REMARKS ON A CONJECTURE OF HUNEKE AND WIEGAND
AND THE VANISHING OF (CO)HOMOLOGY
OLGUR CELIKBAS, UYEN LE, HIROKI MATSUI, AND ARASH SADEGHI

A BSTRACT. In this paper we study a long-standing conjecture of Huneke and Wiegand which is concernedwith the torsion submodule of certain tensor products of modules over one-dimensional local domains. We
utilize Hochster’s theta invariant and show that the conjecture is true for two periodic modules. We also
make use of a result of Orlov and formulate a new condition which, if true over hypersurface rings, forces
the conjecture of Huneke and Wiegand to be true over complete intersection rings of arbitrary codimension.
Along the way we investigate the interaction between the vanishing of Tate (co)homology and torsion in
tensor products of modules, and obtain new results that are of independent interest.

1. I NTRODUCTION
Throughout R denotes a commutative Noetherian local ring with unique maximal ideal m and residue
field k, and all R-modules are assumed to be finitely generated.
In commutative algebra there are several questions about tensor products of modules that are notoriously difficult to solve; see, for example, [26]. A fine example of such a question is the following
long-standing conjecture of Huneke and Wiegand:
Conjecture 1.1. (Huneke-Wiegand; see [34, page 473]) Assume R is one-dimensional and let M bea nonfree and torsion-free R-module. Assume M has rank (e.g., R is a domain). Then the torsion
submodule of M ⊗R M ∗ is nonzero, i.e., M ⊗R M ∗ has (nonzero) torsion, where M ∗ = HomR (M, R).
Conjecture 1.1, over Gorenstein rings, is in fact a special case of a celebrated conjecture of Auslander
and Reiten [6] and is wide open in general, even for two generated ideals over complete intersection
domains of codimension two; see Remark 3.10. On the other hand, besides some other special cases,
Conjecture 1.1 is known to be true over hypersurface rings; see [34] and [32] for the details. In fact,
since maximal Cohen-Macaulay modules (that have no free direct summand) are two-periodic overhypersurface rings, the hypersurface case of Conjecture 1.1 is subsumed by the following result; notethat ΩnR M denotes then nth syzygy of a given module M; see 2.2.
Theorem 1.2. ([16, 4.17]) Assume R is a one-dimensional complete intersection domain and let M bea nonzero R-module. Assume M is two-periodic, i.e., M ∼= Ω2R M. Then M ⊗R M ∗ has torsion.
In this paper we study Conjecture 1.1 for the case where R is a domain. Our aim concerning Theorem
1.2 is twofold. In section 2 we give a proof of Theorem 1.2 – which is entirely different from the one
given in [16] – by using Tate (co)homology; see Corollary 2.18. This approach motivates us to seek, andhence obtain, new results about the vanishing of Tate (co)homology and torsion in tensor products thatare independent of Conjecture 1.1; see, for example, Corollary 2.12 and Theorem 2.14. Furthermore, we
generalize Theorem 1.2 in section 3. More precisely, we remove the complete intersection hypothesis
from Theorem 1.2 and hence prove:2010 Mathematics Subject Classification. Primary 13D07; Secondary 13H10, 13D05, 13C12.
Key words and phrases. complexity, Tate (co)homology, Tor-rigidity, vanishing of Ext and Tor, tensor products, torsion.
Celikbas was partly supported by WVU Mathematics Excellence and Research Funds (MERF). Matsui was partly supported
by JSPS Grant-in-Aid for JSPS Fellows 19J00158.
1

2

OLGUR CELIKBAS, UYEN LE, HIROKI MATSUI, AND ARASH SADEGHI

Theorem 1.3. Assume R is a one-dimensional domain and let M be a nonzero R-module that is twoperiodic. Then M ⊗R M ∗ has torsion.
The proof of Theorem 1.3 relies upon an invariant, referred to as the Hochster’s theta invariant, andis established in the paragraph following Proposition 3.9. In section 3, motivated by Theorems 1.2 and
1.3, we also provide an example of a two-periodic module over a one-dimensional local ring that is not
a complete intersection; see Example 3.11.
A remarkable theorem of Orlov [38] determines an equivalence between the singularity category of
R and that of Proj A for the generic hypersurface A of R. The gist of our work in Section 4 is to exploit
Orlov’s theorem and show that Conjecture 1.1 holds over all one-dimensional complete intersection
domains in case a certain condition we formulate holds for all hypersurface domains. Our approach touse Orlov’s theorem to attack Conjecture 1.1 seems to be new and it establishes the following theorem;
see the paragraph following Remark 4.13.
Theorem 1.4. Conjecture 1.1 is true over each one-dimensional complete intersection domain provided
that the following condition holds:
Whenever R is a hypersurface domain, c is a positive integer, M and M ⊗R Extc−1
R (M, R) are CohenMacaulay R-modules, both of which have codimension c − 1, it follows that the projective dimension of
M is finite.
Note that the case where c = 1 of the condition stated in Theorem 1.4 is nothing but the conditionof Conjecture 1.1. We do not know whether or not each hypersurface domain satisfies the condition
stated in Theorem 1.4, but we are now able to translate the problem of Conjecture 1.1 to a problem overhypersurface rings; the new advantage we have is that homological algebra is better understood overhypersurface rings than over complete intersection rings.
2. A PROOF OF T HEOREM 1.2 VIA TATE HOMOLOGY
In this section we use Tate (co)homology and give a proof of Theorem 1.2 that is distinct from theone obtained in [16]. Along the way we obtain general results that should be useful to further understandthe vanishing of Tate (co)homology and torsion; see, for example, Corollary 2.11 and Theorem 2.14.
We recall several definitions and preliminary results that are necessary for our arguments (we record
some of these results within a number without labeling them as proposition or theorem). We start withthe definitions of the Gorenstein and the complete intersection dimension:
2.1. ([3, 8, 9]) Let M be an R-module. Then M is said to be totally reflexive provided that the natural
map M → M ∗∗ is bijective and ExtiR (M, R) = 0 = ExtiR (M ∗ , R) for all i ≥ 1. The infimum of n for which
there exists an exact sequence 0 → Xn → · · · → X0 → M → 0 such that each Xi is totally reflexive iscalled the Gorenstein dimension of M, and is denoted by G-dimR (M). Therefore M is totally reflexive
if and only if G-dimR (M) ≤ 0, where it follows by convention that G-dimR (0) = −∞.
A diagram of local ring homomorphisms of local rings R → R0  S is called a quasi-deformation
if R → R0 is flat and the kernel of the surjection S  R0 is generated by a regular sequence on S. The
complete intersection dimension of M is defined as follows
CI-dimR (M) = inf{pdS (M ⊗R R0 ) − pdS (R0 ) : R → R0  S is a quasi-deformation},where pdS (M) denotes the projective dimension of M over S.
In general it follows G-dimR (M) ≤ CI-dimR (M) ≤ pdR (M).
Also we have G-dimR (M) = CI-dimR (M) if CI-dimR (M) < ∞, and G-dimR (M) = CI-dimR (M) =pdR (M) if pdR (M) < ∞.
2.2. Let M be an R-module and let n be an integer. If n > 0, then ΩnR M denotes the nth syzygy of M,
that is, the image of the nth differential map in a minimal free resolution of M. Also, if M is totally

REMARKS ON A CONJECTURE OF HUNEKE AND WIEGAND AND THE VANISHING OF (CO)HOMOLOGY

3

reflexive and n < 0, then ΩnR M denotes the nth cosyzygy of M, that is, the image of the R-dual of the nthdifferential map in a minimal free resolution of M ∗ . Note, by convention, we have Ω0R M = M.
2.3. Let M be an R-module. Then the Auslander transpose of M, denoted by Tr R M, is the cokernel offthe map f ∗ = HomR ( f , R), where F1 −→ F0 → M → 0 is part of the minimal free resolution of M; see[3]. Note that M ∗ ∼= Ω2R Tr R M, up to free modules. Note also that Tr R M is unique, up to isomorphism,
since so are minimal free resolutions.
The following fact is used for 2.10 and Propositon 3.9,
2.4. If M is an R-module such that TorR1 (M, Tr R M) = 0, then M is free; see [5, A1] and also [41, 3.9].
Next we recall the definitions of Tate homology and cohomology. Although their definitions do not
require the ring to be local, we keep the local setting for simplicity.
2.5. Let M be an R-module. A complex T of free R-modules is said to be totally acyclic provided thatϑπ
Hn (T) = 0 = Hn (HomR (T, R)) for all n ∈ Z. A complete resolution of M is a diagram T −→ P −→ M,
where P is a projective resolution, T is a totally acyclic complex, and ϑ is a morphism of complexes
such that ϑi is an isomorphism for all i  0.
Assume T → P → M is a complete resolution of M. Then, for an R-module N and for i ∈ Z, thec Ri (M, N) and the Tate cohomology Ext
c iR (M, N) of M and N over R are defined as
Tate homology Tor
Hi (T ⊗R N) and Hi (HomR (T, N)), respectively.
It is known that G-dimR (M) < ∞ if and only if M has a complete resolution; see [10, 3.1]. HenceRcc iR (M, N) are defined for each R-module N in case G-dimR (M) < ∞.
Tori (M, N) and Ext
The following are some of the fundamental properties of Tate (co)homology modules; see, for example [8, 10, 22] or [20, 2.11 and 2.12].
2.6. Let M and N be R-modules such that G-dimR (M) < ∞.
c Ri (M, N) ∼c iR (M, N) ∼(i) If i > G-dimR (M), then it follows Tor= TorRi (M, N) and Ext= ExtiR (M, N).
000(ii) If 0 → M → M → M → 0 is a short exact sequence of R-modules, where G-dimR (M 0 ) < ∞ or
G-dimR (M 00 ) < ∞, then, for each i ∈ Z, there is an exact sequence of cohomology of the form:
c iR (M 00 , N) → Ext
c iR (M, N) → Ext
c iR (M 0 , N) → Extc Ri+1 (M 00 , N),Ext
and also of homology of the form:c Ri (M 0 , N) → Torc Ri (M, N) → Torc Ri (M 00 , N) → Torc Ri−1 (M 0 , N).
Tor∼ ci nc Ri+n (M, N) ∼c Ri (Ωn M, N) and Extc i+n(iii) Tor= Tor
R (M, N) = ExtR (ΩR M, N) for all i, n ∈ Z.
R
c iR (M, N) = 0 = Torc Ri (M, N) for all i ∈ Z.
(iv) If pdR (M) < ∞, then Ext
R

−i−1

cR
c i (M, N) ∼(v) If M is totally reflexive, then Tor
= Ext

−i+1

cR
(M ∗ , N) ∼= Ext

(Tr R M, N) for all i ∈ Z.

In the following we collect some results that are used in the proof of Proposition 2.9.
2.7. Let M and N be R-modules such that n ≥ 1.
(i) If depthR (M ⊗R N) ≥ n, depthR (N) ≥ n − 1 and lengthR (ExtiR (Tr R M, N)) < ∞ for all i = 1, . . . , n,then it follows that ExtiR (Tr R M, N) = 0 for all i = 1, . . . , n; see [20, 4.2(ii)].
(ii) If ExtiR (Tr R M, N) = 0 for all i = 1, . . . , n, then it follows that depthR (M ⊗R N) ≥ min{n, depthR (N)};see [20, 4.2(iii)].
(iii) If pdR (M) ≤ depthR (N) and lengthR (TorRi (M, N)) < ∞ for all i ≥ 1, then it follows TorRi (M, N) = 0
for all i ≥ 1; see [35, 2.2].
(iv) If pdR (M) < ∞ and TorRi (M, N) = 0 for all i ≥ 1, then the depth formula for M and N holds, thatis, depthR (M ⊗R N) = depthR (N) − pdR (M); see [2, 1.2].

4

OLGUR CELIKBAS, UYEN LE, HIROKI MATSUI, AND ARASH SADEGHI

The following approximation results are classical; see [4, 1.1] and also [21, 3.1 and 3.3].
2.8. Let M be an R-module such that G-dimR (M) = n < ∞.
(i) There is a short exact sequence of R-modules 0 → M → X → G → 0, where pdR (X) = n and G is
totally reflexive. Such an exact sequence is called a finite projective hull of M.
(ii) There is a short exact sequence of R-modules 0 → Y → X → M → 0, where pdR (Y ) = n − 1 and
X is totally reflexive. Such an exact sequence is called a Cohen-Macaulay approximation of M.
The next result can be deduced from [1, 7.1(a)], which is a more general result. Here we provide a
different and self-contained proof for the convenience of the reader.
Proposition 2.9. Let M and N be R-modules. Assume n ≥ 1 is an integer and the following conditions
hold:(i) lengthR (TorRi (M, N)) < ∞ for all i ≥ 1.
c Ri (M, N)) < ∞ for all i ∈ Z.
(ii) lengthR (Tor(iii) G-dimR (M) ≤ depthR (N) − n.
Rc i (M, N) = 0 for all i = −n + 1, . . . , 0 if and only if depthR (M ⊗R N) ≥ n.
Then Tor
Proof. We start by considering a finite projective hull of M, that is, a short exact sequence of R-modules(2.9.1)

0 → M → X → G → 0,

where pdR (X) = G-dimR (M) and G is totally reflexive; see 2.8(i). Then, by 2.6(v), the following holds
for all i ∈ Z:
(2.9.2)c Ri (G, N) ∼c R−i−1 (G∗ , N) ∼c −i+1
Tor(TrG, N).
= Ext
= Ext
R

c Ri (X, N) = 0 for all i ∈ Z; see 2.6(iv). So, in view of 2.6(ii), theIt follows, since pdR (X) < ∞, that Tor
short exact sequence in (2.9.1) yields the following isomorphisms for all i ∈ Z:
(2.9.3)c Ri (M, N) ∼c Ri+1 (G, N).
Tor
= Tor

c Ri (M, N)) < ∞ for all i ∈ Z, it follows from (2.9.2) and (2.9.3) that both
Note that, as lengthR (Torc Ri (G, N)) and lengthR (Extc iR (Tr R G, N)) are finite for each i ∈ Z. Therefore, since G is tolengthR (Tortally reflexive, 2.6(i) shows, for each i ≥ 1, that:(2.9.4)length(TorRi (G, N)) < ∞ and lengthR (ExtiR (Tr R G, N)) < ∞.

Recall that we assume lengthR (TorRi (M, N)) < ∞ for all i ≥ 1. So, by (2.9.1) and (2.9.4), it follows
that lengthR (TorRi (X, N)) < ∞ for all i ≥ 1. As pdR (X) = G-dimR (M) ≤ depthR (N) − n < depthR (N),we conclude by 2.7(iii) that TorRi (X, N) = 0 for all i ≥ 1. Hence, by tensoring (2.9.1) with N, we obtainthe following exact sequence:(2.9.5)

0 → TorR1 (G, N) → M ⊗R N → X ⊗R N → G ⊗R N → 0.

Furthermore, the depth formula 2.7(iv), in view of the vanishing of TorRi (X, N) for all i ≥ 1, shows:(2.9.6)depthR (X ⊗R N) = depthR (N) − G-dimR (M) ≥ n.
R

c i (M, N) = 0 for all i = −n + 1, . . . , 0.
Now we assume depthR (M ⊗R N) ≥ n and proceed to prove Tor
Note length(TorR1 (G, N)) < ∞; see (2.9.4). Therefore, (2.9.5) shows that TorR1 (G, N) = 0 and also
depthR (G ⊗R N) ≥ n − 1 since both depthR (X ⊗R N) and depthR (M ⊗R N) are at least n; see (2.9.6).
We know from (2.9.4) that lengthR (ExtiR (Tr R G, N)) < ∞ for all i = 1, . . . , n − 1. Hence 2.7(i) implies∼ Exti (Tr R G, N) = 0 for all i = 1, . . . , n − 1. Consequently, (2.9.2) and (2.9.3) yieldc iR (Tr R G, N) =that ExtR
the required vanishing of Tate Tor modules.

REMARKS ON A CONJECTURE OF HUNEKE AND WIEGAND AND THE VANISHING OF (CO)HOMOLOGY

5

c Ri (M, N) = 0 for all i = −n + 1, . . . , 0 and proceed to prove depthR (M ⊗R N) ≥ n.
Next we assume Tor
c R0 (M, N) and Exti (Tr R G, N) ∼c R1 (G, N) ∼c R−i (M, N) forc iR (Tr R G, N) ∼Note TorR (G, N) ∼
= Tor
= Ext
= Tor
= Tor
1

R

all i = 1, . . . , n − 1; see (2.9.2) and (2.9.3). Therefore, we conclude TorR1 (G, N) = 0 = ExtiR (Tr R G, N)
for all i = 1, . . . , n − 1. Now, since depthR (N) ≥ n and ExtiR (Tr R G, N) = 0 for all i = 1, . . . , n − 1, we use
2.7(ii) and deduce that depthR (G⊗R N) ≥ min{n−1, depthR (N)} ≥ n−1. Recall that depthR (X ⊗R N) ≥n; see (2.9.6). Thus, since TorR1 (G, N) = 0, we obtain, by the depth lemma applied to the exact sequence(2.9.5), that depthR (M ⊗R N) ≥ n.

2.10. Let M and N be R-modules. Assume CI-dimR (M) < ∞ or CI-dimR (N) < ∞. Then the following
conditions are equivalent; see [8, 4.9].
(i) TorRi (M, N) = 0 for all i  0.
(ii) TorRi (M, N) = 0 for all i > CI-dimR (M).
c Ri (M, N) = 0 for all i ∈ Z.
(iii) Tor
Moreover, if CI-dimR (M) = 0 and N = M ∗ , then M is free if and only if one of the above equivalentc Ri (M, M ∗ ) = 0 for all i ∈ Z, then, sinceconditions holds: this is because, if CI-dimR (M) = 0 and TorM∗ ∼= Ω2R Tr R M, it follows that TorR1 (M, Tr R M) = 0, which forces M to be free; see 2.4 and 2.6(i).
We should also note, if CI-dimR (M) < ∞, then the conditions (i), (ii) and (iii) stated above are also
equivalent for Ext and Tate cohomology modules; see [8, 4.7].
Corollary 2.11. Assume R is a d-dimensional Gorenstein ring and let M and N be maximal CohenMacaulay R-modules such that pdRp (Mp ) < ∞ for all p ∈ Spec(R) − {m}. Then M ⊗R N is maximalRc i (M, N) = 0 for all i = −d + 1, . . . , 0.
Cohen-Macaulay if and only if Tor
Proof. Let p ∈ Spec(R) − {m}. Then it follows that CI-dimRp (Mp ) = 0. Therefore, by 2.10, we havec Rj (M, N)p = 0 for all i ≥ 1 and for all j ∈ Z. Now the claim follows from
TorRi (M, N)p = 0 and Tor
Proposition 2.9.
In passing we record an immediate consequence of Corollary 2.11; it yields a new characterizationof torsionfreeness of tensor products in terms of the vanishing of Tate homology:
Corollary 2.12. Assume R is a one-dimensional Gorenstein domain and let M and N be torsion-freec R0 (M, N) = 0.
R-modules. Then M ⊗R N is torsion-free if and only if Tor
Let us note that, in view of Corollary 2.12, Conjecture 1.1 can be stated over Gorenstein rings asfollows: if R is a one-dimensional Gorenstein domain and M is a torsion-free R-module such thatc R0 (M, M ∗ ) = 0, then M is free.
Tor
Our main result in this section is Theorem 2.14 which yields a new criterion for the vanishing of
Tate (co)homology. More precisely, Theorem 2.14 is an extension of [18, 4.8], which considers thevanishing of (absolute) cohomology under the same hypotheses. To prove the theorem, we record a few
more preliminary definitions and results.
2.13. Let M be an R-module. Then the complexity cxR (M) of M is defined as follows; see [7]:inf{r ∈ N ∪ {0} : there exists A ∈ R such that dimk (ExtnR (M, k)) ≤ A · nr−1 for all n  0}.
Note, it follows from the definition that, cxR (M) = 0 if and only if pdR (M) < ∞, and cxR (M) ≤ 1 if andonly if M has bounded Betti numbers. Furthermore, the following properties hold:(i) If CI-dimR (M) < ∞, then it follows that cxR (M) ≤ embdim(R) − depth(R); see [9, 5.6].
(ii) If CI-dimR (M) = 0, then it follows CI-dimR (M ∗ ) = 0 and cxR (M) = cxR (M ∗ ); see [11, 4.2].

6

OLGUR CELIKBAS, UYEN LE, HIROKI MATSUI, AND ARASH SADEGHI

In the following G(R)Q denotes the reduced Grothendieck group with rational coefficients, that is,
G(R)Q = (G(R)/Z·[R])⊗Z Q, where G(R) is the Grothendieck group of (finitely generated) R-modules.
Also, [N] denotes the class of a given R-module N in G(R)Q .
Theorem 2.14. Let M and N be R-modules. Assume:(i) cxR (M) = c.
(ii) CI-dimR (M) < ∞.
(iii) pdRp (Mp ) < ∞ for all p ∈ Spec(R) − {m}.
(iv) [N] = 0 in G(R)Q .
Then, given an integer n, the following hold:c iR (M, N) = 0 for all i = n, . . . , n + c − 1, then Ext
c iR (M, N) = 0 for all i ∈ Z.
(a) If Ext
c Ri (M, N) = 0 for all i = n, . . . , n + c − 1, then Tor
c Ri (M, N) = 0 for all i ∈ Z.
(b) If Tor

Proof. We proceed and prove the statement in part (a) first. Set dim(R) = d and X = Ωn−d−1ΩdR M.
R
Then X is totally reflexive and 2.6(iii) yields the following isomorphisms for all i ∈ Z:
(2.14.1)

c iR (M, N) ∼c Ri−d (ΩdR M, N) ∼c Ri−n+1 (X, N).
Ext
= Ext
= Ext

Therefore, for all j ≥ 1, we obtain:(2.14.2)c Rj (X, N) ∼c Rj+n−1 (M, N).
ExtRj (X, N) ∼
= Ext
= Ext

Here the first and the second isomorphisms are due to 2.6(i) and (2.14.1), respectively. Hence, (2.14.2)
and our assumption give:(2.14.3)

ExtRj (X, N) = 0 for all j = 1, . . . , c.

Note that CI-dimR (X) = 0 and cxR (X) = cxR (M) = c; see [9, 1.9.1] and [40, 3.6]. As X is locally
free on the punctured spectrum of R, we use [18, 4.8] and conclude from (2.14.3) that ExtRj (X, N) = 0
for all j ≥ 1, that is, ExtRj (M, N) = 0 for all j  0. Now the required vanishing follows from 2.10.
Next we prove part (b). Note 2.6(iii) yields the following isomorphisms for all i ∈ Z:
(2.14.4)

c Ri (M, N) ∼c Ri−d (ΩdR M, N) ∼c Ri−n+1 (X, N).
Tor
= Tor
= Tor

Then our assumption and (2.14.4) yield:(2.14.5)c Rj (X, N) = 0 for all j = 1, . . . , c.
Tor

Note, as X is totally reflexive, 2.6(v) gives the following isomorphism for all i ∈ Z:
(2.14.6)j−1
c−c Rj (X, N).
Ext(X ∗ , N) ∼= Tor
R

Now (2.14.5) and (2.14.6) show that:(2.14.7)

j−1
c−
Ext
(X ∗ , N) = 0 for all j = 1, . . . , c.
R

It follows, since CI-dimR (X) = 0, that CI-dimR (X ∗ ) = CI-dimR (X) = 0 and cxR (X ∗ ) = cxR (X) = c;j−1
c−see 2.13(ii). Now we use part (a) of the theorem with (2.14.7) and conclude that Ext(X ∗ , N) = 0
R
for all j ∈ Z. Then (2.14.4) and (2.14.6) establish the required vanishing of Tate homology.

We proceed and collect several examples of rings for which hypothesis (iv) of Theorem 2.14 holds;
see [18, 2.5, 2.6 and 4.11] and also Example 3.6.
2.15. Let N be an R-module. Then [N] = 0 in G(R)Q for each one of the following cases:(i) N = ΩnR M for some n ≥ 0 and for some R-module M such that lengthR (M) < ∞.
(ii) R is Artinian.
(iii) R has dimension one and N has rank.

REMARKS ON A CONJECTURE OF HUNEKE AND WIEGAND AND THE VANISHING OF (CO)HOMOLOGY

7(iv) R is a one-dimensional domain.
(v) R is a two-dimensional normal domain with torsion class group.
(vi) R is a two-dimensional complete rational singularity such that k is algebraically closed of characteristic zero, or k is finite of positive characteristic, or k is the algebraic closure of a finite field
which has positive characteristic.
Corollary 2.16. Assume R has positive depth and let M be a nonfree R-module. Assume the following
conditions hold:(i) CI-dimR (M) = 0 and cx(M) ≤ 1.
(ii) lengthR (TorRi (M, M ∗ )) < ∞ for all i  0.
(iii) [M] or [M ∗ ] is zero in G(R)Q .
Then it follows that depthR (M ⊗R M ∗ ) = 0.
Proof. We start by noting that cxR (M) = 1: this is because we assume M is a nonfree R-module suchthat depth(M) = depth(R) and cxR (M) ≤ 1. Note also we have that CI-dimR (M ∗ ) = 0 and cxR (M ∗ ) = 1;
see 2.13(ii). Furthermore, if p ∈ Spec(R) − {m}, then our assumption (ii) and 2.10 imply that Mp is freeover Rp . In other words, M is locally free on the punctured spectrum of R.
Now suppose depthR (M ⊗R M ∗ ) ≥ 1. Then, setting n = 1, we conclude from Proposition 2.9 thatc R0 (M, M ∗ ) = 0. So Theorem 2.14 yields the vanishing of Tor
c Ri (M, M ∗ ) for all i ∈ Z. As we knowTor
CI-dimR (M) = 0, this implies that M is free; see 2.10. Therefore, the depth of M ⊗R M ∗ must bezero.

We should note that, concerning the hypothesis in part (iii) of Corollary 2.16, we are not aware ofan example of a ring R and an R-module M such that CI-dimR (M) = 0 and [M ∗ ] 6= [M] = 0 in G(R)Q .
However, we observe next that such an example cannot occur over Gorenstein rings.
Remark 2.17. Assume R is Gorenstein. It is well-known that the Grothendieck group G(R) is generated
by maximal Cohen-Macaulay R-modules. In fact, given an R-module M with a minimal free resolutionidd
F = (Fi ), it follows that [M] = ∑d−1
i=0 (−1) [Fi ] + (−1) [ΩR M], where d is the dimension of R.
Now we define a map Φ : G(R) → G(R) as follows:dΦ([M]) = ∑ (−1)i−1 [ExtiR (M, R)].
i=0

One can check that Φ is a well-defined group homomorphism. Let M be a maximal Cohen-Macaulay Rmodule. As ExtiR (M, R) = 0 for all i ≥ 1, it follows that Φ([M]) = [M ∗ ] and so Φ2 ([M]) = [M ∗∗ ] = [M].
This shows that Φ is an isomorphism as G(R) is generated by maximal Cohen-Macaulay R-modules.
Since Φ sends [R] to [R], it induces an isomorphism Φ : G(R)Q → G(R)Q . Hence, if M is a maximal
Cohen-Macaulay R-module, then [M] = 0 in G(R)Q if and only if Φ([M]) = [M ∗ ] = 0 in G(R)Q .
If R is a one-dimensional ring and M is a torsion-free R-module such that CI-dimR (M) < ∞ andcxR (M) ≤ 1 (for example, R is a hypersurface ring, or R is a complete intersection ring and M has
bounded Betti numbers), then it follows that M ∼= Ω2R M (for our purpose we may assume M has no free
direct summand); see [9, 7.3] for the details. Hence Theorem 1.2 is subsumed by the next result which
was proved in [16] via different techniques.
Corollary 2.18. ([17, 4.10]) Assume R is one-dimensional and let M be a nonfree torsion-free R-module.
Assume M has rank (e.g., R is a domain). Assume further CI-dimR (M) < ∞ and cxR (M) ≤ 1. Then it
follows that M ⊗R M ∗ has torsion.
Proof. It follows, as R is a one-dimensional ring and M has rank, that [M] = 0 in G(R)Q ; see 2.15.
Hence the claim follows from Corollary 2.16.
8

OLGUR CELIKBAS, UYEN LE, HIROKI MATSUI, AND ARASH SADEGHI

Note that, in the next section we will prove Theorem 1.3 and hence generalize Corollary 2.18. We
finish this section by pointing out some flexibility about the hypotheses of Corollary 2.18.
Remark 2.19. Assume R is a one-dimensional domain and let M be a nonzero torsion-free R-module.
If CI-dimR (M) < ∞ or CI-dimR (M ∗ ) < ∞, then it follows that CI-dimR (M) = 0 = CI-dimR (M ∗ ); see
2.13(ii) and [17, 4.5(i)]. So, if CI-dimR (M) < ∞ or CI-dimR (M ∗ ) < ∞, and cxR (M) ≤ 1 or cxR (M ∗ ) ≤ 1(for example, if CI-dimR (M ∗ ) < ∞ and cxR (M) ≤ 1, or CI-dimR (M) < ∞ and cxR (M ∗ ) ≤ 1), then it
follows that CI-dimR (M) < ∞ and cxR (M) ≤ 1.
3. C ONJECTURE 1.1 FOR TWO - PERIODIC MODULES OVER ONE - DIMENSIONAL DOMAINS
In this section we give a proof of Theorem 1.3; see the paragraph following Proposition 3.9. The
primary result of the section is Theorem 3.2 which relies upon a slightly modified version of the theta
invariant. This invariant was initially defined by Hochster [31] to study the direct summand conjecture;
it was further developed by Dao, for example, to study the vanishing of Tor; see [23, 24, 25].
We start by recalling the definition of the theta invariant; see also [17, 2.13] and [24, section 2].
Definition 3.1. ([31]) Let M be an R-module. Assume the following hold:(i) M ∼
= Ω2R M.
(ii) pdRp (Mp ) < ∞ for all p ∈ Spec(R) − {m}.
Then, for each i ≥ 1, it follows that TorRi (M, N) ∼= TorRi+2 (M, N) and lengthR (TorRi (M, N)) < ∞. Therefore, given an R-module N, the theta invariant for the pair (M, N) is defined as follows:
(3.1.1)
θ R (M, N) = lengthR TorR2n (M, N) − lengthR TorR2n−1 (M, N) for some n ≥ 1.
Note that θ R (M, N) is well-defined, i.e., its value is independent of the integer n used in the definition.
Next we show that the theta function is additive on short exact sequences.
Theorem 3.2. Let M be an R-module. Assume the following hold:(i) M ∼
= Ω2R M.
(ii) pdRp (Mp ) < ∞ for all p ∈ Spec(R) − {m}.
fg(iii) 0 → X −→ Y −→ Z → 0 is a short exact sequence of R-modules.
Then it follows that θ R (M,Y ) = θ R (M, X) + θ R (M, Z).
Proof. Note that, since M ∼= Ω2R M, we can pick a minimal free resolution F• = (Fn , ∂n ) of M such that
Fn = Fn+2 for each n ≥ 0 and ∂n = ∂n+2 for all n ≥ 1.
Fix an integer n ≥ 1. Then, by tensoring f with Fn , we get the following commutative diagram:Fn ⊗R X

Fn+2 ⊗R X

1Fn ⊗ f/ Fn ⊗R Y

1Fn+2 ⊗ f

/ Fn+2 ⊗R Y

The diagram above yields the following commutative diagram on homologies:
TorRn (M, X)∼=

TorRn (M, f )/ TorRn (M,Y )∼=
TorRn+2 (M, f )/ TorR (M,Y )TorRn+2 (M, X)
n+2
R
Now set Xn = ker Torn (M, f ) . Then it follows, since the diagram above involving the Tor modules,
is commutative with vertical maps being isomorphisms, that Xn ∼= Xn+2 .

REMARKS ON A CONJECTURE OF HUNEKE AND WIEGAND AND THE VANISHING OF (CO)HOMOLOGY

f

9

g

Note that the short exact sequence 0 → X −→ Y −→ Z → 0 gives rise to the following exact sequence:(3.2.1)

0 → Xn+2 → TorRn+2 (M, X) → TorRn+2 (M,Y ) → TorRn+2 (M, Z) →TorRn+1 (M, X) → TorRn+2 (M,Y ) → TorRn+1 (M, Z) → Xn → 0.

As we have Xn ∼= Xn+2 , by taking the alternating sum of lengths of the terms in the exact sequence(3.2.1), we conclude that θ R (M,Y ) = θ R (M, X) + θ R (M, Z).

Recall that G(R)Q denotes the reduced Grothendieck group with rational coefficients.
Corollary 3.3. Let M be an R-module. Assume the following hold:∼ Ω2 M.
(i) M =R(ii) pdRp (Mp ) < ∞ for all p ∈ Spec(R) − {m}.
Then θ R (M, −) : G(R)Q → Q is a well-defined function.
Proof. It is clear from Theorem 3.2 that θ R (M, −) is a well-defined function on G(R). As θ R (M, R) = 0,
the result follows; see Definition 3.1.

3.4. If M and N are R-modules, then the pair (M, N) is said to be n-Tor-rigid for some n ≥ 1 provided
that the following condition holds: if TorRi (M, N) = 0 for all i = r, r + 1, . . . , r + n − 1 for some r ≥ 1,then TorRj (M, N) = 0 for all j ≥ r.
An R-module M is called Tor-rigid if (M, N) is 1-Tor-rigid for each R-module N. For example, if R
is a regular ring, or a hypersurface which is quotient of an unramified regular ring, then each R-moduleof finite projective dimension is Tor-rigid; see [2, 2.2] and [36, Cor. 1 and Thm. 3].
Corollary 3.5. Let M and N be R-modules. Assume:(i) pdRp (Mp ) < ∞ for all p ∈ Spec(R) − {m}.
(ii) [N] = 0 in G(R)Q .
q(iii) M ∼= ΩR M for some even integer q ≥ 2.
Then the pair (M, N) is (q − 1)-Tor-rigid.
q
Proof. Set X = M ⊕ Ω2R (M) ⊕ · · · ⊕ ΩRq−2 (M). Then, since M ∼= ΩR M, it follows that X ∼= Ω2R X. Notealso, if r ≥ 1, then by the definition of X, we have:(3.5.1)

TorRr (X, N) = 0 if and only if TorRr+2i (M, N) = 0 for all i = 0, . . . , (q/2) − 1.

Next, to show (M, N) is (q − 1)-Tor-rigid, suppose TorRi (M, N) = 0 for all i = n, n + 1, . . . , n + q − 2
for some n ≥ 1. Then, by (3.5.1), it follows that TorRn (X, N) = 0. Hence, if we show TorRn+1 (X, N) = 0,
then, as X ∼= Ω2R X, we conclude that TorRi (X, N) = 0 for all i ≥ 1; this yields the vanishing of TorRj (M, N)
for all j ≥ 1 and establishes that (M, N) is (q − 1)-Tor-rigid.
Notice pdRp (Xp ) < ∞, for all p ∈ Spec(R) − {m}. So, Corollary 3.3 shows θ R (X, −) : G(R)Q → Q
is well-defined. Thus θ R (X, N) = 0 because [N] = 0 in G(R)Q . Consequently, as we know TorRn (X, N)
vanishes, we deduce by the definition of the theta function that TorRn+1 (X, N) = 0; see Definition 3.1. 
The following example points out that hypothesis (ii) of Corollary 3.5 is needed, even over isolated
hypersurface singularities.
Example 3.6. Let R = C[[x, y, z, w]]/(xw − yz), M = R/(x, z) and N = R/(x, y). Then one can check that
R is a three-dimensional (A1 ) singularity (and hence is a domain) and TorR2 (M, N) = 0 6= TorR3 (M, N).
Thus we have θ R (M, N) = −1. This implies that [N] 6= 0 in G(R)Q . Let us also note, since G(S)Q ∼=Q
for each one-dimensional (A1 ) singularity S, it follows from Knörrer periodicity that G(R)Q ∼= Q; see[41, 12.10 and 13.10].

10

OLGUR CELIKBAS, UYEN LE, HIROKI MATSUI, AND ARASH SADEGHI

As Example 3.6 shows, the conclusion of Corollary 3.5 may fail if hypothesis (ii) is not satisfied.
However, even without this assumption, the corollary can be useful to produce Tor-rigid modules:
Corollary 3.7. Let M be an R-module, and let x ∈ m be a non zero-divisor on R. Assume that thefollowing hold:(i) pdRp (Mp ) < ∞ for all p ∈ Spec(R) − {m}.
(ii) M ∼= Ω2R M.
Then the pair (M/xM, N/xN) is Tor-rigid over R/xR for each torsion-free R-module N. Therefore, thepair (M/xM, M/xM) is Tor-rigid over R/xR.
x

Proof. Let N be a torsion-free R-module. As there is an exact sequence 0 → N → N → N/xN → 0,
it follows that [N/xN] = [N] − [N] = 0 in G(R)Q . Hence Corollary 3.5 shows that the pair (M, N/xN)
R/xR
is Tor-rigid over R. Now, for an integer n ≥ 0, it follows Torn (M/xM, N/xN) = 0 if and only if
R/xR
TorRn (M, N/xN) = 0 if and only if Tori (M/xM, N/xN) = 0 for all i ≥ n; see [37, Lemma 2, page140].
In the following we record some classes of rings over which periodic modules have certain Torrigidity property; see also A.1 for a characterization of Tor-rigid modules over one-dimensional Gorenstein domains.
Corollary 3.8. Assume that one of the following holds:(i) R is Artinian.
(ii) R is a one-dimensional domain.
(iii) R is a two-dimensional normal domain with torsion class group.
q
If M is an R-module such that M ∼= ΩR M for some even integer q ≥ 2, then M is (q − 1)-Tor-rigid.
Therefore, if M ∼= Ω2R M, then M is Tor-rigid.
Proof. It is known that, if R is as in part (i), (ii) or (iii), then G(R)Q = 0; see 2.15. Therefore the result
follows from Corollary 3.5.
Proposition 3.9. Let M be a nonzero R-module. Assume:(i) Mp is free for each associated prime ideal of R.
q(ii) M ∼= ΩR M for some integer q ≥ 1.
(iii) M is Tor-rigid.
Then M ⊗R M ∗ has torsion.
Proof. Note that M ∗ 6= 0 since M is nonzero and torsion-free. Hence there exists a short exact sequenceof R-modules 0 → M ∗ → F → C → 0, where F is free. Tensoring this exact sequence with M, we
see that TorR1 (C, M) ,→ M ⊗R M ∗ . Note that TorR1 (C, M) is a torsion module since Mp is free for eachassociated prime ideal p of R.
Now assume M ⊗R M ∗ is torsion-free. Then TorR1 (C, M), and hence TorRi (C, M) for each i ≥ 1,
vanishes. This yields TorRi (M, M ∗ ) = 0 for all i ≥ 1. As M ∗ ∼= Ω2R Tr R M (up to free summands), we
2q
2q
R
R
R
∼
∼
conclude Tor1 (M, Tr R M) = Tor1 (ΩR M, Tr R M) = Tor1 (M, ΩR Tr R M) ∼= TorR1+2q−2 (M, M ∗ ) = 0. Thisq
implies M is free and hence M = 0 since M ∼= ΩR M; see 2.4. Therefore, M ⊗R M ∗ must have torsion. We are now ready to prove Theorem 1.3 advertised in the introduction. Recall that our argument
removes the complete intersection hypothesis from Theorem 1.2.
Proof of Theorem 1.3. Note that the module M considered in the theorem is Tor-rigid; see Corollary
3.8. Therefore, M ⊗R M ∗ has torsion by Proposition 3.9.

Remark 3.10. Assume R is a one-dimensional domain and let M be a torsion-free R-module.

REMARKS ON A CONJECTURE OF HUNEKE AND WIEGAND AND THE VANISHING OF (CO)HOMOLOGY

11(i) If R is Gorenstein, then it follows that M ⊗R M ∗ is torsion-free if and only if Ext1R (M, M) = 0; see[33, 5.3]. So conjecture 1.1 predicts that the celebrated conjecture of Auslander and Reiten is true
over one-dimensional Gorenstein rings, even if only one Ext module vanishes; see [6] and [33] forthe details. However, if R is not Gorenstein, then torsion-freeness and the vanishing of Ext may
not be equivalent: for example, if R is Cohen-Macaulay with a canonical module ω  R and if R
has minimal multiplicity, then it follows that Ext1R (ω, ω) = 0, but ω ⊗R ω ∗ has torsion; see [32,3.6].
(ii) It seems interesting that, even if R is not Gorenstein, when M ∼= Ω2R M, it still follows that M ⊗R M ∗1
is torsion-free if and only if ExtR (M, M) = 0, since either of these two conditions forces M to bezero; this follows from Theorem 1.3 and the fact that, when M is a Tor-rigid module over a ring R
and ExtnR (M, M) = 0 for some n ≥ 0, then pdR (M) < n; see, for example, [26, 3.1.2].
In general there exist two-periodic modules that do not have finite complete intersection dimension.
We build on an example of Gasharov and Peeva and construct such a module in the next example (notethe ring considered in the example is not a complete intersection); cf. Theorems 1.2 and 1.3.
Example 3.11. ([27, 3.10]) Let k be a field and fix α ∈ k such that α 4 = 1 6= α 3 . Let R = k[[x1 , x2 , x3 , x4 ]]/Iα ,where Iα is the ideal of R generated by the elements x12 − x22 , x32 , x42 , x3 x4 , x1 x4 + x2 x4 , α · x1 x3 + x2 x3 .
Then the complex
∂

∂

∂

3
2
1
· · · −→
R⊕2 −→
R⊕2 −→
R⊕2 → N → 0

is exact, where
x1 α n · x3 + x4
N = coker(∂1 ) and ∂n =for each n ≥ 1.
0
x2
It follows Ω4R N ∼= coker(∂5 ) = coker(∂1 ) ∼= N and that N  ΩiR N for each i = 1, 2, 3. Then, by setting
22∼ Ω X and hence cxR (X) ≤ 1. Moreover, we have CI-dimR (X) = ∞:X = N ⊕ ΩR N, we see that X =R∼ Ω2 N since cxR (N) ≤ 1.
otherwise we would have CI-dimR (N) < ∞ and so N =R
Next we set T = R[[t]] and M = X ⊗R T . Then, since T is a faithfully flat extension of R, we conclude
that T is a one-dimensional Gorenstein ring, M ∼= Ω2T M, and also CI-dimT (M) = ∞.
The ring T in Example 3.11 is not a domain and also the T -module M does not have rank. This raises
the following question which, in view of Conjecture 1.1 and Theorem 1.3, should be important:
Question 3.12. Is there a one-dimensional ring T and a T -module M such that M has rank over T (e.g.,
T is a domain), M ∼= Ω2T M, and CI-dimT (M) = ∞?
We finish this section by recording some further observations concerning the torsion submoduleof tensor products of the form M ⊗R M ∗ . Let us point out that the conclusion of Theorem 1.3 may
fail if the ring in question is not one-dimensional; for example, [19, 3.5] provides such an example
of a two-dimensional hypersurface domain. In the aforementioned example, the module considered is
torsion-free but it is not maximal Cohen-Macaulay. On the other hand, when maximal Cohen-Macaulay
modules are considered, there is a partial result, which we recall next:
3.13. If R is an even-dimensional hypersurface and M is a nonfree maximal Cohen-Macaulay R-modulethat is locally free on the punctured spectrum of R, then M ⊗R M ∗ has torsion; see [19, 3.7].
Note that the module M in 3.13 is two-periodic since it is maximal Cohen-Macaulay over a hypersurface ring. As we are concerned with periodic modules of even period, we proceed and investigate
whether there is an extension of 3.13 for such modules over rings that are not necessarily hypersurfaces.
For that we first prove:
Proposition 3.14. Assume R is a d-dimensional Cohen-Macaulay ring with canonical module ω andlet M be a nonzero R-module that is locally free on the punctured spectrum of R. Assume d = 2nr and
†
†
M∼
= Ω2n
R M for some positive integers n and r. Then M ⊗R M has torsion, where M = HomR (M, ω).

12

OLGUR CELIKBAS, UYEN LE, HIROKI MATSUI, AND ARASH SADEGHI

Proof. Suppose M ⊗R M † is torsion-free and seek a contradiction. Note that it follows:
0 = ExtdR (M ⊗R M † , ω)
∼ Extd (M, M)=R

2n(r−1)∼
M, M= Ext2nR ΩR
2n∼ Ext (M, M)
=



R

∼
M, Ω2n= Ext1R (Ω2n−1
R M).
RHere the equality follows by [13, 3.5.11] since depthR (M ⊗R M † ) > 0, and the first isomorphism is due
to [28, 2.3].
2n⊕r → Ω2n−1 M → 0 splits. ThereAs Ext1R (Ω2n−1
M, Ω2nR M) = 0, the syzygy sequence 0 → ΩR M → R
R
R
2n
2n
∼
fore ΩR M is free, and hence so is M since M = ΩR M. This implies M = 0 since a (minimal) syzygy
of a free module is zero; see 2.2. This gives the required contradiction and proves M ⊗R M † must havetorsion.

The next corollary of Proposition 3.14 yields an extension of 3.13.
Corollary 3.15. Assume R is Gorenstein and has positive even dimension. Let M be a nonzero R-modulethat is locally free on the punctured spectrum of R. If M ∼= Ω2R M, then M ⊗R M ∗ has torsion.
We should note that Corollary 3.15 may fail if the dimension of R is odd an at least three; see [19,3.12].
4. A CONDITION IMPLYING C ONJECTURE 1.1 OVER COMPLETE INTERSECTION RINGSIn this section throughout R denotes a complete intersection ring of codimension c which equals
S/(x) for some commutative Noetherian regular local ring (S, n) and for some S-regular sequence x =x1 , . . . , xc ⊆ n2 . Such a complete intersection ring is called a hypersurface when c = 1.
As mentioned in the introduction, although Conjecture 1.1 is true over hypersurface rings, it is wide
open for complete intersection rings that have codimension at least two. The aim of this section is
to formulate a condition over hypersurfaces, which, if true, forces Conjecture 1.1 to be true over allcomplete intersection rings.
We start with a setup and then recall a theorem of Orlov [38, Section 2] which plays a key role forour argument. For the basic definitions that are not defined in this section, we refer the reader to [30].
4.1. We let mod R to denote the category of (finitely generated) R-modules. Similarly, for a scheme X,
coh X denotes the category of coherent OX -modules. It follows that there is an equivalence of categories:
e see, for example, [30, II 5.5].
mod R ∼= coh(Spec(R)) given by M 7→ M;
4.2. Let X be a scheme. A perfect complex on X is a bounded complex of coherent sheaves on X which
has finite flat dimension as a complex.
The singularity category is defined as the quotient:Dsg (X) = Db (coh X)/ Dperf (X),
where Dperf (X) denotes the full subcategory of the bounded derived category Db (coh X) consisting of
all perfect complexes on X.
Note that a perfect complex on Spec(R) is isomorphic in Db (mod R) to a bounded complex of finitely
generated projective modules via the equivalence mod R ∼= coh(Spec(R)) mentioned in 4.1. Moreover,
if R is Gorenstein, then the singularity category of R is equivalent to the stable category of maximal
Cohen-Macaulay R-modules by [14, 4.4.1].

REMARKS ON A CONJECTURE OF HUNEKE AND WIEGAND AND THE VANISHING OF (CO)HOMOLOGY

13

4.3. Recall that R = S/(x). Let t = t1 , . . . ,tc be indeterminates over S. Then we define the gradedhypersurface ring c
A = S[t]/ ∑ xiti ,i=1

where the grading is given by deg s = 0 for all s ∈ S, and degti = 1 for each i.
Now we consider the natural surjections:S[t]  A  A/(x) = R[t].
These surjections yield the following commutative diagram of schemes:
Z := Pc−1
Ri/ Y := Proj(A) u/ X := Pc−1
S

q

p



Spec(R) j/ Spec(S).

In the above diagram, the morphisms i, u, and j are closed immersions that are induced by the surjections
A  R[t], S[t]  A, S  R, respectively. Also, the morphisms p and q are canonical. We note that i isa regular closed immersion of codimension c − 1, that is, the ideal sheaf of i is locally generated by aregular sequence of length c − 1. Furthermore, the morphism p is flat.
We consider two functors p∗ : mod R → coh Z and i∗ : coh Z → cohY , which are defined as follows:g(i) p∗ : mod R ∼= coh(Spec(R)) → coh Z is the pullback along p, where p∗ (M) is the OZ -module M[t]∼associated with a graded R[t]-module M[t] = M ⊗R R[t] = (M ⊗R A)/x(M ⊗R A); see [30, p116,
Definition].
e for(ii) i∗ : coh Z → cohY is the pushout along i. Note that, every object of coh Z is isomorphic to Lefsome graded R[t]-module L. Then i∗ (L) is isomrphic to LA , where LA is the graded R[t]-module L
considered as a graded A-module via the ring map A  R[t].
Notice, since p is flat and i is closed immersion, it follows that p∗ and i∗ are exact functors, see [39,02N4 and 01QY]. Therefore, by deriving these functors, we also obtain triangle functors:i∗ : Db (coh Z) → Db (cohY ) and p∗ : Db (mod R) → Db (coh Z).
Here, the functors are given by applying p∗ and i∗ component-wise.
4.4. Note that Y = Proj A is an integral scheme and hence every ring of section is an integral domain.
Indeed, since S[t1 , . . . ,tc ] is an integral domain, we can easily check that ∑ci=1 xiti is an irreducible
element. Therefore, it is a prime element as S[t1 , . . . ,tc ] is a UFD and hence A is a domain; see 4.3.
Throughout this section we keep the notations and the setting of 4.3. The following result of Orlov[38] plays a key role in the proof Theorem 4.12.
4.5. ([38, 2.1], see also [15, A.4]) The triangle functor Φ = i∗ p∗ : Db (mod R) → Db (cohY ) induces a∼=triangle equivalence Φ : Dsg (R) −→ Dsg (Y ).
^ ⊗R A) for any R-module M. Moreover, by letting fdO Φ(M) denote
Note, Φ(M) ∼= (M ⊗R A)/x(MY
the flat dimension of Φ(M), we havepdR (M) < ∞ ⇐⇒ M ∼= 0 in Dsg (R) ⇐⇒ Φ(M) ∼= 0 in Dsg (Y )⇐⇒ fdOY Φ(M) < ∞ ⇐⇒ pdOY,y Φ(M)y < ∞ for all y ∈ Y.
Here, the first and third equivalences follow by the definition of singularity categories, the second equivalence follows from Orlov’s theorem 4.5, and [30, III 9.2(e)] proves the last equivalence.

14

OLGUR CELIKBAS, UYEN LE, HIROKI MATSUI, AND ARASH SADEGHI

4.6. Let M and N be R-modules. Then we have the following:^ ⊗ A) ⊗^Φ(M) ⊗OY Φ(N) = (M ⊗R A)/x(MR
OY (N ⊗R A)/x(N ⊗R A)


^


∼
= (M ⊗R A)/x(M ⊗R A) ⊗A (N ⊗R A)/x(N ⊗R A))
=



^


 
(M ⊗R N) ⊗R A /x (M ⊗R N) ⊗R A

= Φ(M ⊗R N).
where the isomorphism follows from [30, Proof of II 5.12(b)].
Next we proceed to determine Φ(M ∗ ). For this, we use the Grothendieck duality theorem [29].
Lemma 4.7. Let F ∈ coh Z and n ∈ Z. Then there is a natural isomorphism as follows:∼ i∗ E xtn−c+1 (F , OZ )(1).
E xtn (i∗ F , OY ) =OYOZProof. Note, by the Grothendieck duality theorem [29, III 6.7], there is a natural isomorphism∼ i∗ RH omO (F , i! OY ),RH omO (i∗ F , OY ) =Y

Z

where i! : Db (cohY ) → Db (coh Z) denotes the right adjoint functor of ibb
∗ : D (coh Z) → D (cohY ). We!
proceed to prove that i OY is isomorphic to OZ (1)[−c + 1].
Note that, by [29, III 7.3], there is an isomorphism i! OY ∼= ωZ/Y [−c + 1]. Here, ωZ/Y is the relative

canonical sheaf of the regular closed immersion i : Z ,→ Y ; see [29, III §1] for its definition. On theother hand, by [29, III 1.5], we have an isomorphism of the form ωZ/Y ∼= ωZ/X ⊗OZ (i∗ ωY /X )∨ . The
ideal sheaves of ui : Z ,→ X and u : Y ,→ X are globally generated by degree 0 and 1 regular sequences,respectively. Therefore, the following isomorphisms hold:ωZ/X ∼= OZ and ωY /X ∼= OY (−1).
Hence we conclude that i! OY ∼= ωZ/Y [−c + 1] ∼= OZ ⊗OZ i∗ (OY (−1))∨ [−c + 1] ∼= OZ (1)[−c + 1].



4.8. Let M be an R-module. Then there are natural isomorphisms
E xtnOY (Φ(M), OY ) = E xtnOY (i∗ p∗ (M), OY )∼= i∗ E xtn−c+1 (p∗ (M), OZ )(1)OZ∼
(M, R))(1)= Φ(Extn−c+1R
Here, the first isomorphism uses Lemma 4.7 and the last isomorphism is due to [30, III 6.5] together
with the following fact: p∗ (P• ) is a resolution of p∗ (M) by locally free sheaves of finite rank for a given
resolution P• by finite free module of M.
Assume further that M is a totally reflexive R-module. Then, by definition, if n 6= 0, it follows that
ExtnR (M, R) ∼= 0. Therefore, the above isomorphisms yield:(0(i 6= c − 1)i∼
E xtOY (Φ(M), OY ) =∗Φ(M )(1) (i = c − 1).
Next we recall the definition of the codimension of a module. From now on T denotes a commutative
Noetherian local ring and all T -modules are assumed to be finitely generated.
4.9. Let N be a T -module. Then the codimension codimT (N) of N is defined as the codimension of its
support SuppT (N) as a closed set in Spec(T ). More precisely, we havecodimT (N) = inf{heightT (p) | p ∈ SuppT (N)}.

REMARKS ON A CONJECTURE OF HUNEKE AND WIEGAND AND THE VANISHING OF (CO)HOMOLOGY

15

In the following we record some properties of the codimension that are needed for our argument; see[13, 2.1.2 and 3.3.10]. Among those is the fact that the codimension of a module does not change whenlocalizing at a prime ideal in its support.
4.10. Assume T is Cohen-Macaulay and let N be a nonzero T -module.
(i) It follows that codimT (N) = dim(T ) − dimT (N) = gradeT (N) = inf{i ∈ Z | ExtiT (N, T ) 6= 0},
where gradeT (N) denotes the grade of N over T .
(ii) Assume T admits a canonical module ωT .
(a) Then N is Cohen-Macaulay of codimension t if and only if ExtiT (N, ωT ) = 0 for i 6= t.
(b) If N is Cohen-Macaulay of codimension t and p ∈ SuppT (N), then Np is Cohen-Macaulay
over Tp such that codimTp (Np ) = t.
(c) If N is Cohen-Macaulay of codimension t, then it follows SuppT (N) = SuppT (ExttT (N, ωT )).
4.11. Assume T is a domain and let c ≥ 1 be an integer. Consider the following for T :(i) For each T -module M, if M and M ⊗T Extc−1
T (M, T ) are Cohen-Macaulay of codimension c − 1,then pdT (M) < ∞.
(ii) For each T -module M, if M and M ⊗T HomT (M, T ) are maximal Cohen-Macaulay (i.e., CohenMacaulay T -modules of codimension 0), then M is free.
Next is the main result of this section.
Theorem 4.12. Let c ≥ 1 be an integer. If each hypersurface local domain satisfies condition (i) of
4.11, then each complete intersection local domain of codimension c satisfies condition (ii) of 4.11.
Remark 4.13. Prior to giving a proof for Theorem 4.12, we note that condition (ii) of 4.11, or equivalently, c = 1 case of condition (i) of 4.11, is precisely the condition of Conjecture 1.1 for onedimensional commutative Noetherian local domains. Recall also that each hypersurface local domain
satisfies condition (ii) of 4.11; see [34, 3.1].
Proof of Theorem 4.12. We assume, for the given integer c, that each hypersurface local domain satisfies condition (i) of 4.11. Let R = S/(x) be a domain, where S is a regular local ring and x is an S-regularsequence of length c. Let M be a maximal Cohen-Macaulay R-module such that M ⊗R M ∗ is maximal
Cohen-Macaulay. We proceed to prove that M is free.
First we prove that SuppY (Φ(M)) = SuppY (E xtc−1
OY (Φ(M), OY )). Fix y ∈ SuppY (Φ(M)). The combination of 4.8 and 4.10(ii)(a) show that Φ(M)y is a Cohen-Macaulay OY,y -module of codimensionc − 1. Therefore, by 4.10(ii)(c), we conclude y ∈ SuppY (E xtc−1
OY (Φ(M), OY )) and hence it follows that
c−1
SuppY (Φ(M)) j SuppY (E xtOY (Φ(M), OY )). The converse inclusion is trivial. Moreover, the supportof Φ(M) equals to the support of X , where X = Φ(M) ⊗OY E xtc−1
OY (Φ(M), OY )(−1).
Note, it follows from 4.6 and 4.8 that Φ(M ⊗R M ∗ ) ∼= Φ(M) ⊗OY Φ(M ∗ ) ∼= X . As M and M ⊗R M ∗are totally reflexive R-module, by 4.8, we obtain the following:(4.12.1)
E xti (X , OY ) ∼= E xti (Φ(M ⊗R M ∗ ), OY ) ∼= Φ(Exti−c+1 (M ⊗R M ∗ , R))(1) = 0
OY

OY

R

for i 6= c − 1.
Let y ∈ SuppY (Φ(M)). Then the module Xy = Φ(M)y ⊗OY,y Extc−1
OY,y (Φ(M)y , OY,y ) is Cohen-Macaulayof codimension c − 1 over the local hypersurface domain OY,y by 4.10(ii)(a) and (4.12.1). Now ourassumption shows that Φ(M)y has finite projective dimension over OY,y . So, by 4.5, we have thatpdR (M) < ∞
Now the proof of Theorem 1.4 follows as an immediate consequence of Theorem 4.12:
Proof of Theorem 1.4. Let c ≥ 1 be an integer. If each hypersurface local domain satisfies condition(i) of 4.11 (for the given c), then Theorem 4.12 implies that Conjecture 1.1 is true over each onedimensional complete intersection local domain.
16

OLGUR CELIKBAS, UYEN LE, HIROKI MATSUI, AND ARASH SADEGHI

Remark 4.14. We note a fact that follows from the proof of Theorem 4.12: if each hypersurface domain
which is quotient of an equi-characteristic regular local ring satisfies condition (i) of Theorem 4.12, thenConjecture 1.1 holds over each one-dimensional complete intersection domain which is quotient of anequi-characteristic regular local ring.
We finish this section by noting that, if we consider Conjecture 1.1 over one-dimensional complete
intersection domains that have algebraically closed residue fields, then the proof of Theorem 4.12 issimplified significantly due to a result in [12]:Remark 4.15. Recall that R = S/(x) for some regular local ring (S, n) and for some S-regular sequencex = x1 , . . . , xc ⊆ n2 . Set I = (x) and assume R is a one-dimensional domain with algebraically closed
residue field. Let M be a torsion-free R-module such that I ⊆ n2 and M ⊗R M ∗ is torsion-free.
Let f ∈ I − nI. Then we can extend { f } to a minimal generating set { f1 , . . . , fc } of I, with f = f1 ,which is necessarily a regular sequence on S. Hence M ⊗S/( f1 ) Extc−1(M, S/( f1 )) ∼= M ⊗R M ∗ is a
S/( f1 )Cohen-Macaulay S/( f1 )-module of codimension c − 1. Assuming the condition 4.11(ii), it follows that
pdS/( f1 ) (M) < ∞. Now [12, 3.3] implies that pdR (M) < ∞.
A PPENDIX A. R EMARKS ON THE RIGIDITY OF T OR
It is known that Tor-rigidity, a subtle property, is a sufficient condition for Conjecture 1.1 to holdover one-dimensional Gorenstein domains; see 3.4 and Remark 3.10(ii). Motivated by this fact, we
examine the vanishing of Tor more closely over Gorenstein rings. The observation we aim to establish
in this appendix is the following, which may be helpful to further study Tor-rigidity. In this appendix R
denotes a d-dimensional Gorenstein local ring.
A.1. Assume R is a domain and d = 1. Let M be an R-module. Then the following conditions areequivalent:(i) M is Tor-rigid over R.
(ii) (M,C) is Tor-rigid for each torsion (or equivalently, finite length) R-module C.
(iii) (M,C) is Tor-rigid for each torsion-free (or equivalently, maximal Cohen-Macaulay) R-module C.
We deduce A.1 from the following more general result:
Proposition A.2. Let M be an R-module. Then the following conditions are equivalent:(i) If C is a torsion R-module and TorRn (M,C) = 0 for some n ≥ d, then it follows TorRi (M,C) = 0 for
all i ≥ n.
(ii) If C is a maximal Cohen-Macaulay R-module that has rank and TorRn (M,C) = 0 for some n ≥ d,then it follows TorRi (M,C) = 0 for all i ≥ n.
(iii) If C is an R-module with rank and TorRn (M,C) = 0 for some n ≥ d, then it follows TorRi (M,C) = 0
for all i ≥ n.
Proof. First we show that parts (ii) and (iii) are equivalent, that is, part (ii) implies part (iii). For thatassume part (ii) holds. Let C be an R-module with rank such that TorRn (M,C) = 0 for some n ≥ d. We
want to show that TorRi (M,C) = 0 for all i ≥ n.
We may assume C is not maximal Cohen-Macaulay. Then we consider a Cohen-Macaulay approximation of C, that is, a short exact sequence of R-modules 0 → L → X → C → 0, where pdR (L) < ∞ and
X is maximal Cohen-Macaulay; see 2.8(ii). Note pdR (L) < d so that TorRn (M, X) = 0. Thus, as X has
rank, it follows from the hypothesis that TorRi (M, X) = 0 for all i ≥ n. This yields TorRi (M,C) = 0 for
all i ≥ n, and hence establishes part (iii).
Next we show that part (i) implies (ii). Assume part (i) holds and let C be a maximal Cohen-Macaulay
R-module with rank such that TorRn (M,C) = 0 for some n ≥ d. We want to show that TorRi (M,C) = 0
for all i ≥ n. As C has rank, there exists a short exact sequence 0 → C → G → Y → 0, where G is free

REMARKS ON A CONJECTURE OF HUNEKE AND WIEGAND AND THE VANISHING OF (CO)HOMOLOGY

17

and Y is torsion; see [34, 1.3]. As TorRn (M,C) = 0, we have that TorRn+1 (M,Y ) = 0; now the hypothesis
implies that TorRi (M,Y ) = 0 for all i ≥ n + 1. Consequently, TorRi (M,C) = 0 for all i ≥ n, as required.
Finally we note a module is torsion if and only if it has rank zero. So part (iii) implies part (i).

ACKNOWLEDGEMENTS
The authors thank Kenta Sato for pointing to them a simpler proof of Lemma 4.7 than the one ina previous version of the manuscript. The authors also thank the referee for useful comments andsuggestions which improved the manuscript.
R EFERENCES[1] Mohsen Asgharzadeh, Olgur Celikbas, and Arash Sadeghi, A study of the cohomological rigidity property, preprint (2020),
posted at arXiv:2009.06481. 4[2] Maurice Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647. 3, 9
[3] Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94,
American Mathematical Society, Providence, R.I., 1969. 2, 3
[4] Maurice Auslander and Ragnar-O. Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Mém.
Soc. Math. France (N.S.) (1989), no. 38, 5–37, Colloque en l’honneur de Pierre Samuel (Orsay, 1987). 4[5] Maurice Auslander and Oscar Goldman, Maximal orders, Trans. Amer. Math. Soc. 97 (1960), 1–24. 3[6] Maurice Auslander and Idun Reiten, On a generalized version of the Nakayama conjecture, Proc. Amer. Math. Soc. 52(1975), 69–74. 1, 11[7] Luchezar L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), no. 1, 71–101. 5[8] Luchezar L. Avramov and Ragnar-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent.
Math. 142 (2000), no. 2, 285–318. 2, 3, 5
[9] Luchezar L. Avramov, Vesselin N. Gasharov, and Irena V. Peeva, Complete intersection dimension, Inst. Hautes Études Sci.
Publ. Math. (1997), no. 86, 67–114 (1998). 2, 5, 6, 7
[10] Luchezar L. Avramov and Alex Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorensteindimension, Proc. London Math. Soc. (3) 85 (2002), no. 2, 393–440. 3[11] Petter Andreas Bergh and David A. Jorgensen, On growth in minimal totally acyclic complexes, J. Commut. Algebra 6(2014), no. 1, 17–31. 5[12] Petter Andreas Bergh and David A. Jorgensen, A generalized Dade’s lemma for local rings, Algebr. Represent. Theory 21(2018), 1369–1380. 16[13] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. 12, 15[14] Ragnar-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, preprint,
http://hdl.handle.net/1807/16682, 1986. 12[15] Jesse Burke and Mark E. Walker, Matrix factorizations in higher codimension, Trans. Amer. Math. Soc. 367 (2015), 3323–
3370. 13[16] Olgur Celikbas, Vanishing of Tor over complete intersections, J. Commut. Algebra 3 (2011), no. 2, 169–206. 1, 2, 7
, On the vanishing of the theta invariant and a conjecture of Huneke and Wiegand, Pacific J. Math. 309 (2020), no. 1,[17]
103–144. 7, 8
[18] Olgur Celikbas and Hailong Dao, Asymptotic behavior of Ext functors for modules of finite complete intersection dimension,
Math. Z. 269 (2011), no. 3-4, 1005–1020. 5, 6
[19]
, Necessary conditions for the depth formula over Cohen-Macaulay local rings, J. Pure Appl. Algebra 218 (2014),
no. 3, 522–530. 11, 12[20] Olgur Celikbas and Arash Sadeghi, Maximal Cohen-Macaulay tensor products, Ann. Mat. Pura Appl. (4) 200 (2021), no. 3,
923–944. 3[21] Lars Winther Christensen and Srikanth Iyengar, Gorenstein dimension of modules over homomorphisms, J. Pure Appl.
Algebra 208 (2007), no. 1, 177–188. 4[22] Lars Winther Christensen and David A. Jorgensen, Vanishing of Tate homology and depth formulas over local rings, J. Pure
Appl. Algebra 219 (2015), no. 3, 464–481. 3[23] Hailong Dao, Asymptotic behaviour of Tor over complete intersections and applications, preprint (2008), posted at
arxiv:07105818. 8[24], Some observations on local and projective hypersurfaces, Math. Res. Lett. 15 (2008), no. 2, 207–219. 8[25], Decent intersection and Tor-rigidity for modules over local hypersurfaces, Transactions of the American Mathematical Society 365 (2013), no. 6, 2803–2821. 8

18

OLGUR CELIKBAS, UYEN LE, HIROKI MATSUI, AND ARASH SADEGHI[26], Some homological properties of modules over a complete intersection, with applications, Commutative algebra,
Springer, New York, 2013, pp. 335–371. 1, 11[27] Vesselin N. Gasharov and Irena V. Peeva, Boundedness versus periodicity over commutative local rings, Trans. Amer. Math.
Soc. 320 (1990), no. 2, 569–580. 11[28] Shiro Goto and Ryo Takahashi, On the Auslander-Reiten conjecture for Cohen-Macaulay local rings, Proc. Amer. Math.
Soc. 145 (2017), no. 8. 12[29] Robin Hartshorne, Residues and duality, Lecture Notes in Math., vol. 20, Springer-Verlag, 1966. 14[30], Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. 12, 13,14[31] Melvin Hochster, The dimension of an intersection in an ambient hypersurface, Algebraic geometry (Chicago, Ill., 1980),
Lecture Notes in Math., vol. 862, Springer, Berlin, 1981, pp. 93–106. 8[32] Craig Huneke, Srikanth B. Iyengar, and Roger Wiegand, Rigid ideals in Gorenstein rings of dimension one, Acta Math.
Vietnam. 44 (2019), no. 1, 31–49. 1, 11[33] Craig Huneke and David A. Jorgensen, Symmetry in the vanishing of Ext over Gorenstein rings, Math. Scand. 93 (2003),
no. 2, 161–184. 11[34] Craig Huneke and Roger Wiegand, Tensor products of modules and the rigidity of Tor, Math. Ann. 299 (1994), no. 3,
449–476. 1, 15, 17[35] David A. Jorgensen, A generalization of the Auslander-Buchsbaum formula, J. Pure Appl. Algebra 144 (1999), no. 2, 145–
155. 3[36] Stephen Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220–226. 9[37] Hideyuki Matsumura, Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989, Translated from the Japanese by M. Reid. 10[38] Dmitri O. Orlov, Triangulated categories of singularities, and equivalences between landau-ginzburg models, Mat. Sb. 197(2006), no. 12, 117–132. 2, 12, 13[39] Stacks Project, available at https://stacks.math.columbia.edu. 13[40] Sean Sather-Wagstaff, Complete intersection dimensions for complexes, J. Pure Appl. Algebra 190 (2004), no. 1-3, 267–290.
6[41] Yuji Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series,
vol. 146, Cambridge University Press, Cambridge, 1990. 3, 9
O LGUR C ELIKBAS , S CHOOL OF M ATHEMATICAL AND DATA S CIENCES , W EST V IRGINIA U NIVERSITY, M ORGANTOWN ,WV 26506-6310, U.S.A
Email address: olgur.celikbas@math.wvu.edu
U YEN L E , S CHOOL OF M ATHEMATICAL AND DATA S CIENCES , W EST V IRGINIA U NIVERSITY , M ORGANTOWN , WV
26506-6310, U.S.A
Email address: hle1@mix.wvu.edu
H IROKI M ATSUI , D EPARTMENT OF M ATHEMATICAL S CIENCES , FACULTY OF S CIENCE AND T ECHNOLOGY, T OKUSHIMAU NIVERSITY, 2-1 M INAMIJOSANJIMA - CHO , T OKUSHIMA 770-8506, JAPAN
Email address: hmatsui@tokushima-u.ac.jp
A RASH S ADEGHI , D OKHANIAT 49179-66686, G ORGAN , IRAN
Email address: sadeghiarash61@gmail.com