1, then the depth formula (*) holds for M and N.

1991 Mathematics Subject Classification. 13C15, 18G15.

Key words and phrases. Auslander-Buschsbaum, depth, complete intersection dimension.

*Email-address: schoi@kkucc.konkuk.ac.kr.

**Current address: Pure Mathematics, University of Sheffield, Sheffield, S3 7RH, UK

Email-address: s.iyengar@shef.ac.uk.

The description of modules of finite CI-dimension, first identified by Avramov, Gasharov, and Peeva, [4], is defered to Definition 2; for now, we note only that if either M has finite projective dimension or R is a complete intersection, then M has finite CI-dimension. In particular, these results extend that of Auslander.

The result above was established by Huneke and R. Wiegand [7, 2.5] in the particular case when the ring R is a complete intersection and s = 0, also confer [8, 4.3]; the general case is due to Araya and Yoshino [3, 2.5].

We add to this list of cases when the depth formula holds. More precisely, we prove that (*) holds also when M has finite CI-dimension and for some prime ideal 𝔭 ∈ Ass Tor_{s}^{R} (M, N) , one has

depth_{R_{𝔭}} M_{𝔭} + depth_{R_{𝔭}} N_{𝔭} − depth R_{𝔭} ≥ depth_{R} M + depth_{R} N − depth R .

This is the content of Corollary 4. This result is deduced from the more general, and easier to digest, Theorem 3: If M has finite CI-dimension and sup{n | Tor_{n}^{R} (M, N) ≠ 0} = s < ∞, then

depth_{R} M + depth_{R} N − depth R ≥ −s ,

with equality if and only if depth_{R} Tor_{s}^{R} (M, N) = 0. This fact is well known in the special case when M has finite projective dimension, and contains a recent result of Jorgensen [9, 2.2].

Now, in [1] Auslander had asked if the depth formula (*) holds for any pair of modules M and N, given that M has finite projective dimension, but Murthy [12, Pg. 565] gave examples to the contrary. Nevertheless, as is explained in Remark 7, for such an M one has always an inequality

depth_{R} M + depth_{R} N − depth R ≥ inf{depth_{R} Tor_{i}^{R} (M, N) − i} .

Thus, it is not unreasonable to ask if there is some integer i such that

depth_{R} M + depth_{R} N − depth R = depth_{R} Tor_{i}^{R} (M, N) − i .

Note that this is a weaker version of the depth formula. Unfortunately, one cannot expect this formula to hold in general, even over regular local rings where all modules have finite projective dimension; this is the content of Proposition 11.

Results and proofs

The reader is referred to Matsumura [11] for basic definitions and notation. In this note, it seems expedient to introduce the following

Notation 1. For modules M and N over a ring R, set

fd_{R}(M, N) = sup{n | Tor_{n}^{R} (M, N) ≠ 0} .

In particular, if Tor_{n}^{R} (M, N) = 0 for all n, then fd_{R}(M, N) = −∞, else 0 ≤ fd_{R}(M, N) ≤ ∞. When R is local, by which we understand that it is also noetherian, and k is its residue field, fd_{R}(M, k) is the flat dimension of M, which, when M is finitely generated, equals its projective dimension pd_{R} M. Moreover, for such an M, the number fd_{R}(M, N) is finite for each finitely generated N.

Next, we recall the following definition, which surfaced in [4]:

Definition 2. A module M over a local ring R is said to be of finite CI-dimension if there is a diagram of local homomorphisms R → R′ ← Q such that R → R′ is flat, R′ = Q/(x_{1}, …, x_{c}) for a regular sequence x_{1}, …, x_{c} in Q and pd_{Q}(M ⊗_{R} R′) < ∞.

There are two important classes of examples: Modules with finite projective dimension, when R′ = Q = R, and modules over complete intersections, when R′ is the completion of R at its maximal ideal and Q is any Cohen presentation of R′.

We are now ready to state our main result; its proof is deferred to 9.

Theorem 3. Let R be a local ring, and let M and N be finitely generated modules such that fd_{R}(M, N) = s < ∞. If M has finite CI-dimension, then

depth_{R} M + depth_{R} N − depth R ≥ −s

with equality if and only if depth_{R} Tor_{s}^{R} (M, N) = 0.

Here is the depth formula advertised in the introduction:

Corollary 4. Let R, M, and N be as above. If for a prime 𝔭 ∈ Ass Tor_{s}^{R} (M, N) depth_{R_{𝔭}} M_{𝔭} + depth_{R_{𝔭}} N_{𝔭} − depth R_{𝔭} ≥ depth_{R} M + depth_{R} N − depth R , then

depth_{R} M + depth_{R} N − depth R = −s and depth_{R} Tor_{s}^{R} (M, N) = 0 .

Proof. Since Tor^{R_{𝔭}} (M_{𝔭}, N_{𝔭}) = Tor^{R} (M, N)_{𝔭}, one has fd_{R_{𝔭}}(M_{𝔭}, N_{𝔭}) = fd_{R}(M, N) and depth_{R_{𝔭}} Tor_{s}^{R_{𝔭}} (M_{𝔭}, N_{𝔭}) = 0. This yields the last of the following series of (in)equalities; the first and the third are by Theorem 3, and the second is the hypothesis.

− fd_{R_{𝔭}}(M_{𝔭}, N_{𝔭}) = depth_{R_{𝔭}} M_{𝔭} + depth_{R_{𝔭}} N_{𝔭} − depth R_{𝔭}
≥ depth_{R} M + depth_{R} N − depth R
≥ − fd_{R}(M, N)
= − fd_{R_{𝔭}}(M_{𝔭}, N_{𝔭})

Therefore, depth_{R} M + depth_{R} N − depth R = − fd_{R}(M, N), and this implies, thanks to Theorem 3, that depth_{R} Tor_{s}^{R} (M, N) = 0

Remark 5. It may be worth pointing out that inequality hypothesised in Corollary 4 is satisfied under either one of the following conditions:

(a) depth N_{𝔭} ≥ depth N for a prime 𝔭 ∈ Ass Tor_{s}^{R} (M, N) ; this holds, for example, when depth N = 0.

(b) G-dim_{R} N, the Gorenstein dimension of N, cf. [2], is finite and depth R_{𝔭} ≥ depth R for a prime 𝔭 ∈ Ass Tor_{s}^{R} (M, N) .

Indeed, since M has finite CI-dimension, depth M_{𝔭} − depth R_{𝔭} ≥ depth_{R} M − depth R, confer [4, 1.4] and [4, 1.6]. Given this fact, it immediate that Condition (a) above implies the desired inequality. In the case of (b), note that

depth R_{𝔭} − depth_{R_{𝔭}} N_{𝔭} = G-dim_{R_{𝔭}} N_{𝔭}  by [2, 4.13b]
≤ G-dim_{R} N  by [2, 4.15]
= depth R − depth_{R} N  by [2, 4.13b]

Rearranging terms yields depth N_{𝔭} −depth_{R} N ≥ depth R_{𝔭} −depth R; in particular depth N_{𝔭} ≥ depth_{R} N, so that (a) holds.

Here is the local version of Theorem 3; it implies that

inf{depth_{R_{𝔭}} M_{𝔭} + depth_{R_{𝔭}} N_{𝔭} − depth R_{𝔭} | 𝔭 ∈ Spec R} = − fd_{R}(M, N) ,

which has been observed by Jorgensen [9, 2.2].

Corollary 6. Let M and N be finitely generated modules over a local ring R. If fd_{R}(M, N) < ∞ and M has finite CI-dimension, then

depth_{R_{𝔭}} M_{𝔭} + depth_{R_{𝔭}} N_{𝔭} − depth R_{𝔭} ≥ − fd_{R}(M, N)  for  𝔭 ∈ Spec R ,

with equality if and only if 𝔭 ∈ Ass Tor_{s}^{R} (M, N) .

Proof. Since M has finite CI-dimension, so does M_{𝔭} for any prime 𝔭 ∈ Spec R, cf. [4, 1.6], and hence

depth_{R_{𝔭}} M_{𝔭} + depth_{R_{𝔭}} N_{𝔭} − depth R_{𝔭} ≥ − fd_{R_{𝔭}}(M_{𝔭}, N_{𝔭}) ≥ − fd_{R}(M, N) ,

where the inequality on the left is the theorem above, and the one on the right is a consequence of the isomorphism Tor^{R_{𝔭}} (M_{𝔭}, N_{𝔭}) = Tor^{R} (M, N)_{𝔭}. Furthermore, equality holds if and only if fd_{R_{𝔭}}(M_{𝔭}, N_{𝔭}) = fd_{R}(M, N) and depth_{R_{𝔭}} M_{𝔭} + depth_{R_{𝔭}} N_{𝔭} − depth R_{𝔭} = − fd_{R_{𝔭}}(M_{𝔭}, N_{𝔭}), which translates to, by Theorem 3, the condition that 𝔭 ∈ Ass Tor_{s}^{R} (M, N) for s = fd_{R}(M, N).

The proof of Theorem 3 is based on the following observations.

Remark 7. Let M and N be finitely generated modules over a local ring R. If pd_{R} M < ∞, then

depth_{R} M + depth_{R} N − depth R ≥ inf{depth_{R} Tor_{i}^{R} (M, N) − i} ,

with equality if the infimum is achieved at i = fd_{R}(M, N). (Note: The depth of the zero module is infinity.)

Indeed, Foxby [6, 12.ai] and Iyengar [8, 2.2] prove that depth_{R}(M ⊗_{R}^{L} N) = depth_{R} N − pd_{R} M , where M ⊗_{R}^{L} N denotes F ⊗_{R} N, for any free resolution F of M; confer [8] for further details. Thanks to the Auslander-Buchsbaum equality this translates to:

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

On the other hand, [8, 2.7.2] applied to the complex F ⊗_{R} N yields an estimate

depth_{R}(M ⊗_{R}^{L} N) ≥ inf{depth_{R} Tor_{i}^{R} (M, N) − i} ,

and [8, 2.3] asserts that equality holds if the infimum is achieved at fd_{R}(M, N). Combining the preceding equations yields the desired result.

It is useful to note the following special case of the previous remark; it has certainly been observed before, confer [6, 12.ag].

Remark 8. If pd_{R} M < ∞, then for s = fd_{R}(M, N), one has

depth_{R} M + depth_{R} N − depth R ≥ −s

with equality if and only if depth_{R} Tor_{s}^{R} (M, N) = 0.

9. Proof of Theorem 3. Let R → R′ ← Q be the diagram of local homomorphisms provided by the finiteness of the CI-dimension of M, cf. 2. Localizing Q and R′ at a minimal prime of Spec(R′/𝔪R′), where 𝔪 is the maximal ideal of R, we may assume that depth(R′/𝔪R′) = 0. This allows us to reduce, by passing to Q, to the case where the projective dimension of M is finite.

Indeed, since R → R′ is flat Tor^{R′} (M′, N′) = Tor^{R} (M, N) ⊗_{R} R′, where M′ = M ⊗_{R} R′ and N′ = N ⊗_{R} R′, so that fd_{R′}(M′, N′) = fd_{R}(M, N), Moreover, the depth of R′/𝔪R′ is 0, hence [11, 23.3] yields

depth R′ = depth R
depth_{R′} M′ = depth_{R} M
depth_{R′} N′ = depth_{R} N
depth_{R′} Tor_{s}^{R′} (M′, N′) = depth_{R} Tor_{s}^{R} (M, N) .

Thus, it suffices to establish the desired result for modules M′ and N′ over the ring R′. Therefore, by passing to R′ we may assume that R = Q/(x_{1}, …, x_{c}) with x_{1}, …, x_{c} a Q-regular sequence, and that pd_{Q} M < ∞.

In this case, as is well known, Tor_{s+c}^{Q} (M, N) = Tor_{s}^{R} (M, N) and fd_{Q}(M, N) = s + c; for example, confer [3, 2.6]. Furthermore, depth Q = depth R + c and depth_{Q} L = depth_{R} L for any R-module L, so it suffices to prove that

depth_{Q} M + depth_{Q} N − depth Q ≥ −s − c

with equality if and only if depth_{Q} Tor_{s+c}^{Q} (M, N) = 0. Since pd_{Q} M < ∞, it remains to invoke Remark 8.

Remark 10. The reduction technique employed in the argument above also yields another proof of the result of Arraya and Yoshino [3, 2.5] mentioned in the introduction. Indeed, the only change required is that in the last paragraph of the proof one invokes Remark 7 instead of Remark 8.

As has been explained in the introduction, confer also Remark 7, it is reasonable to seek conditions under which there is some integer i such that

depth_{R} M + depth_{R} N − depth R = depth_{R} Tor_{i}^{R} (M, N) − i .

One cannot expect this to hold in general, for the next result identifies a class of examples for which the formula above fails for all i, even when both modules have finite projective dimension.

Proposition 11. Let (R, 𝔪, k) be a regular local ring and let x_{1}, …, x_{n}, y be elements in 𝔪 such that the following conditions are satisfied:

(a) x_{1}, …, x_{n} is a regular sequence;

(b) the Cohen-Macaulay defect d of the local ring R/(x_{1}, …, x_{n}, y) is at least 3.

Then for the modules M = R/(x_{1}, …, x_{n}) and N = R/(y), one has

depth_{R} Tor_{0}^{R} (M, N) = dim R − n − d
depth_{R} Tor_{1}^{R} (M, N) = dim R − n − d + 2
depth_{R} M + depth_{R} N − depth R = dim R − n − 1 .

Proof. It is immediate that depth_{R} M = dim R − n and depth_{R} N = dim R − 1; this yields the last equality that we seek. The complex 0 → R ⟶^{y} R → 0 is a free resolution of N, so Tor^{R} (M, N) is the homology of the complex

0 → M ⟶^{y} M → 0 .

Thus, Tor_{0}^{R} (M, N) = R/(x_{1}, …, x_{n}, y), Tor_{1}^{R} (M, N) = {m ∈ M | y.m = 0}, and these fit into exact sequences

0 → K → M → Tor_{0}^{R} (M, N) → 0
0 → Tor_{1}^{R} (M, N) → M → K → 0 .

Since (x_{1}, …, x_{n}, y) is an ideal of grade n, one has dim Tor_{0}^{R} (M, N) = dim R − n; since the Cohen-Macaulay defect d of Tor_{0}^{R} (M, N) is at least 3, we obtain

depth_{R} Tor_{0}^{R} (M, N) = dim Tor_{0}^{R} (M, N) − d
= dim R − n − d
≤ dim R − n − 3 .

Now depth_{R} M = dim R−n, so a routine ‘depth chase’ on the exact sequences above establishes that depth_{R} K = depth_{R} Tor_{0}^{R} (M, N) + 1 and depth_{R} Tor_{1}^{R} (M, N) = depth_{R} Tor_{0}^{R} (M, N) + 2.

Here is a concrete example which ensures that the class identified in the previous proposition is not empty. It was cooked up using the recipe outlined in Burch [5], also confer Kohn [10].

Example 12. Let k be a field of characteristic 0 and let R be the polynomial ring k[a, b, c, d, e] localized at the maximal ideal (a, b, c, d, e). The elements x_{1} = abc, x_{2} = de(bd + ce) and y = ade + bc(bd + ce) are such that

(a) x_{1}, x_{2} is a regular sequence;

(b) pd_{R}(R/(x_{1}, x_{2}, y)) = 5, as is verified by MACAULAY. Therefore, dim R/(x_{1}, x_{2}, y) = 3 and depth R/(x_{1}, x_{2}, y) = 0, so that the Cohen-Macaulay defect of R/(x_{1}, x_{2}, y) is 3.

Acknowledgements. This collaboration owes its existence to Luchezar Avramov and Craig Huneke, who made each of us aware of the other’s work. It is a pleasure to thank them for the same and for their encouragement. Our thanks also to David Jorgensen for his valuable remarks on an earlier version of this paper.

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