Finiteness of homological dimensions of Ext modules

Kaito Kimura

doi:10.4153/S000843952610229X

Finiteness of homological dimensions of Ext modules

Part of: Commutative algebra: Homological methods

Published online by Cambridge University Press: 13 July 2026

Kaito Kimura

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Kaito Kimura*: Affiliation:
Department of Mathematics, Purdue University , West Lafayette, United States
*: e-mail: m21018b@gmail.com

Article contents

Abstract
Introduction
Notation and motivation
Main results
Examples
Footnotes
References

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Abstract

In this article, we study how certain homological properties of modules over a commutative Noetherian ring interact with each other. Specifically, we investigate how the finiteness of the homological dimensions of the Ext modules between two finitely generated modules affects the homological dimensions of the modules themselves. One of our main results shows that if all Ext modules up to a suitably chosen bound have finite projective dimension, then one module has finite projective or injective dimension exactly when the other module does.

Keywords

Ext module (Gorenstein) projective dimension (Gorenstein) injective dimension dualizing complex

MSC classification

Primary: 13D05: Homological dimension 13D07: Homological functors on modules (Tor, Ext, etc.)

Information

Type: Article
Information: Canadian Mathematical Bulletin , First View , pp. 1 - 17

DOI: https://doi.org/10.4153/S000843952610229X [Opens in a new window]
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of Canadian Mathematical Society

1 Introduction

Throughout the present article, all rings are assumed to be commutative and Noetherian. Let R be a local ring and let M and N be nonzero finitely generated R-modules. The characterization of the homological dimensions of two modules in terms of those of the derived Hom complex or its (finitely many) homologies has been actively studied (see [Reference Dey, Holanda and Miranda-Neto7, Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Reference Mendoza-Rubio and Jorge-Pérez11, Reference Zargar and Gheibi12] for instance). Considering a suitable representation of the complex, one sees that the finiteness of the appropriate homological dimensions of the two modules implies that of the derived Hom complex, as follows:

(I) $$ \begin{align} \text{pd}_R M<\infty \ \mathrm{and} \ \text{pd}_R N<\infty &\implies \text{pd}_R (\mathbf{R}\mathrm{Hom}_R(M,N))<\infty, \end{align} $$

(II) $$ \begin{align} \text{id}_R M<\infty \ \mathrm{and} \ \text{id}_R N<\infty &\implies \text{pd}_R (\mathbf{R}\mathrm{Hom}_R(M,N))<\infty, \end{align} $$

(III) $$ \begin{align} \text{pd}_R M<\infty \ \mathrm{and} \ \text{id}_R N<\infty &\implies \text{id}_R (\mathbf{R}\mathrm{Hom}_R(M,N))<\infty, \end{align} $$

where $\mathrm{pd}$ denotes the projective dimension and $\mathrm{id}$ denotes the injective dimension. The formulas for the Poincaré and Bass series given by Foxby [Reference Foxby8] assert that the converse of (III) holds. In particular, when $\mathrm{Ext}^i_R(M,N)$ has finite injective dimension for all $0\le i\le \mathrm{depth} R-\mathrm{depth} M$ , M has finite projective dimension if and only if N has finite injective dimension. (Indeed, if one of the two conditions holds, then by Ischebeck’s formula all the other Ext modules vanish, and hence the other condition follows as well; see [Reference Zargar and Gheibi12, Proposition 4.2] for instance.)

It is a natural question to ask what can be said for (I) and (II) in this context. Holanda et al. [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9] posed the following question: when $\mathrm{Ext}^i_R(M,N)$ has finite projective dimension for all $0\le i\le \mathrm{depth} R-\mathrm{depth} M$ , is it true that

(i) $$ \begin{align} \text{pd}_R M<\infty &\iff \text{pd}_R N<\infty\ \mathrm{and} \end{align} $$

(ii) $$ \begin{align} \text{id}_R N<\infty &\iff \text{id}_R M<\infty? \end{align} $$

They proved that “ $\Rightarrow $ ” holds for both (i) and (ii), and similarly showed the corresponding statements for Gorenstein homological dimensions. Moreover, in the Cohen–Macaulay case, they also resolved “ $\Leftarrow $ ” of (i). However, “ $\Leftarrow $ ” of (ii), and these questions outside the Cohen–Macaulay setting, remain open.

The main results of this article provide an affirmative answer to the above questions. Here, $\mathrm{H}\text{-}\mathrm{dim}$ denotes either the projective dimension or the Gorenstein projective dimension, and correspondingly, $\mathrm{Hid}$ denotes either the injective dimension or the Gorenstein injective dimension. The large restricted flat dimension $\mathrm{Rfd}_R M$ is the same notion as that used in [Reference Christensen, Foxby and Frankild3] and is characterized by $\mathrm { sup}\{\mathrm{depth} R_{\mathfrak {p}}-\mathrm{depth} M_{\mathfrak {p}} \mid \mathfrak {p}\in \mathrm{Spec} R\}$ .

Theorem 1.1 [Corollary 3.5]

Let R be a local ring, and M and N nonzero finitely generated R-modules with $\mathrm{pd}_R N<\infty $ and $\mathrm{H}\text{-}\mathrm{dim}_R\mathrm{Ext}^i_R(M,N)<\infty $ for all $0\le i\le \mathrm{Rfd}_R M$ . Then $\mathrm{H}\text{-}\mathrm{dim}_R M<\infty $ .

Theorem 1.2 [Corollary 3.9]

Let R be a local ring, M and N nonzero finitely generated R-modules with $\mathrm{id}_R M<\infty $ and $\mathrm{H}\text{-}\mathrm{dim}_R\mathrm{Ext}^i_R(M,N)<\infty $ for all $0\le i\le \mathrm{depth} R-\mathrm{depth} M$ . In the case $\mathrm{H}\text{-}\mathrm{dim}=\mathrm{G}\text{-}\mathrm{dim}$ , assume that R admits a dualizing complex. Then $\mathrm{Hid}_R N<\infty $ .

It is worth noting that we have $\mathrm{Rfd}_R M=\mathrm{depth} R-\mathrm{depth} M$ when M has finite (Gorenstein) projective dimension. From this perspective, the above theorems essentially provide affirmative answers to questions (i) and (ii). Holanda et al. discovered a certain compatibility between Ext modules and the tensor product with the canonical module. Our main idea is to treat the situation where higher Ext modules have finite length as the base case, thereby avoiding the complexities that arise when replacing the canonical module with the dualizing complex. There is also a result analogous to Theorem 1.1 for complete intersection dimension (see Proposition 4.6).

Our main results generalize several previous results. When the ring is Cohen–Macaulay, $\mathrm{Rfd}_R M=\mathrm{depth} R-\mathrm{depth} M$ holds, so Theorem 1.1 recovers [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Corollaries 4.7 and 4.8]. Theorem 1.1, when applied with $N=R$ or assuming the vanishing of certain Ext modules, can be seen as a generalization of [Reference Dey, Holanda and Miranda-Neto7, Proposition 13], [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Proposition 5.3], and [Reference Mendoza-Rubio and Jorge-Pérez11, Theorem 1.3] (see Remark 3.6).

What should be of interest next are concrete examples of pairs of modules M and N that satisfy the assumptions of our theorems. A typical pair of M and N for Theorem 1.1 is given by a (G-)perfect module M, that is, a module whose (Gorenstein) projective dimension is equal to the grade of M, and $N=R$ . This article provides several more nontrivial examples that are not (G-)perfect by considering extensions of (G-)perfect modules.

The organization of this article is as follows. Section 2 is devoted to the conventions and to explaining the main motivation behind our research. In Section 3, we provide proofs for all main results of this article, including Theorems 1.1 and 1.2. In Section 4, we give examples in which the hypotheses of the main results are satisfied and briefly discuss complete intersection dimension.

2 Notation and motivation

This section aims to introduce the conventions and explain the main motivation behind our research. We begin by defining the notation that will be used throughout the article.

Definition 2.1 Let R be a ring and $X=(\cdots \to X^n\xrightarrow {d^n} X^{n+1} \xrightarrow {d^{n+1}} \cdots )$ a complex of R-modules. We denote isomorphisms by the symbol $\cong $ and quasi-isomorphisms by the symbol $\simeq $ . We omit subscripts/superscripts if there is no ambiguity.

(1) Let M be an R-module. Denote by $E_R(M)$ the injective hull of M. Following [Reference Christensen, Foxby and Frankild3, Definition 2.1], we use the notation $\mathrm{Rfd}_R M$ for the large restricted flat dimension of an R-module M. If M is finitely generated over R, $\mathrm{Rfd}_R M$ is always finite and coincides with $\mathrm {sup}\{\mathrm{depth} R_{\mathfrak {p}}-\mathrm{depth} M_{\mathfrak {p}} \mid \mathfrak {p}\in \mathrm{Spec} R\}$ (see [Reference Christensen, Foxby and Frankild3, Theorem 2.4(b)]).

(2) For each integer m, $X[m]$ denotes the complex X shifted m degrees, defined by $X[m]^n=X^{n+m}$ . A complex X is called bounded to the left if $X^n=0$ for all $n\ll 0$ ; similarly, X is called bounded to the right if $X^n=0$ for all $n\gg 0$ . A complex which is bounded to the left and to the right is said to be bounded. The category of complexes of R-modules is denoted by $\mathcal {C}(R)$ . Given a full subcategory $\mathcal {A}(R)$ of $\mathcal {C}(R)$ , we write $\mathcal {A}^{\sqsubset }(R)$ , $\mathcal {A}^{\sqsupset }(R)$ , and $\mathcal {A}^{\square }(R)$ for the subcategories of $\mathcal {A}(R)$ consisting of complexes that are bounded to the left, bounded to the right, and bounded, respectively.

(3) Let M be an R-module. If there exists an exact complex X of projective R-modules such that $M\cong \mathrm{Im} d^0$ and the complex $\mathrm{Hom}_R(X,P)$ is exact for all projective R-modules P, then M is called Gorenstein projective. A finitely generated Gorenstein projective module is also called totally reflexive. If there exists an exact sequence X of flat R-modules such that $M\cong \mathrm{Im} d^0$ and the complex $I\otimes _R X$ is exact for all injective R-modules I, then M is said to be Gorenstein flat. If there exists an exact sequence X of injective R-modules such that $M\cong \mathrm{Im} d^0$ and the complex $\mathrm{Hom}_R(I,X)$ is exact for all injective R-modules I, then M is said to be Gorenstein injective. We denote by $\mathcal {C}_{\mathrm {P}}(R)$ , $\mathcal {C}_{\mathrm {F}}(R)$ , $\mathcal {C}_{\mathrm {I}}(R)$ , $\mathcal {G}(R)$ , $\mathcal {C}_{\mathrm {GP}}(R)$ , $\mathcal {C}_{\mathrm {GF}}(R)$ , and $\mathcal {C}_{\mathrm {GI}}(R)$ the full subcategories of $\mathcal {C}(R)$ consisting of complexes whose terms are projective, flat, injective, totally reflexive, Gorenstein projective, Gorenstein flat, and Gorenstein injective modules, respectively. Projective, flat, and injective modules are also Gorenstein projective, Gorenstein flat, and Gorenstein injective, respectively.

(4) Let Y and Z be complexes of R-modules. When Y and Z are quasi-isomorphic to complexes that are bounded to the right and to the left, respectively, we define the following homological dimensions:

• $\mathrm{pd}_R Y:=\mathrm {inf}\{\mathrm{sup}\{n \mid X^{-n}\ne 0\} \mid Y\simeq X\in \mathcal {C}^{\sqsupset }_{\mathrm {P}}(R)\}$ is the projective dimension;
• $\mathrm{fd}_R Y:=\mathrm {inf}\{\mathrm {sup}\{n \mid X^{-n}\ne 0\} \mid Y\simeq X\in \mathcal {C}^{\sqsupset }_{\mathrm {F}}(R)\}$ is the flat dimension;
• $\mathrm{id}_R Z:=\mathrm {inf}\{\mathrm{sup}\{n \mid X^{n}\ne 0\} \mid Z\simeq X\in \mathcal {C}^{\sqsubset }_{\mathrm {I}}(R)\}$ is the injective dimension;
• $\mathrm{G}\text{-}\mathrm{dim}_R Y:=\mathrm {inf}\{\mathrm {sup}\{n \mid X^{-n}\ne 0\} \mid Y\simeq X\in \mathcal {G}^{\sqsupset }(R)\}$ is the G-dimension;
• $\mathrm{Gpd}_R Y:=\mathrm {inf}\{\mathrm {sup}\{n \mid X^{-n}\ne 0\} \mid Y\simeq X\in \mathcal {C}^{\sqsupset }_{\mathrm {GP}}(R)\}$ is the Gorenstein projective dimension;
• $\mathrm{Gfd}_R Y:=\mathrm {inf}\{\mathrm {sup}\{n \mid X^{-n}\ne 0\} \mid Y\simeq X\in \mathcal {C}^{\sqsupset }_{\mathrm {GF}}(R)\}$ is the Gorenstein flat dimension;
• $\mathrm{Gid}_R Z:=\mathrm {inf}\{\mathrm {sup}\{n \mid X^{n}\ne 0\} \mid Z\simeq X\in \mathcal {C}^{\sqsubset }_{\mathrm {GI}}(R)\}$ is the Gorenstein injective dimension.

When Y and Z are R-modules and each homological dimension is zero, it is a projective, flat, injective, totally reflexive, Gorenstein projective, Gorenstein flat, or Gorenstein injective module, respectively.

(5) In this article, there are several assertions and arguments that hold for both the projective dimension and the G-dimension. In such cases, we collectively denote $\mathrm{pd}$ and $\mathrm{G}\text{-}\mathrm{dim}$ by “ $\mathrm{H}\text{-}\mathrm{dim}$ .” Depending on whether $\mathrm{H}\text{-}\mathrm{dim}$ is treated as the projective dimension or the G-dimension, we denote the injective dimension or the Gorenstein injective dimension by “ $\mathrm{Hid}$ ,” respectively. Throughout this article, whenever we say that D is a dualizing complex over R, we take D to be of the following form:

$$ \begin{align*}D^i\cong \bigoplus_{\substack{\mathfrak{p}\in\text{Spec} R \\ \text{dim} R/\mathfrak{p}=d-i}} E_R(R/\mathfrak{p}), \end{align*} $$ upper D Superscript i Baseline approximately equals circled plus Underscript StartLayout 1st Row German p element of Spec upper R 2nd Row dimension upper R divided by German p equals d minus i EndLayout Endscripts upper E Subscript upper R Baseline left parenthesis upper R divided by German p right parenthesis comma

for all $0\le i\le d$ , where $d=\mathrm{dim} R$ .

We note the following well-known facts as a remark and use them hereafter without further mention.

Remark 2.2 Let R be a ring, and W a complex of R-modules.

(1) Let $0\to X\to Y\to Z\to 0$ be a short exact sequence of complexes of R-modules. If at least one of W and Z is a complex of flat modules, then $0\to X\otimes _R W\to Y\otimes _R W\to Z\otimes _R W\to 0$ is exact (see [Reference Christensen, Foxby and Holm4, Example 2.4.17 and Corollary 5.4.3] for instance).

(2) Suppose that R admits a dualizing complex D and that W is quasi-isomorphic to a bounded complex of R-modules. It follows from [Reference Christensen, Frankild and Holm5, Equivalence 1.3, Theorems 4.1 and 4.4] that W has finite (Gorenstein) flat dimension if and only if $W\otimes _R^{\mathbf {L}} D$ has finite (Gorenstein) injective dimension, and that W has finite (Gorenstein) injective dimension if and only if $\mathbf{R}\mathrm{Hom}_R(D,W)$ has finite (Gorenstein) projective dimension. Also, as in [Reference Christensen, Frankild and Holm5, Theorem 4.4], the finiteness of the Gorenstein flat dimension and that of the Gorenstein projective dimension are equivalent. Moreover, if all homologies of W are finitely generated, then $\mathrm{pd}_R W=\mathrm{fd}_R W$ and $\mathrm{G}\text{-}\mathrm{dim}_R W=\mathrm{Gpd}_R W=\mathrm{Gfd}_R W$ hold.

(3) If W has finite projective dimension or finite G-dimension, then so does $W_{\mathfrak {p}}$ for all $\mathfrak {p}\in \mathrm{Spec} R$ . Moreover, if W is a finitely generated R-module, then for each $\mathfrak {p}\in \mathrm{Spec} R$ , the projective dimension and G-dimension of $W_{\mathfrak {p}}$ and W coincide with $\mathrm{depth} R_{\mathfrak {p}}-\mathrm{depth} W_{\mathfrak {p}}$ and $\mathrm{Rfd}_R W$ , respectively (see [Reference Bruns and Herzog1, Theorem 1.3.3] and [Reference Christensen2, Theorem (2.3.13)]). In the same case as in (2), by using the duality observed there, we see that the finiteness of the (Gorenstein) injective dimension is also preserved under localization.

Holanda et al. [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9] suggested the following questions for local rings.

Question 2.3 [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Section 5]

Let R be a local ring of depth t, and let M and N be nonzero finitely generated R-modules with $\mathrm{pd}_R (\mathrm{Ext}^i_R(M,N))<\infty $ for all $0\le i\le t$ . Is the condition $\mathrm{pd}_R M<\infty $ equivalent to $\mathrm{pd}_R N<\infty $ ?

Question 2.4 [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Question 5.1]

Let R be a local ring of depth t, and let M and N be nonzero finitely generated R-modules with $\mathrm{pd}_R N<\infty $ . Do the following hold?

(1) If $\mathrm{H}\text{-}\mathrm{dim}_R (\mathrm{Ext}^i_R(M,N))<\infty $ for all $0\le i\le t$ , then $\mathrm{H}\text{-}\mathrm{dim}_R M<\infty $ .
(2) If $\mathrm{H}\text{-}\mathrm{dim}_R (\mathrm{Hom}_R(M,N))<\infty $ and $\mathrm{Ext}^i_R(M,N)=0$ for all $1\le i\le t$ , then $\mathrm{H}\text{-}\mathrm{dim}_R M=0$ .

Question 2.5 [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Question 5.7]

Let R be a local ring of depth t, and let M and N be nonzero finitely generated R-modules with $\mathrm{pd}_R (\mathrm{Ext}^i_R(M,N))<\infty $ for all $0\le i\le t$ . Is the condition $\mathrm{id}_R M<\infty $ equivalent to $\mathrm{id}_R N<\infty $ ?

One may also consider formulating the hypothesis of the question as $0\le i\le t-\mathrm{depth} M$ instead of $0\le i\le t$ . The reason they raised these questions is that they proved Question 2.4 in the affirmative when R is Cohen–Macaulay. Even for a general R, Question 2.4 holds when $N=R$ ; indeed, part (2) follows from [Reference Dey, Holanda and Miranda-Neto7, Proposition 13], while part (1) follows from [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Corollary 5.4] (see also [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Proposition 5.3] for a stronger statement). Furthermore, Holanda et al. [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9], focusing on the equivalence of the finiteness of the projective dimensions of M and N under the condition $\mathrm{pd}_R (\mathrm{Ext}^i_R(M,N))<\infty $ for all $0\le i\le t$ , have resolved Question 2.3 in several cases (see [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Corollary 4.8 and Proposition 5.2]). Question 2.5 was posed dually to Question 2.3, in light of the quasi-isomorphism in 4.1(2). However, it is shown by [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Theorems 2.2 and 2.5] that the difficulty of this equivalence resides in Question 2.4(1) and in the following Question 2.6(1).

Question 2.6 Let R be a local ring of depth t, and let M and N be nonzero finitely generated R-modules with $\mathrm{id}_R M<\infty $ . Do the following hold?

(1) If $\mathrm{H}\text{-}\mathrm{dim}_R (\mathrm{Ext}^i_R(M,N))<\infty $ for all $0\le i\le t$ , then $\mathrm{Hid}_R N<\infty $ .
(2) If $\mathrm{H}\text{-}\mathrm{dim}_R (\mathrm{Hom}_R(M,N))<\infty $ and $\mathrm{Ext}^i_R(M,N)=0$ for all $1\le i\le t$ , then $\mathrm{Hid}_R N<\infty $ and $\mathrm{depth} M=t$ .

Since a necessary and sufficient condition for $\mathrm{H}\text{-}\mathrm{dim}_R M=0$ is that $\mathrm{H}\text{-}\mathrm{dim}_R M<\infty $ and $\mathrm{depth} M=t$ , and taking into account that depth is preserved under Foxby equivalence, it is seen that Question 2.6 is the dual version of Question 2.4. In the next section, we essentially resolve all of these questions (i.e., by slightly adjusting the degrees of the assumed Ext modules).

3 Main results

In this section, we investigate the equivalence of the finiteness of homological dimensions for pairs of finitely generated modules and their Ext modules and prove the main results of this article. To prove the main results of this article, we prepare several basic lemmas. The following is often referred to as Ischebeck’s formula.

Lemma 3.1 Let $(R,\mathfrak {m})$ be a local ring, M and N nonzero finitely generated R-modules, and D a dualizing complex of R.

(1) If $\mathrm{pd}_R N<\infty $ , then $\mathrm{dim} R-\mathrm{depth} M=\mathrm {sup}\{i \mid H^i(\mathbf{R}\mathrm{Hom}_R(M, N\otimes _R^{\mathbf {L}} D))\ne 0\}$ .
(2) If $\mathrm{id}_R M<\infty $ , then $\mathrm{depth} R-\mathrm{depth} M=\mathrm {sup}\{i \mid H^i(\mathbf{R}\mathrm{Hom}_R(\mathbf{R}\mathrm{Hom}_R(D,M), N))\ne 0\}$ .
(3) If $\mathrm{G}\text{-}\mathrm{dim}_R M<\infty $ and $\mathrm{pd}_R N<\infty $ , then $\mathrm{depth} R-\mathrm{depth} M=\mathrm {sup}\{i \mid \mathrm{Ext}_R^i(M,N)\ne 0\}$ .
(4) If $\mathrm{id}_R M<\infty $ and $\mathrm{Gid}_R N<\infty $ , then $\mathrm{depth} R-\mathrm{depth} M=\mathrm {sup}\{i \mid \mathrm{Ext}_R^i(M,N)\ne0\}$ .

Proof Statements (3) and (4) follow immediately from [Reference Christensen and Holm6, (ii) and (iii) of Theorem 6.3], respectively. Applying [Reference Christensen, Foxby and Frankild3, Theorem 4.13(c)] with $X=M$ and $Y=N\otimes _R^{\mathbf {L}} D$ yields (1). To prove (2), we may assume that M is nonzero and hence R is Cohen–Macaulay (see [Reference Bruns and Herzog1, Corollary 9.6.2 and Remark 9.6.4(a)]). Then (2) follows from [Reference Christensen, Foxby and Holm4, Proposition 16.4.34] and [Reference Christensen, Foxby and Frankild3, Theorem 4.14(b)].

The following theorem is one of the most important results in this article. The approach to the proof takes inspiration from the proof in [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Corollary 4.7]. However, when R is not Cohen–Macaulay, it is necessary to replace the canonical module with the dualizing complex, making it difficult to compute each Ext module exactly. Our main idea is to avoid this complexity by considering the case where the homology has finite length as the base case.

Theorem 3.2 Let $(R,\mathfrak {m})$ be a local ring, M and N nonzero finitely generated R-modules with $\mathrm{pd}_R N<\infty $ , and $r\ge \mathrm{depth} R-\mathrm{depth} M$ a nonnegative integer. Suppose that $\mathrm{H}\text{-}\mathrm{dim}_R\mathrm{Ext}^i_R(M,N)<\infty $ for all $0\le i\le r$ , and $\mathrm{Ext}^j_R(M,N)$ has finite length for all $j>r$ . Then $\mathrm{H}\text{-}\mathrm{dim}_R M<\infty $ .

Proof We may assume that R is complete. Set $d=\mathrm{dim} R$ and $t=\mathrm{depth} R$ . Let D be a dualizing complex of R, and let $F=(\cdots \to F^{-1}\to F^0\to 0)$ and $G=(0\to G^{-n} \to \cdots \to G^0\to 0)$ be minimal free resolutions of M and N. Note that $\mathrm{Hom}_R(F, G)=: H=(0\to H^{-n}\to H^{-n+1} \to \cdots )$ is a complex of finitely generated free R-modules. By [Reference Christensen, Foxby and Holm4, Theorem 4.5.10], there is a quasi-isomorphism

$$ \begin{align*}\mathbf{R}\mathrm{Hom}_R(M, N\otimes_R^{\mathbf{L}} D)\simeq \text{Hom}_R(F, G\otimes_R D)\cong \text{Hom}_R(F, G)\otimes_R D=H\otimes_R D. \end{align*} $$

Noting that $N\otimes _R^{\mathbf {L}} D\ne 0$ and it has finite injective dimension, if $\mathrm{Hid}_R (H\otimes _R D)<\infty $ , we get $\mathrm{H}\text{-}\mathrm{dim}_R M<\infty $ (see [Reference Christensen, Foxby and Holm4, Proposition 16.4.32] and the proof of [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Theorem 2.5(2)]). We prove $\mathrm{Hid}_R (H\otimes _R D)<\infty $ .

Take a homomorphism $\Phi : P\to H$ of complexes such that $H^i(\Phi ):H^i(P) \to H^i(H)\cong \mathrm{Ext}^i_R(M,N)$ is an isomorphism for every $i\le r$ , where $P=(\cdots \to P^{r-1}\to P^r\to 0)$ is a complex of finitely generated free R-modules. Let C be the mapping cone of $\Phi $ . The short exact sequence

Commutative diagram: H = (... → H^i-1 → H^i → ...) maps down to C = (... → C^i-1 = H^i-1 ⊕ P^i → C^i = H^i ⊕ P^i+1 → ...) which maps down to P[1] = (... → P^i → P^i+1 → ...).

of complexes of finitely generated free R-modules induces a short exact sequence $0\to H\otimes _R D\to C\otimes _R D\to (P\otimes _R D)[1]\to 0$ . By assumption, we obtain $\mathrm{H}\text{-}\mathrm{dim}_R P<~\infty $ and thus $\mathrm{Hid}_R (P\otimes _R D)<\infty $ . If $C\otimes _R D\simeq 0$ , then $H\otimes _R D$ is quasi-isomorphic to $P\otimes _R D$ and we reach $\mathrm{Hid}_R (H\otimes _R D)<\infty $ . In the remainder of the proof, we show that $H^i(C\otimes _R D)=0$ for all integers i.

As $H^j(D)=0=H^k(P)$ for all $j>d-t$ and $k>r$ , we have $H^i(P\otimes _R D)=H^i(P\otimes _R^{\mathbf {L}} D)=0$ for all $i>d-t+r$ . It follows from Lemma 3.1(1) that $H^i(H\otimes _R D)\cong H^i(\mathbf{R}\mathrm{Hom}_R(M, N\otimes _R^{\mathbf {L}} D))=0$ for every $i>d-t+r$ , which means that $H^i(C\otimes _R D)=0$ for all $i>d-t+r$ .

We denote the differential map $C^i\to C^{i+1}$ by $f^i$ . By the construction of $\Phi $ and C, if $i>r$ , then $H^i(C)\cong H^i(H)\cong \mathrm{Ext}^i_R(M,N)$ and it has finite length; otherwise, $H^i(C)=0$ . For all non-maximal prime ideals $\mathfrak {p}$ of R, $C_{\mathfrak {p}}$ is exact. Since the complex H is bounded to the left, for $i\ll 0$ , $f^i$ is equal to the differential map $P^{i+1}\to P^{i+2}$ , and thus $\mathrm{Cok} f^i$ is totally reflexive. By this and [Reference Christensen2, Theorem (1.4.8)], for every integer i, $\mathrm{Cok} f^i_{\mathfrak {p}}$ has finite G-dimension and thus totally reflexive, which implies that the $R_{\mathfrak {p}}$ -dual of $C_{\mathfrak {p}}$ is exact. It follows from [Reference Christensen2, Lemma (5.1.10)] that $C\otimes _R E(R/\mathfrak {p})=C_{\mathfrak {p}}\otimes _{R_{\mathfrak {p}}} E(R/\mathfrak {p})\simeq 0$ . We obtain $C\otimes _R D^i\simeq 0$ for $i\ne d$ . For all $0\le i\le d-1$ , the natural short exact sequence

Commutative diagram showing D(i+1) defined as a sequence from 0 to D^d, D(i) defined as a sequence from 0 to D^d, and a third row of 0 to D^i to 0, connected by vertical arrows and equality signs.

implies $C\otimes _R D=C\otimes _R D(0)\simeq C\otimes _R D(1)\simeq \cdots \simeq C\otimes _R D(d)$ , and hence $H^i(C\otimes _R D)\cong H^{i-d}(C\otimes _R D^d)$ for all integers i. Since $H^i(C)=0$ for all $i\le r$ , $(\cdots \to C^{r-1} \to C^{r}\to C^{r+1}\to 0)$ is a free resolution of $\mathrm{Cok} f^r$ . For $i\ll 0$ , $\mathrm{Cok} f^i$ is totally reflexive, and thus $\mathrm{Cok} f^r$ has finite G-dimension. It follows from Lemma 3.1(3) that $\mathrm{Ext}_R^i(\mathrm{Cok} f^r, R)=0$ for every $i>t$ . We see that $(\cdots \to C^{r-t-1}\otimes _R D^d \to C^{r-t}\otimes _R D^d \to C^{r-t+1}\otimes _R D^d)$ is exact since its Matlis dual is exact. Thus, $H^i(C\otimes _R D)\cong H^{i-d}(C\otimes _R D^d)=0$ for all $i\le d-t+r$ .

The following corollary is the most general statement of the results we have obtained. Refer to Remark 3.4 for the reason why conditions (1) and (2) need to be considered for each maximal ideal.

Corollary 3.3 Let R be a ring, and let M and N be finitely generated R-modules. Then $\mathrm{H}\text{-}\mathrm{dim}_R M<\infty $ if for every maximal ideal $\mathfrak {m}$ of R, at least one of the following holds:

(1) $\mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {m}}} M_{\mathfrak {m}}<\infty .$
(2) $N_{\mathfrak {m}}\ne 0$ , $\mathrm{pd}_{R_{\mathfrak {m}}} N_{\mathfrak {m}}<\infty $ , and $\mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {m}}} \mathrm{Ext}^i_{R_{\mathfrak {m}}}(M_{\mathfrak {m}},N_{\mathfrak {m}})<\infty $ for all $0\le i\le \mathrm{Rfd}_R M$ .

Proof Assume that $\mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {m}}} M_{\mathfrak {m}}=\infty $ for some maximal ideal $\mathfrak {m}$ of R. By assumption, $\mathfrak {m}$ belongs to $S:=\{\mathfrak {p}\in \mathrm{Spec} R\mid \mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {p}}} M_{\mathfrak {p}}=\infty , N_{\mathfrak {p}}\ne 0, \mathrm{pd}_{R_{\mathfrak {p}}} N_{\mathfrak {p}}<\infty , \mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {p}}} \mathrm{Ext}^i_{R_{\mathfrak {p}}}(M_{\mathfrak {p}},N_{\mathfrak {p}})<\infty $ for all $0\le i\le \mathrm{Rfd}_R M\}$ , so we can take a minimal element $\mathfrak {p}$ of S. Let $\mathfrak {q}$ be a prime ideal of R with $\mathfrak {q}\subsetneq \mathfrak {p}$ . Since $\mathrm{pd}_{R_{\mathfrak {p}}} N_{\mathfrak {p}}<\infty $ and $\mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {p}}} \mathrm{Ext}^i_{R_{\mathfrak {p}}}(M_{\mathfrak {p}},N_{\mathfrak {p}})<\infty $ for all $0\le i\le \mathrm{Rfd}_R M$ , we have $\mathrm{pd}_{R_{\mathfrak {q}}} N_{\mathfrak {q}}<\infty $ and $\mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {q}}} \mathrm{Ext}^i_{R_{\mathfrak {q}}}(M_{\mathfrak {q}},N_{\mathfrak {q}})<\infty $ for all $0\le i\le \mathrm{Rfd}_R M$ . However, $\mathfrak {q}$ is not in S by the minimality of $\mathfrak {p}$ in S. Thus, $\mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {q}}} M_{\mathfrak {q}}<\infty $ or $N_{\mathfrak {q}}=0$ holds. Lemma 3.1(3) forces $\mathrm{Ext}^j_{R_{\mathfrak {q}}}(M_{\mathfrak {q}},N_{\mathfrak {q}})=0$ for all $j>\mathrm{Rfd}_R M$ . Therefore, $\mathrm{Ext}^j_{R_{\mathfrak {p}}}(M_{\mathfrak {p}},N_{\mathfrak {p}})$ has finite length for all $j>\mathrm{Rfd}_R M$ . Theorem 3.2 concludes $\mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {p}}} M_{\mathfrak {p}}<\infty $ . This contradicts $\mathfrak {p}\in S$ . It follows from the above that $\mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {m}}} M_{\mathfrak {m}}<\infty $ for all maximal ideals $\mathfrak {m}$ of R, which induces $\mathrm{H}\text{-}\mathrm{dim}_R M<\infty $ .

Remark 3.4 The assumptions of Corollary 3.3 concerning localized modules are seen to be essential in the following sense. If R is not a local ring, then for two finitely generated modules M and N, it is possible that $\mathrm{Ext}^i_R(M, N)=0$ for all i. This does not depend on the homological dimensions of M and N. For example, let S be a ring, $\mathfrak {m}$ and $\mathfrak {n}$ distinct maximal ideals of S, $R=S/\mathfrak {m}^2\mathfrak {n}$ , $M=S/\mathfrak {m,}$ and $N=S/\mathfrak {n}$ . Considering the localization at every maximal ideal, we obtain $\mathrm{pd}_R N<\infty $ , and $\mathrm{Ext}^i_R(M, N)=0$ for all i. On the other hand, if $R_{\mathfrak {m}}$ is not a field (i.e., $\mathfrak {m} S_{\mathfrak {m}}\ne 0$ ), then $\mathrm{pd}_R M=\infty $ holds.

The following corollary provides an affirmative answer to Question 2.4(1) under essential assumptions and is an immediate consequence of Corollary 3.3. Indeed, when $\mathrm{H}\text{-}\mathrm{dim}_R M<\infty $ , we have $\mathrm{depth} R-\mathrm{depth} M=\mathrm{H}\text{-}\mathrm{dim}_R M=\mathrm{Rfd}_R M$ , so the hypothesis of Corollary 3.5 is weaker than that of Question 2.4(1).

Corollary 3.5 Let R be a local ring, and M and N nonzero finitely generated R-modules with $\mathrm{pd}_R N<\infty $ and $\mathrm{H}\text{-}\mathrm{dim}_R\mathrm{Ext}^i_R(M,N)<\infty $ for all $0\le i\le \mathrm{Rfd}_R M$ . Then $\mathrm{H}\text{-}\mathrm{dim}_R M<\infty $ .

Lemma 3.1(3) shows that an affirmative answer to Question 2.4(1) yields an affirmative answer to 2.4(2). In particular, Corollary 3.5 also gives an “essentially” affirmative answer to Question 2.4(2).

Question 2.4 had already been resolved in the Cohen–Macaulay case by [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Corollary 4.7]. We recover their results by combining our results with [Reference Christensen, Foxby and Frankild3, Theorem 3.4].

Remark 3.6 (1) Combining the results obtained so far, [Reference Mendoza-Rubio and Jorge-Pérez11, (2) and (3) of Theorem 1.3] are recovered. In fact, if R is Cohen–Macaulay or M is locally totally reflexive on the punctured spectrum, then $\mathrm{Rfd}_R M=\mathrm {max}\{\mathrm{depth} R-\mathrm{depth} M, 0\}$ . Therefore, by combining Corollary 3.5 and Lemma 3.1(3), it is easy to see that a generalized statement of [Reference Mendoza-Rubio and Jorge-Pérez11, (2) and (3) of Theorem 1.3], where “R” is replaced by “a module N of finite projective dimension,” holds true. We also obtain similar generalizations of [Reference Dey, Holanda and Miranda-Neto7, Proposition 13] and [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Proposition 5.3] by imposing assumptions on more Ext modules. The remaining problem is whether Corollary 3.5 can be proved by replacing $\mathrm{Rfd}_R M$ with $\mathrm{depth} R-\mathrm{depth} M$ .

(2) Corollary 3.5 also holds for *local rings in the sense of [Reference Bruns and Herzog1, Definition 1.5.13] and for graded modules over them. Indeed, as in [Reference Bruns and Herzog1, Proposition 1.5.15], the problems can be reduced to the local case. The same observation applies to Corollary 3.9, which appears at the end of this section and is the dual version of Corollary 3.5.

From this point onward, we consider, dually to the preceding discussion, the finiteness of the (Gorenstein) injective dimension of modules. The following is the (Gorenstein) injective dimension analog of Theorem 3.2 and is among the main results of this article. Our approach, which closely examines the structure of the dualizing complex, remains effective even when the ring is Cohen–Macaulay.

Theorem 3.7 Let $(R,\mathfrak {m})$ be a local ring, M and N nonzero finitely generated R-modules with $\mathrm{id}_R M<\infty $ , and $r\ge \mathrm{depth} R-\mathrm{depth} M$ a nonnegative integer. Suppose that $\mathrm{H}\text{-}\mathrm{dim}_R\mathrm{Ext}^i_R(M,N)<\infty $ for all $0\le i\le r$ , and $\mathrm{Ext}^j_R(M,N)$ has finite length for all $j>r$ . In the case $\mathrm{H}\text{-}\mathrm{dim}=\mathrm{G}\text{-}\mathrm{dim}$ , assume further that R admits a dualizing complex. Then $\mathrm{Hid}_R N<\infty $ .

Proof When $\mathrm{H}\text{-}\mathrm{dim}=\mathrm{pd}$ , we can replace R with its completion, so we may assume that R admits a dualizing complex D in order to prove the theorem. Put $d=\mathrm{dim} R$ . Since M is a nonzero finitely generated R-module with finite injective dimension, R is Cohen–Macaulay (see [Reference Bruns and Herzog1, Corollary 9.6.2 and Remark 9.6.4(a)]). Let $\omega $ be the canonical module of R, and let $I=(0\to I^0\to \cdots \to I^d \to 0)$ and $J=(0\to J^0\to J^1\to \cdots )$ be minimal injective resolutions of M and N. Note that $\mathrm{Hom}_R(I,J)=:K=(0\to K^{-d} \to K^{-d+1}\to \cdots )$ is a complex of flat R-modules. By [Reference Christensen, Foxby and Holm4, Theorem 4.5.13], we have a quasi-isomorphism

$$ \begin{align*}\mathbf{R}\mathrm{Hom}_R(\mathbf{R}\mathrm{Hom}_R(\omega, M), N) &=\text{Hom}_R(\text{Hom}_R(\omega, I), J) \\ &\cong \omega\otimes_R \text{Hom}_R(I,J)=\omega\otimes_R K. \end{align*} $$

Noting that $\mathbf{R}\mathrm{Hom}_R(\omega , M)\ne 0$ and it has finite projective dimension, if $\mathrm{Hid}_R (\omega \otimes _R K)<\infty $ , we get $\mathrm{Hid}_R N<\infty $ (see [Reference Christensen, Foxby and Holm4, Proposition 16.4.32] and the proof of [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Theorem 2.2(2)]). We prove $\mathrm{Hid}_R (\omega \otimes _R K)<\infty $ .

Take a homomorphism $\Phi : P\to K$ of complexes such that $H^i(\Phi ):H^i(P) \to H^i(K)\cong \mathrm{Ext}^i_R(M,N)$ is an isomorphism for every $i\le r$ , where $P=(\cdots \to P^{r-1}\to P^r\to 0)$ is a complex of finitely generated free R-modules. Let C be the mapping cone of $\Phi $ . The short exact sequence

Commutative diagram of chain complexes. K = (... → K^i-1 → K^i → ...) maps down to C = (... → C^i-1 = K^i-1 ⊕ P^i → C^i = K^i ⊕ P^i+1 → ...) which maps down to P[1] = (... → P^i → P^i+1 → ...).

of complexes of flat R-modules induces a short exact sequence $0\to \omega \otimes _R K\to \omega \otimes _R C\to (\omega \otimes _R P)[1]\to 0$ . By assumption, we obtain $\mathrm{H}\text{-}\mathrm{dim}_R P<\infty $ and thus $\mathrm{Hid}_R (\omega \otimes _R P)<\infty $ . If $\omega \otimes _R C\simeq 0$ , then $\omega \otimes _R K$ is quasi-isomorphic to $\omega \otimes _R P$ . In the remainder of the proof, we show that $H^i(\omega \otimes _R C)=0$ for all i.

It follows from Lemma 3.1(2) that $H^i(\omega \otimes _R K)\cong H^i(\mathbf{R}\mathrm{Hom}_R(\mathbf{R}\mathrm{Hom}_R(\omega , M),$ $ N))=0=H^i(\omega \otimes _R P)$ for every $i>r$ , which means that $H^i(\omega \otimes _R C)=0$ for all $i>r$ .

Thanks to [Reference Christensen2, Proposition A.11], tensoring C with the mapping cone of a quasi-isomorphism from $\omega $ to D preserves acyclicity. So there is a quasi-isomorphism $\omega \otimes _R C\simeq D\otimes _R C$ . We denote the differential map $C^i\to C^{i+1}$ by $g^i$ . By the construction of $\Phi $ and C, if $i>r$ , then $H^i(C)\cong H^i(K)\cong \mathrm{Ext}^i_R(M,N)$ and it has finite length; otherwise, $H^i(C)=0$ . For all non-maximal prime ideals $\mathfrak {p}$ of R, $C_{\mathfrak {p}}$ is exact. For $i\ll 0$ , $g^i$ is equal to the differential map $P^{i+1}\to P^{i+2}$ , and thus $\mathrm{Cok} g^i$ is totally reflexive. By this, $\mathrm{Cok} g^j_{\mathfrak {p}}$ has finite Gorenstein flat dimension for every integer j. It follows from [Reference Christensen2, Corollary 5.2.15] that $\mathrm{Gfd}_{R_{\mathfrak {p}}}(\mathrm{Cok} g^{j+d}_{\mathfrak {p}}) \le d$ and $0=\mathrm{Tor}_{d+1}^{R_{\mathfrak {p}}}(E(R/\mathfrak {p}), \mathrm{Cok} g^{j+d}_{\mathfrak {p}})=\mathrm{Tor}_1^{R_{\mathfrak {p}}}(E(R/\mathfrak {p}), \mathrm{Cok} g^{j}_{\mathfrak {p}})$ for all integers j, which means that $E(R/\mathfrak {p})\otimes _R C=E(R/\mathfrak {p})\otimes _{R_{\mathfrak {p}}} C_{\mathfrak {p}}\simeq 0$ . An argument similar to the proof of Theorem 3.2 shows that $H^i(\omega \otimes _R C)\cong H^i(D\otimes _R C)\cong H^{i-d}(D^d\otimes _R C)$ for all integers i.

For the same reason as in the previous paragraph, we see that $\mathrm{Cok} g^r$ has finite Gorenstein flat dimension at most d since $H^i(C)=0$ for $i\le r$ . Note that $(\cdots \to C^r\to C^{r+1}\to 0)$ is a flat resolution of $\mathrm{Cok} g^r$ . Again by [Reference Christensen2, Corollary 5.2.15], we get $H^i(\omega \otimes _R C)\cong H^{i-d}(D^d\otimes _R C)\cong \mathrm{Tor}_{r+1-i+d}^R(D^d, \mathrm{Cok} g^r)=0$ for $i \le r$ . The proof is now complete.

The corollary below corresponds to Corollary 3.3. However, a module whose localizations at all prime ideals have finite (Gorenstein) injective dimension does not necessarily have finite (Gorenstein) injective dimension itself. This is due to the fact that the (Gorenstein) injective dimension of a finitely generated module over a local ring coincides with the depth of the ring (see [Reference Bruns and Herzog1, Theorem 3.1.17] and [Reference Christensen2, Theorem 6.2.15]). For example, the injective dimension of an infinite-dimensional Gorenstein ring is infinite.

Corollary 3.8 Let R be a ring, and let M and N be finitely generated R-modules. In the case $\mathrm{H}\text{-}\mathrm{dim}=\mathrm{G}\text{-}\mathrm{dim}$ , assume that $R_{\mathfrak {p}}$ admits a dualizing complex for all $\mathfrak {p}\in \mathrm{Spec} R$ . Then $\mathrm{Hid}_{R_{\mathfrak {p}}} N_{\mathfrak {p}}<\infty $ for all prime ideals $\mathfrak {p}$ of R if at least one of the following holds for every maximal ideal $\mathfrak {m}$ of R:

(1) $\mathrm{Hid}_{R_{\mathfrak {m}}} N_{\mathfrak {m}}<\infty .$
(2) $M_{\mathfrak {m}}\ne 0$ , $\mathrm{id}_{R_{\mathfrak {m}}} M_{\mathfrak {m}}<\infty $ , and $\mathrm{H}\text{-}\mathrm{dim}_{R_{\mathfrak {m}}} \mathrm{Ext}^i_{R_{\mathfrak {m}}}(M_{\mathfrak {m}},N_{\mathfrak {m}})<\infty $ for all $0\le i\le \mathrm{Rfd}_R M$ .

In addition, if R is finite-dimensional, and it admits a dualizing complex when $\mathrm{H}\text{-}\mathrm{dim}=\mathrm{G}\text{-}\mathrm{dim}$ , then $\mathrm{Hid}_R N<\infty $ .

Proof The proof of the former part is the same as that of Corollary 3.3, using Lemma 3.1(4) and Theorem 3.7 in place of Lemma 3.1(3) and Theorem 3.2, respectively. In the case $\mathrm{H}\text{-}\mathrm{dim}=\mathrm{pd}$ , if R is finite-dimensional, then $\mathrm{id}_{R_{\mathfrak {p}}} N_{\mathfrak {p}}\le \mathrm{dim} R$ for all $\mathfrak {p}\in \mathrm{Spec} R$ , which means $\mathrm{id}_R N<\infty $ . Assume that R admits a dualizing complex D when $\mathrm{H}\text{-}\mathrm{dim}=\mathrm{G}\text{-}\mathrm{dim}$ . Then $\mathrm{G}\text{-}\mathrm{dim}_{R_{\mathfrak {p}}} (\mathbf{R}\mathrm{Hom}_R(D, N)_{\mathfrak {p}})<\infty $ for all $\mathfrak {p}\in \mathrm{Spec} R$ , and hence $\mathrm{G}\text{-}\mathrm{dim}_R (\mathbf{R}\mathrm{Hom}_R(D, N))<\infty $ , which gives $\mathrm{Gid}_R N<\infty $ .

The dual version of Corollary 3.5 can be derived in the same way. It states that Question 2.6 (and hence Question 2.5) has an affirmative answer.

Corollary 3.9 Let R be a local ring, and M and N nonzero finitely generated R-modules with $\mathrm{id}_R M<\infty $ and $\mathrm{H}\text{-}\mathrm{dim}_R\mathrm{Ext}^i_R(M,N)<\infty $ for all $0\le i\le \mathrm{depth} R-\mathrm{depth} M$ . In the case $\mathrm{H}\text{-}\mathrm{dim}=\mathrm{G}\text{-}\mathrm{dim}$ , assume that R admits a dualizing complex. Then $\mathrm{Hid}_R N<\infty $ .

Proof As in the proof of Theorem 3.7, R is Cohen–Macaulay. The claim follows from [Reference Christensen, Foxby and Frankild3, Theorem 3.4] and Corollary 3.8.

4 Examples

In the remainder of this article, we present examples related to the results obtained herein. We wish to construct examples confirming the existence of finitely generated modules M and N such that

• $\mathrm{pd}_R N<\infty $ , $\mathrm{H}\text{-}\mathrm{dim}_R M<\infty $ , and $\mathrm{H}\text{-}\mathrm{dim}_R(\mathrm{Ext}_R^i(M,N))<\infty $ for all integers i, or
• $\mathrm{id}_R M<\infty $ , $\mathrm{Hid}_R N<\infty $ , and $\mathrm{H}\text{-}\mathrm{dim}_R(\mathrm{Ext}_R^i(M,N))<\infty $ for all integers i.

Representative examples are provided in [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Section 6]. In this section, we slightly generalize them and provide several nontrivial examples concretely.

4.1 Let R be a ring and let M and N be finitely generated R-modules.

(1) Recall that $\mathrm {inf}\{i \mid \mathrm{Ext}^i_R(M,R)\ne 0 \}$ is called the grade of M and is denoted by $\mathrm{grade}_R M$ . The (nonzero) module M is said to be H-perfect of grade g if $g=\mathrm{grade}_R M=\mathrm{H}\text{-}\mathrm{dim}_R M$ . If M is $\mathrm{H}$ -perfect of grade g, then $\mathrm{H}\text{-}\mathrm{dim}_R (\mathrm{Ext}^g_R(M,R))=g$ and $\mathrm{Ext}^i_R(M,R)=0$ for $i\ne g$ (see [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Example 6.2] for instance). (It is simply called perfect when $\mathrm{H}\text{-}\mathrm{dim}=\mathrm{pd}$ .) For an R-regular sequence $x_1,\ldots ,x_g$ , the quotient $R/(x_1,\ldots ,x_g)$ is perfect of grade g.

(2) Suppose that R is a Cohen–Macaulay ring with canonical module $\omega $ , and that for all integers i, $\mathrm{H}\text{-}\mathrm{dim}_R ({\mathrm{Ext}}^i_R(M,N))<\infty $ , $\widehat {\mathrm{H}}-\mathrm{dim}_R (M)<\infty $ , and $\overline {\mathrm{H}}-\mathrm{dim}_R (N)<\infty $ . (Here, $\widehat {\mathrm{H}}-\mathrm{dim}$ and $\overline {\mathrm{H}}-\mathrm{dim}$ also refer to either the projective dimension or the G-dimension, and the reason for changing the notation is to emphasize that it is harmless for each of them to independently refer to either the projective dimension or the G-dimension.) Then

$$ \begin{align*} \mathbf{R}\mathrm{Hom}_R(M\otimes_R \omega, N\otimes_R \omega)&\simeq \mathbf{R}\mathrm{Hom}_R(M\otimes_R^{\mathbf{L}} \omega, N\otimes_R^{\mathbf{L}} \omega)\\ &\simeq \mathbf{R}\mathrm{Hom}_R(M, \mathbf{R}\mathrm{Hom}_R(\omega, N\otimes_R^{\mathbf{L}} \omega)) \\ &\simeq \mathbf{R}\mathrm{Hom}_R(M,N). \end{align*} $$

StartLayout 1st Row 1st Column bold upper R hom Subscript upper R Baseline left parenthesis upper M circled times Subscript upper R Baseline omega comma upper N circled times Subscript upper R Baseline omega right parenthesis 2nd Column asymptotically equals bold upper R hom Subscript upper R Baseline left parenthesis upper M circled times Subscript upper R Superscript bold upper L Baseline omega comma upper N circled times Subscript upper R Superscript bold upper L Baseline omega right parenthesis 2nd Row 1st Column Blank 2nd Column asymptotically equals bold upper R hom Subscript upper R Baseline left parenthesis upper M comma bold upper R hom Subscript upper R Baseline left parenthesis omega comma upper N circled times Subscript upper R Superscript bold upper L Baseline omega right parenthesis right parenthesis 3rd Row 1st Column Blank 2nd Column asymptotically equals bold upper R hom Subscript upper R Baseline left parenthesis upper M comma upper N right parenthesis period EndLayout

We have ${\widehat {\mathrm{H}}\mathrm{id}}_R (M\otimes _R \omega )<\infty $ , ${\overline {\mathrm{H}}\mathrm{id}}_R (N\otimes _R \omega )<\infty $ , and $\mathrm{H}\text{-}\mathrm{dim}_R (\mathrm{Ext}^i_R(M\otimes _R \omega , N\otimes _R \omega ))<\infty $ for all i. Conversely, if ${\widehat {\mathrm{H}}\mathrm{id}}_R (M)<\infty $ and ${\overline {\mathrm{H}}\mathrm{id}}_R (N)<\infty $ , then there is a quasi-isomorphism

$$ \begin{align*} & \mathbf{R}\mathrm{Hom}_R(\text{Hom}_R(\omega, M), \text{Hom}_R(\omega, N))\\ &\quad \simeq \mathbf{R}\mathrm{Hom}_R(\mathbf{R}\mathrm{Hom}_R(\omega, M), \mathbf{R}\mathrm{Hom}_R(\omega, N))\\ &\quad \simeq \mathbf{R}\mathrm{Hom}_R(\mathbf{R}\mathrm{Hom}_R(\omega, M)\otimes_R^{\mathbf{L}}\omega, N)\\ &\quad \simeq \mathbf{R}\mathrm{Hom}_R(M,N), \end{align*} $$

StartLayout 1st Row 1st Column Blank 2nd Column bold upper R hom Subscript upper R Baseline left parenthesis hom Subscript upper R Baseline left parenthesis omega comma upper M right parenthesis comma hom Subscript upper R Baseline left parenthesis omega comma upper N right parenthesis right parenthesis 2nd Row 1st Column Blank 2nd Column asymptotically equals bold upper R hom Subscript upper R Baseline left parenthesis bold upper R hom Subscript upper R Baseline left parenthesis omega comma upper M right parenthesis comma bold upper R hom Subscript upper R Baseline left parenthesis omega comma upper N right parenthesis right parenthesis 3rd Row 1st Column Blank 2nd Column asymptotically equals bold upper R hom Subscript upper R Baseline left parenthesis bold upper R hom Subscript upper R Baseline left parenthesis omega comma upper M right parenthesis circled times Subscript upper R Superscript bold upper L Baseline omega comma upper N right parenthesis 4th Row 1st Column Blank 2nd Column asymptotically equals bold upper R hom Subscript upper R Baseline left parenthesis upper M comma upper N right parenthesis comma EndLayout

with $\widehat {\mathrm{H}}-\mathrm{dim}_R (\mathrm{Hom}_R(\omega , M))<\infty $ and $\overline {\mathrm{H}}-\mathrm{dim}_R (\mathrm{Hom}_R(\omega , N))<\infty $ . Therefore, constructing examples about (Gorenstein) injective dimension is essentially no different from constructing examples about (Gorenstein) projective dimension. Hence, in this section, we concentrate only on examples related to (Gorenstein) projective dimension.

Note that when M and N do not necessarily have finite projective or injective dimension, $M\otimes _R^{\mathbf {L}} \omega $ and $\mathbf{R}\mathrm{Hom}_R(\omega , N)$ are not even guaranteed to be bounded complexes. This is the reason why Corollaries 3.5 and 3.9 cannot be easily deduced from [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Theorems 2.2 and 2.5].

Now, let us explicitly construct the examples. The examples of quotient rings of polynomial rings appearing below can be treated in the same way even if they are quotients of formal power series rings or localized at the maximal ideal generated by the variables, that is, the following covers the local case.

Example 4.2 Let R be a ring, $x,y$ an R-regular sequence in R, and $z\in R$ a nonzero divisor on R. There is a short exact sequence $0\to R/(x,y) \xrightarrow {z} R/(xz,yz) \to R/(z) \to 0$ . Since $R/(x,y)$ is perfect of grade 2 and $R/(z)$ is perfect of grade 1, we have

$$ \begin{align*}\text{Ext}^i_R(R/(xz,yz), R)= \begin{cases} \text{Ext}^i_R(R/(z), R)\cong R/(z) & (i=1), \\ \text{Ext}^i_R(R/(x,y), R)\cong R/(x,y) & (i=2), \\ 0 & (\mathrm{otherwise}). \end{cases} \end{align*} $$

Ext Subscript upper R Superscript i Baseline left parenthesis upper R divided by left parenthesis x z comma y z right parenthesis comma upper R right parenthesis equals StartLayout Enlarged left brace 1st Row 1st Column Ext Subscript upper R Superscript i Baseline left parenthesis upper R divided by left parenthesis z right parenthesis comma upper R right parenthesis approximately equals upper R divided by left parenthesis z right parenthesis 2nd Column left parenthesis i equals 1 right parenthesis comma 2nd Row 1st Column Ext Subscript upper R Superscript i Baseline left parenthesis upper R divided by left parenthesis x comma y right parenthesis comma upper R right parenthesis approximately equals upper R divided by left parenthesis x comma y right parenthesis 2nd Column left parenthesis i equals 2 right parenthesis comma 3rd Row 1st Column 0 2nd Column left parenthesis otherwise right parenthesis period EndLayout

We see that $0 \to R \xrightarrow {\tiny \begin {pmatrix} -y \\ x \end {pmatrix}} R^2 \xrightarrow {\tiny \begin {pmatrix} xz &\! \! \! \! yz \end {pmatrix}} R \to 0$ is a free resolution of $R/(xz,yz)$ (see [Reference Bruns and Herzog1, Theorem 1.4.13] for instance). In particular, $\mathrm{pd}_R (R/(xz,yz))<\infty $ and $\mathrm{pd}_R(\mathrm{Ext}_R^i(R/(xz,yz),R))<\infty $ for all integers i.

Example 4.3 Let R be a ring, and $x,y$ an R-regular sequence in R. Choose nonzero divisors $a, b$ in $(x,y)R$ and elements $c,d\in R$ . We write $a=a_1 x+a_2 y$ and $b=b_1 x+b_2 y$ for some $a_1,a_2,b_1,b_2\in R$ . Then there exists a commutative diagram

Commutative diagram of modules R, R^2, R^3, and R^4 with horizontal and vertical maps represented by matrices including Psi. The diagram shows exact sequences connecting these modules.

with exact rows and columns (see [Reference Bruns and Herzog1, Theorem 1.4.13] for instance). The exact sequence

$$ \begin{align*}0\to R/(a)\oplus R/(b) \to M\to R/(x,y)\to 0\end{align*} $$

is induced by the Snake Lemma, where $M=\mathrm{Cok}\psi $ . Note that $R/(a)\oplus R/(b)$ is perfect of grade 1 and $R/(x,y)$ is perfect of grade 2. We obtain the following commutative diagram with exact rows by computing the connected homomorphism:

Commutative diagram. Top row: 0 → Ext sub R super 1 (M, R) → Ext sub R super 1 (R/(a) ⊕ R/(b), R) → Ext sub R super 2 (R/(x, y), R) → Ext sub R super 2 (M, R) → 0. Bottom row: R/(a) ⊕ R/(b) → R/(x, y) → R/(x, y, c, d) → 0.

where $\phi (r+(a), s+(b))=cr+ds+(x,y)$ for $r,s\in R$ . Hence, $\mathrm{H}\text{-}\mathrm{dim}_R (R/(x,y,c,d))<\infty $ if and only if $\mathrm{H}\text{-}\mathrm{dim}_R (\mathrm{Ext}_R^i(M,R))<\infty $ for every i.

Let us consider the quotient $R=k[x,y,z,v,w]/(xy,zw,zv)$ of a polynomial ring over a field k and an R-regular sequence $x+y, z+w$ . Then R is of dimension 3 and is not Cohen–Macaulay. Put $S=R/(x+y, z+w)R\cong k[x,z,v]/(x^2,z^2,zv)$ and $X=S/xS\cong R/(x+y, z+w, x)R\cong k[z,v]/(z^2,zv)$ . It is seen that $(\cdots \xrightarrow {x} S \xrightarrow {x} S \xrightarrow {x} S \to X\to 0)$ is exact, and thus $\mathrm{G}\text{-}\mathrm{dim}_R X=\mathrm{G}\text{-}\mathrm{dim}_S X+2=2$ . By the previous paragraph, the cokernel M of the following homomorphism satisfies the condition stated in the third line at the beginning of this section, where $\mathrm{H}\text{-}\mathrm{dim}=\mathrm{G}\text{-}\mathrm{dim}$ and $N=R$ :

$$ \begin{align*}R^4 \xrightarrow{\tiny \begin{pmatrix} x+y& \! \! \! \! 0& \! \! \! \! 0& \! \! \! \! -x \\ 0& \! \! \! \! z+w& \! \! \! \! x& \! \! \! \! 0 \\ 0& \! \! \! \! 0& \! \! \! \! x+y& \! \! \! \! z+w\end{pmatrix}} R^3 \to M \to 0. \end{align*} $$ upper R Superscript 4 Baseline right arrow Overscript Start 3 By 4 Matrix 1st Row 1st Column x plus y 2nd Column 0 3rd Column 0 4th Column negative x 2nd Row 1st Column 0 2nd Column z plus w 3rd Column x 4th Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column x plus y 4th Column z plus w EndMatrix Endscripts upper R cubed right arrow upper M right arrow 0 period

Note that M has finite projective dimension, and thus so does $\mathbf{R}\mathrm{Hom}_R(M,R)$ , but its second Ext module $X=\mathrm{Ext}^2_R(M,R)$ satisfies $\mathrm{pd}_R X=\mathrm{pd}_S X+2=\infty $ since $x+y, z+w\notin (x,y,z,v,w)^2R$ .

We provide an example of a module that has infinite projective dimension but finite G-dimension.

Example 4.4 Let $S=k[x,y,z,v,w]/(v^2, vx+vw, xy+z^2)$ be a quotient of a polynomial ring over a field k and $R=S/(y^2-wz)S$ . Note that $xy+z^2$ is a prime element of $k[x,y,z]$ and that

$$ \begin{align*}(v^2, vx+vw) &=(v^2,x+w)\cap (v) \ \mathrm{and} \ (v^2, vx+vw, xy+z^2)\\ &=(v, xy+z^2)\cap (v^2, x+w, xy+z^2) \end{align*} $$

are the primary decompositions of $(v^2, vx+vw)$ and $(v^2, vx+vw, xy+z^2)$ , respectively. Then $xy+z^2, y^2-wz, w$ is a regular sequence on $k[x,y,z,v,w]/(v^2, vx+vw)$ , and R is a non-Cohen–Macaulay ring of dimension 2. There are exact sequences

$$ \begin{align*} & \cdots S^2 \xrightarrow{\tiny \begin{pmatrix} y& \! \! \! \! \! \!z \\ -z& \! \! \! \! \! \! x \end{pmatrix}} S^2 \xrightarrow{\tiny \begin{pmatrix} x& \! \! \! \! \! \! -z \\ z& \! \! \! \! \! \! y \end{pmatrix}} S^2 \xrightarrow{\tiny \begin{pmatrix} y& \! \! \! \! \! \!z \\ -z& \! \! \! \! \! \! x \end{pmatrix}} S^2 \xrightarrow{\tiny \begin{pmatrix} z& \! \! \! \! \! \! y \end{pmatrix}} S \ \mathrm{and } \\ &\cdots R^2 \xrightarrow{\tiny \begin{pmatrix} w& \! \! \! \! \! \! y \\ y& \! \! \! \! \! \! z \end{pmatrix}} R^2 \xrightarrow{\tiny \begin{pmatrix} z& \! \! \! \! \! \! -y \\ -y& \! \! \! \! \! \! w \end{pmatrix}} R^2 \xrightarrow{\tiny \begin{pmatrix} w& \! \! \! \! \! \! y \\ y& \! \! \! \! \! \! z \end{pmatrix}} R^2 \xrightarrow{\tiny \begin{pmatrix} -y& \! \! \! \! \! \! w \end{pmatrix}} R. \end{align*} $$

StartLayout 1st Row 1st Column Blank 2nd Column midline horizontal ellipsis upper S squared right arrow Overscript Start 2 By 2 Matrix 1st Row 1st Column y 2nd Column z 2nd Row 1st Column negative z 2nd Column x EndMatrix Endscripts upper S squared right arrow Overscript Start 2 By 2 Matrix 1st Row 1st Column x 2nd Column negative z 2nd Row 1st Column z 2nd Column y EndMatrix Endscripts upper S squared right arrow Overscript Start 2 By 2 Matrix 1st Row 1st Column y 2nd Column z 2nd Row 1st Column negative z 2nd Column x EndMatrix Endscripts upper S squared right arrow Overscript Start 1 By 2 Matrix 1st Row 1st Column z 2nd Column y EndMatrix Endscripts upper S and 2nd Row 1st Column Blank 2nd Column midline horizontal ellipsis upper R squared right arrow Overscript Start 2 By 2 Matrix 1st Row 1st Column w 2nd Column y 2nd Row 1st Column y 2nd Column z EndMatrix Endscripts upper R squared right arrow Overscript Start 2 By 2 Matrix 1st Row 1st Column z 2nd Column negative y 2nd Row 1st Column negative y 2nd Column w EndMatrix Endscripts upper R squared right arrow Overscript Start 2 By 2 Matrix 1st Row 1st Column w 2nd Column y 2nd Row 1st Column y 2nd Column z EndMatrix Endscripts upper R squared right arrow Overscript Start 1 By 2 Matrix 1st Row 1st Column negative y 2nd Column w EndMatrix Endscripts upper R period EndLayout

By this, the S-module $S/(y, z)S$ is $\mathrm{G}$ -perfect of grade 1 and the R-module $R/(y, w)R$ is $\mathrm{G}$ -perfect of grade 1. It follows from [Reference Christensen2, Proposition 1.5.3] that $R/(y, z)R\cong S/(y, z)S$ is totally reflexive as an R-module.

The rows and columns of the following commutative diagram are exact:

Commutative diagram with three rows of exact sequences. Top row: 0 to R^2 to R^4 to R^2 to 0. Middle row: 0 to R to R^2 to R to 0. Bottom row: 0 to R/(y, w)R to M to R/(y, z)R to 0. Vertical maps connect rows, ending in 0.

where $M=\mathrm{Cok}\psi $ . Indeed, by diagram chasing, if $\phi (a+(y, w)R)=0$ for $a\in R$ , then there are $b,c,d,e\in R$ such that $a=by+cw-dvz$ and $dy+ez=0$ . As $d\in (zR:_R y) \subseteq (x,y,z)R$ , we have $a=by+cw-dvz\in (y, w)R$ , which means that $\phi $ is injective.

By the $\mathrm{G}$ -perfectness of $R/(y, w)R$ and $R/(y, z)R$ , we have

$$ \begin{align*}\text{Ext}^i_R(M, R)= \begin{cases} \text{Hom}_R(R/(y, z), R) & (i=0), \\ \text{Ext}^1_R(R/(y, w), R) & (i=1), \\ 0 & (\mathrm{otherwise}). \end{cases} \end{align*} $$

Ext Subscript upper R Superscript i Baseline left parenthesis upper M comma upper R right parenthesis equals StartLayout Enlarged left brace 1st Row 1st Column hom Subscript upper R Baseline left parenthesis upper R divided by left parenthesis y comma z right parenthesis comma upper R right parenthesis 2nd Column left parenthesis i equals 0 right parenthesis comma 2nd Row 1st Column Ext Subscript upper R Superscript 1 Baseline left parenthesis upper R divided by left parenthesis y comma w right parenthesis comma upper R right parenthesis 2nd Column left parenthesis i equals 1 right parenthesis comma 3rd Row 1st Column 0 2nd Column left parenthesis otherwise right parenthesis period EndLayout

The short exact sequence $0\to R\xrightarrow {w} R\to R/(w)R\to 0$ induces a long exact sequence

$$ \begin{align*} 0&\to \text{Hom}_R(M, R)\xrightarrow{w} \text{Hom}_R(M, R) \to \text{Hom}_R(M, R/(w)R) \\ &\to \text{Ext}^1_R(M, R)\xrightarrow{w} \text{Ext}^1_R(M, R) \to \text{Ext}^1_R(M, R/(w)R) \to \text{Ext}^2_R(M, R)=0. \end{align*} $$

The multiplication by w on $\mathrm{Ext}^1_R(M, R)$ is the zero map since $\mathrm{Ext}^1_R(M, R)\cong \mathrm{Ext}^1_R(R/(y, w), R)$ . Then

• $0\to \mathrm{Hom}_R(M, R)\xrightarrow {w} \mathrm{Hom}_R(M, R) \to \mathrm{Hom}_R(M, R/(w)R) \to \mathrm{Ext}^1_R(M, R)\to 0$ is exact,
• $\mathrm{Ext}^1_R(M, R/(w)R) \cong \mathrm{Ext}^1_R(M, R)$ ,
• $\mathrm{Ext}^i_R(M, R/(w)R)=0$ for $i\ge 2$ .

As $\mathrm{Hom}_R(M, R)\cong \mathrm{Hom}_R(R/(y, z), R)$ and $\mathrm{Ext}^1_R(M, R)\cong \mathrm{Ext}^1_R(R/(y, w), R)$ have finite G-dimension, so do $\mathrm{Hom}_R(M, R/(w)R)$ and $\mathrm{Ext}^1_R(M, R/(w)R)$ . In particular, M and $N=R/(w)R$ satisfy the condition stated in the third line at the beginning of this section.

Finally, we note that the R-module M has infinite projective dimension. Indeed, if $\mathrm{pd}_R M<\infty $ , then $\mathrm{pd}_S M<\infty $ since $\mathrm{pd}_S R=1$ . As $w, y$ is an S-regular sequence, $S/(y, w)S\cong R/(y, w)R$ has finite projective dimension over S. The exact sequence $0\to R/(y, w)R \to M \to R/(y, z)R \to 0$ yields that $S/(y, z)S\cong R/(y, z)R$ has finite projective dimension over S. However, the complex

$$ \begin{align*}\cdots S^2 \xrightarrow{\tiny \begin{pmatrix} x& \! \! \! \! -z \\ z& \! \! \! \! y \end{pmatrix}} S^2 \xrightarrow{\tiny \begin{pmatrix} y& \! \! \! \! z \\ -z& \! \! \! \! x \end{pmatrix}} S^2 \xrightarrow{\tiny \begin{pmatrix} x& \! \! \! \! -z \\ z& \! \! \! \! y \end{pmatrix}} S^2 \xrightarrow{\tiny \begin{pmatrix} y& \! \! \! \! z \\ -z& \! \! \! \! x \end{pmatrix}} S^2 \xrightarrow{\tiny \begin{pmatrix} z& \! \! \! \! y \end{pmatrix}} S \to 0 \end{align*} $$

is an infinite minimal free resolution of $S/(y, z)S$ , contradicting the finiteness of $\mathrm{pd}_S S/(y,z)S$ .

4.1 Complete intersection dimension

To close, we consider complete intersection dimension briefly.

Definition 4.5 Let R be a local ring. A quasi-deformation of R is a pair of homomorphisms $R\to R' \leftarrow S$ of local rings, where $R\to R'$ is flat and $R' \leftarrow S$ is surjective with the kernel generated by a regular sequence. For a finitely generated R-module M, the complete intersection dimension of M is defined to be $\mathrm{CI-dim}_R M:=\mathrm { inf}\{\mathrm{pd}_S (M\otimes _R R')-\mathrm{pd}_S R' \mid R\to R' \leftarrow S$ is a quasi-deformation of $R\}$ .

Below is the complete intersection dimension version of Corollary 3.5.

Proposition 4.6 Let R be a local ring, and M and N nonzero finitely generated R-modules with $\mathrm{pd}_R N<\infty $ and $\mathrm{CI-dim}_R (\bigoplus _{i=0}^{\mathrm{Rfd}_R M}\mathrm{Ext}^i_R(M,N))<\infty $ . Then $\mathrm{CI-dim}_R M<\infty $ .

We prepare one lemma and prove Proposition 4.6.

Lemma 4.7 Let $(R,\mathfrak {m})$ be a local ring, $x\in \mathfrak {m}$ a nonzero divisor on R, and M and N nonzero finitely generated $R/xR$ -modules. Suppose that S is a ring and R is an S-algebra. If $\mathrm{H}\text{-}\mathrm{dim}_S(\mathbf{R}\mathrm{Hom}_{R/xR}(M,N))<\infty $ , then $\mathrm{H}\text{-}\mathrm{dim}_S(\mathbf{R}\mathrm{Hom}_{R}(M,N))<\infty $ .

Proof Let $F^{R/xR}_M$ be an $(R/xR)$ -free resolution of M and $F^R_M$ an R-free resolution of M. Similarly to the proof of [Reference Kimura, Lyle and Soto Levins10, Lemma 4.2], there is a short exact sequence

$$ \begin{align*}0\to F^{R/xR}_M[-1] \to R/xR\otimes_R F^{R}_M \to F^{R/xR}_M\to 0 \end{align*} $$

of complexes of free $(R/xR)$ -modules. This induces a short exact sequence

$$ \begin{align*}0\to \text{Hom}_{R/xR}(F^{R/xR}_M, N) &\to \text{Hom}_{R/xR}(R/xR\otimes_RF^{R}_M,N)\\ &\to \text{Hom}_{R/xR}(F^{R/xR}_M,N)[1] \to 0. \end{align*} $$

StartLayout 1st Row 1st Column 0 right arrow hom Subscript upper R divided by x upper R Baseline left parenthesis upper F Subscript upper M Superscript upper R divided by x upper R Baseline comma upper N right parenthesis 2nd Column right arrow hom Subscript upper R divided by x upper R Baseline left parenthesis upper R divided by x upper R circled times Subscript upper R Baseline upper F Subscript upper M Superscript upper R Baseline comma upper N right parenthesis 2nd Row 1st Column Blank 2nd Column right arrow hom Subscript upper R divided by x upper R Baseline left parenthesis upper F Subscript upper M Superscript upper R divided by x upper R Baseline comma upper N right parenthesis left bracket 1 right bracket right arrow 0 period EndLayout

From this, the claim holds since $\mathbf{R}\mathrm{Hom}_{R/xR}(M,N)=\mathrm{Hom}_{R/xR}(F^{R/xR}_M, N)$ and $\mathbf{R}\mathrm{Hom}_{R}(M,N)=\mathrm{Hom}_{R}(F^{R}_M,N)\cong \mathrm{Hom}_{R/xR}(R/xR\otimes _RF^{R}_M,N)$ .

Proof of Proposition 4.6

Put $r=\mathrm{Rfd}_R M$ . Suppose that there is a quasi-deformation $R\to R' \leftarrow S$ such that $\mathrm{pd}_S(\mathrm{Ext}^i_{R'}(M',N'))<\infty $ for all $0\le i\le r$ , where $(-)'=(-)\otimes _R R'$ . We prove $\mathrm{pd}_S M'<\infty $ .

For all $0\le i\le r$ , $\mathrm{Ext}^i_R(M,N)$ has finite G-dimension over R since $\mathrm{pd}_S(\mathrm{Ext}^i_{R'}(M',N'))<\infty $ . By Corollary 3.5, we have $\mathrm{G}\text{-}\mathrm{dim}_R M=r$ . Lemma 3.1(3) asserts that $\mathrm{Ext}^i_R(M,N)=0$ for all $i>r$ . Then $\mathrm{Ext}^i_{R'}(M',N')=0$ for all $i>r$ as $R'$ is flat over R, which means $\mathrm{pd}_S (\mathbf{R}\mathrm{Hom}_{R'}(M',N'))<\infty $ . Lemma 4.7 implies $\mathrm{pd}_S (\mathbf{R}\mathrm{Hom}_{S}(M',N'))<\infty $ . As $\mathrm{G}\text{-}\mathrm{dim}_{R'} M'<\infty $ and $\mathrm{pd}_{R'} N'<\infty $ , we obtain $\mathrm{G}\text{-}\mathrm{dim}_S M'<\infty $ and $\mathrm{pd}_S N'<\infty $ . Let $\widehat {S}$ be the completion of S and D the dualizing complex of $\widehat {S}$ . Since $\mathrm{G}\text{-}\mathrm{dim}_{\widehat {S}} (M'\otimes _S \widehat {S})<\infty $ and $\mathrm{pd}_{\widehat {S}} (N'\otimes _S \widehat {S})<\infty $ , $(M'\otimes _S \widehat {S})\otimes _{\widehat {S}}^{\mathbf {L}} D$ is quasi-isomorphic to a bounded complex of finitely generated $\widehat {S}$ -modules and we get

$$ \begin{align*} \mathbf{R}\mathrm{Hom}_{\widehat{S}}(M'\otimes_S \widehat{S},N'\otimes_S \widehat{S})&\simeq \mathbf{R}\mathrm{Hom}_{\widehat{S}}(M'\otimes_S \widehat{S},\mathbf{R}\mathrm{Hom}_{\widehat{S}}(D, (N'\otimes_S \widehat{S})\otimes_{\widehat{S}}^{\mathbf{L}} D)) \\ &\simeq \mathbf{R}\mathrm{Hom}_{\widehat{S}}((M'\otimes_S \widehat{S})\otimes_{\widehat{S}}^{\mathbf{L}} D, (N'\otimes_S \widehat{S})\otimes_{\widehat{S}}^{\mathbf{L}} D). \end{align*} $$

StartLayout 1st Row 1st Column bold upper R hom Subscript ModifyingAbove upper S With caret Baseline left parenthesis upper M prime circled times Subscript upper S Baseline ModifyingAbove upper S With caret comma upper N prime circled times Subscript upper S Baseline ModifyingAbove upper S With caret right parenthesis 2nd Column asymptotically equals bold upper R hom Subscript ModifyingAbove upper S With caret Baseline left parenthesis upper M prime circled times Subscript upper S Baseline ModifyingAbove upper S With caret comma bold upper R hom Subscript ModifyingAbove upper S With caret Baseline left parenthesis upper D comma left parenthesis upper N prime circled times Subscript upper S Baseline ModifyingAbove upper S With caret right parenthesis circled times Subscript ModifyingAbove upper S With caret Superscript bold upper L Baseline upper D right parenthesis right parenthesis 2nd Row 1st Column Blank 2nd Column asymptotically equals bold upper R hom Subscript ModifyingAbove upper S With caret Baseline left parenthesis left parenthesis upper M prime circled times Subscript upper S Baseline ModifyingAbove upper S With caret right parenthesis circled times Subscript ModifyingAbove upper S With caret Superscript bold upper L Baseline upper D comma left parenthesis upper N prime circled times Subscript upper S Baseline ModifyingAbove upper S With caret right parenthesis circled times Subscript ModifyingAbove upper S With caret Superscript bold upper L Baseline upper D right parenthesis period EndLayout

Note that $\mathrm{pd}_{\widehat {S}} (\mathbf{R}\mathrm{Hom}_{\widehat {S}}(M'\otimes _S \widehat {S},N'\otimes _S \widehat {S}))<\infty $ and $\mathrm{id}_{\widehat {S}} ((N'\otimes _S \widehat {S})\otimes _{\widehat {S}}^{\mathbf {L}} D)<\infty $ . It follows from [Reference Christensen, Foxby and Holm4, Proposition 16.4.34] that $\mathrm{id}_{\widehat {S}} ((M'\otimes _S \widehat {S})\otimes _{\widehat {S}}^{\mathbf {L}} D)<\infty $ , and hence $\mathrm{pd}_{\widehat {S}} (M'\otimes _S \widehat {S})<\infty $ . So $\mathrm{pd}_S M'<\infty $ .

Acknowledgements

The author would like to thank his supervisor Ryo Takahashi for valuable suggestions and comments. The author thanks Yuki Mifune for pointing out where a fact used in this article is stated explicitly, Victor Daniel Mendoza Rubio for pointing out several errors in a previous version of this article, and Yuya Otake for doing both. The author is also grateful to the anonymous referee for carefully reading the manuscript and for many helpful suggestions that improved and simplified the exposition.

Footnotes

The author was partly supported by a Grant-in-Aid for JSPS Fellows Grant Number 23KJ1117.

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Article contents

Finiteness of homological dimensions of Ext modules

Abstract

Keywords

MSC classification

Information

1 Introduction

Theorem 1.1 [Corollary 3.5]

Theorem 1.2 [Corollary 3.9]

2 Notation and motivation

Question 2.3 [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Section 5]

Question 2.4 [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Question 5.1]

Question 2.5 [Reference Holanda, Jorge-Pérez and Mendoza-Rubio9, Question 5.7]

3 Main results

4 Examples

4.1 Complete intersection dimension

Proof of Proposition 4.6

Acknowledgements

Footnotes

References

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