On modules whose dual is of finite Gorenstein dimension: The $k$ -torsion, homological criteria, and applications in modules of differentials of order $n$

Victor D. Mendoza-Rubio and Victor H. Jorge-Pérez Universidade de São Paulo - ICMC, Caixa Postal 668, 13560-970, São Carlos-SP, Brazil [email protected] Universidade de São Paulo - ICMC, Caixa Postal 668, 13560-970, São Carlos-SP, Brazil [email protected]

Abstract.

In this paper, we aim to obtain some results under the condition that the dual of a module has finite Gorenstein dimension. In this direction we derive results involving vanishing of Ext as well as the freeness or totally reflexivity of modules. For instance, we provide a generalization of a celebrated theorem by Evans and Griffith, obtain criteria for the totally reflexivity of modules over Cohen-Macaulay rings as well as of locally totally reflexive modules on the punctured spectrum, and recover a result by Araya. Moreover, we prove that the Auslander-Reiten conjecture holds true for all finitely generated modules $M$ over a Noetherian local ring $R$ such that $\operatorname{G-dim}_{R}(\operatorname{Hom}_{R}(M,R))<\infty$ and $\operatorname{pd}_{R}(\operatorname{Hom}_{R}(M,M))<\infty$ . Additionally, we derive Gorenstein criteria under the condition that the dual of certain modules is of finite Gorenstein dimension. Furthermore, we explore some applications in the theory of the modules of Kähler differentials of order $n\geq 1$ , specifically concerning the $k$ -torsionfreeness of these modules and the Herzog-Vasconcelos’s conjecture.

2020 Mathematics Subject Classification:

Primary: 13D45, 13D07, 13C10, 13C14; Secondary: 13D05, 13D02, 13H10, 14B15, 14B05.

1. Introduction

The Gorenstein dimension is a well-developed topic in commutative algebra, as evidenced by various works such as [5, 14, 44, 50, 18, 52]. Despite the extensive research in this area, there are relatively few results concerning the condition of the Gorenstein dimension of the dual of a certains modules being finite. The aim of this work is to obtain results in this direction, including some that involve vanishing of Ext and homological criteria.

We prove the following theorem, which is a generalization of a result of Evans and Griffith ([15, Theorem 3.8]) and is an analogous version of [14, Proposion 2.4] when $C=R$ .

1.1 Theorem.

Let $R$ be a Noetherian local ring and $k$ be a non-negative integer. Let $M$ be a finitely generated non-zero $R$ -module such that $M^{\ast}$ has locally finite Gorenstein dimension on $\widetilde{X}^{k-1}(R)$ . Then:

$M$ is $k$ -torsionfree $\Longleftrightarrow$ $M$ is $k$ -syzygy $\Longleftrightarrow$ $M$ satisfies $(\widetilde{S}_{k})$ .

This theorem provides an answer to [37, Question 1.1]. Additionally, it allows us to recover one of the results described in the main theorem of [52].

1.2 Corollary ([52, Theorem 5.8(1)]).

Let $R$ be a Noetherian local ring, $M$ be a finitely generated $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ and $k$ be a non-negative integer. If $M$ satisfies $(\widetilde{S}_{k})$ , then $M$ is $k$ -torsionfree.

As an application of Corollary 1.2 and [50, Theorem 42], we characterize under certain conditions and in terms of vanishing of Ext modules, the total reflexivity of a finitely generated $R$ -module $M$ with $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . We obtain the following.

1.3 Theorem.

Let $R$ be a Noetherian local ring of depth $t$ , and let $M$ be a finitely generated $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then:

(1)

$M$ satisfies $(\widetilde{S}_{t})$ if and only if $M$ is totally reflexive if and only if $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq t$ .
(2)

Let $R$ be a Cohen-Macaulay local ring and $M$ is non-zero. If $n$ is a non-negative integer such that $\operatorname{depth}_{R}(M)\geq n$ and $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq t-n,$ then $M$ is totally reflexive.
(3)

If $M$ is non-zero and locally totally reflexive on the punctured spectrum of $R$ , then $\operatorname{depth}_{R}(M)\leq t$ with equality if and only if $M$ is totally reflexive.

Additionally, part of this work is motivated by the celebrated Auslander-Reiten conjecture ([6]), which states the following.

1.4 Conjecture.

(Auslander-Reiten) Let $R$ be a Noetherian local ring and let $M$ be a finitely generated $R$ -module. If $\operatorname{Ext}_{R}^{i}(M,R)=\operatorname{Ext}_{R}^{i}(M,M)=0$ for all $i>0$ , then $M$ is free.

This conjecture is still open but is true in several cases. For a list of some of them we refer the reader to [30], it introduction and to the recent work [12]. In [2], Araya proved the following result that provided an positive answer to Auslander-Reiten conjecture, and that implies that it is valid over normal Gorenstein local rings of dimension at least two.

1.5 Theorem ([2, Corollary 10]).

Let $R$ be a Gorenstein local ring of dimension $d\geq 2$ . Let $M$ be a maximal Cohen-Macaulay $R$ -module. If $M$ is locally free on the punctured spectrum of $R$ and $\operatorname{Ext}_{R}^{d-1}(M,M)=0$ , then $M$ is free.

Subsequently, results concerning the context of Gorenstein rings that recover this theorem have appeared. For instance, the last theorem in [42, Section 1] and [3, Corollary 1.6]. It is important to mention that in a more general context, [30, Proposition 2.10(4)] and [8, Proposition 1.7] are improvements of these results. In terms of the finiteness of the Gorenstein dimension of the dual of a module, we achieve the following results that recover the last theorem in [42, Section 1], [3, Corollary 1.6] and of course Theorem 1.5.

1.6 Theorem.

Let $R$ be a Noetherian local ring of depth $t\geq 1$ and let $M$ be a finitely generated $R$ -module. Suppose that $M$ is locally free on the spectrum punctured such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . If $0pt(M)\geq t$ and $\operatorname{Ext}_{R}^{t-1}(M,M)=0$ , then $M$ is free.

1.7 Theorem.

Let $R$ be a Noetherian ring satisfying $(S_{1})$ and $X$ a subset of $\operatorname{Spec}(R)$ containing $X^{1}(R)$ . Let

$s:=\inf\{\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\mid\mathfrak{p}\in\operatorname{Spec}(R)\backslash X\}$ and $t:=\sup\{\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\mid\mathfrak{p}\in\operatorname{Spec}(R)\backslash X\}$ .

Let $M$ be a finitely generated $R$ -module and suppose that $M$ is locally free on $X$ and that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . If $\operatorname{Ext}_{R}^{i}(M,R)=\operatorname{Ext}_{R}^{j}(M,M)=0$ for all $1\leq i\leq t$ and $s-1\leq j\leq t-1$ , then $M$ is projective.

On the other hand, in [12, Corollary 6.9(2)], Dey and Ghosh proved that the Auslander-Reiten conjecture holds for finitely generated $R$ -modules $M$ such that $\operatorname{G-dim}_{R}(M)<\infty$ and $\operatorname{pd}_{R}(\operatorname{Hom}_{R}(M,M))<\infty$ . Motivated by this, we analyze if the conjecture holds if we consider $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ instead of $\operatorname{G-dim}_{R}(M)<\infty$ . Consequently, we get the following result that affirms that it holds.

1.8 Theorem.

Let $R$ be a Noetherian local ring of depth $t$ , and let $M$ be a finitely generated $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then the following conditions are equivalent:

(1)

$M$ is free.
(2)

$\operatorname{Hom}_{R}(M,M)$ is free and $\operatorname{Ext}_{R}^{j}(M,M)=0$ for all $1\leq j\leq t-1$ .
(3)

$\operatorname{Hom}_{R}(M,M)$ has finite projective dimension and $\operatorname{Ext}_{R}^{i}(M,R)=\operatorname{Ext}_{R}^{j}(M,M)=0$ for all $1\leq i\leq t$ and $1\leq j\leq t-1$ .

Additionally, we also derive some Gorenstein criteria involving the condition that the dual of certain $R$ -modules are of finite Gorenstein dimension. We do this essentially by discussing of some questions about that. We obtain the following.

1.9 Theorem.

Let $R$ be a Noetherian local ring.Then $R$ is Gorenstein in each one of the following cases:

(1)
$R$ be a Cohen-Macaulay local ring and there exists an $R$ -module $M$ such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ and it satisfies one of the following:
1. (1.1)
  
  $M$ is Cohen-Macaulay of positive rank and $2\mu(M)>e(R)\operatorname{rank}(M)$ .
2. (2.2)
  
  $M$ is an Ulrich module and $M^{\ast}\not=0$ .
(2)

$\operatorname{G-dim}_{R}(M^{\ast})<\infty$ for all finitely generated $R$ -modules $M$ such that $M^{\ast}\not=0$ .
(3)

$R$ has positive depth, and every finitely generated $R$ -modules $M$ whose dual $M^{\ast}$ is non-zero and of finite Gorenstein dimension, is also of finite Gorenstein dimension.
(4)

$R$ has depth at most one and $\operatorname{G-dim}_{R}(\mathfrak{m}^{\ast})<\infty$ .

In the last part of this paper, we focus on the Kähler differential modules $\Omega_{X/Y}^{(n)}$ and the derivation modules $\mathcal{D}er_{Y}^{n}(X)$ defined over locally Noetherian schemes and analytic spaces, where $n\geq 1$ is an integer. The motivation for studying these modules arises from the Zariski-Lipman conjecture [33], which states: Let $X$ be a complex variety such that the tangent sheaf $\mathcal{T}_{X}:=\mathcal{H}om_{\mathcal{O}_{X}}(\Omega_{X}^{(1)},\mathcal{O}_{X})$ is locally free. Then $X$ is smooth. This conjecture, proposed about 50 years ago, is still open, although it has been resolved in some special cases, see, for instance, [39, Section 4]. Following the same direction as the Zariski-Lipman Conjecture, there is also a homological version of it, independently proposed by Herzog and Vasconcelos, which predicts that if ${\rm pd}_{R}{\rm Der}_{k}(R)\,<\,\infty$ , then ${\rm Der}_{k}(R)$ is free. Unlike the former, this problem seems to be widely open (with exceptions in specific cases; see, for example, [26, Section 4] and [40]). There is also another conjecture called the Strong Zariski-Lipman Conjecture: If ${\rm pd}_{R}{\rm Der}_{k}(R)<\infty$ , then $R$ is regular. Due to these conjectures, the following questions arise:

1.10 Questions.

(i)

(Generalizations of Herzog-Vasconcelos’s conjecture (GHVC)) For some integer $n\geq 1$ , we have ${\rm pd}_{\mathcal{O}_{X}}{\mathcal{D}er}_{Y}^{n}(X)\,<\,\infty$ . Under what assumptions on $X$ and $Y$ , and for which values of $n$ , does this imply that ${\mathcal{D}er}_{Y}^{n}(X)$ is locally free?
(ii)

(Strong generalizations of Herzog-Vasconcelos’s conjecture (SGHVC)) For some integer $n\geq 1$ , we have ${\rm pd}_{\mathcal{O}_{X}}{\mathcal{D}er}_{Y}^{n}(X)\,<\,\infty$ . Under what assumptions on $X$ and $Y$ , and for which values of $n$ , does this imply that $X$ is locally smooth?

It is worth noting that these questions and others appear in the works of Graf [22] (in the case of algebraic varieties and analytic spaces) and in the case of affine schemes in [35]. Historically, to answer these questions, it is necessary to study the concepts of torsion, cotorsion, torsion-free, and reflexive of the Kähler differential module (see, for example, a summary in [39, p. 133] and [22]). For this reason,in the Sections 8 and 9, we obtain some direct applications connecting these concepts with the concepts of $k$ -torsion-free and $k$ -syzygy. Consequently, we also obtain some special answers for GHVC and SGHVC.

The organization of this paper is as follows. In Section 2 we provide definitions, notations, and some results that are considered in this paper. In Section 3 we prove Theorem 1.1 and explore some of its consequences. In Section 4, we prove Theorem 1.3, and in Section 5 we provide freeness criteria including Theorem 1.7 and 1.8. In Section 6, we discuss some questions about of the Gorensteiness of a ring involving Gorenstein dimension of the dual of modules, and prove Theorem 1.9. In Section 7, we introduce notation and review established results regarding Kähler differential modules and differential modules. Then, in Section 8, we investigate the properties of the Kähler differential module $\Omega_{X/Y}^{(n)}$ , examining its local freeness, reflexivity, $k$ -torsion-freeness, and $k$ -syzygy properties, specially when $k=1,2$ . Finally, in the last section, we provide a partial response to the SGHV and SGHV conjectures.

2. Setup and preliminaries

Throughout this paper, $R$ is a Noetherian ring, and all $R$ -modules are considered to be finitely generated. Whenever $R$ is local, we denote by $\mathfrak{m}$ and $k$ its maximal ideal and its residual field respectively.

2.1 Definition.

Let $M,N$ be two $R$ -modules, and $k$ be a non-negative integer.

(1)

For an $R$ -module $M$ , we set $M^{\ast}=\operatorname{Hom}_{R}(M,R)$ and $M^{\ast\ast}=(M^{\ast})^{\ast}$ .

(2)

Let

\boldsymbol{P}:\cdots\longrightarrow P_{i}\stackrel{{\scriptstyle\varphi_{i}}}{{\longrightarrow}}P_{i-1}\longrightarrow\cdots\longrightarrow P_{1}\stackrel{{\scriptstyle\varphi_{1}}}{{\longrightarrow}}P_{0}\stackrel{{\scriptstyle\varphi_{0}}}{{\longrightarrow}}M\longrightarrow 0

be a projective resolution of resolution of $M$ .

(2.a)

For $k\geq 1$ , the $k$ -syzygy of $M$ , denoted by $\Omega^{k}(M)$ , is defined as the kernel of the map $\varphi_{k-1}$ . When $k=0$ , we set $\Omega^{0}(M)=M$ .
(2.b)

The Auslander Transpose of $M$ , denoted by $\operatorname{Tr}(M)$ , is defined as the cokernel of the induced map $\varphi_{1}^{\ast}:P_{0}^{\ast}\to P_{1}^{\ast}.$ When $k\geq 1$ , we set

$\mathcal{T}_{k}(M)=\operatorname{Tr}(\Omega^{k-1}(M)).$

(3)

We write $M\approx N$ if there exist projective $R$ -modules $F,G$ such that $M\oplus F\cong N\oplus G$ .
(4)

$M$ is a $k$ -syzygy if there exists an exact sequence

$0\rightarrow M\rightarrow P_{k-1}\rightarrow P_{k-2}\rightarrow\cdots\rightarrow P_{1}\rightarrow P_{0},$

where each $P_{i}$ is of projective dimension.
(5)

$M$ is $k$ -torsionfree if $\operatorname{Ext}_{R}^{i}(\operatorname{Tr}(M),R)=0$ for all $1\leq i\leq k$ .
(6)

We set $X^{k}(R)$ (resp. $\widetilde{X}^{k}(R)$ ) to the set of all prime ideals $\mathfrak{p}$ of $R$ such that $\operatorname{ht}(\mathfrak{p})\leq k$ (resp. $\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\leq k)$ .
(7)

We say that $M$ satisfies $(S_{k})$ (resp. $(\widetilde{S}_{k})$ ) if $\operatorname{depth}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq\min\{k,\operatorname{ht}(\mathfrak{p})\}$ (resp. $\operatorname{depth}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq\min\{k,0pt_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\})$ .
(8)

We say that $R$ satisfies $(G_{k})$ if the local ring $R_{\mathfrak{p}}$ is Gorenstein for all $\mathfrak{p}\in X^{k}(R)$ .

The $R$ -modules $\operatorname{Tr}(M)$ and $\Omega^{k}(M)$ ( $k\geq 0$ ) are uniquely determined up to projective summands. It is easy to see that $\operatorname{Tr}(\operatorname{Tr}(M))\approx M$ . Note that $M$ is a $k$ -syzygy if and only if there exists an $R$ -module $L$ such that $M\approx\Omega^{k}(L)$ . Besides, for every $R$ -module $N$ , there exists an exact sequence of functors (see [5, Theorem 2.8]).

(2.1)

0\rightarrow\operatorname{Ext}_{R}^{1}\left(\mathcal{T}_{k+1}(M),-\right)\rightarrow\operatorname{Tor}_{k}^{R}(M,-)\rightarrow\operatorname{Hom}_{R}(\operatorname{Ext}_{R}^{k}(M,R),-)\rightarrow\operatorname{Ext}_{R}^{2}\left(\mathcal{T}_{k+1}(M),-\right).

Whenever $R$ is local, we consider the Auslander tranpose and the syzygies (of an $R$ -module) defined using minimal free resolutions. In this case, these $R$ -modules are defined uniquely up to isomorphism (rather than projective equivalence).

The notion of Gorenstein dimension was introduced by Auslander [36] and developed by Auslander and Bridger in [5].

2.2 Definition.

Let $M$ be an $R$ -module.

(1)

We say that $M$ is totally reflexive if $M$ is reflexive and $\operatorname{Ext}_{R}^{i}(M,R)=\operatorname{Ext}_{R}^{i}(M^{\ast},R)=0$ for all $i>0$ .
(2)

The Gorenstein dimension of $M$ , denoted by $\operatorname{G-dim}_{R}(M)$ , is defined to be the infimum of all non-negative integers $k$ such that there exists an exact sequence

$0\rightarrow G_{k}\rightarrow\cdots\rightarrow G_{0}\rightarrow M\rightarrow 0,$

where each $G_{i}$ is totally reflexive.

We can observe that $\operatorname{G-dim}_{R}(M)=0$ if and only if $M$ is totally reflexive. Below, we collect some facts related to the Gorenstein dimension.

2.3 Facts.

Let $M$ be an $R$ -module.

(1)

([9, Theorem 1.2.7]) If $M\not=0$ and $\operatorname{G-dim}_{R}(M)<\infty$ , then

$\operatorname{G-dim}_{R}(M)=\sup\{i\geq 0:\operatorname{Ext}_{R}^{i}(M,R)\not=0\}.$
(2)

([5, Lemma 3.19(1)]) $M$ is totally reflexive if and only if $\operatorname{Tr}_{R}(M)$ is totally reflexive.
(3)

([9, Corollary 1.2.9]) Let $0\to M\to N\to L\to 0$ be an exact sequence of $R$ -modules. If two $R$ -modules of the sequence have finite Gorenstein dimension, then so has the third.
(4)

([9, Proposition 1.2.10]) $\operatorname{G-dim}_{R}(M)\leq\operatorname{pd}_{R}(M)$ with equality if $\operatorname{pd}_{R}(M)<\infty$ .

Now, assume that $R$ is local.

(5)

([9, Theorem 1.4.8]) If $M\not=0$ , then $\operatorname{G-dim}_{R}(M)<\infty$ , then

$\operatorname{G-dim}_{R}(M)+\operatorname{depth}_{R}(M)=\operatorname{depth}_{R}(R).$

In particular, $\operatorname{G-dim}_{R}(M)<\infty$ .
(6)
([9, Theorem 1.4.9]) The following conditions are equivalent:
1. (6.1)
  
  $R$ is Gorenstein.
2. (6.2)
  
  $\operatorname{G-dim}_{R}(M)<\infty$ for all (finitely) $R$ -modules $M$ .
3. (6.3)
  
  $\operatorname{G-dim}_{R}(k)<\infty$ .

The formula given in Fact 2.3(5) is known as the Auslander-Bridger formula. From the definition of the Auslander transpose and Fact 2.3(3) we can derive the following remark.

2.4 Remark.

Let $R$ be a ring and let $M$ be an $R$ -module. Then $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ if and only if $\operatorname{G-dim}_{R}(\operatorname{Tr}_{R}(M))<\infty$ .

3. A generalization of a theorem of Evans and Griffith

The main goal of this section is to provide a generalization of a celebrated result by Evans and Griffith ([15, Theorem 3.8]), which characterizes a module to be $k$ -torsionfree, $k$ -syzygy, involving the condition $(\widetilde{S}_{k})$ .

Before presenting such a generalization, we provide the following result, which will be crucial for the proof of our main theorem in this section.

3.1 Lemma.

Let $R$ be a local ring of depth $t$ , $M$ be a non-zero $R$ -module and $0\leq k\leq t$ be an integer. If $M$ is a $k$ -syzygy of an $R$ -module of depth $0$ , then $\operatorname{depth}_{R}(M)=k$ .

Proof.

It is sufficient to consider $k\geq 1$ . We proceed by induction on $k$ . Suppose $k=1$ . Then $t\geq 1$ and there exists an exact sequence $0\to M\to F_{0}\to L\to 0,$ where $F_{0}$ is free $R$ -module and $L$ has of depth $0$ . The depth lemma gives us the inequalities

(3.1)

0=\operatorname{depth}_{R}(L)\geq\min\{\operatorname{depth}_{R}(M)-1,t\}

and

(3.2)

\operatorname{depth}_{R}(M)\geq\min\{t,1\}=1.

Since $t\geq 1$ , it follows from (3.1) that $\operatorname{depth}_{R}(M)\leq 1$ . Thus (3.2) say us that $\operatorname{depth}_{R}(M)=1$ .

Now suppose $1<k\leq t$ . Since $M$ is a $k$ -syzygy, there exists an exact sequence

0\to M\to F_{k-1}\to\cdots\to F_{0}\to L\to 0,

where $\operatorname{depth}_{R}(L)=0$ . Then it induces exact sequences

(3.3)

0\to M\to F_{k-1}\to N\to 0

and

(3.4)

0\to N\to F_{k-2}\to\cdots\to F_{0}\to L\to 0.

If $N=0$ , then (3.4) says that $\operatorname{pd}_{R}(L)\leq k-2$ and, by the Auslander-Buchsbaum formula, $t\leq k-2,$ which does not occurs by assumption. So, $N\not=0$ . Then by induction, we have that $\operatorname{depth}_{R}(N)=k-1$ . Applying the depth formula in (3.3) we get that

\operatorname{depth}_{R}(M)\geq\min\{t,\operatorname{depth}_{R}(N)+1\}=\min\{t,k\}=k

and

k-1=\operatorname{depth}_{R}(N)\geq\min\{\operatorname{depth}_{R}(M)-1,t\}.

Thus, if $t\leq\operatorname{depth}_{R}(M)-1$ , then $k-1\geq t$ , which contradicts our assumption. Therefore, we must have $t>\operatorname{depth}_{R}(M)-1$ . Consequently, $\operatorname{depth}_{R}(M)-1\leq t$ , and thus $\operatorname{depth}_{R}(M)-1\leq k-1$ , which implies $\operatorname{depth}_{R}(M)\leq k$ . Therefore, $k=\operatorname{depth}_{R}(M)$ . ∎

Below, we provide a characterization of a module to be $k$ -torsionfree and $k$ -syzygy, involving the condition $(\widetilde{S}_{k})$ . This characterization generalizes a celebrated result by Evans and Griffith ([15, Theorem 3.8]).

3.2 Theorem.

Let $R$ be a ring and let $k$ be a non-negative integer. Let $M$ be a non-zero $R$ -module such that $M^{\ast}$ has locally finite Gorenstein dimension on $\widetilde{X}^{k-1}(R)$ . Then the following conditions are equivalent:

(1)

$M$ is $k$ -torsionfree.
(2)

$M$ is $k$ -syzygy.
(3)

$M$ satisfies $(\widetilde{S}_{k})$ .

Proof.

The implications $(1)\Rightarrow(2)\Rightarrow(3)$ follow from [50, Proposition 38]. Now let us prove $(3)\Rightarrow(1)$ . We prove this by induction on $k$ . If $k=0$ , there is nothing to prove. Assume that $k\geq 1$ . By induction, $M$ is $(k-1)$ -torsionfree, that is $\operatorname{Ext}_{R}^{i}(\operatorname{Tr}(M),R)=0$ for all $1\leq i\leq k-1$ , and so it remains to conclude that $\operatorname{Ext}_{R}^{k}(\operatorname{Tr}(M),R)=0$ .

By contradiction, assume that $\operatorname{Ext}_{R}^{k}(\operatorname{Tr}(M),R)\not=0$ . Let $\mathfrak{p}\in\operatorname{Ass}_{R}(\operatorname{Ext}_{R}^{k}(\operatorname{Tr}(M),R))$ . Then $\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Ass}_{R_{\mathfrak{p}}}\left(\operatorname{Ext}_{R_{\mathfrak{p}}}^{k}\left(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}),R_{\mathfrak{p}}\right)\right)$ . From (2.1), we have an exact sequence

0\to\operatorname{Ext}_{R_{\mathfrak{p}}}^{k}\left(\operatorname{Tr_{R_{\mathfrak{p}}}}(M_{\mathfrak{p}}),R_{\mathfrak{p}}\right)\to\mathcal{T}_{k}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})),

which implies that $\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Ass}_{R_{\mathfrak{p}}}(\mathcal{T}_{k}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}))$ and hence

\operatorname{depth}_{R_{\mathfrak{p}}}(\mathcal{T}_{k}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}))=0.

Since $\operatorname{Ext}_{R_{\mathfrak{p}}}^{i}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}),R_{\mathfrak{p}})=0$ for all $1\leq i\leq k-1$ , we see from [52, 5.6] that

M_{\mathfrak{p}}\approx\Omega_{R_{\mathfrak{p}}}^{k-1}\mathcal{T}_{k}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})).

We claim that $k\leq 0pt_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})$ . Indeed, if $k>\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})$ , then by assumption $\operatorname{G-dim}_{R_{\mathfrak{p}}}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}))<\infty$ , and from the Auslander-Bridger formula, $\operatorname{G-dim}_{R_{\mathfrak{p}}}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}))<k$ . But since $\operatorname{Ext}_{R_{\mathfrak{p}}}^{i}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}),R_{\mathfrak{p}})=0$ for all $1\leq i\leq k-1$ , we get that $\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})$ is a totally reflexive. This shows that $\operatorname{Ext}_{R_{\mathfrak{p}}}^{k}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}),R_{\mathfrak{p}})=0$ , which does not occurs.

Now, as $M_{\mathfrak{p}}\approx\Omega_{R_{\mathfrak{p}}}^{k-1}\mathcal{T}_{k}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}))$ , it follows from Lemma 3.1 that $\operatorname{depth}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})=k-1$ . But since $M$ satisfies $(\widetilde{S}_{k})$ , we get that $\operatorname{depth}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq\min\{k,\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\}\linebreak=k$ , a contradiction. ∎

Observe that Theorem 3.2 is the version with the dual of an $R$ -module being of finite Gorenstein dimension of [14, Proposition 2.4] when $C=R$ . Now, we explore some consequences of Theorem 3.2.

3.3 Corollary.

Let $R$ be a ring and let $M$ be an $R$ -module. Suppose that $\operatorname{G-dim}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}^{\ast})<\infty$ for all $\mathfrak{p}\in\operatorname{Ass}_{R}(R)$ . Then the following conditions are equivalent:

(1)

$M$ is $1$ -torsionfree.
(2)

$M$ is $1$ -syzygy.
(3)

$M$ satisfies $(\widetilde{S}_{1})$ .

3.4 Corollary.

Let $R$ be a ring and let $M$ be an $R$ -module. Suppose that $\operatorname{G-dim}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}^{\ast})<\infty$ for all $\mathfrak{p}\in\widetilde{X}^{1}(R)$ . Then the following conditions are equivalent:

(1)

$M$ is $2$ -torsionfree.
(2)

$M$ is $2$ -syzygy.
(3)

$M$ satisfies $(\widetilde{S}_{2})$ .
(4)

$M$ is reflexive.

Proof.

The equivalences $(1)\Leftrightarrow(2)\Leftrightarrow(3)$ follow from Theorem 3.2. The equivalence $(2)\Leftrightarrow(4)$ follows from (2.1). ∎

This theorem allows us to recover one of the results described in the main result of [52].

3.5 Corollary ([52, Theorem 5.8(1)]).

Let $R$ be a local ring, $M$ be an $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ and $k$ be a non-negative integer. If $M$ satisfies $(\widetilde{S}_{k})$ , then $M$ is $k$ -torsionfree.

Note that Theorem 3.2 provides an answer to the following question posed in [37].

3.6 Question ([37, Question 1.1]).

When $k$ -syzygy modules are $k$ -torsionfree?

The following result is an analogous version of [44, Theorem 4.5], which is one of the main results of that paper.

3.7 Theorem.

Let $R$ be a ring, and let $M$ be an $R$ -module such that $M^{\ast}$ has locally finite Gorenstein dimension on $\widetilde{X}^{k-1}(R)$ . Then $M$ is $k$ -torsionfree if and only if the following hold:

(1)

$M_{\mathfrak{p}}$ is $(k-1)$ -torsionfree for all $\mathfrak{p}\in\widetilde{X}^{k-1}(R)$ .
(2)

$\operatorname{depth}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq k$ for all $\mathfrak{p}\in\operatorname{Spec}(R)$ with $\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\geq k$ .

Proof.

If $M$ is $k$ -torsionfree, then it is clear that (1) is true. Besides, by Theorem 3.2, $M$ satisfies $(\widetilde{S}_{k})$ , which shows that (2) holds.

Reciprocally, suppose that (1) and (2) hold. By Theorem 3.2, it is sufficient to show that $M$ satisfies $(\widetilde{S}_{k}),$ that is $\operatorname{depth}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq\min\{k,\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\}$ for all $\mathfrak{p}\in\operatorname{Spec}(R)$ . By (2), it holds if $\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\geq k$ . Now, let $0pt_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\leq k-1$ . Then by the Auslander-Bridger formula, $\operatorname{G-dim}_{R_{\mathfrak{p}}}(\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}))\leq k-1$ . Thus, by (1), $\operatorname{Tr}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})$ is totally reflexive, and hence $M_{\mathfrak{p}}$ as well. In particular, $\operatorname{depth}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})=\min\{k,\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\}$ . ∎

4. Characterization of totally reflexives

In this section, $R$ is a local ring. Next, we derive some criteria of totally reflexive modules, assuming the condition $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ , and involving the vanishing of the Ext functor. We first provide a general criterion via the vanishing of $\operatorname{Ext}_{R}^{1\leq i\leq\operatorname{depth}_{R}(R)}(M,R)$ , and subsequently, we provide criteria over Cohen-Macaulay local rings and later for locally totally reflexive modules on the punctured spectrum.

4.1 Theorem.

Let $R$ be a local ring of depth $t$ . Let $M$ be an $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then the following are equivalent:

(1)

$M$ satisfies $(\widetilde{S}_{t})$ .
(2)

$M$ is totally reflexive.
(3)

$\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq t$ .

Proof.

$(1)\Rightarrow(2)$ . According to Corollary 3.5, $M$ is $t$ -torsion-free, meaning that $\operatorname{Ext}_{R}^{i}(\operatorname{Tr}(M),R)=0$ for all $1\leq i\leq t$ . Since $\operatorname{G-dim}_{R}(\operatorname{Tr}(M))\leq t,$ it follows that $\operatorname{Tr}(M)$ is totally reflexive. Hence, $M$ is totally reflexive.

$(2)\Rightarrow(3)$ . This follows from definition of totally reflexives modules.

$(3)\Rightarrow(1)$ . Suppose that $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq t$ . Since $M\approx\operatorname{Tr}(\operatorname{Tr}(M))$ , this implies that $\operatorname{Tr}(M)$ is $t$ -torsion-free. Given that $\operatorname{G-dim}_{R}(\operatorname{Tr}(M))<\infty$ , according to [50, Theorem 42], $\operatorname{Tr}(M)$ satisfies $(\widetilde{S}_{t})$ . Thus, $\operatorname{depth}_{R}(\operatorname{Tr}_{R}(M))\geq t$ since $0pt_{R}(R)=t$ . By the Auslander-Bridger formula, $\operatorname{Tr}(M)$ is totally reflexive. Thus, $\operatorname{Ext}_{R}^{i}(\operatorname{Tr}_{R}(M),R)=0$ for all $i>0$ . In particular, $M$ is $t$ -torsionfree, and by [50, Proposition 11], $M$ satisfies $(\widetilde{S}_{t})$ . ∎

Over Cohen-Macaulay local rings

As an immediate application of Theorem 4.1, we obtain the following characterization for a module over a Cohen-Macaulay ring to be totally reflexive, involving the vanishing of the Ext functor.

4.2 Theorem.

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and let $M$ be a non-zero $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then the following are equivalents:

(1)

$M$ is maximal Cohen-Macaulay.
(2)

$M$ is totally reflexive.
(3)

$\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq d$ .

Proof.

Since $R$ is Cohen-Macaulay, it is clear that $M$ is maximal Cohen-Macaulay if and only if $M$ satisfies $(\widetilde{S}_{d})$ . Thus, the result follows immediately from Theorem 4.1. ∎

4.3 Remark.

It is known that if $R$ is a Gorenstein local ring and $M$ is maximal Cohen-Macaulay, then $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $i>0$ . We can observe from Theorem 4.2, that in this fact instead of $R$ being Gorenstein, we can consider the weaker condition of $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ .

We recall that the grade of an $R$ -module $M$ is defined as:

\operatorname{grade}_{R}(M)=\inf\{i\geq 0:\operatorname{Ext}_{R}^{i}(M,R)\not=0\}.

4.4 Corollary.

Let $R$ be a Cohen-Macaulay local ring and let $M$ be a Cohen-Macaulay $R$ -module such that $M^{\ast}\not=0$ and $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then $M$ is totally reflexive.

Proof.

Since $M^{\ast}\not=0$ , note that $\operatorname{grade}_{R}(M)=0$ . Moreover, by [7, Corollary 2.1.4], $\operatorname{grade}_{R}(M)=\operatorname{dim}_{R}(R)-\operatorname{dim}_{R}(M)$ and hence $\operatorname{dim}_{R}(M)=\operatorname{dim}_{R}(R).$ Thus from Cohen-Macaulayness of $M$ , we see that $M$ is maximal Cohen-Macaulay. Therefore, Theorem 4.2 asserts that $M$ is totally reflexive. ∎

Now, as an application of Theorem 4.2, we extend the formula presented in [13, Lemma 4.1] to all $R$ -modules $M$ such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . But before, we recall the definition of reduced grade. The reduced grade of an $R$ -module $M$ is defined as

\operatorname{r.grade}_{R}(M)=\inf\{i>0:\operatorname{Ext}_{R}^{i}(M,R)\not=0\}.

4.5 Remark.

Let $R$ be a ring. If $\operatorname{grade}_{R}(M)>0,$ then $\operatorname{grade}_{R}(M)=\operatorname{r.grade}_{R}(M).$

4.6 Theorem.

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ , and let $M$ be an $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then the equality

(4.1)

\sup\{i\geq 0:H_{\mathfrak{m}}^{i}(M)\neq 0\text{ and }i\neq d\}=d-\operatorname{r.grade}(M)

holds in $\mathbb{Z}\cup\{\pm\infty\}$ .

Proof.

We may assume that $R$ is complete, and hence $R$ has a canonical module $\omega$ . We prove the equality by considering the following cases:

(1)

$M$ is not maximal Cohen-Macaulay and $\operatorname{dim}(M)=\operatorname{dim}(R)$ .
(2)

$M$ is not maximal Cohen-Macaulay and $\operatorname{dim}(M)\not=\operatorname{dim}(R)$ .
(3)

$M$ is maximal Cohen-Macaulay.

The equality in the case (1) was proved in [13, Lemma 4.1].

Now, consider the case (2). Since $\operatorname{dim}(M)\not=\dim(R)$ , we see from [7, Theorem 3.5.7] that $\sup\{i\geq 0:H_{\mathfrak{m}}^{i}(M)\not=0\text{ and }i\not=d\}=\operatorname{dim}(M).$ Moreover, by [7, Corollary 2.14], we get that $\operatorname{grade}_{R}(M)=d-\operatorname{dim}_{R}(M)>0$ . Then by Remark 4.5, $\operatorname{r.grade}_{R}(M)=d-\operatorname{dim}(M)$ . Now observe that the desired equality holds.

Finally, let us prove the equality in the case (3). In this case, by [7, Theorem 3.5.7], $\sup\{i\geq 0:H_{\mathfrak{m}}^{i}(M)\not=0\text{ and }i\not=d\}=-\infty$ . On the other hand, by Theorem 4.2, $M$ is totally reflexive and hence $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $i\geq 1$ . Thus $\operatorname{r.grade}_{R}(M)=\infty$ and we can see that the equality holds. ∎

4.7 Corollary.

Let $R$ be a Cohen-Macaulay local ring with a canonical module $\omega$ , and let $M$ be an $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then:

\operatorname{r.grade}(M)=\inf\{i>0:\operatorname{Ext}_{R}^{i}(M,\omega)\not=0\}.

Proof.

We may assume that $R$ is complete. Because of [7, Theorem 3.5.8], the desired equality is a reformulation of the equality (4.1). ∎

4.8 Theorem.

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and let $M$ be an $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Let $n$ be a non-negative integer such that $\operatorname{depth}_{R}(M)\geq n$ . If $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq d-n,$ then $M$ is totally reflexive.

Proof.

We may assume that $M\not=0$ and (passing to the completion of $R$ if necessary) that $R$ has a canonical module $\omega$ .

Since $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq d-n$ and $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ , by Corollary 4.7, $\operatorname{Ext}_{R}^{i}(M,\omega)=0$ for all $1\leq i\leq d-n$ . Since [7, Exercise 3.1.24] say us that

d-\operatorname{depth}_{R}(M)=\sup\{i\geq 0:\operatorname{Ext}_{R}^{i}(M,\omega)\not=0\},

then $d-\operatorname{depth}_{R}(M)=0$ or $d-\operatorname{depth}_{R}(M)>d-n$ . The second case does not occur because $\operatorname{depth}_{R}(M)\geq n$ by assumption. Thus $d=\operatorname{depth}_{R}(M)$ and hence $M$ is maximal Cohen-Macaulay. Thus, by Theorem 4.2, $M$ is totally reflexive.

∎

Over locally totally reflexive modules on the punctured spectrum

The following result provides a reflexivity criterion for $R$ -modules $M$ with $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ of locally totally reflexives on the punctured spectrum of $R$ .

4.9 Theorem.

Let $R$ be a local ring of depth $t$ and let $M$ be a non-zero $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Suppose that $M$ is locally totally reflexive on the punctured spectrum of $R$ . Then $0pt_{R}(M)\leq t$ with equality if and only if $M$ is totally reflexive.

Proof.

First we prove that $\operatorname{depth}_{R}(M)\leq t$ . Set $r=\operatorname{depth}_{R}(M)$ and suppose that $r>t$ . Then $M$ satisfies $(\widetilde{S}_{r})$ since $M$ is locally totally reflexive on the punctured spectrum of $R$ . In particular, $M$ satisfies $(\widetilde{S}_{t})$ . By Theorem 4.1, $M$ is totally reflexive, which contradicts the fact of $r>t$ .

Now, we prove the second part of the result. It is clear that if $M$ is totally reflexive, then $\operatorname{depth}_{R}(M)=t$ . Suppose now that $\operatorname{depth}_{R}(M)=t$ . Since $M$ is locally totally reflexive on the punctured spectrum of $R$ and $\operatorname{depth}_{R}(M)=t$ , note that $M$ satisfies $(\widetilde{S}_{t})$ . Hence, by Theorem 4.1, $M$ is totally reflexive. ∎

4.10 Corollary.

Let $R$ be a local ring of depth $t$ and let $M$ be a non-zero $R$ module. Suppose that $M$ is locally totally reflexive on the punctured spectrum of $R$ . Suppose that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then the following are equivalent:

(1)

$\operatorname{depth}_{R}(M)\geq t$ .
(2)

$\operatorname{depth}_{R}(M)=t$ .
(3)

$M$ is totally reflexive.
(4)

$\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq t$ .

Proof.

The equivalences $(1)\Leftrightarrow(2)\Leftrightarrow(3)$ follow from Theorem 4.9, while $(3)\Leftrightarrow(1)$ follow from Theorem 4.1. ∎

In this section, we have derived a number of criteria for an $R$ -module $M$ with $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ to be totally reflexive. Thus, we close this section with the following question.

4.11 Question.

Let $R$ be a ring and let $M$ be an $R$ -module. When does $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ imply that $\operatorname{G-dim}_{R}(M)<\infty$ ?

5. Freeness Criteria

In this section, we aim to obtain freeness criteria by developing the following topics:

(I)

To derive freeness criteria from Theorem 4.8.
(II)

To provide generalizations of a celebrated result of Araya (Theorem 1.5) with the condition that the dual of a module is of finite Gorenstein dimension.
(III)

To discuss the Auslander-Reiten conjecture for $R$ -modules such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ and $\operatorname{pd}_{R}(\operatorname{Hom}_{R}(M,M))<\infty$ .

Freeness criteria from Theorem 4.8

As an application of Theorem 4.8, we obtain the following result:

5.1 Theorem.

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ , and let $M$ be an $R$ -module such that $\operatorname{pd}_{R}(M^{\ast})<\infty$ . Let $n$ be a non-negative integer such that $\operatorname{depth}_{R}(M)\geq n$ and $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq d-n$ . Then $M$ is free.

Proof.

As $\operatorname{pd}_{R}(M^{\ast})<\infty$ , then $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . By Theorem 4.8, $M$ is totally reflexive. Thus $M$ is reflexive and $\operatorname{pd}_{R}(M^{\ast})=\operatorname{G-dim}_{R}(M^{\ast})=0$ . Therefore, $M$ is free. ∎

5.2 Corollary.

Let $R$ be a Cohen-Macaulay local ring and let $M$ be an $R$ -module such that $\operatorname{pd}_{R}(M^{\ast})<\infty$ . If $M$ is maximal Cohen-Macaulay, then $M$ is free.

Proof.

This follows immediately from Theorem 5.1 by taking $n=\operatorname{dim}(R)$ . ∎

5.3 Corollary.

Let $R$ be a Cohen-Macaulay local ring and let $M$ be an $R$ -module such that $M^{\ast}\not=0$ and $\operatorname{pd}_{R}(M^{\ast})<\infty$ . If $M$ is Cohen-Macaulay, then $M$ is free.

Proof.

Since $M^{\ast}\not=0$ , note that $\operatorname{grade}_{R}(M)=0$ . Moreover, by [7, Corollary 2.1.4], $\operatorname{grade}_{R}(M)=\operatorname{dim}_{R}(R)-\operatorname{dim}_{R}(M)$ and hence $\operatorname{dim}_{R}(M)=\operatorname{dim}_{R}(R).$ Thus from Cohen-Macaulayness of $M$ , we see that $M$ is maximal Cohen-Macaulay. Hence, it follows from Corollary 5.2 that $M$ is free. ∎

As an application of Corollary 5.2, we will provide an answer to the Generalization of Herzog-Vasconcelos’s conjecture (see Proposition 9.4). Another application is given below.

5.4 Theorem.

Let $R$ be a local Cohen-Macaulay. The maximal ideal $\mathfrak{m}$ is Cohen-Macaulay if and only $\operatorname{dim}(R)\leq 1$ .

Proof.

Assume that $\mathfrak{m}$ is Cohen-Macaulay and that $\operatorname{dim}(R)=\operatorname{depth}_{R}(R)\geq 2$ . Consider the exact sequence

(5.1)

0\to\mathfrak{m}\to R\to k\to 0.

It induces an exact sequence

0\to k^{\ast}\to R\to\mathfrak{m}^{\ast}\to\operatorname{Ext}_{R}^{1}(k,R).

Since $\operatorname{depth}_{R}(R)\geq 2,$ then $k^{\ast}=0=\operatorname{Ext}_{R}^{1}(k,R)$ and $\mathfrak{m}^{\ast}\cong R$ . Therefore, by Corollary 5.3, $\mathfrak{m}$ is free, and hence $\mathfrak{m}\cong R$ . From the exact sequence (5.1), we obtain an exact sequence

0\to\operatorname{Hom}_{R}(k,\mathfrak{m})\to\operatorname{Hom}_{R}(k,R)\to\operatorname{Hom}_{R}(k,k)\to\operatorname{Ext}_{R}^{1}(k,\mathfrak{m}).

As $\operatorname{depth}_{R}(R)\geq 2,$ then $\operatorname{Hom}_{R}(k,\mathfrak{m})\cong\operatorname{Hom}_{R}(k,R)=0=\operatorname{Ext}_{R}^{1}(k,R)\cong\operatorname{Ext}_{R}^{1}(k,\mathfrak{m})$ . Consequently, $\operatorname{Hom}_{R}(k,k)=0,$ which does not occurs.

Conversely, assume that $\operatorname{dim}(R)\leq 1$ . If $\dim(R)=0$ it is clear that all $R$ -modules are Cohen-Macaulay. Suppose $\operatorname{dim}R(R)=1$ . Since $R$ is Cohen-Macaulay, then there exists an $R$ -regular element $x\in\mathfrak{m}$ . Note that $x$ is also an $\mathfrak{m}$ -regular element. Therefore $1\leq\operatorname{depth}_{R}(\mathfrak{m})\leq\operatorname{dim}_{R}(R)=1,$ which shows that $\mathfrak{m}$ is maximal Cohen-Macaulay. ∎

Some generalizations of a result of Araya

As mentioned in the introduction, after the publication of the Araya’s result referenced in Theorem 1.5, results have appeared that generalize or recover such theorem, first in the context of Gorenstein rings and later in more general contexts. Motivated by this, we provide some generalizations of that celebrated result in terms of the dual of an $R$ -module is of finite Gorenstein dimension.

5.5 Proposition.

Let $R$ be a local ring of depth $t\geq 1$ . Let $M$ be a locally free $R$ -module on the spectrum punctured such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . If $\operatorname{Ext}_{R}^{i}(M,R)=\operatorname{Ext}_{R}^{t-1}(M,M)=0$ for all $1\leq i\leq t$ , then $M$ is free.

Proof.

If $t=1$ , then $\operatorname{Hom}_{R}(M,M)\cong\operatorname{Ext}_{R}^{0}(M,M)=0$ , whence $M$ is zero and hence free. Suppose that $t\geq 2$ . By Theorem 4.1, $M$ is totally reflexive and hence $\operatorname{Tr}(M)$ as well. Thus, $\operatorname{Ext}_{R}^{i}(M,R)=\operatorname{Ext}_{R}^{i}(\operatorname{Tr}(M),R)=0$ for all $i>0$ . Therefore, the result follows from [30, Proposition 2.8]. ∎

5.6 Corollary.

Let $R$ be a local ring of depth $t\geq 1$ . Let $M$ be a locally free $R$ -module on the spectrum punctured such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . If $0pt(M)\geq t$ and $\operatorname{Ext}_{R}^{t-1}(M,M)=0$ , then $M$ is free.

Proof.

This follows from Proposition 5.5 and Corollary 4.10. ∎

5.7 Theorem.

Let $R$ be a ring satisfying $(S_{1})$ and $X$ a subset of $\operatorname{Spec}(R)$ containing $X^{1}(R)$ . Let

$s:=\inf\left\{\operatorname{depth}R_{\mathfrak{p}}\mid\mathfrak{p}\in\operatorname{Spec}(R)\backslash X\right\}$ and $t:=\sup\left\{\operatorname{depth}R_{\mathfrak{p}}\mid\mathfrak{p}\in\operatorname{Spec}(R)\backslash X\right\}$ .

Suppose that $M$ is locally free on $X$ and that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . If $\operatorname{Ext}_{R}^{i}(M,R)=\operatorname{Ext}_{R}^{j}(M,M)=0$ for all $1\leq i\leq t$ and $s-1\leq j\leq t-1$ , then $M$ is projective.

Proof.

We may suppose that $X\not=\operatorname{Spec}(R)$ . We show that $M_{\mathfrak{p}}$ is free for all $\mathfrak{p}\in\operatorname{Spec}(R)$ . By assumption this is true if $\mathfrak{p}\in X$ . So, suppose that $\mathfrak{p}\in\operatorname{Spec}(R)\backslash X$ . Therefore $\operatorname{ht}(\mathfrak{p})\geq 2$ and $\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\geq 1$ since $R$ satisfies $(S_{1})$ . By definition of $s$ and $t$ , note that $s\leq\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\leq t$ , so $\operatorname{Ext}_{R_{\mathfrak{p}}}^{i}(M_{\mathfrak{p}},R_{\mathfrak{p}})=0$ for all $1\leq i\leq\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})$ and $\operatorname{Ext}_{R_{\mathfrak{p}}}^{\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})-1}(M_{\mathfrak{p}},M_{\mathfrak{p}})=0$ . Hence, by Proposition 5.5, $M_{\mathfrak{p}}$ is free. ∎

Observe that Proposition 5.5, Corollary 5.6 and Theorem 5.7 are generalizations of Theorem 1.5.

Now, we explore some consequences of Theorem 5.7 for Cohen-Macaulay local rings.

5.8 Remark.

([30, Remark 2.11]) Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and $n$ be an integer such that $1\leq n\leq d-1$ . Then

\inf\{\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\mid\mathfrak{p}\in\operatorname{Spec}(R)\backslash\mathrm{X}^{n}(R)\}=n+1\,\,\mbox{ and }\,\,

\sup\{\operatorname{depth}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})\mid\mathfrak{p}\in\operatorname{Spec}(R)\backslash\mathrm{X}^{n}(R)\}=d.

5.9 Theorem.

Let $R$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ , $M$ be an $R$ -module and $n$ be a positive integer. Suppose that $M$ is locally of finite projective dimension on $X^{n}(R)$ and that $\operatorname{Ext}_{R}^{j}(M,M)=0$ for all $n\leq j\leq d-1$ . Then the following hold:

(1)

If $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ and $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq d$ , then $M$ is free.
(2)

If $\operatorname{G-dim}_{R}(M)<n,$ then $\operatorname{pd}_{R}(M)<\infty$ .

Proof.

(1) We claim that $M$ is locally free on $X^{n}(R)$ . In fact, given $\mathfrak{p}\in X^{n}(R)$ , we have by hypothesis that $\operatorname{pd}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty$ and $\operatorname{Ext}_{R_{\mathfrak{p}}}^{i}(M_{\mathfrak{p}},R_{\mathfrak{p}})=0$ for all $1\leq i\leq d$ . Hence, by [38, p. 154, Lemma 1(iii)], $M_{\mathfrak{p}}$ is free as $R_{\mathfrak{p}}$ -module. Thus item (1) follows immediately from Theorem 5.7 and Remark 5.8.

(2) We separate the proof in two cases. Let $p=\operatorname{G-dim}_{R}(M)$ . We prove the result by considering two cases.

(2.a) Suppose $p=0$ . In this case, $M$ is totally reflexive. Hence $M^{\ast}$ is totally reflexive and $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $i\geq 1$ . So, it follows from (1) that $M$ is free. In particular, $\operatorname{pd}_{R}(M)<\infty$ .

(2.b) Suppose $p>0$ . We have $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $i\geq p+1$ . By [30, Lemma 2.6], we get

\operatorname{Ext}_{R}^{i}(M,M)\cong\operatorname{Ext}_{R}^{i}\left(\Omega^{p}(M),\Omega^{p}(M)\right)

for all $i\geq n\geq p+1$ . Hence $\operatorname{Ext}_{R}^{i}\left(\Omega^{p}(M),\Omega^{p}(M)\right)=0$ for all $n\leq i\leq d-1$ . Note that $\operatorname{G-dim}_{R}(\Omega^{p}(M))=0$ and that $\Omega^{p}(M)$ is locally of finite projective dimension on $X^{n}(R)$ . Hence, by (1), $\Omega^{p}(M)$ is free and therefore $\operatorname{pd}_{R}(M)<\infty$ . ∎

5.10 Corollary.

Let $R$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ , and let $M$ be a maximal Cohen-Macaulay $R$ -module such that $\operatorname{G-dim}_{R}(M)<\infty$ . Let $1\leq n\leq d-1$ be an integer such that $M$ is locally of finite projective dimension on $X^{n}(R)$ . If $\operatorname{Ext}_{R}^{i}(M,M)=0$ for all $n\leq i\leq d-1$ . Then $M$ is free.

Proof.

As $\operatorname{G-dim}_{R}(M)<\infty$ and $M$ is maximal Cohen-Macaulay, it follows from the Auslander-Bridger Formula that $\operatorname{G-dim}_{R}(M)=0$ . Then, by Theorem 5.9(2), $\operatorname{pd}_{R}(M)<\infty$ , and hence $\operatorname{pd}_{R}(M)=\operatorname{G-dim}_{R}(M)=0$ . This yields that $M$ is free. ∎

5.11 Corollary.

Let $R$ be a Cohen-Macaulay normal local ring of dimension $d\geq 2$ , and let $M$ be an $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Suppose that $\operatorname{Ext}_{R}^{i}(M,R)=\operatorname{Ext}_{R}^{j}(M,M)=0$ for all $1\leq i\leq d$ and $1\leq j\leq d-1$ . Then $M$ is free.

Proof.

As $R$ is normal, then $R_{\mathfrak{p}}$ is a regular local ring for all $\mathfrak{p}\in X^{1}(R)$ . Hence, $M$ is locally of finite projective dimension on $X^{1}(R)$ . Thus, the result follows from Theorem 5.9. ∎

On the Auslander-Reiten conjecture, and a result of Dey and Ghosh

In [12, Corollary 6.9(2)], Dey and Ghosh demonstrated that the Auslander-Reiten conjecture holds for (finitely generated) $R$ -modules $M$ satisfying $\operatorname{G-dim}_{R}(M)<\infty$ and $\operatorname{pd}_{R}(\operatorname{Hom}_{R}(M,M))<\infty$ . Motivated by this, we investigate whether the conjecture remains valid when we consider $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ instead of $\operatorname{G-dim}_{R}(M)<\infty$ in the referenced case.

5.12 Theorem.

Let $R$ be a local ring of depth $t$ , and let $M$ be an $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then the following conditions are equivalent.

(1)

$M$ is free.
(2)

$\operatorname{Hom}_{R}(M,M)$ is free and $\operatorname{Ext}_{R}^{j}(M,M)=0$ for all $1\leq j\leq t-1$ .
(3)

$\operatorname{Hom}_{R}(M,M)$ has finite projective dimension and $\operatorname{Ext}_{R}^{i}(M,R)=\operatorname{Ext}_{R}^{j}(M,M)=0$ for all $1\leq i\leq t$ and $1\leq j\leq t-1$ .

Proof.

$(1)\Rightarrow(2)$ is trivial.

$(2)\Rightarrow(3)$ . We only need to prove that $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq t$ . For this, we will show that $M$ is totally reflexive. In view of [12, Corollary 7.6(2)], $M$ satisfies $(\widetilde{S}_{t})$ . Therefore, $M$ is totally reflexive by Theorem 4.1.

$(3)\Rightarrow(1)$ . By Theorem 4.1, $M$ is totally reflexive. Therefore by [12, Corollary 6.8], $M$ is free. ∎

5.13 Corollary.

Let $R$ be local ring. The Auslander-Reiten conjecture holds true for all (finitely generated) $R$ -modules $M$ such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ and $\operatorname{pd}_{R}(\operatorname{Hom}_{R}(M,M))<\infty$ .

An $R$ -module $C$ is said to be semidualizing if the natural map $R\rightarrow\operatorname{Hom}_{R}(C,C)$ is an isomorphism and $\operatorname{Ext}_{R}^{i}(C,C)=0$ for all $i>0$ .

5.14 Corollary.

Let $R$ be a local ring, and let $C$ be a semidualizing $R$ -module. If $\operatorname{G-dim}_{R}(C^{\ast})<\infty$ , then $C\cong R$ .

6. Gorenstein criteria and related questions

Throughout this section, let $R$ be a local ring. In this section, we aim to provide a number of criteria for a local ring $R$ to be Gorenstein in terms of the dual of certain modules having finite Gorenstein dimension. In this sense, we discuss some related questions to this subject.

The first criterion that we present in this section is an application of Theorem 4.2. Before stating it, first recall that if $R$ is a ring with total quotient ring $Q$ , then an $R$ -module $M$ is said to have a (generic) rank, denoted by $\operatorname{rank}(M)$ , if $M\otimes_{R}Q$ is a free $Q$ -module of rank $\operatorname{rank}(M)$ . Let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ and $\mu(M)$ denote the minimum number of generators of $M$ . Moreover, if $M$ is Cohen-Macaulay and $e(M)=\mu(M)$ , then $M$ it is said to be an Ulrich module ([21, Definition 2.1]).

6.1 Proposition.

Let $R$ be a Cohen-Macaulay local ring, and let $M$ be an $R$ -module such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then $R$ is Gorenstein in each one of the following cases.

(1)

$M$ is Cohen-Macaulay with positive rank and $2\mu(M)>e(R)\operatorname{rank}(M)$ .
(2)

$M$ is an Ulrich module and $M^{\ast}\not=0$ .

Proof.

(1) The positivity of the rank of $M$ implies that $M^{\ast}\neq 0$ . Since $M$ is Cohen-Macaulay and $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ , according to Corollary 4.4, we conclude that $M$ is totally reflexive. Consequently, $\operatorname{Ext}_{R}^{i}(M,R)=0$ for all $1\leq i\leq d$ . Thus, by assumption and [48, Theorem 3.1], we obtain the result.

(2) By definition, $M$ is Cohen-Macaulay, and by Theorem 4.4, $M$ is totally reflexive. Hence $R$ is Gorenstein by [11, Proposition 2.19]. ∎

Due to Proposition 6.1, is natural to ask the following.

6.2 Question.

Let $R$ be a local ring. Suppose there exists a non-free (finitely generated) $R$ -module $M$ such that $M^{\ast}\not=0$ and $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Then, is $R$ Gorenstein?

6.3 Remark.

This question is a refined version of Question 4.5 of the second version of this paper, which was answered negatively in [19, Example 3.9 and Remark 3.10] by Ghosh and Samanta. The same example shows that Question 6.2 is false in general.

On the other hand, it is known that if $R$ is a local ring such that $\operatorname{G-dim}_{R}(M)<\infty$ for all $R$ -modules, then $R$ is Gorenstein ([9, Theorem 1.4.9]). Thus, by considering the dual of $R$ -modules it is natural to ask the following.

6.4 Question.

Let $R$ be a local ring. If $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ for all (finitely generated) $R$ -modules $M$ , then is $R$ Gorenstein?

Then motivated by Question 6.4, we obtain the following result:

6.5 Theorem.

Let $R$ be a local ring. If $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ for all (finitely generated) $R$ -modules $M$ such that $M^{\ast}\not=0,$ then $R$ is Gorenstein.

Proof.

First, suppose that $\operatorname{depth}_{R}(R)=0$ . Then $k^{\ast}\not=0$ and by assumption $\operatorname{G-dim}_{R}(k^{\ast})<\infty$ . Since $k^{\ast}\cong k^{n}$ for some $n\geq 1$ , note that $\operatorname{G-dim}_{R}(k)<\infty$ . Hence $R$ is Gorenstein.

Now, suppose that $\operatorname{depth}_{R}(R)\geq 1$ . We may write $\mathfrak{m}=(x_{1},\ldots,x_{n})$ . Let $\varphi:R\to R^{n}$ the homomorphism defined by $\varphi(r)=(rx_{1},\ldots,rx_{n})$ . Since $\operatorname{depth}_{R}(R)>0$ , we see that $\varphi$ is injective. Thus we have an exact sequence

0\rightarrow R\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}R^{n}\rightarrow M\rightarrow 0.

Dualizing this exact sequence we obtain a sequence

(6.1)

0\rightarrow M^{\ast}\to R^{n}\stackrel{{\scriptstyle\varphi^{\ast}}}{{\rightarrow}}R\to\operatorname{coker}(\varphi^{\ast})=k\to 0.

If $M^{\ast}=0,$ we see from (6.1) that $\operatorname{pd}_{R}(k)<\infty,$ which shows that $R$ is regular, and consequently Gorenstein. If $M^{\ast}\not=0,$ by assumption, $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ , so from (6.1), $\operatorname{G-dim}_{R}(k)<\infty,$ concluding that $R$ is Gorenstein. ∎

As an immediate consequence we obtain the following corollary that answer positively Question 6.4

6.6 Corollary.

Let $R$ be a local ring such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ for all (finitely generated) $R$ -modules $M$ . Then $R$ is Gorenstein.

6.7 Remark.

Corollary 6.6 can be obtained in other ways. For instance, it could be derived from [44, Corollary 3.5]. Additionally, this corollary can be obtained as an application of Theorem 3.2. Indeed, setting $d=\dim_{R}(R)$ , by the assumption and Theorem 3.2, all (finitely generated) $R$ -modules satisfy the following equivalences:

$M$ is $(d+1)$ -torsionfree $\Longleftrightarrow$ $M$ is $(d+1)$ -syzygy $\Longleftrightarrow$ $M$ satisfies $(\widetilde{S}_{d+1}).$

Then by [37, Theorem 1.4], $R$ satisfies $(G_{d})$ , concluding that $R$ is Gorenstein.

For a local ring of depth at most one, the finiteness of the Gorenstein dimension of the dual of its maximal ideal characterizes its Gorensteiness.

6.8 Proposition.

Let $R$ be a local ring of depth at most one. If $\operatorname{G-dim}_{R}(\mathfrak{m}^{\ast})<\infty$ , then $R$ is Gorenstein.

Proof.

First, suppose $\operatorname{depth}_{R}(R)=0$ . Since $\operatorname{G-dim}_{R}(\mathfrak{m}^{\ast})<\infty,$ then $\operatorname{G-dim}_{R}(\operatorname{Tr}_{R}(\mathfrak{m}))<\infty$ . Since $\operatorname{depth}_{R}(R)=0$ , the Auslander-Bridger formula shows that $\operatorname{Tr}_{R}(\mathfrak{m})$ is totally reflexive, and hence $\mathfrak{m}$ as well. Now, by considering the exact sequence $0\to\mathfrak{m}\to R\to k\to 0,$ we see that $\operatorname{G-dim}_{R}(k)<\infty$ . Thus, we conclude that $R$ is Gorenstein.

Now, suppose that $\operatorname{depth}_{R}(R)=1$ . Then the exact sequence $0\to\mathfrak{m}\to R\to k\to 0,$ induces an exact sequence

0\to k^{\ast}\to R\to\mathfrak{m}^{\ast}\to\operatorname{Ext}_{R}^{1}(k,R)\to 0.

Since $\operatorname{depth}_{R}(R)=1$ , we see that $k^{\ast}=0$ and $\operatorname{Ext}_{R}^{1}(k,R)\cong k^{n}$ for some $n\geq 1$ . Thus, we obtain an exact sequence

0\to R\to\mathfrak{m}^{\ast}\to k^{n}\to 0.

In view of that $\operatorname{G-dim}_{R}(\mathfrak{m}^{\ast})<\infty,$ we see that $\operatorname{G-dim}_{R}(k)<\infty$ . This shows that $R$ is Gorenstein. ∎

The next question that we will present is motivated by Question 4.11. Consider the following condition:

(GDUAL):: Every (finitely generated) $R$ -module $M$ whose dual $M^{\ast}$ is finite Gorenstein dimension, is also of finite Gorenstein dimension.

6.9 Question.

Let $R$ be a local ring. If $R$ satisfies (GDUAL), then is $R$ Gorenstein?

6.10 Remark.

Let $R$ be a local ring. Suppose $\operatorname{depth}_{R}(R)=0$ . By the Auslander-Bridger formula, all $R$ -modules of finite Gorenstein dimension are totally reflexive. Moreover of for any $R$ -module $M$ we have that the following implications hold:

	$\displaystyle\operatorname{G-dim}_{R}(M^{\ast})<\infty$	$\displaystyle\Rightarrow\operatorname{G-dim}_{R}(\operatorname{Tr}_{R}(M))<\infty$
		$\displaystyle\Rightarrow\operatorname{G-dim}_{R}(\operatorname{Tr}_{R}(M))=0$
		$\displaystyle\Rightarrow\operatorname{G-dim}_{R}(M)=0.$

In particular, $R$ satisfies (GDUAL). Hence the existence of non-Gorenstein local rings of depth zero implies that Question 6.9 is false in general.

Motivated by condition $(\operatorname{GDUAL})$ , we define a weaker condition than it, and prove that if $R$ is of positive depth and satisfies such condition, then $R$ is Gorenstein. This will show that Question 6.9 is positive for local rings of positive depth.

(GDUAL*):: Every (finitely generated) $R$ -module $M$ whose dual $M^{\ast}$ is non-zero and of finite Gorenstein dimension, is also of finite Gorenstein dimension.

6.11 Proposition.

Let $R$ be a local ring of depth at least one satisfying $\operatorname{(GDUAL*)}$ . Then $R$ is Gorenstein.

Proof.

Since $R$ has positive depth, note that $k^{\ast}=0$ . Let $M=R\oplus k$ . Then $M^{\ast}=R$ and hence $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ . Thus $\operatorname{G-dim}_{R}(k)<\infty$ by assumption. Hence, $R$ is Gorenstein. ∎

Now, motivated by Corollary 5.14, we will discuss about a question that was addressed by Holanda and Miranda-Neto in [28]. The question is as follows:

6.12 Question.

([28, Question 5.24]) Let $R$ be a Cohen-Macaulay local ring with a canonical module $\omega$ . If $\operatorname{pd}_{R}(\omega^{\ast})<\infty$ , then must $R$ be Gorenstein?

Firstly, we observe that the answer to Question 6.12 is positive and follows directly from the Corollary 5.14. However, it is worth highlighting that this question had already been answered affirmatively, albeit through a different proof, by Asgharzadeh in [4, Theorem 11.3]. Additionally, it is worth noting that long before Holanda and Miranda-Neto posed this question, the same result could be obtained directly from the Foxby equivalence (see [9, Theorem 3.4.11]).

We finish this section by proposing the following question, which is motivated by a celebrated result of Foxby [17] that asserts that the existence of a non-zero (finitely generated) $R$ -module $M$ of projective dimension and injective dimension both finite implies that $R$ is Gorenstein.

6.13 Question.

If there exists a (finitely generated) $R$ -module $M$ such that $\operatorname{G-dim}_{R}(M^{\ast})<\infty$ and $\operatorname{id}_{R}(M)<\infty$ , then is $R$ Gorenstein?

7. Preliminary facts on Kähler differentials

In this section, we establish our notation and, for the reader’s convenience, we recall some known results concerning Kähler differential modules and differential modules, both in the context of algebras and locally ringed spaces.

Affine derivation modules and Kähler fifferentials $n$ -th order

Let $R$ be a $S$ -algebra and $M$ be an $R$ -module. Recall that a $S$ -linear map $D:R\rightarrow M$ is said to be a derivation if for any two elements $x_{0},x_{1}\in R$ , the following identity holds:

D(x_{0}x_{1})=x_{0}D(x_{1})+x_{1}D(x_{0}).

A derivation of order $n$ can be defined generalizing the previous identity as follows.

A $S$ -linear map $D:R\rightarrow M$ is said to be a $n$ -th order derivation if for any $x_{0},\ldots,x_{n}\in R$ , the following identity holds:

D(x_{0}\cdots x_{n})=\sum_{s=1}^{n}(-1)^{s-1}\sum_{i_{1}<\cdots<i_{s}}x_{i_{1}}\cdots x_{i_{s}}D(x_{0}\ldots\hat{x}_{i_{1}}\cdots\hat{x}_{i_{s}}\cdots x_{n}),

where $\hat{x}_{i_{j}}$ means that this element does not appear in the product. The set of $n$ -th order derivations of an $S$ -algebra $R$ into an $R$ -module $M$ over $S$ will be denoted by $\text{Der}^{n}_{S}(R,M)$ . When $M=R$ , we shall use the notation $\text{Der}^{n}_{S}(R)$ in place of $\text{Der}^{n}_{S}(R,R)$ .

The module of derivations of order $n$ can be represented as follows. Let $I$ denote the kernel of the homomorphism $R\otimes_{S}R\rightarrow R$ , $a\otimes b\mapsto ab$ . Giving structure of $R$ -module to $R\otimes_{S}R$ by multiplying on the left, we define the $R$ -module

\Omega^{(n)}_{R/S}:=I/I^{n+1}.

Define the map $d_{n}^{R}:R\rightarrow\Omega^{(n)}_{R/S}$ , $a\mapsto(1\otimes a-a\otimes 1)+I^{n+1}$ . This map is a derivation of order $n$ , and its image generates $\Omega^{(n)}_{R/S}$ as an $R$ -module (see [41, Chapter II-1]).

The $R$ -module $\Omega^{(n)}_{R/S}$ is called the module of Kähler differentials of order $n$ of $R$ over $S$ . The map $d_{n}^{R}$ is called the canonical derivation of $R$ in $\Omega^{(n)}_{R/S}$ . It comes equipped with a universal derivation $d_{S/R}\in\text{Der}_{R}^{n}(S,\Omega^{(n)}_{S/R})$ with the property that composition with $d_{S/R}$ yields an isomorphism $\operatorname{Hom}_{R}(\Omega^{(n)}_{R/S},R)\cong\text{Der}_{S}^{n}(R)$ ([43, Proposition 1.6]). To see more properties regarding modules of derivations and Kähler differentials, we recommend [43, 51, 41].

7.1 Remark.

It is important to note that the differential module and Kähler differentials may not be finitely generated. However, they are finitely generated in certain cases. For instance: If $R$ is essentially of finite type over $S$ . If $S=k$ is a field with a valuation and $R$ is an analytic $k$ -algebra, meaning $R$ is module-finite over a convergent power series ring $k\{x_{1},\ldots,x_{n}\}$ . If $S=k$ is a field, $(R,\mathfrak{m})$ is a complete local ring, and $R/\mathfrak{m}$ is a finite extension of $k$ . For this reason, in this article, for each $n$ , we will consider $\Omega^{(n)}_{R/S}$ to be finitely generated as an $R$ -module. In particular, by the universal property, $\text{Der}^{n}_{S}(R)$ is also finitely generated as an $R$ -module.

Next, we collect some definitions and properties that will play a central role in this paper.

7.2 Definition.

A ringed space $(X,\mathcal{O}_{X})$ consists of a topological space $X$ paired with a sheaf of rings $\mathcal{O}_{X}$ , known as the structure sheaf of $X$ . If $\mathcal{O}_{X,x}$ is a local ring for every $x\in X$ , $(X,\mathcal{O}_{X})$ is termed a locally ringed space. Let $\mathfrak{m}_{x}$ denote the maximal ideal of $\mathcal{O}_{X,x}$ ; $k(x):=\mathcal{O}_{X,x}/\mathfrak{m}_{x}$ is referred to as the residue field of $X$ at $x$ . We consider the following locally ringed spaces: locally Noetherian schemes (i.e., a scheme is locally Noetherian if it has a covering by open affine subsets $\operatorname{Spec}R_{i}$ , where each $R_{i}$ is a Noetherian ring) and analytic spaces (the ringed space $(X,\mathcal{O}_{X})$ is called an analytic space if every point $x\in X$ has a neighborhood $V$ such that $(V,\mathcal{O}_{X}|_{V})$ is isomorphic to some analytic model space, where $\mathcal{O}_{X,x}\cong\frac{\mathcal{O}_{\mathbb{C}^{n},x}}{{\mathcal{I}}_{x}}$ for all $x\in X$ , and $\frac{\mathcal{O}_{\mathbb{C}^{n},x}}{{\mathcal{I}}_{x}}$ is Noetherian and reduced; for further details, e.g., see [29], [23]).

7.1. Global Kähler differentials

Next, we will recall the globalized definition of the constructions given above. To do this, instead of gluing together the modules $\Omega^{(1)}_{S/R}$ , we provide a global definition, and then point out that it reduces to the original definition given above. Before defining it, it is worth noting that the fiber product exists in our context (see, for example [10, Corollary 0.32] and [20, Theorem 4.18]). Let $\varphi:X\to Y$ be a morphism of space. This induces a morphism $\Delta:X\to X\times_{Y}X$ , called the diagonal morphism. The map $\Delta$ comes with a map of sheaves $\Delta^{\#}:\mathcal{O}_{X\times_{Y}X}\to\Delta_{*}\mathcal{O}_{X}$ on $X\times_{Y}X$ . Although really a map of sheaves of rings, we will regard it as a map of sheaves of $\mathcal{O}_{X\times_{Y}X}$ -modules. Let $\mathcal{I}$ be its kernel, again regarded as an $\mathcal{O}_{X\times_{Y}X}$ -module.

7.3 Definition.

The sheaf relative Kähler differential of $\mathcal{O}_{X}$ -module is defined as:

\Omega^{(1)}_{X/Y}=\Delta^{\ast}(\mathcal{I}/\mathcal{I}^{2}).

Thus, more generally, the sheaf of relative Kähler $n$ -differentials of $X$ over $Y$ is defined by

\Omega_{X/Y}^{(n)}:=\begin{cases}\bigwedge^{n}\Omega_{X/Y}^{(1)},&\text{if }n\geq 1,\\ \mathcal{O}_{X},&\text{if }n=0,\\ 0,&\text{if }n<0.\end{cases}

Note that if $Y=\operatorname{Spec}(k)$ , where $k$ is a field, $\Omega_{X/Y}^{(n)}$ is usually denoted by $\Omega_{X}^{(n)}$ . For further details and properties, see for example, [1, Section I.6.6, p. 34].

7.4 Facts.

Let $X$ and $Y$ be (locally) Noetherian spaces.

(1)

([34, Proposition 6.1.20]). If $\varphi:X\to Y$ is locally of finite type, then $\Omega_{X/Y}^{(1)}$ is coherent. Additionally, by [20, Proposition 7.48], $\Omega_{X/Y}^{(n)}$ is also $\mathcal{O}_{X}$ -coherent for all $n>1$ .
(2)

([34, Proposition 1.17]). If $\varphi:X\to Y$ is a morphism of spaces, then for $x\in X$ there is a canonical isomorphism of $\mathcal{O}_{X,x}$ -modules

$\left(\Omega_{X/Y}^{(1)}\right)_{x}=\Omega_{\mathcal{O}_{X,x}/\mathcal{O}_{Y,\varphi(x)}}^{(1)}.$

Since the stalk commutes with exterior power, more generally, one obtains $(\Omega_{X/Y}^{(n)})_{x}\cong\Omega_{\mathcal{O}_{X,x}/\mathcal{O}_{Y,\varphi(x)}}^{(n)}=:\Omega_{(X,x)/(Y,\varphi(x)}^{(n)}$ .
(3)

([1, Corollaire (I6.5.5)]). There is also a definition of the sheaf of $Y$ -derivations over $X$ , denoted by $\mathcal{D}er_{Y}^{1}(X)$ . Moreover, the universal property of $\Omega_{X/Y}^{(1)}$ comes down to saying that there is an isomorphism of $\mathcal{O}_{X}$ -modules $\mathcal{D}er_{Y}^{1}(X)\cong\mathcal{H}om_{\mathcal{O}_{X}}\left(\Omega_{X/Y}^{(1)},\mathcal{O}_{X}\right).$ Naturally, is defined the sheaf of $n$ -th order $Y$ -derivations over $X$ .

$\mathcal{D}er_{Y}^{n}(X):=\mathcal{H}om_{\mathcal{O}_{X}}\left(\Omega_{X/Y}^{(n)},\mathcal{O}_{X}\right).$
(4)

[46, Lemma 28.20.2, Section 05P1] Let $X$ be a locally Noetherian scheme, and let $\mathcal{F}$ be a coherent $\mathcal{O}_{X}$ -module. Then $\mathcal{F}$ is locally free if and only if for all $x\in X$ , the stalk $\mathcal{F}_{x}$ is a free $\mathcal{O}_{X,x}$ -module.

Based on Remarks 7.1 and Facts 7.4, the schemes considered in this section will be locally Noetherian, and the $\mathcal{O}_{X}$ -module $\mathcal{D}er_{Y}^{n}(X)$ will also always be considered a non-zero coherent sheaf for each integer $n\geq 1$ . Additionally, we assume that the sheaves have non-zero stalks, implying that the chosen elements belong to the support. For simplicity, the stalk of $\Omega_{(X,x)/(Y,\varphi(x))}^{(n)}$ , as defined in Facts 7.4(2), will be denoted by $\Omega_{(X,x)/(Y,y)}^{(n)}$ , where $y=\varphi(x)$ .

7.5 Lemma.

Let $X$ and $Y$ be locally Noetherian spaces, then the following properties hold:

(i)

$\mathcal{D}er_{Y}^{n}(X)$ is a $\mathcal{O}_{X}$ -coherent.
(ii)

for $x\in X$ there is a canonical isomorphism of $\mathcal{O}_{X,x}$ -modules

$(\mathcal{D}er_{Y}^{n}(X))_{x}\cong{\rm Der}^{n}_{\mathcal{O}_{Y,y}}(\mathcal{O}_{X,x}).$

Thus, ${\rm Der}^{n}_{\mathcal{O}_{Y,y}}(\mathcal{O}_{X,x})$ will be denoted by ${\rm Der}^{n}_{{(Y,y)}}(X,x).$

Proof.

(i) From Fact 7.4(1), we establish the coherence of $\Omega_{X/Y}^{(n)}$ . Subsequently, from that $\mathcal{D}er_{Y}^{n}(X)=\mathcal{H}om_{\mathcal{O}_{X}}\left(\Omega_{X/Y}^{(n)},\mathcal{O}_{X}\right)$ and by [34, Exercise 5.1.6(b)], we deduce the result.

(ii)

	$\displaystyle(\mathcal{D}er_{Y}^{n}(X))_{x}$	$\displaystyle\cong\left(\mathcal{H}om_{\mathcal{O}_{X}}\left(\Omega_{X/Y}^{(n)},\mathcal{O}_{X}\right)\right)_{x}\,\,\,\text{from Fact \ref{fact1}(3)}$
		$\displaystyle\cong\operatorname{Hom}_{\mathcal{O}_{X,x}}\left(\Omega_{\mathcal{O}_{X,x}/\mathcal{O}_{Y,\varphi(x)}}^{(n)},\mathcal{O}_{X,x}\right)$
		$\displaystyle\cong{\rm Der}^{n}_{\mathcal{O}_{Y,\varphi(x)}}(\mathcal{O}_{X,x}).$

Since $\Omega_{X/Y}^{(n)}$ is $\mathcal{O}_{X}$ -coherent (Facts 7.4(1)), so $\mathcal{H}om$ commutes with the stalk [20, Proposition 7.27]. Now from Facts 7.4(2)(3) from the universal property ( [27, Theorem 2.2.6] or [43, Proposition 1.6]) we get the last isomorphism. ∎

8. Applications on the $k$ -torsion of the modules of differentials

Next, we study when the Kähler differential module $\Omega_{X/Y}^{(n)}$ of order $n$ is locally free, reflexive, $k$ -torsion-free and $k$ -syzygy, specially when $k=1,2$ . The study of these properties in the modules of Kähler differential of order $n$ concerning the regularity questions of affine rings has a long history; when $n=1$ and $Y$ is a Spec of a field (see Kunz-Waldi’s book [31]). Next, we provide a summary given by Milher in [39], which is relevant for motivating the study of these concepts. Let $d:=\dim_{x}X=\dim(\mathcal{O}_{X,x})$ and $s:=\operatorname{codim}_{x}\operatorname{Sing}X$ (definition for ${\rm Sing}$ and $\operatorname{codim}$ see 8.6). Lipman (see also Suzuki [47]) showed that for an affine complete intersection germ over a field of characteristic zero, $\Omega^{(1)}_{X,x}$ is torsion-free if and only if $X$ is normal at $x\in X$ , while $\Omega^{(1)}_{X,x}$ is reflexive if and only if $s>3$ (see [33]). Vetter proved in [49] that for a non-smooth reduced complete intersection singularity and for $1\leq n\leq d$ , the module $\Omega^{(n)}_{X,x}$ is torsion-free (resp., reflexive) if $n<s$ (resp., $n<s-1$ ) and has torsion (resp., cotorsion) if $n=s$ (resp., if $n=\max\{1,s-1\}$ ). Lebelt strengthened Vetter’s result by showing that if $\Omega^{(n)}_{X,x}$ is torsion-free (resp., reflexive), then $n<s$ (resp., $n<s-1$ ) (see [32]). Greuel in [24] show that $\Omega^{(n)}_{X,x}$ has torsion for any $n>d$ ; (for any reduced complex analytic space, [16]). This fact implies that $(\Omega^{(n)}_{X,x})^{**}=0$ since $d>0$ and consequently results in $\text{cotor}\Omega^{(n)}_{X,x}=0$ for $n>d$ . Also, new results on the connection between torsion and cotorsion and reflexivity of Kähler differential modules can be found in [22, 39]. Due to it and the results mentioned above, we obtain the following results:

8.1 Theorem.

Let $X$ and $Y$ be locally Noetherian spaces and let $k$ be a non-negative integer, $x\in X$ , such that ${\rm Der}^{n}_{{(Y,y)}}(X,x)$ has locally finite Gorenstein dimension on $\widetilde{X}^{k-1}(\mathcal{O}_{X,x})$ . Then the following conditions are equivalent:

(1)

$\Omega_{(X,x)/(Y,y)}^{(n)}$ is $k$ -torsionfree.
(2)

$\Omega_{(X,x)/(Y,y)}^{(n)}$ is $k$ -syzygy.
(3)

$\Omega_{(X,x)/(Y,y)}^{(n)}$ satisfies $(\widetilde{S}_{k})$ .

Proof.

By Fact 7.4(2), we have $\Omega_{(X,x)/(Y,\varphi(x)}^{(n)}=\Omega_{\mathcal{O}_{X,x}/\mathcal{O}_{Y,\varphi(x)}}^{(n)}$ . Therefore, $\Omega_{(X,x)/(Y,\varphi(x))}^{(n)}$ is an $\mathcal{O}_{X,x}$ -module. Moreover, we know that ${\rm Der}^{n}_{{(Y,\varphi(x))}}(X,x)=(\Omega_{(X,x)/(Y,\varphi(x))}^{(n)})^{\ast}$ . Now, the shows of the equivalences follow from Theorem 3.2. ∎

As an immediate consequence of Theorem 8.1 and Corollaries 3.3 and 3.4, we have the following corollaries.

8.2 Corollary.

Let $X$ and $Y$ be locally Noetherian spaces and let $x\in X$ , such that ${\rm Der}^{n}_{{(Y,y)}}(X,x)$ has locally finite Gorenstein dimension on ${\rm Ass}_{\mathcal{O}_{X,x}}(\mathcal{O}_{X,x})$ . Then the following conditions are equivalent:

(1)

$\Omega_{(X,x)/(Y,y)}^{(n)}$ is $1$ -torsionfree.
(2)

$\Omega_{(X,x)/(Y,y)}^{(n)}$ is $1$ -syzygy.
(3)

$\Omega_{(X,x)/(Y,y)}^{(n)}$ satisfies $(\widetilde{S}_{1})$ .

8.3 Corollary.

Let $X$ and $Y$ be locally Noetherian spaces and let $x\in X$ such that ${\rm Der}^{n}_{{(Y,y)}}(X,x)$ has locally finite Gorenstein dimension on $\widetilde{X}^{1}(\mathcal{O}_{X,x})$ . Then the following conditions are equivalent:

(1)

$\Omega_{(X,x)/(Y,y)}^{(n)}$ is $2$ -torsionfree.
(2)

$\Omega_{(X,x)/(Y,y)}^{(n)}$ is $2$ -syzygy.
(3)

$\Omega_{(X,x)/(Y,y)}^{(n)}$ satisfies $(\widetilde{S}_{2})$ .
(4)

$\Omega_{(X,x)/(Y,y)}^{(n)}$ is reflexive.

8.4 Remark.

Let $\varphi:X\rightarrow Y$ be a morphism of schemes which is locally of finite type. If we consider $X$ and $Y$ as affine Noetherian schemes, that is, $X=\operatorname{Spec}(R)$ and $Y=\operatorname{Spec}(S)$ where $R$ and $S$ are local Noetherian rings. By Fact 7.4(2) and Lemma 7.5(ii), if we consider $x\in X$ and $\varphi(x)\in Y$ closed points respectively, then we have $\Omega_{(X,x)/(Y,\varphi(x))}^{(n)}=\Omega^{(n)}_{R/S}$ and ${\rm Der}^{n}_{{(Y,\varphi(x))}}(X,x)={\rm Der}^{n}_{S}(R)$ . Moreover, these are finitely generated $R$ -modules.

From Remark 8.4, Theorem 8.1, and Corollaries 8.2 and 8.3, we obtain similar results in the affine context, that is, the derivation modules and Kähler differential modules defined over algebras.

Some application in algebraic variety and analytic variety

Before establishing our significant results that connect algebraic geometry and analytic geometry, let’s recall the following definitions: Let $R$ be a (Noetherian) ring, $M$ be an (finitely generated) $R$ -module and let $f_{M}:M\to M^{\ast\ast}$ be an $R$ -homomorphism. We say that $M$ is torsionless, if the map $f_{M}$ is injective, and $M$ is reflexive if $f_{M}$ is an isomorphism. If $f_{M}$ fails to be surjective, we say that $M$ has cotorsion and we denote the cotorsion by $\text{cotor }M:=\text{coker }(f_{M})$ . Note that, by Definition 2.1(5), torsionless is equivalent to 1-torsionfree, and reflexive is equivalent to 2-torsionfree ([50, Proposition 5]).

The torsion submodule of $M$ is defined as the kernel of the natural map $\theta:M\rightarrow M\otimes_{R}{Q(R)}$ where $Q(R)$ is the total quotient ring of $R$ . One says that $M$ is torsion-free if $\text{tor}M=0$ , and $M$ is a torsion module if $\text{tor}M=M$ .

8.5 Remark.

Clearly the torsion submodule $\text{tor}M$ is contained in $\text{ker }f_{M}$ . If $R$ is a Noetherian domain and $M$ is finitely generated, one can show that $\text{ker }f_{M}=\text{tor}M$ . Thus, the concepts of torsionless and torsion-free are equivalent ([23, p.70]). If $R$ is only a reduced, Noetherian ring and $M$ is finitely generated, still the concepts are equivalent ([45]). Therefore, in the same situations as above, torsionless, torsion-free, and 1-torsionfree are equivalent.

We shall apply these notions to $(X,\mathcal{O}_{X})$ , ringed spaces locally Noetherian (or analytic space), and the stalk at a point $x\in X$ of a coherent sheaf $\mathcal{F}$ viewed as an $\mathcal{O}_{X,x}$ -module. For a coherent sheaf $\mathcal{F}$ , define the torsion and cotorsion sheaves, $\text{tor }\mathcal{F}$ and $\text{cotor }\mathcal{F}$ , as the kernel and cokernel sheaves of the natural map $f_{\mathcal{F}}:\mathcal{F}\rightarrow\mathcal{F}^{**}$ . At the stalk level, this gives an exact sequence of $\mathcal{O}_{X,x}$ -modules

0\rightarrow\text{tor }\mathcal{F}_{x}\rightarrow\mathcal{F}_{x}\rightarrow(\mathcal{F}^{**})_{x}\rightarrow\text{cotor }\mathcal{F}_{x}\rightarrow 0,

where the middle map is the evaluation map $f_{{\mathcal{F}}_{x}}$ upon the identification $(\mathcal{F}^{**})_{x}\cong(\mathcal{F}_{x})^{**}$ .

8.6 Definition.

Let $X$ be a scheme(or analytic space), and let $x\in X$ . Then $\mathfrak{m}_{X,x}/\mathfrak{m}_{X,x}^{2}$ is a vector space over $k(x):=\mathcal{O}_{X,x}/\mathfrak{m}_{X,x}$ , called Zariski cotangent space to $X$ at $x$ . The Zariski tangent space of $X$ in $x$ is, by definition, the dual vector space

T_{x}X:=(\mathfrak{m}_{X,x}/\mathfrak{m}_{X,x}^{2})^{\vee}.

Recall that a locally Noetherian $X$ is said to be nonsingular (or regular) at $x\in X$ if the Zariski tangent space to $X$ at $x$ has dimension equal to $\dim(X,x)$ , that is, $\mathcal{O}_{X,x}$ is regular; otherwise, we say that $X$ is singular at $x$ . We say that $X$ is regular (or smooth) if it is regular at all of its points. Denote by $\operatorname{Reg}(X)$ and $\operatorname{Sing}(X)$ the regular locus and the singular locus of $X$ , respectively. For $x\in X$ , the codimension of the singular locus of $X$ at $x$ is defined as $\text{codim}_{x}\text{Sing}\,X:=\text{dim}_{x}X-\text{dim}_{x}\text{Sing}\,X$ .

We say that $X$ has isolated singularity if $\operatorname{Sing}(X)$ has dimension at most zero (i.e., equal to $0$ or $-\infty$ ).

8.7 Definition (Cohen-Macaulay Sheaf).

A coherent sheaf $\mathcal{F}$ over a scheme $X$ is said to be (maximal) Cohen-Macaulay if, for every point $x\in X$ , the stalk $\mathcal{F}_{x}$ is a (maximal) Cohen-Macaulay module over the local ring $\mathcal{O}_{X,x}$ . $X$ is a complete intersection at $x\in X$ , if the stalk $\mathcal{O}_{X,x}$ is complete intersection ring. A point $x$ in $X$ is said to be normal if $\mathcal{O}_{X,x}$ is a normal local ring.

In the last part, we considered either one of the following two possible settings, one analytic and one algebraic. More specifically, we consider a reduced complex analytic variety $X$ with the structure sheaf $\mathcal{O}_{X}$ , such that, at $x\in X$ , the stalk $\mathcal{O}_{X,x}$ is isomorphic to a reduced local $\mathbb{C}$ -algebra, which is a quotient of a convergent power series ring $\mathbb{C}\{z_{1},\dots,z_{n}\}$ . We also consider an algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$ with the structure sheaf $\mathcal{O}_{X}$ , such that, at $x\in X$ , the stalk $\mathcal{O}_{X,x}$ is isomorphic as a $k$ -algebra, reduced to a quotient of a regular local ring by an ideal.

Recent results on the connection between torsion and cotorsion and reflexivity of Kähler differential modules on algebraic and analytic varieties can be found in the papers by Graf and Miller-Vassiliadou ([22, 39]). Next, we establish a connection between the results of Graf and Miller-Vassiliadou with $k$ -torsionfree, $k$ -syzygy, and $(\widetilde{S}_{k})$ where $k=1,2$ .

8.8 Proposition.

Let $X$ be either a complex analytic variety or an algebraic variety over an algebraically closed field of characteristic zero. Assume that $X$ is a complete intersection at a normal point $x\in X$ of dimension $\dim_{x}X=d>0$ . Suppose $x$ is a singular point of $X$ . Then, for $1\leq n\leq d$ , the following conditions are equivalent:

(1)

$\operatorname{codim}_{x}(\operatorname{Sing}(X))\leq n\leq d.$
(2)

$\Omega^{(n)}_{X,x}$ is not $1$ -torsionfree.
(3)

$\Omega^{(n)}_{X,x}$ is not $1$ -syzygy.
(4)

$\Omega^{(n)}_{X,x}$ does not satisfy $(\widetilde{S}_{1})$ .

Proof.

The equivalences $(2)\Leftrightarrow(3)\Leftrightarrow(4)$ follow from Corollary 8.2. Since $X$ is reduced, by hypothesis, moreover, in this case, as noted in Remark 8.5, $1$ -torsionfree and torsion-free are equivalent. Thus, the equivalence $(2)\Leftrightarrow(1)$ follows from [39, Corollary 3.1]. ∎

8.9 Proposition.

(1)

$\operatorname{codim}_{x}(\operatorname{Sing}(X))-1\leq n\leq d$ .
(2)

$\Omega^{(n)}_{X,x}$ is not $2$ -torsionfree.
(3)

$\Omega^{(n)}_{X,x}$ is not $2$ -syzygy.
(4)

$\Omega^{(n)}_{X,x}$ does not satisfy $(\widetilde{S}_{2})$ .
(5)

$\Omega^{(n)}_{X,x}$ is not reflexive.

Proof.

The equivalences $(2)\Leftrightarrow(3)\Leftrightarrow(4)\Leftrightarrow(5)$ follow from Corollary 8.3.

Since reflexive is equivalent to 2-torsionfree ([50, Proposition 5]) and $X$ is reduced, the equivalence $(5)\Leftrightarrow(1)$ follows from [39, Corollary 3.1]. ∎

9. Applications on Herzog-Vasconcelos’s Conjecture

One of the main conjectures, which may have motivated the study of properties of the Kähler differential modules and the regularity of schemes or analytic spaces, is the famous Lipman-Zariski conjecture [33], which states the following: Let $X$ be a complex variety such that the tangent sheaf $\mathcal{T}_{X}:=\mathcal{H}om_{\mathcal{O}_{X}}(\Omega_{X}^{(1)},\mathcal{O}_{X})$ is locally free. Then $X$ is smooth. This conjecture remains widely open. Following the same direction as the Zariski-Lipman Conjecture, there is also a homological version proposed by Herzog and Vasconcelos as:

9.1 Conjecture (Herzog-Vasconcelos’s conjecture).

Let $R$ be a local Noetherian ring. If ${\rm pd}_{R}{\rm Der}_{k}(R)<\infty$ , then ${\rm Der}_{k}(R)$ is a free $R$ -module.

9.2 Conjecture (Strong Zariski-Lipman Conjecture).

Let $R$ be a local Noetherian ring. If ${\rm pd}_{R}{\rm Der}_{k}(R)<\infty$ , then $R$ is regular.

These conjectures, particularly the Lipman-Zariski conjecture when $n=1$ , have been extensively studied and resolved in special cases. References of Lipman-Zariski Conjecture can be found in Miller-Vassiliadou [39, Section 4]. While progress has been made for $n>1$ , it has been to a lesser extent compared to $n=1$ . Further investigations can be found in Graf [22] and Miller-Vassiliadou [39, Section 4]. It’s worth noting that these conjectures are generally false for positive characteristic. For instance, for $n=1$ , consider the surface $X\subset\mathbb{A}^{3}_{k}$ over a perfect field $k$ of characteristic $p$ defined by the equation $xy-z^{n}=0$ , where $p$ divides $n$ . Then one can see that $\mathcal{T}_{X}$ is free (see [33, p. 892]). Example for $n>1$ , see [35]. Due to this, from now on our spaces or rings will be equicharacteristic and of characteristic zero.

Next, we present conjectures which are generalizations of 9.1 and 9.2, where SGHVC was introduced by Ludington [35] and GHVC by Graf [22] in the analytical context. Le $M$ be an $R$ -module. Since, $\operatorname{pd}_{R}(M)=\sup\{\operatorname{pd}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})|\mathfrak{p}\in\operatorname{Spec}(R)\}.$ Then, the projective dimension of a coherent sheaf over $X$ is defined as

\operatorname{pd}_{\mathcal{O}_{X}}\mathfrak{F}={\rm sup}\{\operatorname{pd}_{\mathcal{O}_{X,x}}{\mathcal{F}_{x}}|x\in X\}.

9.3 Questions.

(i)

(Generalizations of Herzog-Vasconcelos’s conjecture (GHVC)) For some integer $n\geq 1$ , ${\rm pd}_{\mathcal{O}_{X}}{\mathcal{D}er}_{Y}^{n}(X)\,<\,\infty$ . Under what assumptions on $X$ and $Y$ , and for which values of $n$ , does this imply that ${\mathcal{D}er}_{Y}^{n}(X)$ is locally free?
(i)

(Strong generalizations of Herzog-Vasconcelos’s conjecture (SGHVC)) For some integer $n\geq 1$ , ${\rm pd}_{\mathcal{O}_{X}}{\mathcal{D}er}_{Y}^{n}(X)\,<\,\infty$ . Under what assumptions on $X$ and $Y$ , and for which values of $n$ , does this imply that $X$ is locally smooth?
(iii)

In particular, consider $X=\operatorname{Spec}(R)$ and $Y=\operatorname{Spec}(k)$ as in Remark 8.4. Assume that for some integer $n\geq 1$ , ${\rm pd}_{R}\operatorname{Der}_{k}^{n}(R)<\infty$ . Under what assumptions on $R$ , and for which values of $n$ does this imply that $\operatorname{Der}_{k}^{n}(R)$ is free (resp. smooth)?

The following results provide a partial answer to the SGHV conjecture when $X$ is Cohen-Macaulay and the Kähler differential module $\Omega_{X/Y}^{(n)}$ is Cohen-Macaulay or maximal Cohen-Macaulay.

9.4 Proposition.

Let $X$ and $Y$ be locally Noetherian spaces, with $X$ being Cohen-Macaulay, and suppose that ${\rm pd}_{\mathcal{O}_{X}}{\mathcal{D}er}_{Y}^{n}(X)\,<\,\infty$ for some $n\geq 1$ . Then ${\mathcal{D}er}_{Y}^{n}(X)$ is locally free if one of the following conditions holds:

(1)

If $\Omega_{X/Y}^{(n)}$ is maximal Cohen-Macaulay.
(2)

If $\Omega_{X/Y}^{(n)}$ is Cohen-Macaulay.

Proof.

It is suffices to show it at the level of stalks. Note that, ${\rm pd}_{\mathcal{O}_{X}}{\mathcal{D}er}_{Y}^{n}(X)\,<\,\infty$ if and only if $\operatorname{pd}_{\mathcal{O}_{X,x}}\left({\rm Der}^{n}_{\mathcal{O}_{Y,\varphi(x)}}(\mathcal{O}_{X,x})\right)<\infty$ for all $x\in X$ . Moreover, $\Omega_{X/Y}^{(n)}$ is Cohen-Macaulay (maximal) if and only if $\Omega_{(X,x)/(Y,y)}^{(n)}$ is Cohen-Macaulay (maximal) for all $x\in X$ .

Now, by Corollary 5.2 or Corollary 5.3, $\Omega_{(X,x)/(Y,y)}^{(n)}$ is an $\mathcal{O}_{X,x}$ -free module for all $x\in X$ . Thus, by the universal property, we obtain that ${\rm Der}^{n}_{{(Y,y)}}(X,x)$ is free for all $x\in X$ . ∎

Note that in Proposition 9.4 for $n=1$ , if we consider $X=\operatorname{Spec}(R)$ , $Y=\operatorname{Spec}(k)$ as in Remark 8.4, the assumption ${\rm pd}_{R}({\rm Der}_{k}(R))<\infty$ implies that $R$ is Gorenstein (see [25]). Thus, by Theorem 4.2 and Corollary 4.4, ${\rm Der}_{k}(R)$ is totally reflexive, hence ${\rm Der}_{k}(R)$ is free.

The following results provide a partial answer to the SGHV conjecture for complete intersections.

9.5 Proposition.

Let $X$ be either a complex analytic variety or an algebraic variety over an algebraically closed field of characteristic zero. Assume that $X$ is a complete intersection at a point $x\in X$ of dimension $\text{dim}_{x}X=d>0$ and $\Omega_{X,x}^{(n)}$ is (maximal) Cohen-Macaulay for some $1\leq n\leq d-1$ . Let the codimension of the singular locus of $X$ at $x$ be at least three. If $\operatorname{pd}_{\mathcal{O}_{X,x}}\left({\rm Der}^{n}_{k}(\mathcal{O}_{X,x})\right)<\infty$ at $x$ , then $x$ is a smooth point of $X$ .

Proof.

In the proof of Theorem 9.4, it is noted that such a result holds at the level of stalks. Additionally, in the same proof or by Corollary 5.2, Corollary 5.3, we show that $\Omega_{X,x}^{(n)}$ is free as an $\mathcal{O}_{X,x}$ -module. Therefore, $\Omega_{X,x}^{[n]}$ is a $\mathcal{O}_{X,x}$ -free module, because $\Omega_{X,x}^{[n]}=\left(\Omega_{X,x}^{(n)}\right)^{\ast\ast}$ . Now by hypothesis and from [39, Theorem 4.1], we get the result. ∎

Acknowledgement.

We thank Rafael Holanda for his valuable comments on the manuscript. The first author was supported by grant 2022/03372-5, São Paulo Research Foundation (FAPESP). The second author was supported by grant 2019/21181-0, São Paulo Research Foundation (FAPESP)

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On modules whose dual is of finite Gorenstein dimension: The kk-torsion, homological criteria, and applications in modules of differentials of order nn

Abstract.

2020 Mathematics Subject Classification:

1. Introduction

1.1 Theorem.

1.2 Corollary ([52, Theorem 5.8(1)]).

1.3 Theorem.

1.4 Conjecture.

1.5 Theorem ([2, Corollary 10]).

1.6 Theorem.

1.7 Theorem.

1.8 Theorem.

1.9 Theorem.

1.10 Questions.

2. Setup and preliminaries

2.1 Definition.

2.2 Definition.

2.3 Facts.

2.4 Remark.

3. A generalization of a theorem of Evans and Griffith

3.1 Lemma.

Proof.

3.2 Theorem.

Proof.

3.3 Corollary.

3.4 Corollary.

Proof.

3.5 Corollary ([52, Theorem 5.8(1)]).

3.6 Question ([37, Question 1.1]).

3.7 Theorem.

Proof.

4. Characterization of totally reflexives

4.1 Theorem.

Proof.

Over Cohen-Macaulay local rings

4.2 Theorem.

Proof.

4.3 Remark.

4.4 Corollary.

Proof.

4.5 Remark.

4.6 Theorem.

Proof.

4.7 Corollary.

Proof.

4.8 Theorem.

Proof.

Over locally totally reflexive modules on the punctured spectrum

4.9 Theorem.

Proof.

4.10 Corollary.

Proof.

4.11 Question.

5. Freeness Criteria

Freeness criteria from Theorem 4.8

5.1 Theorem.

Proof.

5.2 Corollary.

Proof.

5.3 Corollary.

Proof.

5.4 Theorem.

Proof.

Some generalizations of a result of Araya

5.5 Proposition.

Proof.

5.6 Corollary.

Proof.

5.7 Theorem.

Proof.

5.8 Remark.

5.9 Theorem.

Proof.

5.10 Corollary.

Proof.

5.11 Corollary.

Proof.

On the Auslander-Reiten conjecture, and a result of Dey and Ghosh

5.12 Theorem.

Proof.

On modules whose dual is of finite Gorenstein dimension: The $k$ -torsion, homological criteria, and applications in modules of differentials of order $n$

Affine derivation modules and Kähler fifferentials $n$ -th order

8. Applications on the $k$ -torsion of the modules of differentials