Tensor Products of Approximately Cohen-Macaulay Rings^{*}

M. R. Pournaki, M. Tousi, S. Yassemi

Abstract Our aim in this article is to study a problem originally raised by Grothendieck. We show that the approximately Cohen-Macaulay property is preserved for the tensor product of algebras over a field k. We also discuss the converse problem.

Keywords: Approximately Cohen-Macaulay ring, Flat homomorphism of rings, Tensor product. 2000 Mathematics Subject Classification: Primary 13H10, 13C15; Secondary 13D03.

1 Introduction

All rings and algebras considered in this article are commutative Noetherian with identity element, and all ring homomorphisms are unital. Throughout, k stands for a field. Let (R, 𝔪) be a local ring with dim(R) = d. Recall that R is a Gorenstein ring if and only if there is an element a of 𝔪 such that R/a^{n}R is a Gorenstein ring of dimension d − 1 for every integer n > 0 (cf. [7]). Clearly, this is not true for Cohen-Macaulay rings. The local ring R is called an approximately Cohen-Macaulay ring if either dim(R) = 0 or there exists an element a of 𝔪 such that R/a^{n}R is a Cohen-Macaulay ring of dimension d−1 for every integer n > 0 (cf. [5]). It is shown that if R is an approximately Cohen-Macaulay ring, then so is the ring R_{𝔭} for any prime ideal 𝔭 (see Theorem 2). Therefore, the concept of approximately Cohen-Macaulay is extended to nonlocal rings as follows. A ring R is an approximately Cohen-Macaulay ring if for all prime ideals 𝔭 of R, the ring R_{𝔭} is an approximately Cohen-Macaulay ring. It is well known that the tensor product R ⊗_{A} S of regular rings is not regular in general, even if we assume R and S are A-algebras, where A is a field (cf. [10, Remark 7]). In [11, Remark 1.7], Watanabe, Ishikawa, Tachibana and Otsuka showed that under a suitable condition, tensor products of regular rings are complete intersections. It is proven in [6] that the tensor product R ⊗_{A} S of Cohen-Macaulay rings are again Cohen-Macaulay, if we assume that R is a flat Amodule and S is a finitely generated A-module, and in [11], it is shown that the same is true for Gorenstein rings. In [1], Bouchiba and Kabbaj showed that if R and S are k-algebras such that R ⊗_{k} S is Noetherian, then R ⊗_{k} S is a Cohen-Macaulay ring if and only if R and S are Cohen-Macaulay rings. Recently, in [10], Tousi and Yassemi showed that if R and S are nonzero k-algebras such that R ⊗_{k} S is Noetherian, then R ⊗_{k} S is a locally complete intersection (resp. Gorenstein, Cohen-Macaulay) if and only if R and S are locally complete intersections (resp. Gorenstein, Cohen-Macaulay).

*The research of the first author was in part supported by a grant from IPM.

The research of the second author was in part supported by grant no. 83130212 from IPM.

The research of the third author was in part supported by grant no. 83130214 from IPM.

In this article we shall investigate if the approximately Cohen-Macaulay property is conserved under tensor product operations. It is shown that if φ : (R, 𝔪) ⟶ (S, 𝔫) is a flat local homomorphism and R is not a Cohen-Macaulay ring, then the following are equivalent (see Theorem 6):

(a) R is an approximately Cohen-Macaulay ring and S/𝔪S is a Cohen-Macaulay ring.

(b) S is an approximately Cohen-Macaulay ring and Ass_{S}(S/𝔭S) = Assh_{S}(S/𝔭S) for every 𝔭 ∈ Assh(R).

Further, if R is a homomorphic image of a Cohen-Macaulay local ring, then the next condition is also equivalent:

(c) S is an approximately Cohen-Macaulay ring.

We will also prove the following result. Let R and S be nonzero k-algebras such that T := R ⊗_{k} S is Noetherian. Assume that R is not a Cohen-Macaulay ring. Then the following hold (see Theorem 10):

(i) If R is an approximately Cohen-Macaulay ring and S is a Cohen-Macaulay ring, then T is an approximately Cohen-Macaulay ring.

(ii) If T is an approximately Cohen-Macaulay ring, then S is a Cohen-Macaulay ring.

(iii) If R is a homomorphic image of a Cohen-Macaulay ring or k is algebraically closed, then the following conditions are equivalent:

(a) T is an approximately Cohen-Macaulay ring.

(b) R is an approximately Cohen-Macaulay ring and S is a Cohen-Macaulay ring.

2 Main Results

For a finitely generated R-module M of finite Krull dimension, recall that

Assh_{R}(M) = {𝔭 ∈ Supp_{R}(M)| dim(R/𝔭) = dim(M)},

and denote Assh_{R}(R) = Assh(R). Let U_{R}(0) = _{⋂𝔭∈Assh(R)}I(𝔭), where (0) = ⋂𝔭∈Ass(R)I(𝔭) denotes a minimal primary decomposition of the zero ideal of R (cf. [5]).

Let M be a finitely generated R-module and I an ideal of R such that IM ≠ M. Then the common length of the maximal M-sequences in I is called the grade of I on M, denoted by grade_{M}(I). If (R, 𝔪) is a local ring, and M is a finitely generated nonzero R-module, then the grade of 𝔪 on M is called the depth of M, denoted by depth(M).

Theorem 1 (see [5]) Let R be a local ring with maximal ideal 𝔪 and dim(R) = d. Suppose that R is not a Cohen-Macaulay ring. Then the following conditions are equivalent:

(i) R is an approximately Cohen-Macaulay ring.

(ii) R contains an ideal I such that I is a Cohen-Macaulay R-module of dimension d − 1 and R/I is a Cohen-Macaulay ring of dimension d.

(iii) R/U_{R}(0) is a Cohen-Macaulay ring and depth(R) = d − 1.

(iv) (a) H_{𝔪}^{i}(R) = (0) for i ≠ d − 1, d.

(b) Hom_{R}(H_{𝔪}^{d−1}(R), E_{R}(R/𝔪)) is a Cohen-Macaulay R̂-module of dimension d − 1.

(c) The local ring R/𝔭 is unmixed for every 𝔭 ∈ Assh(R), i.e., the equality dim(R̂/P) = d holds for every P ∈ AssR̂(R̂/𝔭R̂) and for every 𝔭 ∈ Assh(R).

In this case, the ideal I appearing in assertion (ii) is uniquely determined and equals U_{R}(0). Here R̂ (resp. E_{R}(R/𝔪)) denotes the 𝔪-adic completion of R (resp. the injective hull of R/𝔪).

In the following theorem we consider the behavior of approximately Cohen-Macaulay property by passing to localizations.

Theorem 2 Let (R, 𝔪) be an approximately Cohen-Macaulay ring. Then (i) For any 𝔭 ∈ Spec(R), dim(R_{𝔭}) − depth(R_{𝔭}) ≤ 1. (ii) Suppose R is not a Cohen-Macaulay ring. Then for any 𝔭 ∈ Spec(R) such

that R_{𝔭} is not a Cohen-Macaulay ring, ht(𝔭) + dim(R/𝔭) = dim(R).

(iii) Suppose R is not a Cohen-Macaulay ring. Then for any 𝔭 ∈ Spec(R) such that R_{𝔭} is not a Cohen-Macaulay ring, U_{R_{𝔭}}(0) = U_{R}(0)R_{𝔭}.

(iv) For any 𝔭 ∈ Spec(R), R_{𝔭} is an approximately Cohen-Macaulay ring.

Proof. (i): This follows from the fact that dim(R_{𝔭}) − depth(R_{𝔭}) ≤ dim(R) − depth(R) for any 𝔭 ∈ Spec(R) (see [8, Exercise 17.5(ii)]).

(ii): Let 𝔭 ∈ Spec(R) such that R_{𝔭} is not a Cohen-Macaulay ring. By (i), we have depth(R_{𝔭}) = ht(𝔭) − 1. Also, in view of [8, Exercise 17.5(i)], we have

ht(𝔭) + dim(R/𝔭) − 1 ≤ dim(R) − 1
= depth(R)
≤ grade_{R}(𝔭) + dim(R/𝔭)
≤ depth(R_{𝔭}) + dim(R/𝔭).

Therefore, ht(𝔭) + dim(R/𝔭) = dim(R).

(iii): Let 𝔭 ∈ Spec(R) such that R_{𝔭} is not a Cohen-Macaulay ring. We claim that U_{R}(0)R_{𝔭} = U_{R_{𝔭}}(0). If (0) = _{⋂𝔮∈Ass(R)}I(𝔮) is an irredundant primary decomposition for the zero ideal of R, then

(0) = ⋂_{\begin{matrix}𝔮 ∈ Ass(R) \\ 𝔮 ⊆ 𝔭 \end{matrix}} I(𝔮)R_{𝔭}

is a minimal primary decomposition for the zero ideal of R_{𝔭}. Thus it is enough to show that

Assh(R_{𝔭}) = {𝔮R_{𝔭}| 𝔮 ∈ Assh(R), 𝔮 ⊆ 𝔭}.

Let 𝔮 ∈ Spec(R). We have

𝔮R_{𝔭} ∈ Assh(R_{𝔭}) ⟺ 𝔮 ⊆ 𝔭 and dim(R_{𝔭}/𝔮R_{𝔭}) = dim(R_{𝔭})
⟺ 𝔮 ⊆ 𝔭 and ht(𝔭/𝔮) = ht(𝔭)
⟺ 𝔮 ⊆ 𝔭 and ht(𝔭/𝔮) + dim(R/𝔭) = dim(R).

Let 𝔮R_{𝔭} ∈ Assh(R_{𝔭}). Since

dim(R/𝔮) ≥ ht(𝔭/𝔮) + dim(R/𝔭)
= dim(R),

dim(R/𝔮) = dim(R) and hence 𝔮 ∈ Assh(R).

Now, let 𝔮 ⊆ 𝔭 and 𝔮 ∈ Assh(R). Since R/U_{R}(0) is a catenary ring, R/𝔮 is also catenary and hence by [8, Theorem 31.4], ht(𝔭/𝔮) + dim(R/𝔭) = dim(R/𝔮). Thus ht(𝔭/𝔮) + dim(R/𝔭) = dim(R). Therefore, 𝔮R_{𝔭} ∈ Assh(R_{𝔭}).

(iv): Let 𝔭 ∈ Spec(R). If R_{𝔭} is a Cohen-Macaulay ring, then R_{𝔭} is an approximately Cohen-Macaulay ring. If R_{𝔭} is not a Cohen-Macaulay ring, then R is not a Cohen-Macaulay ring, and so R/U_{R}(0) is a Cohen-Macaulay ring. Now, by (iii), R_{𝔭}/U_{R_{𝔭}}(0) is Cohen-Macaulay. Thus the assertion follows from (i) and Theorem 1(iii). □

By using Theorem 2, the concept of approximately Cohen-Macaulay ring can be extended to nonlocal rings by defining that a ring R is approximately Cohen-Macaulay if for all prime ideals 𝔭 of R, the ring R_{𝔭} is an approximately Cohen-Macaulay ring.

In the following result we consider the behavior of Assh(−) and the primary decomposition of the zero submodule under base change.

Proposition 3 Let φ : (R, 𝔪) ⟶ (S, 𝔫) be a flat local homomorphism. Then the following hold:

(i) Assh(R) = {𝔮 ∩ R| 𝔮 ∈ Assh(S)}.

(ii) U_{R}(0)S ⊆ U_{S}(0).

(iii) The following conditions are equivalent:

(a) U_{R}(0)S = U_{S}(0).

(b) Assh(S) = {𝔮 ∈ Ass(S)| 𝔮 ∩ R ∈ Assh(R)}.

(c) Ass_{S}(S/𝔭S) = Assh_{S}(S/𝔭S) for every 𝔭 ∈ Assh(R). Proof. (i): Let 𝔭 ∈ Assh(R). Then dim(R) = dim(R/𝔭). Consider the flat local

homomorphism φ̃ : R/𝔭 → S/𝔭S. Hence,

dim(R) = dim(S/𝔭S) − dim(S/𝔪S),

and therefore dim(S) = dim(S/𝔭S). So there exists a minimal prime ideal of 𝔭S, say 𝔮, such that dim(S) = dim(S/𝔮). Since 𝔮 is a minimal prime ideal of 𝔭S, 𝔮 ∩ R = 𝔭.

Let 𝔮 ∈ Assh(S) and 𝔮 ∩ R = 𝔭. We have dim(S/𝔮) = dim(S) and 𝔭S ⊆ 𝔮; that means dim(S/𝔭S) = dim(S). Thus,

dim(R/𝔭) + dim(S/𝔪S) = dim(S),

and so dim(R/𝔭) = dim(R).

(ii): It is enough to show that U_{R}(0)S_{𝔮} = (0)S_{𝔮} for every 𝔮 ∈ Assh(S). Indeed, since S is flat,

U_{R}(0)S_{𝔮} = ⋂_{𝔭∈Assh(R)}I(𝔭)S_{𝔮}

= ⋂_{\begin{matrix}𝔭 ∈ Assh(R) \\ 𝔭 ⊆ 𝔮 ∩ R \end{matrix}} I(𝔭)S_{𝔮}.

If 𝔮 ∈ Assh(S), then 𝔮∩R = 𝔭. We know that (I(𝔭)R_{𝔭})S_{𝔮} and (0)S_{𝔮} are 𝔭R_{𝔭}-primary submodules of the R_{𝔭}-module S_{𝔮}. Thus I(𝔭)S_{𝔮} = (I(𝔭)R_{𝔭})S_{𝔮} = (0)S_{𝔮}, as required.

(iii): It is known that (cf. [8, Theorem 23.2])

{𝔭} = {φ^{−1}(𝔮)| 𝔮 ∈ Ass_{S}(S/𝔭S)} for each 𝔭 ∈ Spec(R),

Ass(S) = ⋃_{𝔭∈Ass(R)}Ass_{S}(S/𝔭S),

Ass_{S}(S/U_{R}(0)S) = ⋃_{𝔭∈Assh(R)}Ass_{S}(S/𝔭S),

Assh(S) = Ass_{S}(S/U_{S}(0)) = ⋃_{𝔭∈Assh(R)}Assh_{S}(S/𝔭S).

Also note that ∩_{𝔭∈Ass(R)}I(𝔭)S = (0) is a minimal primary decomposition of the zero submodule of the R-module S and if ∩_{1≤i≤n}Q_{i} = (0) is a minimal primary decomposition of the zero ideal in S, then ∩_{1≤i≤n}Q_{i} = (0) is a primary decomposition of the zero submodule of the R-module S. We can now easily obtain (iii). □

The next result shows that the approximately Cohen-Macaulay property is stable under specialization.

Lemma 4 Let (R, 𝔪) be a Noetherian local ring with dim(R) = d. Let x ∈ 𝔪\Z(R). If R is an approximately Cohen-Macaulay ring, then R/xR is an approximately Cohen-Macaulay ring.

Proof. We may assume that R is not a Cohen-Macaulay ring. Then there exists an ideal I of R such that I is a Cohen-Macaulay R-module, dim(I) = d−1, and R/I is a Cohen-Macaulay ring of dimension d. Therefore, I/xI is a Cohen-Macaulay R/xRmodule of dimension d − 2. Since Ass_{R}(R/I) = {𝔮 ∈ Assh(R)| I ⊆ 𝔮}, x ∉ Z_{R}(R/I) and hence R/(I +xR) is a Cohen-Macaulay ring of dimension d−1 and I ∩xR = xI. By using the isomorphism (I + xR)/xR ≅ I/(I ∩ xR), we obtain that (I + xR)/xR is a Cohen-Macaulay R/xR-module of dimension d − 2. The assertion now follows from Theorem 1. □

Lemma 5 Let φ : (R, 𝔪) ⟶ (S, 𝔫) be a flat local homomorphism. Let S be an approximately Cohen-Macaulay ring. Then either R or S/𝔪S is Cohen-Macaulay.

Proof. Assume that R is not Cohen-Macaulay. Then we have

depth(R) = depth(S) − depth(S/𝔪S)
= dim(S) − 1 − depth(S/𝔪S)
≥ dim(S) − 1 − dim(S/𝔪S)
= dim(R) − 1.

Since R is not Cohen-Macaulay, depth(R) = dim(R) − 1 and hence S/𝔪S is Cohen-Macaulay. □

We are now ready to prove that the approximately Cohen-Macaulay property is stable (in some sense) under change of ring. This result is somehow parallel to the results on properties like regular, complete intersection and Cohen-Macaulay (cf. [10, Theorem 1]).

Theorem 6 Let φ : (R, 𝔪) ⟶ (S, 𝔫) be a flat local homomorphism. Assume that R is not a Cohen-Macaulay ring. Then the following are equivalent:

(a) R is an approximately Cohen-Macaulay ring and S/𝔪S is a Cohen-Macaulay ring.

(b) S is an approximately Cohen-Macaulay ring and Ass_{S}(S/𝔭S) = Assh_{S}(S/𝔭S) for every 𝔭 ∈ Assh(R).

Further, if R is a homomorphic image of a Cohen-Macaulay local ring, then the next condition is also equivalent:

(c) S is an approximately Cohen-Macaulay ring.

Proof. Consider the induced flat local homomorphism φ̃ : R/U_{R}(0) ⟶ S/U_{R}(0)S.

(a) ⟹ (b): By [3, Theorem 2.1.7], S/U_{R}(0)S is a Cohen-Macaulay ring. Also the following (in)equalities hold by Proposition 3(ii):

dim(S) ≥ dim(S/U_{R}(0)S)
≥ dim(S/U_{S}(0))
= dim(S).

On the other hand, U_{R}(0) is a Cohen-Macaulay R-module of dimension dim(R) − 1. Thus U_{R}(0)S ≅ U_{R}(0) ⊗_{R} S is a Cohen-Macaulay S-module of dimension dim(S)−1, because

dim(S) − 1 = dim(R) − 1 + dim(S/𝔪S)
= dim(U_{R}(0)) + dim(S/𝔪S)
= dim(U_{R}(0) ⊗_{R} S).

The last paragraph of Theorem 1 implies that U_{R}(0)S = U_{S}(0). Now, the assertions follow from Proposition 3 and Theorem 1(ii).

(b) ⟹ (a): By Lemma 5, S/𝔪S is a Cohen-Macaulay ring. We have U_{S}(0) = U_{R}(0)S by Proposition 3(iii). On the other hand, S is not a Cohen-Macaulay ring and so S/U_{R}(0)S is a Cohen-Macaulay ring. Therefore, R/U_{R}(0) is a Cohen-Macaulay ring. Now, the assertion follows from the following equalities:

dim(R) = dim(S) − dim(S/𝔪S)
= depth(S) + 1 − depth(S/𝔪S)
= depth(R) + 1.

(b) ⟹ (c): It is clear.

(c) ⟹ (a): By Lemma 5, we know that S/𝔪S is Cohen-Macaulay. Assume that 𝔮 is a minimal ideal of V(𝔪S). Then 𝔮 ∩ R = 𝔪 and dim(S_{𝔮}/𝔪S_{𝔮}) = 0. By considering the induced homomorphism φ̂ : R ⟶ S_{𝔮} one can reduce to the case where dim(S/𝔪S) = 0. By using “(a) ⟹ (b)” and [5, Corollary 2.6], we may assume that R and S are complete. Note that S is not a Cohen-Macaulay ring. We use induction on dim(S) = n. If n = 1, then dim(R) + dim(S/𝔪S) = 1 and hence dim(R) = 1. Thus, R is an approximately Cohen-Macaulay ring. Now suppose, inductively, that n ≥ 2 and we have established the result for n − 1. Set N = Hom_{R}(H_{𝔪}^{n−1}(R), E_{R}(R/𝔪)). Since

depth(R) = depth(S)
= dim(S) − 1
= dim(R) − 1,

H_{𝔪}^{i}(R) = (0) for i ∉ {n − 1, n}. Therefore, it is enough to show that N is a Cohen-Macaulay R-module of dimension n − 1.

We claim that 𝔪 ∉ Ass_{R}(N). Otherwise 𝔪 ∈ Att_{R}(H_{𝔪}^{n−1}(R)) and so by [2, Exercise 11.3.7], 𝔫 ∈ Att_{S}(H_{𝔫}^{n−1}(S)). Therefore, 𝔫 ∈ Ass_{S}(Hom_{S}(H_{𝔫}^{n−1}(S), E_{S}(S/𝔫))). But Hom_{S}(H_{𝔫}^{n−1}(S), E_{S}(S/𝔫)) is a Cohen-Macaulay S-module of dimension n−1 ≥ 1. That is a contradiction.

Since N is a finitely generated R-module, the set Ass_{R}(N) is finite and hence there exists x ∈ 𝔪 \ (Z(R) ∪ Z_{R}(N)). Consider the induced flat local homomorphism φ̂ : R/xR ⟶ S/φ(x)S. Since φ(x) ∉ Z(S), by Lemma 4, S/φ(x)S is an approximately Cohen-Macaulay ring of dimension n−1. Therefore, by the inductive hypothesis R/xR is an approximately Cohen-Macaulay module of dimension n − 1. Set R̄ = R/xR and 𝔪̄ = 𝔪/xR. The R/xR-module HomR̄(Hn−2 ; 𝔪̄(R̄), ER̄(R̄/𝔪̄)) is a Cohen-Macaulay module of dimension n − 2. The exact sequence

0 → R →^{x.} R → R/xR → 0

induces the exact sequence

0 → H_{𝔪}^{n−2}(R/xR) → H_{𝔪}^{n−1}(R) →^{x.} H_{𝔪}^{n−1}(R).

Therefore, Hn−2 ; 𝔪̄(R̄) ≅ Hom_{R}(R/xR, H_{𝔪}^{n−1}(R)). By [2, Lemma 10.1.15],

HomR̄(Hn−2 ; 𝔪̄(R̄), ER̄(R̄/𝔪̄)) ≅ HomR̄(Hom_{R}(R̄, H_{𝔪}^{n−1}(R)), Hom_{R}(R̄, E_{R}(R/𝔪)))
≅ Hom_{R}(Hom_{R}(R̄, H_{𝔪}^{n−1}(R)), E_{R}(R/𝔪))
≅ R̄ ⊗_{R} Hom_{R}(H_{𝔪}^{n−1}(R), E_{R}(R/𝔪)).

Thus N/xN is a Cohen-Macaulay R/xR-module of dimension n − 2. Since x ∉ Z_{R}(N), N is a Cohen-Macaulay R-module of dimension n − 1. □

Note that in Theorem 6, the condition “Ass_{S}(S/𝔭S) = Assh_{S}(S/𝔭S) for every 𝔭 ∈ Assh(R)” is not superficial, as the following example shows.

Example 7 (See [9]) Let (R, 𝔪) be a 2-dimensional local domain for which the 𝔪-adic completion R̂ ≅ k[[x, y, z]]/(xy, xz). Put S = R̂. Let φ : R ⟶ S be a natural ring homomorphism. Then

(i) R is not approximately Cohen-Macaulay local domain. In particular, Assh(R) = {(0)}.

(ii) S/𝔪S = k is regular, and thus is Cohen-Macaulay.

(iii) S is approximately Cohen-Macaulay, but not unmixed.

(iv) R is not a homomorphic image of a Cohen-Macaulay local ring.

Corollary 8 Let φ : R ⟶ S be a flat homomorphism. If R is an approximately Cohen-Macaulay ring and (R_{𝔭}/𝔭R_{𝔭}) ⊗_{R} S is a Cohen-Macaulay ring for every 𝔭 ∈ Spec(R), then S is an approximately Cohen-Macaulay ring.

Proof. Let 𝔮 ∈ Spec(S). Set 𝔭 = 𝔮 ∩ R ∈ Spec(R). The induced homomorphism φ̃ : R_{𝔭} ⟶ S_{𝔮} is a flat local homomorphism. It is clear that S_{𝔮}/(𝔭R_{𝔭})S_{𝔮} is a localization of (R_{𝔭}/𝔭R_{𝔭}) ⊗_{R} S. Now, the assertion follows from Theorem 6. □

Corollary 9 Let φ : R ⟶ S is a faithfully flat homomorphism. Suppose that R is not a Cohen-Macaulay ring, but a homomorphic image of a Cohen-Macaulay ring. If S is an approximately Cohen-Macaulay ring, then R is an approximately Cohen-Macaulay ring.

Proof. Assume that 𝔭 ∈ Spec(R) and 𝔮 is a minimal ideal of V(𝔭S). Then 𝔮∩R = 𝔭 and dim(S_{𝔮}/𝔭S_{𝔮}) = 0. Consider the induced homomorphism φ̂ : R_{𝔭} ⟶ S_{𝔮}. The assertion now follows from Theorem 6. □

Theorem 10 Let R and S be nonzero k-algebras such that T := R ⊗_{k} S is Noetherian. Assume that R is not a Cohen-Macaulay ring. Then the following hold:

(i) If R is an approximately Cohen-Macaulay ring and S is a Cohen-Macaulay ring, then T is an approximately Cohen-Macaulay ring.

(ii) If T is an approximately Cohen-Macaulay ring, then S is a Cohen-Macaulay ring.

(iii) If R is a homomorphic image of a Cohen-Macaulay ring or k is algebraically closed, then the following conditions are equivalent:

(a) T is an approximately Cohen-Macaulay ring.

(b) R is an approximately Cohen-Macaulay ring and S is a Cohen-Macaulay ring.

Proof. (i): Consider the faithfully flat homomorphism φ : R ⟶ (R ⊗_{k} S). It is enough to show that the fibers (R_{𝔭}/𝔭R_{𝔭}) ⊗_{R} (R ⊗_{k} S) ≅ (R_{𝔭}/𝔭R_{𝔭}) ⊗_{k} S over every prime ideal 𝔭 of R are Cohen-Macaulay rings. Since R_{𝔭}/𝔭R_{𝔭} is a Cohen-Macaulay ring (it is actually a field), (R_{𝔭}/𝔭R_{𝔭}) ⊗_{k} S is also a Cohen-Macaulay ring by [10, Theorem 6].

(ii): Assume that S is not a Cohen-Macaulay ring. Then there exist 𝔭 ∈ Spec(R) and 𝔮 ∈ Spec(S) such that R_{𝔭} and S_{𝔮} are not Cohen-Macaulay rings, and hence grade_{R_{𝔭}}(𝔭R_{𝔭}) ≤ ht(𝔭) − 1 and grade_{S_{𝔮}}(𝔮S_{𝔮}) ≤ ht(𝔮) − 1. Therefore,

grade_{R_{𝔭}}(𝔭R_{𝔭}) + grade_{S_{𝔮}}(𝔮S_{𝔮}) ≤ ht(𝔭) + ht(𝔮) − 2.  (*)

There exists Q ∈ Spec(R ⊗_{k} S) such that Q ∩ R = 𝔭 and Q ∩ S = 𝔮. On the other hand, by [1, Proposition 2.3],

ht(Q) = ht(𝔭) + ht(𝔮) + ht(Q/((𝔭 ⊗_{k} S) + (R ⊗_{k} 𝔮)))

and

grade_{(R ⊗_{k} S)_{Q}}(Q(R ⊗_{k} S)_{Q}) = grade_{R_{𝔭}}(𝔭R_{𝔭}) + grade_{S_{𝔮}}(𝔮S_{𝔮})
+ht(Q/((𝔭 ⊗_{k} S) + (R ⊗_{k} 𝔮))).

But

grade_{(R ⊗_{k} S)_{Q}}(Q(R ⊗_{k} S)_{Q}) ≥ ht(Q) − 1,

so we have

grade_{R_{𝔭}}(𝔭R_{𝔭}) + grade_{S_{𝔮}}(𝔮S_{𝔮}) ≥ ht(𝔭) + ht(𝔮) − 1

and by using (*), ht(𝔭) + ht(𝔮) − 1 ≤ ht(𝔭) + ht(𝔮) − 2. That is a contradiction.

(iii): In fact, (b) ⟹ (a) is just (i), and for proving (a) ⟹ (b); by (ii), it is enough to show that R is an approximately Cohen-Macaulay ring.

First, let R be a homomorphic image of a Cohen-Macaulay ring. Consider the faithfully flat homomorphism φ : R ⟶ (R ⊗_{k} S). The assertion now follows form Corollary 9.

Next, let k be algebraically closed. Let 𝔭 ∈ Spec(R). Then there exists 𝔮 ∈ Spec(R ⊗_{k} S) with 𝔮 ∩ R = 𝔭. Consider the induced flat local homomorphism φ̃ : R_{𝔭} ⟶ (R ⊗_{k} S)_{𝔮}. Let 𝔮′ ∈ Ass(R ⊗_{k} S) with 𝔮′ ⊆ 𝔮 and ht(𝔭/(𝔮′∩R)) = ht(𝔭). By using Proposition 3 and Theorem 6, it is enough to show that ht(𝔮/𝔮′) = ht(𝔮). Set 𝔮′ ∩ R = 𝔭′, 𝔮 ∩ S = 𝔭_{2}, and 𝔮′ ∩ S = 𝔭_{1}. Then by [8, Theorem 23.2], 𝔮′ ∈ Ass_{T} (T/𝔭′T). On the other hand, S is Cohen-Macaulay and hence by [10, Theorem 6], T/𝔭′T = (R/𝔭′) ⊗_{k} S satisfies Serre’s condition (S_{1}). Therefore, 𝔮′ ∈ Min(𝔭′T) and so 𝔮′ ∈ Min(𝔭′T + 𝔭_{1}T). On the other hand, T/(𝔭′T + 𝔭_{1}T) ≅ (R/𝔭′) ⊗_{k} (S/𝔭_{1}) is an integral domain (see [4, Exercise A1.2(a), p. 562]). Thus 𝔮′ = 𝔭′T + 𝔭_{1}T. Now, the following equalities hold:

ht(𝔮/𝔮′) = ht(𝔮/(𝔭′T + 𝔭_{1}T))
= ht(𝔭/𝔭′) + ht(𝔭_{2}/𝔭_{1}) + ht(𝔮/(𝔭T + 𝔭_{2}T))
= ht(𝔭) + ht(𝔭_{2}) + ht(𝔮/(𝔭T + 𝔭_{2}T))
= ht(𝔮),

where the second and the last equalities hold by [1, Proposition 2.3] and the third one uses the fact that 𝔭_{1} ∈ Ass(S). □

Acknowledgment: This work was done while the first author was a Postdoctoral Research Associate at the School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics (IPM). He would like to thank the IPM for the financial support.

References

[1] S. Bouchiba, S. E. Kabbaj, Tensor Products of Cohen-Macaulay Rings: Solution to a Problem of Grothendieck, J. Algebra 252 (2002), no. 1, 65-73.

[2] M. P. Brodmann, R. Y. Sharp, “Local Cohomology: An Algebraic Introduction with Geometric Applications”, Cambridge Studies in Advanced Mathematics, 60. Cambridge University Press, Cambridge, 1998.

[3] W. Bruns, J. Herzog, “Cohen-Macaulay Rings”, Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993.

[4] D. Eisenbud, “Commutative Algebra with a View Toward Algebraic Geometry”, Springer-Verlag, 1995.

[5] S. Goto, Approximately Cohen-Macaulay Rings, J. Algebra 76 (1982), no. 1, 214-225.

[6] A. Grothendieck, “Éléments de Géométrie Algébrique” IV. Étude Locale des Schémas et des Morphismes de Schémas. II. (French) Inst. Hautes Études Sci. Publ. Math., No. 24, 1965.

[7] M. Hochster, Cyclic Purity Versus Purity in Excellent Noetherian Rings, Trans. Amer. Math. Soc. 231 (1977), no. 2, 463-488.

[8] H. Matsumura, “Commutative Ring Theory”, Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1986.

[9] M. Nagata, “Local Rings”, Interscience Tracts in Pure and Applied Mathematics, No. 13, New York, 1962.

[10] M. Tousi, S. Yassemi, Tensor Products of Some Special Rings, J. Algebra 268 (2003), no. 2, 672-676.

[11] K. Watanabe, T. Ishikawa, S. Tachibana, K. Otsuka, On Tensor Products of Gorenstein Rings, J. Math. Kyoto Univ. 9 (1969), 413-423.

The Authors’ Addresses

M. R. Pournaki, School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: pournaki@ipm.ir

M. Tousi, School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran, and Department of Mathematics, Shahid Beheshti University, Evin, Tehran 19834, Iran. E-mail address: mtousi@ipm.ir

S. Yassemi, Center of Excellence in Biomathematics, School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran, and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran.

E-mail address: yassemi@ipm.ir