ON THE DEPTH AND REFLEXIVITY OF TENSOR PRODUCTS
OLGUR CELIKBAS, UYEN LE, AND HIROKI MATSUIA BSTRACT. In this paper we study the depth of tensor products of homologically finite complexes overcommutative Noetherian local rings. As an application of our main result, we determine new conditions under which the factors of a nonzero reflexive tensor product of finitely generated modules over hypersurface
rings can be reflexive.
A result of Asgharzadeh shows that nonzero symbolic powers of prime ideals in a local ring cannot
have finite projective dimension, unless the ring in question is a domain. We make use of this fact in the
appendix and consider the reflexivity of tensor products of prime ideals over hypersurface rings.

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.
The results in this paper are motivated by the following beautiful result of Huneke and Wiegand; see[8, 1.3 and 4.4], [9, 1.1], [16, 2.7], and [17, 1.9].
Theorem 1.1. (Huneke and Wiegand [16, 17]) Let R be a local hypersurface ring, and let M and N be
nonzero finitely generated R-modules. Assume N has rank and M ⊗R N is reflexive. Then TorRi (M, N) = 0
for all i ≥ 1, M is reflexive, N is torsion-free, and pdR (M) < ∞ or pdR (N) < ∞.
It has been an open problem for quite some time whether or not the module N in Theorem 1.1 mustalso be reflexive. This problem occurs since M may not have full support; note, on the contrary, N has
full support since it has rank and the tensor product M ⊗R N is torsion-free; see [8, 16, 18] for the details.
In 2019 Celikbas and Takahashi [10] constructed examples settling this query; one of their examples isthe following:
Example 1.2. ([10, 2.5]) Let R = C[|x, y, z, w]]/(xy), M = R/(x) and let N be the Auslander transposeof R/p, where p = (y, z, w) ∈ Spec(R). Then R is a reduced local hypersurface ring and pdR (N) < ∞ (sothat N has rank). Moreover, M and M ⊗R N are both reflexive, but N is not reflexive.
In this paper, motivated by Theorem 1.1 and Example 1.2, we study the depth of tensor products
and determine new conditions that force both of the modules considered in Theorem 1.1 to be reflexive.
To faciliate the discussion, let us note, in Example 1.2, the sequence {y, z, w} is M-regular, but it is not
R-regular. We prove, if such sequences do not exist locally in the support of the module M consideredin Theorem 1.1, then both of the modules in question must be reflexive. More precisely, we prove:
Theorem 1.3. Let R be a local hypersurface ring, and let M and N be nonzero R-modules such that
M ⊗R N is reflexive. Assume the following conditions hold:(i) N has rank (e.g., pdR (N) < ∞).
(ii) Each Mp -regular sequence is Rp -regular for all p ∈ SuppR (M).
Then M and N are both reflexive.
2010 Mathematics Subject Classification. Primary 13D07; Secondary 13H10, 13D05, 13C12.
Key words and phrases. Complexes, depth formula, reflexivity of tensor products, Serre’s condition, vanishing of Tor.
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O. CELIKBAS, U. LE, AND H. MATSUI

It is worth pointing out that the condition in part (ii) of Theorem 1.3 holds provided that the module
M in question has full support; see Corollary 3.8 (note the module M considered in Example 1.2 does
not have full support). On the other hand, there are examples of modules – without full support – thatsatisfy the aforementioned condition of Theorem 1.3; see Examples 4.1 and 4.2.
In Section 3 we establish Theorem 1.3 as a consequence of our main result, namely Theorem 3.1,
which concerns the depth of (derived) tensor products of homologically finite complexes that have finite
complete intersection dimension over local rings. The proof of Theorem 1.3 relies upon a relation
between the condition in part (ii) of Theorem 1.3 and a certain depth inequality, which does not hold forthe module M in Example 1.2; see Corollary 3.4, Example 3.5, and Proposition 3.6.
In Section 4 we compare Theorem 1.3 with the main result of [9], in which the reflexivity of tensor
products of modules under the setting of Theorem 1.1 is also studied. We give examples and highlightthat the condition we consider in part (ii) of Theorem 1.3 is independent of the main tool used in [9];see the examples and the first paragraph in Section 4.
In the appendix we give an application of Theorem 1.1: we prove that, if the tensor product of two
prime ideals is reflexive over a hypersurface ring that is not a domain, then both of the primes consideredmust be minimal. In fact, due to the work of Asgharzadeh [2], we are able to state our result in terms ofthe tensor product of symbolic powers of prime ideals; see Remark A.1 and Corollary A.4.
2. P RELIMINARIES
We start by recording some definitions and preliminary results that are needed for our arguments.
2.1. Complexes ([11]). Throughout, by an R-complex X, we mean a chain complex of R-modules which
has homological differentials ∂iX : Xi → Xi−1 , and which is homologically degreewise finite (i.e., Hi (X)is finitely generated for all i ∈ Z) and homologically bounded below (i.e., Hi (X) = 0 for all i ≪ 0).
We say that an R-complex is bounded if it is homologically bounded (i.e., Hi (X) = 0 for all |i| ≫ 0).
If X is an R-complex, then we set:
sup X = sup{i ∈ Z | Hi (X) ̸= 0} and inf X = inf{i ∈ Z | Hi (X) ̸= 0}.
Note we have that sup X = −∞ if and only if H(X) = 0 if and only if inf X = ∞.
2.2. Let X and Y be R-complexes. Then the derived tensor product X ⊗LR Y of X and Y is an R-complex.
Note, since R is local, it follows that inf(X ⊗RL Y ) = inf(X) + inf(Y ); see [11, (A.4.15) and (A.4.16)].
2.3. Annihilator and Support ([11, A.8.4]). Let X be an R-complex. Then the annihilator and supportof X are defined as follows:
AnnR (X) =∩i∈Z

AnnR (Hi (X)) and SuppR (X) =∪SuppR (Hi (X)).

i∈Z

Note, if X is bounded, then it follows that SuppR (X) = V(AnnR (X)). It seems worth mentioning that
this equality may fail if X is not bounded. For example, if x ∈ m is an element that is not nilpotent and
X is an R-complex with zero differentials such that Xi = R/xi R for each i ≥ 1 and X j = 0 for each j ≤ 0,∩then one can see that X is not bounded, AnnR (X) = i≥0 xi R = 0, and SuppR (X) = V(x) ̸= V(0).
2.4. Depth of complexes ([11, 19]). Let X be an R-complex, I an ideal of R, x = x1 , . . . , xn is a generating
set of I, and K(x) denotes the Koszul complex on x. Then the I-depth of X is defined as:
depthR (I, X) = n − sup(K(x) ⊗LR X).
This depth definition, which is known to be independent of the choice of generators of I, is from [19,§2]; see also [11, (A.6.1)] for another definition and [15, 2.1] for the equivalence of these two depth
definitions. Note that it follows −∞ < depthR (I, X) ≤ n − sup X.
We set depthR (X) = depthR (m, X). Then, by our convention for complexes, depthR (X) is finite
provided that H(X) ̸= 0; see [19, Observation on page 549]. Note also that depthR (0) = ∞.

ON THE DEPTH AND REFLEXIVITY OF TENSOR PRODUCTS

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The following facts play an important role in the proof of Theorem 3.1.
2.5. Let X be a bounded R-complex and let I be an ideal of R. Then the following hold:(i) depthRp (Xp ) ≥ depthRq (Xq ) − dim(Rq /pRq ) for each p, q ∈ Spec(R) with p ⊆ q; see [11, (A.6.2)].
(ii) depthR (I, X) = inf{depthRp (Xp ) | p ∈ V(I)}; see [15, 2.10].
√(iii) depthR (I, X) = depthR ( I, X).
(iv) depthR (I + AnnR (X), X) = depthR (I, X).
√()
Note that, as V(I) = V( I) and V I + AnnR (X) = V(I) ∩ SuppR (X), part (ii) yields parts (iii) and (iv).
2.6. Serre’s condition for complexes ([8]). Let X be a bounded R-complex and let n ≥ 0 be an integer.
X is said to satisfy Serre’s condition (Sn ) if the following inequality holds for each prime ideal p of R:
depthRp (Xp ) + inf(Xp ) ≥ min{n, dim(Rp )}.
2.7. Complete intersection dimension of complexes ([4, 24]). The complete intersection dimension ofa bounded R-complex X is defined as:
CI-dimR (X) = inf{pdS (X ⊗LR R′ ) − pdS (R′ ) : R → R′ ↞ S is a quasi-deformation},where R → R′ ↞ S is called a quasi-deformation provided that it is a diagram of local ring maps, R → R′
is flat, and the kernel of the surjection R′ ↞ S is generated by a regular sequence on S.
Some facts about the complete intersection dimension are recorded next:
2.8. Let X be a bounded R-complex. Then the following hold:(i) CI-dimR (X) ∈ {−∞} ∪ Z ∪ {∞}; see [24, 3.2.1].
(ii) CI-dimR (X) = −∞ if and only if H(X) = 0; see [24, 3.2.2].
(iii) inf(X) ≤ sup(X) ≤ CI-dimR (X); see [24, 3.3].
(iv) If CI-dimR (X) < ∞, then CI-dimR (X) = depth(R) − depthR (X); see [24, 3.3].
(v) If R is a complete intersection, then CI-dimR (X) < ∞; see [24, 3.5].
2.9. Derived Depth Formula. Let X and Y be bounded R-complexes. If CI-dimR (X) < ∞ and X ⊗LR Y
is bounded, then the equality depthR (X) + depthR (Y ) = depth(R) + depthR (X ⊗LR Y ) holds, i.e., the pair
(X,Y ) satisfies the derived depth formula; see [12, 4.4] and also [25].
We need a few arguments from the proof of [8, 3.1] to establish our main result. In the following, forthe sake of completeness, we include the arguments we need, along with a few additional details thatare not explicitly stated in [8].
2.10. ([8, see the proof of 3.1]) Let X and Y be bounded R-complexes such that H(X) ̸= 0 ̸= H(Y ).
Assume CI-dimR (X) < ∞. Assume further X ⊗LR Y is bounded and satisfies (Sn ) for some n ≥ 0.
Let p ∈ SuppR (Y ). We proceed and look at depthRp (Yp ) + inf(Yp ). We pick a minimal prime ideal
q of p + AnnR (X) and consider the following three cases separately: dim(Rq ) ≤ n, dim(Rq ) > n, andp ∈ SuppR (X). Note that, by the choice, we have that q ∈ SuppR (X ⊗LR Y ).

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O. CELIKBAS, U. LE, AND H. MATSUI

Case 1. Assume dim(Rq ) ≤ n. Then it follows:()
depthRp (Yp ) + inf(Yp ) = depth(Rq )pRq (Yq )pRq + inf((Yq )pRq )[]≥ depthRq (Yq ) − dim(Rq /pRq ) + inf(Yq )][= depthRq (Xq ⊗LRq Yq ) + depth(Rq ) − depthRq (Xq ) − dim(Rq /pRq ) + inf(Yq )[]= depthRq (Xq ⊗LRq Yq ) + CI-dimRq (Xq ) − dim(Rq /pRq ) + inf(Yq )][(2.10.1)≥ depthRq (Xq ⊗LRq Yq ) + inf(Xq ) − dim(Rq /pRq ) + inf(Yq )= depthRq (Xq ⊗LRq Yq ) + inf(Xq ⊗LRq Yq ) − dim(Rq /pRq )≥ min{n, dim(Rq )} − dim(Rq /pRq )≥ dim(Rq ) − dim(Rq /pRq )≥ dim(Rp )≥ min{n, dim(Rp )}.
Here, in (2.10.1), the first inequality follows from 2.5(i) and the definition of inf, the second inequality
follows from 2.8(iii), and the third inequaliy is due to the fact that X ⊗LR Y satisfies (Sn ); note that theother inequalities are standard. For the equalities in (2.10.1), the first one is due to localization, thesecond one follows from 2.9, the third one is due to 2.8(iv), and the fourth one can be obtained by 2.2.
Case 2. Assume dim(Rq ) > n, and set t = depthRq (Xq ⊗LRq Yq ) + inf(Xq ⊗LRq Yq ). Then it follows:()
depthRp (Yp ) + inf(Yp ) ≥ depthRq Yq ) + inf(Yq ) − dim(Rq /pRq()
= depth(Rq ) − depthRq (Xq ) + inf(Xq ) + t − dim(Rq /pRq )()(2.10.2)≥ depth(Rq ) − depthRq (Xq ) + inf(Xq ) + n − dim(Rq /pRq )≥ depth(Rq ) − (depthRq (Xq ) + inf(Xq )) + n + dim(Rp ) − dim(Rq ).
Here, in (2.10.2), the first inequality is due to 2.5(i) (as discussed in Case 1), the second inequality is dueto the fact that X ⊗LR Y satisfies (Sn ) and dim(Rq ) > n, and the third inequality is standard. Moreover,the equality in (2.10.2) follows from 2.2 and 2.9.
Case 3. Assume p ∈ SuppR (X). Then one has q = p. If dim(Rp ) ≤ n, then it follows by Case 1 that
depthRp (Yp ) + inf(Yp ) ≥ min{n, dim(Rp )}. Next assume dim(Rp ) > n. Then it follows that:()
depthRp (Yp ) + inf(Yp ) ≥ depth(Rp ) − depthRp (Xp ) + inf(Xp ) + n + dim(Rp ) − dim(Rp )(2.10.3)= CI-dimRp (Xp ) − inf(Xp ) + n≥ n ≥ min{n, dim(Rp )}.

Here, in (2.10.3), the first inequality follows by Case 2, the second inequality is due to 2.8(iii), and theequality is due to 2.8(iv). So, if p ∈ SuppR (X), we have that depthRp (Yp ) + infYp ≥ min{n, dim(Rp )}.
3. P ROOF OF THE MAIN THEOREM AND COROLLARIESWe are now ready to prove the main result in this paper, namely Theorem 3.1. The proof of thetheorem is motivated by the results given in 2.10, but the gist of our argument is different: the finishing
touch of the proof of Theorem 3.1 relies upon an application of the properties stated in 2.5.
We set, for q ∈ Spec(R), that U(q) = {p ∈ Spec(R) : p ⊆ q where heightR (p) > 0}.
Theorem 3.1. Let R be a Cohen-Macaulay local ring, and let X and Y be bounded R-complexes suchthat H(X) ̸= 0 ̸= H(Y ). Assume m and n are nonnegative integers and the following conditions hold:(i) CI-dimR (X) < ∞.
(ii) X ⊗LR Y is bounded.

ON THE DEPTH AND REFLEXIVITY OF TENSOR PRODUCTS

5(iii) X ⊗LR Y satisfies Serre’s condition (Sn ).
(iv) If q ∈ SuppR (X ⊗LR Y ), then depthRq (pRq , Xq ) + inf(Xq ) ≤ depthRq (pRq , Rq ) + m for all p ∈ U(q).
Then Y satisfies Serre’s condition (Sn−m ).
Proof. Let p ∈ SuppR (Y ). We want to show that the following inequality holds:(3.1.1)depthRp (Yp ) + inf(Yp ) ≥ min{n − m, dim(Rp )}.

If dim(Rp ) = 0, then (3.1.1) holds trivially. Moreover, if p ∈ SuppR (X), then the inequality (3.1.1)
holds by Case 3 of 2.10. Hence we assume dim(Rp ) > 0, p ∈/ SuppR (X), and pick a prime ideal q of R
which is minimal over p + AnnR (X). Then it follows that q ∈ SuppR (X ⊗LR Y ).
If dim(Rq ) ≤ n, then (3.1.1) holds by Case 1 of 2.10. Hence, we further assume that dim(Rq ) > n.
Therefore, Case 2 of 2.10 yields:(3.1.2)depthRp (Yp ) + inf(Yq ) ≥ depth(Rq ) − (depthRq (Xq ) + inf(Xq )) + n + dim(Rp ) − dim(Rq )= n + dim(Rp ) − (depthRq (Xq ) + inf(Xq )).

Now we suppose depthRp (Yp ) + inf(Yp ) < min{n − m, dim(Rp )} and look for a contradiction. Notethat we have n − m > depthRp (Yp ) + inf(Yp ) and so (3.1.2) shows:dim(Rp ) < depthRq (Xq ) + inf(Xq ) − m.

(3.1.3)Note that the following inequalities hold:(3.1.4)depthRq (pRq , Xq ) + inf(Xq ) − m ≤ depthRq (pRq , Rq )= dim(Rp )< depthRq (Xq ) + inf(Xq ) − m.

In (3.1.4), the first inequality is due to the hypothesis (iv) since q ∈ SuppR (X ⊗LR Y ) and p ∈ U(q). Moreover, the equality of (3.1.4) follows from the fact that R is Cohen-Macaulay, and the second inequality
is due to (3.1.3). Hence (3.1.4) gives:
depthRq (pRq , Xq ) < depthRq (Xq ).

(3.1.5)On the other hand, we have:(3.1.6)depthRq (Xq ) = depthRq (qRq , Xq )(√)= depthRq
pRq + AnnRq (Xq ), Xq= depthRq (pRq + AnnRq (Xq ), Xq )= depthRq (pRq , Xq ).

In (3.1.6), the first equality follows from the definition of depth, the second one is due to the fact thatpRq + AnnRq (Xq ) is qRq -primary, and the last two equalities follow from 2.5(iii) and 2.5(iv), respectively.
Consequently, in view of (3.1.5) and (3.1.6), we obtain a contradiction. This contradiction implies
that the inequality (3.1.1) holds, and hence completes the proof.
□
We proceed by recording some consequences of Theorem 3.1.
Corollary 3.2. Let R be a Cohen-Macaulay local ring, M and N be finitely generated R-modules, andlet m and n be nonnegative integers. Assume the following hold:(i) CI-dimR (M) < ∞.
(ii) TorRi (M, N) = 0 for all i ≥ 1.
(iii) M ⊗R N satisfies Serre’s condition (Sn ).
(iv) If q ∈ SuppR (M ⊗R N), then depthRq (pRq , Mq ) ≤ depthRq (pRq , Rq ) + m for all p ∈ U(q).
Then N satisfies Serre’s condition (Sn−m ).

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O. CELIKBAS, U. LE, AND H. MATSUIProof. Note that M ⊗LR N ∼= M ⊗R N if and only if TorRi (M, N) = 0 for all i ≥ 1. Therefore, it follows by□
Theorem 3.1 that N satisfies Serre’s condition (Sn−m ).
Remark 3.3. It seems worth mentioning that the depth inequality stated in part (iv) of Corollary 3.2
always holds when m = 1 and R is a hypersurface. More precisely, if R is a hypersurface ring, I is an
ideal of R, and M is a nonzero R-module, then the inequality depth(I, M) ≤ depthR (I, R) + 1 holds; see[3, 2.2 and 4.3] and [20, Cor. 1]. In [7] we extend the aforementioned depth inequality for modules withbounded Betti numbers over local complete intersection rings of codimension at most 2.
□
Modules over hypersurface rings are reflexive if and only if they satisfy Serre’s condition (S2 ); see,for example, [8, 2.5]. This fact is used in the next corollary of Theorem 3.1.
Corollary 3.4. Let R be a local hypersurface ring, and let M and N be nonzero R-modules. Assume thefollowing conditions hold:(i) N has rank.
(ii) If q ∈ SuppR (M), then depthRq (pRq , Mq ) ≤ heightR (p) for all p ∈ U(q).
If M ⊗R N is reflexive, then M and N are both reflexive.
Proof. Assume M ⊗R N is reflexive. Then it follows from Theorem 1.1 that TorRi (M, N) = 0 for all i ≥ 1
and M is reflexive. Moreover, since R is Cohen-Macaulay, the equality depthRq (pRq , Rq ) = heightR (p)holds for each p, q ∈ Spec(R) with p ⊆ q. Note also CI-dimR (M) < ∞ as R is a hypersurface; see 2.8(v).
Therefore, setting m = 0 and n = 2, we conclude from Corollary 3.2 that N is reflexive.
□
Recall that the module N in Example 1.2 is not reflexive. Hence, it is worth pointing out that thedepth inequality in part (ii) of Corollary 3.4 does not hold for the module M in the example.
Example 3.5. Let R, M, and N be as in Example 1.2, i.e., R = C[|x, y, z, w]]/(xy), M = R/(x) and let
N be the Auslander transpose of R/p, where p = (y, z, w) ∈ Spec(R). Let q = m. Then it follows that
depthRq (pRq , Mq ) = depthR (p, M) = 3 > heightR (p) = 2.
□
Now our aim is to establish Theorem 1.3, advertised in the introduction. First we prove the following
general result which seems to be of independent interest.
Proposition 3.6. Let R be a local ring and let M be a nonzero R-module. Then the following conditions
are equivalent:(i) Each M-regular sequence is R-regular.
(ii) depthR (I, M) ≤ depthR (I, R) for each ideal I of R.
(iii) depthR (p, M) ≤ depthR (p, R) for each p ∈ Spec(R).
Proof. It follows by definition
that (i) =⇒ (ii) =⇒ (iii). So we proceed and show (iii) =⇒ (ii) =⇒ (i).
√(iii) =⇒ (ii): Write I = p1 ∩ . . . ∩ pn for some prime ideals pi . Then it follows that√
depthR (I, M) = depthR ( I, M)= inf{depthR (p1 , M), . . . , depthR (pn , M)}(3.6.1)≤ inf{depthR (p1 , R), . . . , depthR (pn , R)}√= depthR ( I, R)= depthR (I, R).

Here in (3.6.1), the first and the fourth equalities are due to [6, 1.2.10(b)] (see also 2.5(ii)), the second
and third equalities are due to [6, 1.2.10(c)], and the inequality follows by assumption.
(ii) =⇒ (i): Let x = x1 , . . . , xn ⊆ m be an M-regular sequence. We will show that this sequence is
R-regular by induction on n.

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If n = 1, then we have 1 ≤ depthR ((x1 ), M) ≤ depthR ((x1 ), R), which implies that x1 is a non zerodivisor on R. Hence we assume n ≥ 2. Then, by the induction hypothesis, it follows that x′ = x1 , . . . , xn−1
is R-regular. Thus, we have:
depthR ((xn ), M/(x′ )M) + (n − 1) = depthR ((x), M)(3.6.2)≤ depthR ((x), R)= depthR ((xn ), R/(x′ )) + (n − 1)

Here in (3.6.2), the equalities are due to [6, 1.2.10(d)], while the inequality follows by assumption.
Therefore, we have1 ≤ depthR ((xn ), M/(x′ )M) ≤ depthR ((xn ), R/(x′ )R),
which implies that xn is a non zero-divisor on R/(x′ )R, as required.

□

Remark 3.7. The equivalent conditions in Proposition 3.6 hold if and only if, whenever x = x1 , . . . , xn
is a sequence of elements in m with TorR1 (M, R/xR) = 0, it follows TorR2 (M, R/xR) = 0; see [5, 2.2]. □The following result is an immediate consequence of Proposition 3.6:
Corollary 3.8. Let R be a local ring and let M be a nonzero R-module such that SuppR (M) = Spec(R).
If depthRp (Mp ) ≤ depth(Rp ) for all p ∈ Spec(R) (e.g., R is Cohen-Macaulay, or CI-dimR (M) < ∞), theneach Mp -regular sequence is Rp -regular for all p ∈ Spec(R).
Proof. Let p ∈ Spec(R) and let J be an ideal of Rp . Then it follows that:
depthRp (J, Mp ) = inf{depthRq (Mq ) | qRp ∈ V(J)} ≤ inf{depth(Rq ) | qRp ∈ V(J)} = depth(J, Rp ).
Here both equalities follow due to 2.5(ii). Hence the result follows from Proposition 3.6.

□

We are now ready to prove Theorem 1.3. Let us first note a fact proved in [16, 2.6]: if the module M
in Theorem 1.3 has full support, then M and N have full support so that a quick application of the depth
formula shows that both M and N satisfy (S2 ), i.e., both M and N are reflexive; see also 2.9, and [8, 1.3]
for the details. Therefore, the gist of Theorem 1.3 is the case where SuppR (M) ̸= Spec(R).
Proof of Theorem 1.3. Let q ∈ SuppR (M). Then, since each Mq -regular sequence is Rq -regular by assumption, it follows from Proposition 3.6 that depthRq (pRq , Mq ) ≤ heightR (p) for all p ∈ Spec(R) withp ⊆ q. Therefore, we have that depthRq (pRq , Mq ) ≤ heightR (p) for all p ∈ U(q). Consequently, Corollary 3.4 implies that both M and N are reflexive.
□
It is interesting to note that Theorem 1.3 (and also Theorem 1.1) can fail over rings that are not
hypersurfaces. For example, if R = k[[t 3 ,t 4 ,t 5 ]] and N = (t 3 ,t 4 ), the canonical module of R, Huneke
and Wiegand [16, 4.8] constructs an R-module M such that M ⊗R N is reflexive, but neither M nor N isreflexive; note that the hypotheses in parts (i) and (ii) of Theorem 1.3 hold for these modules M and N.
In [13, 2.1] one can find a similar example over a Gorenstein ring that is not a hypersurface.
4. F URTHER REMARKS ON T HEOREM 1.3
Recall that an R-module M is called Tor-rigid provided that the following condition holds: for each
R-module N satisfying TorR1 (M, N) = 0, one has that TorR2 (M, N) = 0. Examples of Tor-rigid modules
are abundant in the literature. For example, if R is hypersurface, that is quotient of an unramified regular
local ring, and M is an R-module such that lengthR (M) < ∞ or pdR (M) < ∞, then M is Tor-rigid; see[16, 2.4] and [20, Theorem 3], respectively. Tor-rigidity condition can impose certain restrictions on thering in question. For example, Auslander [3, 4.3] proved that, if M is a nonzero Tor-rigid module overa local ring R, then each M-regular sequence is an R-regular sequence. Note, this fact implies that thedepth of a nonzero Tor-rigid module is always bounded by the depth of the ring considered.

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Celikbas, Matsui, and Sadeghi [9] examined the conclusion of Theorem 1.1 and studied the reflexivity of tensor products of modules over local hypersurface rings in terms of the Tor-rigidity. Their main
result establishes the same conclusion of Theorem 1.3 for Tor-rigid modules. More precisely, the main
result of [9] shows that, if M and N are nonzero modules over a local hypersurface ring R such that
M ⊗R N is reflexive, M is Tor-rigid, and N has rank, then M and N are both reflexive; see Theorem
1.1 and [9, 3.1]. Therefore, we next give examples and highlight that the Tor-rigidity condition and thecondition we study in this paper, namely the condition in part (ii) of Theorem 1.3, are independent ofeach other, in general.
Example 4.1. Let R = k[[x, y, z]]/(xy) and let M = R/(x2 ). Then it follows that SuppR (M) ̸= Spec(R),M is not Tor-rigid, and each Mp -regular sequence is Rp -regular for all p ∈ SuppR (M). We justify theseproperties as follows:(i) SuppR (M) ̸= Spec(R): this is clear since (y) ∈/ Supp(R (M).
) In( fact, )since {(x),( )(x, y)} is the set of
all associated primes of M, it follows that SuppR (M) = V (x) ∪ V (x, y) = V (x) .
(ii) M is not Tor-rigid: setting N = R/(y), one can check that TorR1 (M, N) = 0 ̸= TorR2 (M, N).
(iii) Each Mp -regular sequence is Rp -regular for all p ∈ SuppR (M): note, to justify this claim, due to
Proposition 3.6, we proceed to prove the following claim:(4.1.1)

If I is an ideal of R such that I ⊆ p ∈ SuppR (M), then depthRp (IRp , Mp ) ≤ heightRp (IRp ).

Let p ∈ SuppR (M) and let I be an ideal of R such that I ⊆ p. We look at the height of p, i.e., dim(Rp ).
Case 1: Assume heightR (p) = 0. Then the claim in (4.1.1) holds as depthRp (IRp , Mp ) ≤ dim(Rp ) = 0.
Case 2: Assume heightR (p) = 1. We first consider the case where p = (x, y). As p is an associatedprime of M, it follows that depthRp (IRp , Mp ) ≤ depthRp (Mp ) = 0, and so the claim in (4.1.1) holds.
Next, we consider the case where p ̸= (x, y). Note, as p ∈ SuppR (M), we have that (x) ⊆ p. Moreover,in case I ⊆ r ⫋ p for some r ∈ Spec(R), one can observe that I ⊆ (x). As depthR(x) (M(x) ) = 0, theaforementioned observation and 2.5(ii) yield:{
depthRp (Mp ) if I ⊈ (x)(4.1.2)depthRp (IRp , Mp ) = inf{depthRq (Mq ) | I ⊆ q ⊆ p} =0
if I ⊆ (x)As the equalities in (4.1.2) also hold when M is replaced with R, the claim in (4.1.1) holds.
Case 3: Assume heightR (p) = 2, i.e., p = m. As depthR (I, M) ≤ depthR (M) = 1, to establish theclaim in (4.1.1), it suffices to assume heightR (I) = 0 and show that depthR (I, M) = 0.
Note that I is contained in a minimal prime ideal of R, namely in (x) or (y). Therefore, I ⊆ (x, y) andthis shows:depthR (I, M) = inf{depthRr (Mr ) | r ∈ V(I)} ≤ depthR(x,y) (M(x,y) ) = 0.
This completes the proof of Case 3.

□

Example 4.2. Let R = k[[x, y, z, u]]/(xy) and let M = N ⊕ T , where N = R/(x) and T is an R-module
such that dimR (T ) = 0 and pdR (T ) = ∞ (e.g., T = k). Then it follows that SuppR (M) ̸= Spec(R), M
is Tor-rigid, and there is an Mp -regular sequence which is not Rp -regular for some p ∈ SuppR (M). We
justify these properties as follows:(i) SuppR (M) ̸= Spec(R): this is clear since (y) ∈/ SuppR (M).
(ii) M is Tor-rigid: to see this assume TorR1 (M, X) = 0 for some R-module X. Then TorR1 (T, X) = 0,
and since T is Tor-rigid [16, 2.4], we have that TorRi (T, X) = 0 for all i ≥ 1. This implies X is free andhence TorRi (M, X) = 0 for all i ≥ 1; see [16, 2.5] and [17, 1.9]. Therefore, M is Tor-rigid (note also that
N is not Tor-rigid since TorR1 (N, R/yR) = 0 ̸= R/p = TorR2 (N, R/yR)).
(iii) There is an Mp -regular sequence which is not Rp -regular for some p ∈ SuppR (M): for this part,let p = (x, y). Then it follows that p ∈ SuppR (M) and Mp ∼= Np . Hence, y is a non zero-divisor on Mp .
On the other hand, as x ̸= 0, y ̸= 0 and xy = 0 in Rp , we see that y is a zero-divisor on Rp . Thus, {y} isan Mp -regular sequence which is not Rp -regular.
□

ON THE DEPTH AND REFLEXIVITY OF TENSOR PRODUCTS

9

The modules considered in Examples 4.1 and 4.2 do not have full support. Next, in Example 4.3, welook at a module M that has full support and observe, even for such a module, Tor-rigidity condition is
distinct from the condition stated in part (ii) of Theorem 1.3.
Example 4.3. Let R = k[[x, y, z, u]]/(xu − yz) and consider the ideal M = (x, y) of R. Then each Mp regular sequence is Rp -regular since SuppR (M) = Spec(R); see Corollary 3.8. Furthermore, it follows
that TorR1 (M, M) = 0 ̸= k = TorR2 (M, M), and hence M is not Tor-rigid.
□
One can also construct examples similar to Example 4.3 over rings that are not hypersurfaces.
Example 4.4. Let R be a Cohen-Macaulay local ring with canonical module ω such that ω ≇ R; forexample, R = k[[t 3 ,t 4 ,t 5 ]] and ω = (t 3 ,t 4 ). Then ω is not Tor-rigid; see, for example, [26, 4.13(i)]. On
the other hand, each ωp -regular sequence is Rp -regular for all p ∈ Spec(R); see Corollary 3.8.
□
It is well-known that the Tor-rigidity property does not localize, in general. For example, if M is
the module considered in Example 4.2, then M is Tor-rigid over R, but Mp is not Tor-rigid over Rp .
Furhermore, the same example also shows that the condition we consider in part (ii) of Theorem 1.3
does not localize in general, too. It seems worth summarizing these observations as a separate remark;see also Proposition 3.6.
Remark 4.5. Let R be a local ring and let M be a nonzero R-module. Consider the following conditions.
(i) M is Tor-rigid over R.
(ii) Mp is Tor-rigid over Rp for all p ∈ SuppR (M).
(iii) Each M-regular sequence is R-regular.
(iv) Each Mp -regular sequence is Rp -regular for all p ∈ SuppR (M).
Then we have:+3 (iii)
KS
(2)
_
(4)
(3)
(5) _ (6)(7)
 +3 (iv)(ii) ks
(i)
KS ks

(1)



(8)

The implications in the above diagram can be justified as follows:(1) and (7): see [3, 4.3].
(2) and (8): see the module M in Example 4.3.
(3): see the module M in Example 4.2.
(4) and (5): these follow by definition.
(6): in view of [3, 4.3], see the module M in Example 4.2.
A PPENDIX A. A N APPLICATION OF T HEOREM 1.1
In this appendix, we give an application of Theorem 1.1, and provide a criterion for tensor productsof prime ideals to be reflexive over hypersurface rings. More precisely, we prove in Corollary A.4 that,
if R be a hypersurface ring that is not a domain, and the tensor product of two prime ideals is reflexive,
then both of the primes considered must be minimal. Our result, which seems to be new, is based on thefollowing observations of Asgharzadeh [2, 5.1 and 5.5]; see also [21, II.3.3].
Remark A.1. Let R be a commutative Noetherian ring, and let p ∈ Spec(R). Assume p(n) ̸= 0 for somen ≥ 1, where p(n) = pn Rp ∩ R denotes the nth symbolic power of p. Set M = R/p(n) .
(i) Assume M is Tor-rigid over R. Then each non zero-divisor on M is a non zero-divisor on R so thatthe canonical map R → Rp is injective; see Remark 4.5. Hence, R is a domain if Rp is a domain.
(ii) Assume pdR (M) < ∞. Then each non zero-divisor on M is a non zero-divisor on R so that the
canonical map R → Rp is injective; see [22], [23, 6.2.3]. Also, R is a domain as Rp is regular.

10

O. CELIKBAS, U. LE, AND H. MATSUI(iii) Assume idR (M) < ∞. Then it follows that R is Gorenstein [21, II.5.3] so that pdR (M) < ∞. Hence,part (ii) implies that R is a domain.
Proposition A.2. Let R be a local hypersurface ring which is not a domain, M a nonzero R-module,(n)and let p ∈ Spec(R)( such that) heightR (p) ≥ 1. Let r ≥ 0 and N denote the rth syzygy of R/p for somer(n)n ≥ 1, i.e., N = Ω R/p . Assume M ⊗R N is reflexive. Then it follows that r ≥ 1, both M and N arereflexive, and pdR (M) < ∞ = pdR (N). Moreover, if M is not free, then M has rank at least two.
Proof. As M ⊗R N is a nonzero torsion-free R-module, we observe that neither M nor N can be torsion.
This implies that r ≥ 1. Note, since p has positive height, it follows that p(n) has positive height.
Therefore, R/p(n) is torsion, i.e., R/p(n) has rank zero. Thus N has rank, and so the conclusions ofTheorem 1.1 hold.
We know, by Theorem 1.1, that pdR (M) < ∞ or pdR (N) < ∞. However, if pdR (N) < ∞, thenpdR (R/p(n) ) < ∞ and this forces R to be a domain; see part (ii) of Remark A.1. Therefore, we havethat pdR (M) < ∞ = pdR (N).
Notice, since both M and N have rank, it follows from Theorem 1.1 that both M and N are reflexive.
Now assume M is not free. Then, since M is reflexive, it follows that M = Ω2 (X) for some R-module
X such that 3 ≤ pdR (X) < ∞. Hence, the syzygy theorem of Evans and Griffith [14, 1.1] (see also [6,9.5.6]) applied to X implies that M has rank at least two.1□
The positive height assumption on the prime ideal considered in Proposition A.2 cannot be removed.
Example A.3. Let R = k[[x, y]]/(xy), p = (x), M = R/(x2 ), and let N = Ω(R/p). Then p is a minimal
prime ideal of R, pdR (M) = ∞, but M ⊗R N ∼= N is a reflexive R-module.
The next corollary of Proposition A.2 yields the criterion we seek concerning the tensor products ofprime ideals over local hypersurface rings:
Corollary A.4. Let R be a local hypersurface ring that is not a domain, and let p, q ∈ Spec(R). Assume
either p or q is not a minimal prime. Then p(r) ⊗R q(s) is not a reflexive R-module. In particular, p ⊗R q
is not a reflexive R-module.
In view of Corollary A.4, it is worth noting that the tensor product of two minimal prime ideals overa non-domain hypersurface ring may, or may not, be reflexive.
Example A.5. Let R and p be as in Example A.3, and let q = (y). Then q is also a minimal prime idealof R. It follows that p ∼= R/(y) and p ⊗R p ∼= p. Therefore, p ⊗R p is a reflexive R-module. On the other
hand p ⊗R q ∼= k so that p ⊗R q is not reflexive.
ACKNOWLEDGEMENTS
The authors are grateful to Mohsen Asgharzadeh for his valuable comments on a previous version of
the manuscript, Thomas Polstra for useful and valuable discussions about Remark A.1, Arash Sadeghi
for showing them the rank conclusion in Proposition A.2, and Tirdad Sharif for pointing out the reference [25]. The authors are also grateful to Greg Piepmeyer for discussions on previous versions of
the manuscript, especially for his help and suggestions about the results stated in 2.5 and 2.10 and theirusage in the proof of Theorem 3.1.
The authors thank the referee for his/her/their valuable comments and suggestions.
Matsui was partly supported by JSPS Grant-in-Aid for JSPS Fellows 19J00158.
1Note that, due to Andre’s result [1], the syzygy theorem of Evans and Griffith does not require the ring in question to containa field.

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O. CELIKBAS, U. LE, AND H. MATSUIO 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 , G RADUATE S CHOOL OF M ATHEMATICAL S CIENCES , U NIVERSITY OF T OKYO , 3-8-1 KOMABA , M EGURO KU , T OKYO 153-8914, JAPANCURRENT ADDRESS : 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