On the depth and reflexivity of tensor products
aa r X i v : . [ m a t h . A C ] F e b 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 conditionsunder which nonzero tensor products of finitely generated modules over hypersurface rings can be reflexiveonly if both of their factors are reflexive.A result of Asgharzadeh shows that nonzero symbolic powers of prime ideals in a local ring cannothave finite projective dimension, unless the ring in question is a domain. We make use of this fact in theappendix 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 residuefield 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 benonzero finitely generated R-modules. Assume N has rank and M ⊗ R N is reflexive. Then
Tor Ri ( M , N ) = for all i ≥ , M is reflexive, N is torsion-free, Supp R ( N ) = Spec ( R ) , and pd R ( M ) < ∞ or pd R ( N ) < ∞ . It has been an open problem for quite some time whether or not the module N in Theorem 1.1must also be reflexive; see [8, 16, 18] for the details. In 2019 Celikbas and Takahashi [10] constructedexamples settling this query; one of their examples is the 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 pd R ( 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 productsand 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 thatM ⊗ R N is reflexive. Assume the following conditions hold: (i)
N has rank (e.g., pd R ( N ) < ∞ ). (ii) Each M p -regular sequence is R p -regular for all p ∈ Supp R ( M ) .Then M and N are both reflexive. 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.Matsui was partly supported by JSPS Grant-in-Aid for JSPS Fellows 19J00158.
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 doesnot 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 finitecomplete intersection dimension over local rings. The proof of Theorem 1.3 relies upon a relationbetween 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 tensorproducts 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 twoprime 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. [11] ). Throughout, by an R -complex X , we mean a chain complex of R -modules whichhas homological differentials ∂ Xi : X i → X i − , and which is homologically finite, i.e., H i ( X ) = | i | ≫ i ( X ) is a finitely generated R -module.If X is a (not necessarily homologically finite) R -complex, we set:sup X = sup { i ∈ Z | H i ( X ) = } and inf X = inf { i ∈ Z | H i ( X ) = } . Note we have that sup X = − ∞ if and only if H ( X ) = X = ∞ . If X and Y are R -complexes, then it follows that inf ( X ⊗ L R Y ) = inf ( X ) + inf ( Y ) ; see [11, (A.4.11)and (A.4.16)]. Here, X ⊗ L R Y denotes the derived tensor product of X and Y . [11, A.8.4] ). The annihilator and support of an R -complex X is:Ann R ( X ) = \ i ∈ Z Ann R ( H i ( X )) and Supp R ( X ) = [ i ∈ Z Supp R ( H i ( X )) . Note that the equality Supp R ( X ) = V ( Ann R ( X )) holds. [11, 19] ). Let X be an R -complex, I an ideal of R , and let x = x , . . . , x n be agenerating set of I . Then the I-depth of X is defined as:depth R ( I , X ) = n − sup ( K ( x ) ⊗ L R X ) . Here K ( x ) is the Koszul complex on x ; see [19, §2] and [11, (A.6.1)]. It is known that this definition isindependent of the choice of generators of I [19, 1.3]. It follows that − ∞ < depth R ( I , X ) ≤ n − sup X .We set depth R ( X ) = depth R ( m , X ) . Then, by our convention for complexes, depth R ( X ) is finiteprovided that H ( X ) =
0; see [19, Observation on page 549]. Note also that depth R ( ) = ∞ .The following facts play an important role in the proof of Theorem 3.1. Let X be an R -complex and I an ideal of R . Then the following hold:(i) depth R p ( X p ) ≥ depth R q ( X q ) − dim ( R q / p R q ) for each p , q ∈ Spec ( R ) with p ⊆ q ; see [11, (A.6.2)].(ii) depth R ( I , X ) = inf { depth R p ( X p ) | p ∈ V ( I ) } ; see [15, 2.10].(iii) depth R ( I , X ) = depth R ( √ I , X ) . N THE DEPTH AND REFLEXIVITY OF TENSOR PRODUCTS 3 (iv) depth R ( I + Ann R ( X ) , X ) = depth R ( I , X ) .Note that, as V ( I ) = V ( √ I ) and V (cid:0) I + Ann R ( X ) (cid:1) = V ( I ) ∩ Supp R ( X ) , part (ii) yields parts (iii) and (iv). [8] ). Let X be an R -complex and let n ≥ X is said to satisfy Serre’s condition ( S n ) if the following inequality holds for each prime ideal p of R :depth R p ( X p ) + inf ( X p ) ≥ min { n , dim ( R p ) } . [4, 25] ). Let X be an R -complex. A diagram oflocal ring maps R → R ′ և S is called a quasi-deformation provided that R → R ′ is flat and the kernel ofthe surjection R ′ և S is generated by a regular sequence on S . The complete intersection dimension of X is defined as:CI-dim R ( X ) = inf { pd S ( X ⊗ L R R ′ ) − pd S ( R ′ ) : R → R ′ և S is a quasi-deformation } . Some facts about the complete intersection dimension are recorded next:
Let X be an R -complex. Then the following hold:(i) CI-dim R ( X ) ∈ {− ∞ } ∪ Z ∪ { ∞ } ; see [25, 3.2.1].(ii) CI-dim R ( X ) = − ∞ if and only if H ( X ) =
0; see [25, 3.2.2].(iii) inf ( X ) ≤ sup ( X ) ≤ CI-dim R ( X ) ; see [25, 3.3].(iv) If CI-dim R ( X ) < ∞ , then CI-dim R ( X ) = depth ( R ) − depth R ( X ) ; see [25, 3.3].(v) If R is a complete intersection, then CI-dim R ( X ) < ∞ ; see [25, 3.5]. Let X and Y be R -complexes. If CI-dim R ( X ) < ∞ and X ⊗ L R Y is bounded,i.e., Tor Ri ( X , Y ) = i ≫
0, then the equality depth R ( X )+ depth R ( Y ) = depth ( R )+ depth R ( X ⊗ L R Y ) holds, i.e., the pair ( X , Y ) satisfies the derived depth formula ; see [12, 4.4].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]. ([8, see the proof of 3.1]) Let X and Y be R -complexes such that H ( X ) = = H ( Y ) . AssumeCI-dim R ( X ) < ∞ . Assume further X ⊗ L R Y is bounded and satisfies ( S n ) for some n ≥ p ∈ Supp R ( Y ) . We proceed and look at depth R p ( Y p ) + inf ( Y p ) . We pick a minimal prime ideal q of p + Ann R ( X ) and consider the following three cases separately: dim ( R q ) ≤ n , dim ( R q ) > n , and p ∈ Supp R ( X ) . Note that, by the choice, we have that q ∈ Supp R ( X ⊗ L R Y ) . Case 1 . Assume dim ( R q ) ≤ n . Then it follows:depth R p ( Y p ) + inf ( Y p ) = depth ( R q ) p R q (cid:0) ( Y q ) p R q (cid:1) + inf (( Y q ) p R q ) ≥ (cid:2) depth R q ( Y q ) − dim ( R q / p R q ) (cid:3) + inf ( Y q )= (cid:2) depth R q ( X q ⊗ L R q Y q ) + depth ( R q ) − depth R q ( X q ) (cid:3) − dim ( R q / p R q ) + inf ( Y q )= (cid:2) depth R q ( X q ⊗ L R q Y q ) + CI-dim R q ( X q ) (cid:3) − dim ( R q / p R q ) + inf ( Y q ) ≥ (cid:2) depth R q ( X q ⊗ L R q Y q ) + inf ( X q ) (cid:3) − dim ( R q / p R q ) + inf ( Y q ) (2.10.1) = depth R q ( X q ⊗ L R q Y q ) + inf ( X q ⊗ L R q Y q ) − dim ( R q / p R q ) ≥ min { n , dim ( R q ) } − dim ( R q / p R q ) ≥ dim ( R q ) − dim ( R q / p R q ) ≥ dim ( R p ) ≥ min { n , dim ( R p ) } . O. CELIKBAS, U. LE, AND H. MATSUI
Here, in (2.10.1), the first inequality follows from 2.5(i) and the definition of inf, the second inequalityfollows from 2.8(iii), and the third inequaliy is due to the fact that X ⊗ L R Y satisfies ( S n ) ; 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 ( R q ) > n , and set t = depth R q ( X q ⊗ L R q Y q ) + inf ( X q ⊗ L R q Y q ) . Then it follows:depth R p ( Y p ) + inf ( Y p ) ≥ depth R q (cid:0) Y q ) + inf ( Y q ) − dim ( R q / p R q (cid:1) = depth ( R q ) − (cid:0) depth R q ( X q ) + inf ( X q ) (cid:1) + t − dim ( R q / p R q ) ≥ depth ( R q ) − (cid:0) depth R q ( X q ) + inf ( X q ) (cid:1) + n − dim ( R q / p R q ) (2.10.2) ≥ depth ( R q ) − ( depth R q ( X q ) + inf ( X q )) + n + dim ( R p ) − dim ( R q ) . Here, in (2.10.2), the first inequality is due to 2.5(i) and the definition of inf, the second inequality is dueto the fact that X ⊗ L R Y satisfies ( S n ) and dim ( R q ) > 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 ∈ Supp R ( X ) . Then one has q = p . Set dim ( R p ) = l . If l ≤ n , then it follows byCase 1 that depth R p ( Y p ) + inf ( Y p ) ≥ min { n , dim ( R p ) } . Next assume l > n . Then it follows that:depth R p ( Y p ) + inf ( Y p ) ≥ depth ( R p ) − (cid:0) depth R p ( X p ) + inf ( X p ) (cid:1) + n + dim ( R p ) − dim ( R p )= CI-dim R p ( X p ) − inf ( X p ) + n (2.10.3) ≥ n ≥ min { n , dim ( R p ) } . 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 ∈ Supp R ( X ) , we have that depth R p ( Y p ) + inf Y p ≥ min { n , dim ( R p ) } .3. P ROOF OF THE MAIN THEOREM AND COROLLARIES
We 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 finishingtouch 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 height R ( p ) > } . Theorem 3.1.
Let R be a local ring, and let X and Y be R-complexes such that H ( X ) = = H ( Y ) .Assume m and n are nonnegative integers and the following conditions hold: (i) CI-dim R ( X ) < ∞ . (ii) X ⊗ L R Y is bounded, i.e.,
Tor Ri ( X , Y ) = for all i ≫ . (iii) X ⊗ L R Y satisfies Serre’s condition ( S n ) . (iv) If q ∈ Supp R ( X ⊗ L R Y ) , then depth R q ( p R q , X q ) + inf ( X q ) ≤ depth R q ( p R q , R q ) + m for all p ∈ U ( q ) .Then Y satisfies Serre’s condition ( S n − m ) .Proof. Let p ∈ Supp R ( Y ) . We want to show that the following inequality holds:depth R p ( Y p ) + inf ( Y p ) ≥ min { n − m , dim ( R p ) } . (3.1.1)If dim ( R p ) =
0, then (3.1.1) holds trivially. Moreover, if p ∈ Supp R ( X ) , then the inequality (3.1.1)holds by Case 3 of 2.10. Hence we assume dim ( R p ) > p / ∈ Supp R ( X ) , and pick a prime ideal q of R which is minimal over p + Ann R ( X ) . Then it follows that q ∈ Supp R ( X ⊗ L R Y ) .If dim ( R q ) ≤ n , then (3.1.1) holds by Case 1 of 2.10. Hence, we further assume that dim ( R q ) > n .Therefore, Case 2 of 2.10 yields:depth R p ( Y p ) + inf ( Y q ) ≥ depth ( R q ) − ( depth R q ( X q ) + inf ( X q )) + n + dim ( R p ) − dim ( R q )= n + dim ( R p ) − ( depth R q ( X q ) + inf ( X q )) . (3.1.2) N THE DEPTH AND REFLEXIVITY OF TENSOR PRODUCTS 5
Now we suppose depth R p ( Y p ) + inf ( Y p ) < min { n − m , dim ( R p ) } and look for a contradiction. Notethat we have n − m > depth R p ( Y p ) + inf ( Y p ) and so (3.1.2) shows:dim ( R p ) < depth R q ( X q ) + inf ( X q ) − m . (3.1.3)Note that the following inequalities hold:(3.1.4) depth R q ( p R q , X q ) + inf ( X q ) − m ≤ depth R q ( p R q , R q ) ≤ depth (( R q ) p R q )= depth ( R p ) ≤ dim ( R p ) < depth R q ( X q ) + inf ( X q ) − m . In (3.1.4), the first inequality is due to the hypothesis (iv) since q ∈ Supp R ( X ⊗ L R Y ) and p ∈ U ( q ) .Moreover, the second inequality of (3.1.4) follows from 2.5(ii), the third one is by [6, 1.2.12], and theforth one is due to (3.1.3). Hence (3.1.4) gives:(3.1.5) depth R q ( p R q , X q ) < depth R q ( X q ) . On the other hand, we have:(3.1.6) depth R q ( X q ) = depth R q ( q R q , X q )= depth R q (cid:16)q p R q + Ann R q ( X q ) , X q (cid:17) = depth R q ( p R q + Ann R q ( X q ) , X q )= depth R q ( p R q , X q ) In (3.1.6), the first equality follows from 2.4, the second one is due to the fact that p R q + Ann R q ( X q ) is q R q -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 impliesthat the inequality (3.1.1) holds, and hence completes the proof. (cid:3) We proceed by recording some consequences of Theorem 3.1.
Corollary 3.2.
Let R be a local ring, M and N be finitely generated R-modules, and let m and n benonnegative integers. Assume the following hold: (i) CI-dim R ( M ) < ∞ . (ii) Tor Ri ( M , N ) = for all i ≥ . (iii) M ⊗ R N satisfies Serre’s condition ( S n ) . (iv) If q ∈ Supp R ( M ⊗ R N ) , then depth R q ( p R q , M q ) ≤ depth R q ( p R q , R q ) + m for all p ∈ U ( q ) .Then N satisfies Serre’s condition ( S n − m ) .Proof. Note that M ⊗ L R N ∼ = M ⊗ R N if and only if Tor Ri ( M , N ) = i ≥
1. Therefore, it follows byTheorem 3.1 that N satisfies Serre’s condition ( S n − m ) . (cid:3) Remark 3.3.
In [7] one can find further results concerning the depth inequality stated in part (iv) ofCorollary 3.2; see also Proposition 3.6. In fact, when m =
1, it is proved in [7] that the aforementionedinequality always holds over hypersurface rings. More precisely, if R is a hypersurface ring, I an idealof R , and M is a non-zero torsion-free R -module which is generically free, then it follows from a resultin [7] that depth ( I , M ) ≤ depth R ( I , R ) + (cid:3) Modules over hypersurface rings are reflexive if and only if they satisfy Serre’s condition ( S ) ; 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:
O. CELIKBAS, U. LE, AND H. MATSUI (i)
N has rank. (ii) If q ∈ Supp R ( M ) , then depth R q ( p R q , M q ) ≤ height R ( 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 Tor Ri ( M , N ) = i ≥ M is reflexive. Moreover, since R is Cohen-Macaulay, the equality depth R q ( p R q , R q ) = height R ( p ) holds for each p , q ∈ Spec ( R ) with p ⊆ q . Note also CI-dim R ( M ) < ∞ as R is a hypersurface; see 2.8(v).Therefore, setting m = n =
2, we conclude from Corollary 3.2 that N is reflexive. (cid:3) 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 thatdepth R q ( p R q , M q ) = depth R ( p , M ) = > height R ( p ) = (cid:3) Now our aim is to establish Theorem 1.3, advertised in the introduction. First we prove the followinggeneral 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 conditionsare equivalent: (i)
Each M-regular sequence is R-regular. (ii) depth R ( I , M ) ≤ depth R ( I , R ) for each ideal I of R. (iii) depth R ( p , M ) ≤ depth R ( 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 = p ∩ . . . ∩ p n for some prime ideals p i . Then it follows thatdepth R ( I , M ) = depth R ( √ I , M )= inf { depth R ( p , M ) , . . . , depth R ( p n , M ) }≤ inf { depth R ( p , R ) , . . . , depth R ( p n , R ) } (3.6.1) = depth R ( √ I , R )= depth R ( 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 secondand third equalities are due to [6, 1.2.10(c)], and the inequality follows by assumption. ( ii ) = ⇒ ( i ) : Let x = x , . . . , x n ⊆ m be an M -regular sequence. We will show that this sequence is R -regular by induction on n .If n =
1, then we have 1 ≤ depth R (( x ) , M ) ≤ depth R (( x ) , R ) , which implies that x is a non zero-divisor on R . Hence we assume n ≥
2. Then, by the induction hypothesis, it follows that x ′ = x , . . . , x n − is R -regular. Thus, we have:depth R (( x n ) , M / ( x ′ ) M ) + ( n − ) = depth R (( x ) , M ) ≤ depth R (( x ) , R ) (3.6.2) = depth R (( x n ) , R / ( x ′ )) + ( n − ) Here in (3.6.2), the equalities are due to [6, 1.2.10(d)], while the inequality follows by assumption.Therefore, we have 1 ≤ depth R (( x n ) , M / ( x ′ ) M ) ≤ depth R (( x n ) , R / ( x ′ ) R ) , which implies that x n is a non zero-divisor on R / ( x ′ ) R , as required. (cid:3) N THE DEPTH AND REFLEXIVITY OF TENSOR PRODUCTS 7
Remark 3.7.
The equivalent conditions in Proposition 3.6 hold if and only if, whenever x = x , . . . , x n is a sequence of elements in m with Tor R ( M , R / xR ) =
0, it follows Tor R ( M , R / xR ) =
0; see [5, 2.2]. (cid:3)
The following result is an immediate consequence of Proposition 3.6:
Corollary 3.8.
Let R be a local ring, I an ideal of R, and let M be a nonzero R-module such that
Supp R ( M ) = Spec ( R ) . If depth R p ( M p ) ≤ depth ( R p ) for all p ∈ Spec ( R ) (e.g., R is Cohen-Macaulay, or CI-dim R ( M ) < ∞ ), then each M p -regular sequence is R p -regular for all p ∈ Spec ( R ) .Proof. We have depth R ( I , M ) = inf { depth R p ( M p ) | p ∈ V ( I ) } ≤ inf { depth ( R p ) | p ∈ V ( I ) } = depth ( I , R ) ;see 2.5(ii). Hence the result follows from Proposition 3.6. (cid:3) 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 depthformula shows that both M and N satisfy ( S ) , 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 Supp R ( M ) = Spec ( R ) . Proof of Theorem 1.3.
Let q ∈ Supp R ( M ) . Then, since each M q -regular sequence is R q -regular by as-sumption, it follows from Proposition 3.6 that depth R q ( p R q , M q ) ≤ height R ( p ) for all p ∈ Spec ( R ) with p ⊆ q . Therefore, we have that depth R q ( p R q , M q ) ≤ height R ( p ) for all p ∈ U ( q ) . Consequently, Corol-lary 3.4 implies that both M and N are reflexive. (cid:3) It is interesting to note that Theorem 1.3 (and also Theorem 1.1) can fail over rings that are nothypersurfaces. For example, if R = k [[ t , t , t ]] and N = ( t , t ) , the canonical module of R , Hunekeand 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 R -module M is called Tor-rigid provided that the following condition holds: for each R -module N satisfying Tor R ( M , N ) =
0, one has that Tor R ( M , N ) =
0. Examples of Tor-rigid modulesare abundant in the literature. For example, if R is hypersurface, that is quotient of an unramified regularlocal ring, and M is an R -module such that length R ( M ) < ∞ or pd R ( 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.In 2019 Celikbas, Matsui and Sadeghi [9] examined the conclusion of Theorem 1.1 and studied thereflexivity of tensor products of modules over local hypersurface rings in terms of the Tor-rigidity. Theirmain result establishes the same conclusion of Theorem 1.3 for Tor-rigid modules. More precisely, themain result of [9] shows that, if M and N are nonzero modules over a local hypersurface ring R suchthat M ⊗ R N is reflexive, M is Tor-rigid, and N has rank, then M and N are both reflexive; see Theorem1.1 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 / ( x ) . Then it follows that Supp R ( M ) = Spec ( R ) , M is not Tor-rigid, and each M p -regular sequence is R p -regular for all p ∈ Supp R ( M ) . We justify theseproperties as follows:(i) Supp R ( M ) = Spec ( R ) : this is clear since ( y ) / ∈ Supp R ( M ) . In fact, since { ( x ) , ( x , y ) } is the set ofall associated primes of M , it follows that Supp R ( M ) = V (cid:0) ( x ) (cid:1) ∪ V (cid:0) ( x , y ) (cid:1) .(ii) M is not Tor-rigid: setting N = R / ( y ) , one can check that Tor R ( M , N ) = = Tor R ( M , N ) . O. CELIKBAS, U. LE, AND H. MATSUI (iii) Each M p -regular sequence is R p -regular for all p ∈ Supp R ( M ) : note, to justify this claim, due toProposition 3.6, we proceed to prove the following claim:(4.1.1) If I is an ideal of R such that I ⊆ p ∈ Supp R ( M ) , then depth R p ( IR p , M p ) ≤ height R p ( IR p ) . Let p ∈ Supp R ( M ) and let I be an ideal of R such that I ⊆ p . We look at the height of p , i.e., dim ( R p ) . Case 1 : Assume height R ( p ) =
0. In this case the claim in (4.1.1) holds as height R p ( IR p ) ≤ dim ( R p ) and depth R p ( IR p , M p ) ≤ depth R p ( M p ) ≤ dim ( R p ) . Case 2 : Assume height R ( p ) =
1. We first consider the case where p = ( x , y ) . As p is an associatedprime of M , it follows that depth R p ( IR p , M p ) ≤ depth R p ( M p ) =
0, and so the claim in (4.1.1) holds.Next, we consider the case where p = ( x , y ) . Note, as p ∈ Supp R ( M ) , we have that ( x ) ⊆ p . Moreover,in case I ⊆ r $ p for some r ∈ Spec ( R ) , one can observe that I ⊆ ( x ) . As depth R ( x ) ( M ( x ) ) =
0, theaforementioned observation and 2.5(ii) yield:(4.1.2) depth R p ( IR p , M p ) = inf { depth R q ( M q ) | I ⊆ q ⊆ p } = ( depth R p ( M p ) if I * ( x ) 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 height R ( p ) =
2, i.e., p = m . As depth R ( I , M ) ≤ depth R ( M ) =
1, to establish theclaim in (4.1.1), it suffices to assume height R ( I ) = R ( I , M ) =
0. We observe, aseach element of I is a zero-divisor on R , that I ⊆ ( x ) or I ⊆ ( y ) . Thus, we have I ⊆ ( x , y ) and hence:depth R ( I , M ) = inf { depth R r ( M r ) | r ∈ V ( I ) } ≤ depth R ( x , y ) ( M ( x , y ) ) = . This completes the proof of Case 3. (cid: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 -modulesuch that dim R ( T ) = R ( T ) = ∞ (e.g., T = k ). Then it follows that Supp R ( M ) = Spec ( R ) , M is Tor-rigid, and there is an M p -regular sequence which is not R p -regular for some p ∈ Supp R ( M ) . Wejustify these properties as follows:(i) Supp R ( M ) = Spec ( R ) : this is clear since ( y ) / ∈ Supp R ( M ) .(ii) M is Tor-rigid: to see this assume Tor R ( M , X ) = R -module X . Then Tor R ( T , X ) = T is Tor-rigid [16, 2.4], we have that Tor Ri ( T , X ) = i ≥
1. This implies X is free andhence Tor Ri ( M , X ) = i ≥
1; see [16, 2.5] (or see 2.8(v) and 2.9). Therefore, M is Tor-rigid (notealso that N is not Tor-rigid since Tor R ( N , R / yR ) = = R / p = Tor R ( N , R / yR ) ).(iii) There is an M p -regular sequence which is not R p -regular for some p ∈ Supp R ( M ) : for this part,let p = ( x , y ) . Then it follows that p ∈ Supp R ( M ) and M p ∼ = N p . Hence, y is a non zero-divisor on M p .On the other hand, as x = y = xy = R p , we see that y is a zero-divisor on R p . Thus, { y } isan M p -regular sequence which is not R p -regular. (cid:3) 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 isdistinct from the condition stated in part (ii) of Theorem 1.3. Example 4.3.
Let R = k [[ x , y , z , u ]] / ( xu − yz ) and let M = ( x , y ) = Ω (cid:0) R / ( x , y ) (cid:1) . Then each M p -regularsequence is R p -regular because Supp R ( M ) = Spec ( R ) ; see Corollary 3.8. Furthermore, it follows thatTor R ( M , M ) = = k = Tor R ( M , M ) , and hence M is not Tor-rigid. (cid:3) 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 , t , t ]] and ω = ( t , t ) . Then M is not Tor-rigid; see, for example, [26, 4.13(i)]. Onthe other hand, each M p -regular sequence is R p -regular for all p ∈ Spec ( R ) ; see Corollary 3.8. (cid:3) N THE DEPTH AND REFLEXIVITY OF TENSOR PRODUCTS 9
It is well-known that the Tor-rigidity property does not localize, in general. For example, if M isthe module considered in Example 4.2, then M is Tor-rigid over R , but M p is not Tor-rigid over R p .Furhermore, the same example also shows that the condition we consider in part (ii) of Theorem 1.3does 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) M p is Tor-rigid over R p for all p ∈ Supp R ( M ) .(iii) Each M -regular sequence is R -regular.(iv) Each M p -regular sequence is R p -regular for all p ∈ Supp R ( M ) .Then we have: (i) ( ) + ❴ ( ) (cid:11) (cid:19) (iii) ✤ ( ) k s ❴ ( ) (cid:11) (cid:19) (ii) ( ) K S ( ) + (iv) ✤ ( ) k s ( ) K S 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 R be a hypersurface ring that is not a domain, and the tensor product of two prime ideals is refexive,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 [22, II.3.3]. Remark A.1.
Let R be a commutative Noetherian ring, and let p ∈ Spec ( R ) . Assume p ( n ) = n ≥
1, where p ( n ) = p n R p ∩ R denotes the n th 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 → R p is injective; see Remark 4.5. Hence, R is a domain if R p is a domain.(ii) Assume pd R ( M ) < ∞ . Then each non zero-divisor on M is a non zero-divisor on R so that thecanonical map R → R p is injective; see [23], [24, 6.2.3]. Also, R is a domain as R p is regular.(iii) Assume id R ( M ) < ∞ . Then it follows that R is Gorenstein [22, II.5.3] so that pd R ( 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, and let M be a nonzeroR-module. Let p ∈ Spec ( R ) such that height R ( p ) ≥ and p ( n ) = for some n ≥ . Set N = Ω r (cid:0) R / p ( n ) (cid:1) for some r ≥ . Assume M ⊗ R N is reflexive. Then it follows that r ≥ , both M and N are reflexive, and pd R ( M ) < ∞ = pd R ( 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 R / p is torsion, i.e., R / p hasrank zero. Thus N has rank, and so the conclusions of Theorem 1.1 hold. We know, by Theorem 1.1, that pd R ( M ) < ∞ or pd R ( N ) < ∞ . However, if pd R ( N ) < ∞ , thenpd R ( R / p ( n ) ) < ∞ and this forces R to be a domain; see part (ii) of Remark A.1. Therefore, we havethat pd R ( M ) < ∞ = pd R ( N ) .Notice, since both M and N have rank, both of these modules have full support. Consequently,Theorem 1.1 implies that both M and N are reflexive; see [8, 1.3].Now assume M is not free. Then, since M is reflexive, it follows that pd R ( M ) ≥
3. Hence, as M is asecond syzygy module, the syzygy theorem of Evans and Griffith [14, 1.1] (see also [1], [6, 9.5.6], and[21]) forces M to have rank at least two. (cid:3) 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 ) , and let q = ( y ) . Then p and q are the minimal prime idealsof R . Set M = R / ( x ) and N = Ω ( R / p ) . Then pd R ( 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 p ( r ) = and p ( s ) = for some r ≥ and s ≥ . If p or q has positive height, then p ( r ) ⊗ R q ( s ) is not areflexive R-module. Therefore, if p ⊗ R q is reflexive, then both p and q are minimal primes. 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 , p and q be as in Example A.3. Then p and q are the minimal prime ideals of R .It follows that p ∼ = R / ( y ) and p ⊗ R p ∼ = p are reflexive R -modules. On the other hand, the tensor product p ⊗ R q ∼ = k is not reflexive. A CKNOWLEDGEMENTS
The authors thank Mohsen Asgharzadeh and Greg Piepmeyer for their comments and suggestions onan earlier version of the manuscript. We also thank Arash Sadeghi for pointing us the rank conclusionin Proposition A.2. R
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