PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0

VANISHING OF COHOMOLOGY OVER GORENSTEIN RINGS OF SMALL CODIMENSION

LIANA M. ŞEGA

(Communicated by Wolmer Vasconcelos)

Abstract. We prove that if M, N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext_{R}^{n}(M, N) for n ≫ 0 is equivalent to the vanishing of Ext_{R}^{n}(N, M) for n ≫ 0. Furthermore, if R̂ has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.

Introduction

Let (R, 𝔪, k) be a commutative noetherian local ring with maximal ideal 𝔪 and residue field k. The codimension of R is the number codim R = edim R − dim R, where edim R denotes the minimal number of generators of 𝔪.

Let M, N be finite R-modules. We study the vanishing of Ext_{R}^{n}(M, N) for all n ≫ 0 and the vanishing of Tor_{n}^{R}(M, N) for all n ≫ 0, assuming that R is Gorenstein and M, N are finite R-modules.

It is convenient to state the vanishing properties of (co)homology in terms of numbers e_{R}(M, N) and t^{R}(M, N), defined as follows:

e_{R}(M, N) = sup{n ∈ ℕ | Ext_{R}^{n}(M, N) ≠ 0}
t^{R} (M, N) = sup{n ∈ ℕ | Tor_{n}^{R}(M, N) ≠ 0}

A local ring (Q, 𝔫, k) is called an (embedded) deformation of R if R ≅ Q/(f), where f is a Q-regular sequence (contained in 𝔫^{2}). The ring R is said to be a complete intersection if its 𝔪-adic completion R̂ has an embedded deformation which is a regular ring. Vanishing of (co)homology over complete intersections has remarkable properties, as proved by Avramov and Buchweitz [4, Theorem III]. They relate vanishing properties to homogeneous algebraic varieties V_{R}^{∗}(M) ⊆ k̃^{c}, defined in [3], where c = codim R and k̃ denotes the algebraic closure of k.

Theorem AB. Let R be a complete intersection local ring. One of the numbers e_{R}(M, N), e_{R}(N, M), t^{R}(M, N) is finite if and only if all of them are less than or equal to dim R, if and only if V_{R}^{∗}(M) ∩ V_{R}^{∗}(N) = 0.

An interesting consequence of Theorem AB is that the vanishing of Ext_{R}^{n}(M, N) for all n ≫ 0 is symmetric in the two module variables. It is clear that a ring with this property is Gorenstein. To study the symmetry in the vanishing of Ext,

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

1991 Mathematics Subject Classification. Primary 13D07, 13H10; Secondary 13D40.

Key words and phrases. Gorenstein rings, vanishing of Ext, CI-dimension.

Huneke and Jorgensen [8] define a class of Gorenstein rings, called AB rings, by the condition that sup{e_{R}(M, N)} is finite when the supremum is taken over all finite R-modules M, N with e_{R}(M, N) < ∞. In [8, 3.1, 3.3(1), 4.1] they prove:

Theorem HJ. The ring R is AB if and only if every pair (M, N) of finite Rmodules with e_{R}(M, N) < ∞ (respectively t^{R}(M, N) < ∞) satisfies e_{R}(M, N) ≤ dim R and e_{R}(N, M) ≤ dim R (respectively t^{R}(M, N) ≤ dim R).

Theorem AB shows that a complete intersection local ring is an AB ring. A noncomplete intersection example of AB rings is given by Gorenstein rings of minimal multiplicity and codimension at least 3: It is proved in [8, 3.5] that over such rings the finiteness of e_{R}(M, N) or t^{R}(M, N) implies one of the modules has finite projective dimension.

In Theorem 1 we identify another class of rings with this property.

Theorem 1. Let R be a local Gorenstein ring such that codim R ≤ 4 and R̂ admits no embedded deformation. If one of the numbers e_{R}(M, N), t^{R}(M, N) is finite, then M or N has finite projective dimension.

The theorem is proved in Section 2 as Theorem 2.3. The proof uses the fact that, for the rings in Theorem 1, all finite R-modules have rational Poincaré series which share an explicitly known denominator: this is due to Jacobsson, Kustin, Miller [13], [14], [15]. Preliminaries on rational Poincaré series are presented in Section 1, where we also give a new proof of the result on Gorenstein rings of minimal multiplicity. Clearly, such an approach is not applicable to rings that have finite modules with irrational Poincaré series. Such rings have been constructed by Bøgvad [6] in codimension 12 and higher.

To describe the aymptotic vanishing of cohomology over all Gorenstein rings with codim R ≤ 4, we use the notion of finite complete intersection dimension introduced by Avramov, Gasharov and Peeva [5]. An R-module M is said to have finite CIdimension if there exist a flat ring homomorphism R → R′ and a deformation Q of R′ such that pd_{Q}(M ⊗_{R} R′) < ∞. Note that any module of finite projective dimension has finite CI-dimension, and that if R is a complete intersection, then every finite R-module has finite CI-dimension. The next statement is derived from the theorems presented above, as well as from expressions for e_{R}(M, N) and t^{R}(M, N) obtained by Araya, Yoshino, Avramov, Buchweitz [1], [4]. It shows, in particular, that all Gorenstein rings of codimension at most 4 are AB:

Theorem 2. Let R be a local Gorenstein ring with codim R ≤ 4.

If e_{R}(M, N) < ∞, then M or N has finite CI-dimension and

e_{R}(M, N) = dim R − depth_{R} M
e_{R}(N, M) = dim R − depth_{R} N

If t^{R}(M, N) < ∞, then M or N has finite CI-dimension and

t^{R}(M, N) ≤ max{dim R − depth_{R} M, dim R − depth_{R} N} .

The theorem is proved as Theorem 3.4 in Section 3. An example is provided there to show that the conclusion regarding CI-dimension cannot be extended to high codimensions. However, it is not known (to the author) whether the other conclusions can be extended or not.

Under the hypotheses of Theorem 1, asymptotic vanishing of Tor or Ext occurs only for trivial reasons: one of the modules has finite projective dimension. The situation is different if R is a complete intersection with codim R ≥ 2. Over such rings, Avramov [3, 6.5] constructs for every linear subspace H ⊆ k̃^{c} an R-module M of finite length with V_{R}^{∗}(M) = H; taking modules M, N that correspond to nontrivial subspaces with zero intersection, one sees from Theorem AB that the numbers e_{R}(M, N), e_{R}(N, M) and t^{R}(M, N) are all finite.

We show that (co)homology may vanish for nontrivial reasons whenever R̂ has a non-regular embedded deformation:

Theorem 3. Let R be a local Gorenstein ring with codim R ≥ 2. If R̂ has an embedded deformation, then there exist R-modules M, N of finite length with pd_{R} M = pd_{R} N = ∞ and e_{R}(M, N) = e_{R}(N, M) = t_{R}(M, N) = dim R.

This result is contained in the more general Corollary 4.4 proved in Section 4. The hypothesis on the codimension is necessary. Indeed, if R is a Gorenstein ring with codim R ≤ 1, then R̂ = Q/(f) for a regular local ring Q and a Q-regular element f. For such rings, t^{R}(M, N) < ∞ or e_{R}(M, N) < ∞ implies that one of the modules M or N has finite projective dimension, cf. Huneke and Wiegand [9, 1.9], respectively Avramov and Buchweitz [4, 5.12].

1. Rational Poincaré series

In this section (R, 𝔪, k) is a local ring and M, N denote finite R-modules. We use several formal power series with integer coefficients associated to a finite R-module.

We let P_{M}^{R}(t) denote the Poincaré series of M over R

P_{M}^{R}(t) = ∑_{i=0}^{∞} rank_{k} Tor_{i}^{R}(M, k)t^{i} ∈ ℤ[[t]]

and I_{R}^{M}(M) denote the Bass series of M over R

I_{R}^{M}(t) = ∑_{i=0}^{∞} rank_{k} Ext_{R}^{i}(k, M)t^{i} ∈ ℤ[[t]] .

We also use the Hilbert series of M over R, defined to be the series

H_{M}^{R}(t) = ∑_{i=0}^{∞} rank_{k}(𝔪^{i}M/𝔪^{i+1}M)t^{i} ∈ ℤ[[t]] .

Recall that H_{M}^{R}(t) represents a rational function with denominator (1 − t)^{dim M}.

1.1. Let R → R′ be a flat homomorphism of local rings such that 𝔪R′ is the maximal ideal of R′. For M′ = M ⊗_{R} R′ and N′ = N ⊗_{R} R′ there are equalities P_{M}^{R}(t) = P_{M′}^{R′}(t) and I_{R}^{M}(t) = I_{R′}^{M′}(t), e_{R}(M, N) = e_{R′}(M′, N′) and t^{R}(M, N) = t^{R′}(M′, N′), pd_{R′}(M′) = pd_{R} M and id_{R′}(N′) = id_{R} N. This situation applies in particular to the completion map R → R̂.

1.2. Lemma. Assume there exists a polynomial c(t) such that c(t) P_{M}^{R}(t) ∈ ℤ[t] for each finite R-module M. If c(t) P_{k}^{R}(t) = (1 + t)^{e}a(t) for an integer e ≥ 0 and a polynomial a(t) ∈ ℤ[t] with a(−1) ≠ 0, then c(t)(1 + t)^{m} I_{R}^{N}(t) ∈ ℤ[t] for each finite R-module N, where m = max{0, dim R − e}.

Proof. Set d = dim R. By Foxby [7, 3.10] there exists a finite R̂-module L and a polynomial g(t) with integer coefficients and degree less than d such that

I_{R}^{N}(t) = g(t) + t^{d} PR̂ ; L(t) .

As shown by Levin [16, p. 8] there exists an integer n such that M = L/𝔪^{n}L satisfies

PR̂ ; L(t) = PR̂ ; M(t) − t HR̂ ; 𝔪^{n}L(−t) PR̂ ; k(t) .

The R̂-module M has finite length, so considered as an R-module it has finite length and is complete. From 1.1 we get P_{M}^{R}(t) = PR̂ ; M(t) and P_{k}^{R}(t) = PR̂ ; k(t). Since dimR̂(𝔪^{n}L) ≤ d, we have (1 − t)^{d} HR̂ ; 𝔪^{n}L(t) ∈ ℤ[t] and the conclusion follows. □

The next result connects vanishing of (co)homology to Bass series and Poincaré series. Part (2) is due to C. Miller, cf. the proof of [17, 1.1]; part (1) is [4, 1.5(1)].

1.3. For finite R-modules M, N the following hold:

(1) If e_{R}(M, N) = 0, then I_{R}^{Hom_{R}(M,N)}(t) = P_{M}^{R}(t) I_{R}^{N}(t)

(2) If t^{R}(M, N) = 0, then P_{M⊗_{R}N}^{R}(t) = P_{M}^{R}(t) P_{N}^{R}(t).

1.4. Definition. A factorization c(t) = p(t) · q(t) · r(t) in ℤ[t] is said to be good if p(t) = 1 or p(t) is irreducible, q(t) has non-negative coefficients, and r(t) = 1 or r(t) is irreducible and has no positive real root among its complex roots of minimal absolute value.

1.5. Proposition. Let R be a local ring for which there exists a polynomial c(t) such that c(t) P_{L}^{R}(t) ∈ ℤ[t] for each finite R-module L. When c(t) has a good factorization the following hold for all finite R-modules M, N:

(1) If e_{R}(M, N) < ∞, then pd_{R} M < ∞ or id_{R} N < ∞.

(2) If t^{R}(M, N) < ∞, then pd_{R} M < ∞ or pd_{R} N < ∞.

Proof. We only give a proof of (1); the proof of (2) is similar.

Replacing M by a syzygy, if necessary, we may assume e_{R}(M, N) = 0. Set b(t) = (1 + t)^{d}c(t), where d = dim R. By hypothesis and Lemma 1.2 we have

b(t) P_{M}^{R}(t) = m(t) ∈ ℤ[t] and b(t) I_{R}^{N}(t) = n(t) ∈ ℤ[t] .

Since (1 + t)^{d} is in ℕ[t] it follows that b(t) has a good factorization b(t) = p(t) · q(t) · r(t) as in Definition 1.4. Using 1.3(1) we obtain:

b(t) I_{R}^{Hom_{R}(M,N)}(t) = b(t) P_{M}^{R}(t) I_{R}^{N}(t) = \frac{m(t)n(t)}{p(t)q(t)r(t)} .

By Lemma 1.2 the expression on the left is a polynomial, so p(t) divides m(t)n(t). Since p(t) = 1 or p(t) is irreducible, two cases arise.

Case 1. m(t) = p(t)m_{1}(t) with m_{1}(t) ∈ ℤ[t]. We then have an equality

q(t) P_{M}^{R}(t) = \frac{m_{1}(t)}{r(t)} .

Assume r(t) does not divide m_{1}(t). This means that the radius of convergence ρ of these power series is finite. The series q(t) P_{M}^{R}(t) has non-negative coefficients, so by the Pringsheim Principle, cf. [22, 7.2], ρ is a singular point for it. On the other hand, the hypothesis on r(t) implies that ρ is not a singular point for m_{1}(t)/r(t). This contradiction shows that r(t) divides m_{1}(t), so q(t) P_{M}^{R}(t) is a polynomial. As both factors have non-negative coefficients, we conclude that P_{M}^{R}(t) is a polynomial, that is, pd_{R}(M) < ∞.

Case 2. n(t) = p(t)n_{1}(t) with n_{1}(t) ∈ ℤ[t]. An argument similar to the one above shows that I_{R}^{N}(t) is a polynomial, hence id_{R} N < ∞. □

The two parts of the proposition can be combined when the ring is Gorenstein:

1.6. If R is Gorenstein, then id_{R} N < ∞ is equivalent to pd_{R} N < ∞.

Let R be a Gorenstein ring with codim R ≥ 2. It is known that R has multiplicity at least codim R + 2, cf. [19, 3.2]. When equality holds, R is said to be Gorenstein of minimal multiplicity.

1.7. Remark. If R is Gorenstein of minimal multiplicity with codim R ≥ 2, then c(t) P_{M}^{R}(t) ∈ ℤ[t] for each finite R-module M, where c(t) = 1 − (codim R)t + t^{2}.

To see this, we may assume that k is infinite: if not, then R′ = R[t]_{𝔪[t]} has mult R′ = mult R and we apply 1.1. It also suffices to consider maximal Cohen-Macaulay modules M. If dim R = 0, then 𝔪^{3} = 0 and the result is proved by Sjödin [20]. If dim R > 0, then there exists a maximal R-regular sequence g such that codim R = codim R/(g) and mult R = mult R/(g). Since M is maximal Cohen-Macaulay, this sequence is also M-regular. The isomorphisms Tor_{n}^{R}(M, k) ≅ Tor_{n}^{R/(g)}(M/gM, k) then yield P_{M}^{R}(t) = P_{M/gM}^{R/(g)}(t). The conclusion follows by the dimension 0 case.

If codim R > 2, then the polynomial c(t) = 1 − (codim R)t + t^{2} is irreducible, hence c(t) = c(t) · 1 · 1 is a good factorization. From Proposition 1.5 and Remark 1.7, we obtain a new proof of [8, 3.5]:

1.8. Corollary. Let R be a local Gorenstein ring of minimal multiplicity with codim R ≥ 3. If M, N are finite R-modules such that one of the numbers e_{R}(M, N), t^{R}(M, N) is finite, then M or N has finite projective dimension. □

2. Gorenstein rings of small codimension

In this section R denotes a Gorenstein ring of codimension at most 4 which is not a complete intersection. Over such rings, the Poincaré series of all finite Rmodules are rational and share a common denominator. This result, and the form of the denominator were obtained by Jacobsson [13] in codimension 3, by Jacobsson, Kustin and Miller [14, 2.3] in codimension 4, characteristic different from 2 and by Kustin [15] in codimension 4 and characteristic different from 3. We collect below the relevant information; we refer to [2, §3] for details.

2.1. There exists a polynomial c(t) ∈ ℤ[t] such that c(t) P_{M}^{R}(t) ∈ ℤ[t] for all finite Rmodules M. This polynomial has the form c(t) = d(t)(1+t)^{m}, where the polynomial d(t) ∈ ℤ[t] and the non-negative integer m are as follows:

type codim R  d(t)  m restrictions
G(l + 1)  3  1 − t − lt^{2} − t^{3} + t^{4}  1  l ≥ 4
GTE  4  1 − 2t − (l − 2)t^{2} + t^{3} + t^{4} − t^{5}  2  l ≥ 5
GGO  4  1 − 2t − (l − 2)t^{2} − 2t^{3} + t^{4}  2  l ≥ 5
GH(p)  4  1 − 2t − (l − 2)t^{2} + (p − 2)t^{3} + 2t^{4} − t^{5} 2 1 ≤ p ≤ l ≥ 5

Avramov [2, 3.1] determines when R̂ has an embedded deformation:

2.2. The ring R̂ admits an embedded deformation if and only if d(1) = 0, if and only if R is of type GH(p), with p = l.

The proof of the next result goes through a careful examination of the polynomials in the chart above. Earlier, Sun [21] used a case by case analysis of these polynomials to prove that the Betti numbers of finite modules over a Gorenstein ring of codimension at most 4 are eventually non-decreasing.

2.3. Theorem. Let R be a local Gorenstein ring such that codim R ≤ 4 and R̂ admits no embedded deformation. If M, N are finite R-modules and one of the numbers e_{R}(M, N), t^{R}(M, N) is finite, then M or N has finite projective dimension.

Proof. By Proposition 1.5 and 1.6, it suffices to show that the common denominator c(t) has a good factorization. Since (1 + t)^{m} has non-negative coefficients, this will follow once we prove that d(t) has a good factorization.

If d(t) is irreducible, then d(t) = d(t) · 1 · 1 is a good factorization. For the rest of the proof we assume that d(t) is reducible. If d(t) has a linear factor, then d(−1) = 0. Indeed, the only possible rational roots of d(t) are ±1, and d(1) = 0 is excluded by 2.2. For each type of polynomial in the chart, we study the factorization of d(t) in the two remaining cases: d(−1) = 0 and d(t) has no linear factor.

G(l + 1) or GGO: If d(−1) = 0, then l = 4 or l = 8. Thus, we have good factorizations d(t) = (1 − 3t + t^{2}) · (1 + t)^{2} · 1 or d(t) = (1 − 4t + t^{2}) · (1 + t)^{2} · 1 .

If d(t) has no linear factor, then d(t) = (1 + at + εt^{2})(1 + bt + εt^{2}) with a, b ∈ ℤ, both factors irreducible, and ε = ±1. If ε = −1, then comparison of the coefficients of t and t^{3} gives a + b < 0 and −a − b < 0, so this case does not occur. If ε = 1, then comparison of the coefficients of t^{2} gives ab < 0, hence we may assume b > 0, and then d(t) = (1 + at + t^{2}) · (1 + bt + t^{2}) · 1 is a good factorization.

GTE: If d(−1) = 0, then l = 6 and d(t) = (1 − 4t + 3t^{2} − t^{3}) · (1 + t)^{2} · 1 is a good factorization.

If d(t) has no linear factor, then d(t) = (1 + at + εt^{2})(1 + bt + ct^{2} − εt^{3}) with a, b, c ∈ ℤ, both factors irreducible, and ε = ±1. Comparing coefficients, we get:

a + b = −2
ac + εb − ε = 1
εc − εa = 1

Using the first and the last equality to eliminate b and c from the middle, we obtain a^{2} = 1 + 3ε. If ε = −1, then a^{2} = −2, which is not possible. If ε = 1, then a^{2} = 4, hence a = ±2, and this contradicts the assumption that 1 + at + εt^{2} is irreducible.

GH(p): If d(−1) = 0, then p + l = 10 and d(t) = e(t)(1 + t), where e(t) = 1−3t+(p−5)t^{2} +3t^{3} −t^{4}. Since 1+t has non-negative coefficients, the polynomial d(t) has a good factorization if and only if e(t) has a good factorization. If e(t) is irreducible, then e(t) = e(t) · 1 · 1 is a good factorization, so we assume e(t) is reducible. If e(t) has a linear factor, then e(−1) = 0, hence p = 5. It follows that p = l = 5, and this is ruled out by 2.2. If e(t) has no linear factor, then e(t) = (1+at−t^{2})(1+bt+t^{2}) with both factors irreducible and a, b ∈ ℤ. Comparing the coefficients of t and t^{3}, we get a + b = −3 and a − b = 3, hence a = 0. This contradicts the hypothesis that 1 + at − t^{2} is irreducible.

If d(t) has no linear factor, then d(t) = (1 + at + εt^{2})(1 + bt + ct^{2} − εt^{3}) with a, b, c ∈ ℤ, both factors irreducible, and ε = ±1. Comparison of coefficients yields:

a + b = −2
ab + c + ε = −l + 2
ac + εb − ε = p − 2
εc − εa = 2

These equalities yield −a^{2} − a = −l + 2 − 3ε and a^{2} + εa = p − 2 + 3ε, so εa − a = p − l. By 2.1 and 2.2 we have p < l, hence ε = −1 and 2a = l − p > 0. Thus, d(t) = (1 + bt + ct^{2} + t^{3}) · 1 · (1 + at − t^{2}) is a good factorization. □

3. Finite CI-dimension

In this section we let R denote a local ring and let M, N be finite R-modules.

3.1. We refer to the introduction for the definition of finite CI-dimension and we recall below the basic examples:

3.1.1. If R is a complete intersection, then M has finite CI-dimension.

3.1.2. If pd_{R} M < ∞, then M has finite CI-dimension.

3.2. When one of the modules M or N has finite CI-dimension, several (in)equalities involving the numbers e_{R}(M, N) and t^{R}(M, N) are known.

The inequality below follows from [4, 4.9] and [5, 1.4]:

3.2.1. If t^{R}(M, N) < ∞ and M has finite CI-dimension, then

t^{R}(M, N) ≤ depth R − depth_{R} M .

A formula for e_{R}(M, N) is given by Araya and Yoshino [1, 4.2]:

3.2.2. If e_{R}(M, N) < ∞ and M has finite CI-dimension, then

e_{R}(M, N) = depth R − depth_{R} M .

Over Gorenstein rings the equality of 3.2.2 is valid more generally:

3.3. Lemma. If R is Gorenstein, e_{R}(M, N) is finite and N has finite CI-dimension, then

e_{R}(M, N) = depth R − depth_{R} M .

Proof. If pd_{R} N < ∞, then id_{R} N < ∞ (see 1.6), so the equality is given by a result of Ischebeck [10, 2.6]. In general, there exists a flat ring homomorphism R → R′ and a deformation Q of R′ such that pd_{Q}(M ⊗_{R} R′) < ∞. By 1.1 we may assume that R = R′ and then e_{Q}(M, N) = depth Q − depth_{Q} M. A standard argument, cf. [1, 2.6], then gives

e_{R}(M, N) = e_{Q}(M, N) − (depth_{Q} R − depth Q) = depth R − depth_{R} M . □

The next theorem is the main result of this section. It shows, in particular, that every Gorenstein ring of codimension at most 4 is AB.

3.4. Theorem. Let R be a local Gorenstein ring with codim R ≤ 4 and let M, N be finite R-modules.

If e_{R}(M, N) is finite, then M or N has finite CI-dimension and

e_{R}(M, N) = dim R − depth_{R} M
e_{R}(N, M) = dim R − depth_{R} N

If t^{R}(M, N) is finite, then M or N has finite CI-dimension and

t^{R}(M, N) ≤ max{dim R − depth_{R} M, dim R − depth_{R} N} .

Proof. In view of the results recalled in 3.2, Lemma 3.3, and Theorem HJ stated in the introduction, it suffices to prove the following claim: If e_{R}(M, N) or t^{R}(M, N) is finite, then one of the modules M, N has finite CI-dimension.

By 3.1.1, the claim holds when R is a complete intersection. If R has no embedded deformation, then the statement results from Theorem 2.3, in view of 3.1.2.

It remains thus to treat the case when R is not complete intersection and has an embedded deformation Q. This only happens when codim R = 4 and Q has no embedded deformation (otherwise, R̂ deforms to a Gorenstein ring Q′ with codim Q′ ≤ 2, and such a ring is a complete intersection). Standard arguments (see [1, 2.6] for example) show that e_{R}(M, N) < ∞ implies e_{Q}(M̂, N̂) < ∞ and t^{R}(M, N) < ∞ implies t^{Q}(M̂, N̂) < ∞. By Theorem 1, M̂ or N̂ has then finite projective dimension over Q, and thus the corresponding module has finite CIdimension over R. □

We note some further applications of the conclusion on finite CI-dimension of Theorem 3.4.

Araya and Yoshino [1, 4.2] give a self-test for finite projective dimension when the CI-dimension is finite: If M has finite CI-dimension, then there is an equality e_{R}(M, M) = pd_{R}(M). In view of Theorem 3.4 we have thus:

3.5. Corollary. There is an equality e_{R}(M, M) = pd_{R}(M). □

When M or N has finite CI-dimension, (1) below is proved by Jorgensen [13, 2.2] and (2) is proved by Araya and Yoshino [1, 2.5] (see also Iyengar [11, 4.3] for the case q = 0) over any local ring R. By Theorem 3.4 we have thus:

3.6. Corollary. Set d_{R}(M, N) = depth R − depth_{R} M − depth_{R} N.

If t^{R}(M, N) = q < ∞, then the following hold:

(1) t^{R}(M, N) = sup{d_{R_{𝔭}}(M_{𝔭}, N_{𝔭}) | 𝔭 ∈ Supp M ∩ Supp N}.

(2) If q = 0 or depth_{R} Tor_{q}^{R}(M, N) ≤ 1, then

t^{R}(M, N) = d_{R}(M, N) + depth_{R} Tor_{q}^{R}(M, N) .  □

The conclusion on CI-dimension of Theorem 3.4 does not extend to higher codimensions: There exist rings of any codimension greater than or equal to 6 and R-modules M, N of infinite CI-dimension such that e_{R}(M, N) < ∞. The example below is based on a construction in [8, 4.3].

3.7. Example. Let (S, 𝔰, k) and (T, 𝔱, k) be two Gorenstein rings that are essentially of finite type over a field k and none of them is a complete intersection, so they have codimension at least 3. Set 𝔭 = 𝔰 ⊗_{k} T + S ⊗_{k} 𝔱. The local ring R = (S ⊗_{k} T)_{𝔭} is then Gorenstein and has codim R = codim T + codim S ≥ 6. The R-modules M = S ⊗_{k} k and N = k ⊗_{k} T satisfy e_{R}(M, N) ≤ dim R by [8, 4.3]. Since S is not a complete intersection, k has infinite CI-dimension over it, cf. [5, 1.3]. The ring S is faithfully flat as a k-module, so [5, 1.13] yields CI-dim_{R} M ≥ CI-dim_{S} k = ∞. By symmetry, N also has infinite CI-dimension over R.

4. Embedded deformations

In this section we construct examples to show that vanishing of (co)homology can occur for nontrivial reasons.

Throughout the section, S denotes a local commutative noetherian ring. Our examples are based on the existence of modules with periodic resolutions over rings with embedded deformations. We recall the relevant definitions and results:

Let a ≥ 1 be an integer. The minimal free resolution of a finite S-module U is said to be periodic of period a if Syz_{n}^{S}(U) ≅ Syz_{n+a}^{S}(U) for all n ≥ 0. The following result is due to Avramov, Gasharov and Peeva [5, 3.2]:

4.1. If S ≅ Q/(f) for a non-regular local ring (Q, 𝔫) and a Q-regular element f ∈ 𝔫^{2}, then there exists a finite S-module U whose minimal S-free resolution is periodic of period 2 and pd_{Q} U = 1. In particular, pd_{S} U = ∞ and depth_{S} U = depth S.

4.2. Theorem. If S ≅ Q/(f) for a non-regular local ring (Q, 𝔫) and a Q-regular element f ∈ 𝔫^{2}, and U is the S-module from 4.1, then there exists a finite S-module V with depth_{S} V = depth S such that

id_{S} V = ∞ = pd_{S} V and e_{S}(U, V) = 0 = t_{S}(U, V).

If, furthermore, S is Gorenstein, then also e_{S}(V, U) = 0.

Proof. Since Q is not regular, we have pd_{Q} k = ∞. Set s = depth Q + 2. By [18, §2] we have depth_{Q} Syz_{i}^{Q}(k) = depth Q for each i ≥ s. If Q is Gorenstein, then id_{Q} Syz_{i}^{Q}(k) = ∞ for each i ≥ 0 by 1.6 and we set V′ = Syz_{s}^{Q}(k). Otherwise, one of the syzygy modules in the short exact sequence 0 → Syz_{s+1}^{Q}(k) → Q^{b} → Syz_{s}^{Q}(k) → 0 has infinite injective dimension and we let V′ be this module.

Set V = V′/fV′. Note that f is a V′-regular element and depth_{S} V = depth S. The isomorphisms Ext_{S}^{n}(k, V) ≅ Ext_{Q}^{n+1}(k, V′) and Tor_{n}^{S}(k, V) ≅ Tor_{n}^{Q}(k, V′) show that id_{S} V = pd_{S} V = ∞. The isomorphisms Ext_{S}^{n}(U, V) ≅ Ext_{Q}^{n+1}(U, V′) and Tor_{n}^{S}(U, V) ≅ Tor_{n}^{Q}(U, V′) yield e_{S}(U, V) = 0 and t^{S}(U, V) ≤ 1, because pd_{Q} U = 1. The periodicity of the minimal free resolution of U gives Tor_{n}^{S}(U, V) ≅ Tor_{n+2}^{S}(U, V) for all n > 0, hence t^{S}(U, V) = 0.

If S is a Gorenstein ring, then Q is Gorenstein, hence id_{Q} U < ∞ by 1.6. The isomorphisms Ext_{S}^{n}(V, U) ≅ Ext_{Q}^{n}(V′, U) then show e_{S}(V, U) < ∞. Since the Smodule U has finite CI-dimension, Lemma 3.3 yields e_{S}(V, U) = 0. □

The next lemma is an extension of the fact that if U is a finite S-module and \overline{U} = U/gU for an U-regular element g, then pd_{S} \overline{U} = pd_{S} U + 1 and id_{S} \overline{U} = id_{S} U.

4.3. Lemma. Let S be a local ring, U, V be finite S-modules and g ∈ S an Uregular element. For the S-module \overline{U} = U/gU the following hold:

(1) e_{S}(\overline{U}, V) = e_{S}(U, V) + 1.

(2) e_{S}(V, \overline{U}) = e_{S}(V, U).

(3) t^{S}(U, V) ≤ t^{S}(\overline{U}, V) ≤ t^{S}(U, V) + 1.

If gV = 0 or V has finite length, then t^{S}(\overline{U}, V) = t^{S}(U, V) + 1.

(4) If the minimal free resolution of Syz_{s}^{S}(U) is periodic of period 2 for an integer s ≥ 0, then the minimal free resolution of Syz_{s+1}^{S}(\overline{U}) is periodic of period 2.

Proof. Properties (1) to (3) are deduced using the long exact sequences induced by the exact sequence 0 → U ⟶^{g} U → \overline{U} → 0.

We only give the proof of (3). In this case, the long exact sequence is:

… → Tor_{n+1}^{S}(\overline{U}, V) → Tor_{n}^{S}(U, V) ⟶^{g} Tor_{n}^{S}(U, V) →
→ Tor_{n}^{S}(\overline{U}, V) → Tor_{n−1}^{S}(U, V) → …

Nakayama’s Lemma shows that if Tor_{n}^{S}(U, V) ≠ 0 then Tor_{n}^{S}(\overline{U}, V) ≠ 0, hence t^{S}(U, V) ≤ t^{S}(\overline{U}, V). To prove the remaining statements it suffices to assume t^{S}(U, V) = p < ∞. The long exact sequence shows that if Tor_{i}^{S}(U, V) = 0 for i = n and i = n − 1 then Tor_{n}^{S}(\overline{U}, V) = 0. We conclude t^{S}(\overline{U}, V) ≤ p + 1. If gV = 0 or V has finite length, then multiplication by g on Tor_{p}^{S}(U, V) has non-zero kernel, hence Tor_{p+1}^{S}(\overline{U}, V) ≠ 0 and thus t^{S}(\overline{U}, V) = p + 1.

(4) Let F be a minimal free resolution of U. The mapping cone of the homomorphism F ⟶^{g} F is a minimal free resolution of \overline{U}, hence we have Syz_{n}^{S}(\overline{U}) ≅ Syz_{n}^{S}(U)/g Syz_{n}^{S}(U) ⊕ Syz_{n−1}^{S}(U) for all n ≥ 0, and the conclusion follows. □

4.4. Corollary. Let R be a d-dimensional Cohen-Macaulay local ring such that codim R ≥ 2 and R̂ ≅ Q/(f) for a local ring (Q, 𝔫) and a Q-regular element f ∈ 𝔫^{2}.

There exist R-modules M, N of finite length such that pd_{Q} M = d + 1, the minimal R-free resolution of Syz_{d}^{R}(M) is periodic of period 2 and

id_{R} N = ∞ = pd_{R} N and e_{R}(M, N) = d = t^{R}(M, N).

If, furthermore, R is Gorenstein, then e_{R}(N, M) = d.

Proof. Since codim R ≥ 2, the ring Q is not regular. Set S = R̂ and let U and V be as in Theorem 4.2. Choose a U- and V -regular sequence g of length d and set M = U/gU and N = V/gV. By Lemma 4.3 we have id_{S} N = ∞ = pd_{S} N and the minimal S-free resolution of Syz_{d}^{S}(M) is periodic of period 2. Using Theorem 4.2 and Lemma 4.3 we conclude e_{S}(M, N) = e_{S}(U, V) + d = d and t^{S}(M, N) = t^{S}(U, N) + d < ∞. Since the minimal free resolution of U is periodic of period 2, we have Tor_{n}^{S}(U, N) ≅ Tor_{n+2}^{S}(U, N) for all n > 0, hence t^{S}(U, N) = 0 and thus t^{S}(M, N) = d. If S is a Gorenstein ring, then e_{S}(N, M) = e_{S}(V, U) + d = d by Theorem 4.2 and Lemma 4.3.

It remains to notice that M and N are R̂-modules of finite length, hence, considered as R-modules, they have finite length and are complete. The desired conclusion now follows by applying 1.1 □

Acknowledgment

I wish to thank my thesis advisor Luchezar Avramov for inspiring the development of this paper and for his constant help.

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Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

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