Remarks on a conjecture of Huneke and Wiegand
and the vanishing of (co)homology

Let $R$ be a local ring, $M$ be an $R$-module and let $n$ be an integer. If $n>0$, then $\Omega^{n}_{R}M$ denotes the $n$th syzygy of $M$, that is, the image of the $n$th differential map in a minimal free resolution of $M$. Also, if $M$ is totally reflexive and $n<0$, then $\Omega^{n}_{R}M$ denotes the $n$th cosyzygy of $M$, that is, the image of the $R$-dual of the $n$th differential map in a minimal free resolution of $M^{\ast}$. Note, by convention, we have $\Omega^{0}_{R}(M)=M$.

Let $R$ be a local ring and let $M$ be an $R$-module. Then the Auslander transpose of $M$, denoted by $\mathsf{Tr}\hskip 0.72229ptM$, is the cokernel of the map $f^{\ast}=\operatorname{Hom}_{R}(f,R)$, where $F_{1}\stackrel{{\scriptstyle f}}{{\longrightarrow}}F_{0}\to M\to 0$ is part of the minimal free resolution of $M$; see [3]. Note that $M^{\ast}\cong\Omega^{2}_{R}\mathsf{Tr}\hskip 0.72229ptM$. Note also that $\mathsf{Tr}\hskip 0.72229ptM$ is unique, up to isomorphism, since so are minimal free resolutions.

If $R$ is a local ring and $M$ is an $R$-module such that $\operatorname{Tor}_{1}^{R}(M,\mathsf{Tr}\hskip 0.72229ptM)=0$, then $M$ is free; see [5, A1] and also [41, 3.9].

Let $R$ be a local ring and let $M$ be an $R$-module. A complex $\mathbf{T}$ of free $R$-modules is said to be totally acyclic provided that $\operatorname{H}_{n}(\mathbf{T})=0=\operatorname{H}_{n}(\operatorname{Hom}_{R}(\mathbf{T},R))$ for all $n\in\mathbb{Z}$. A complete resolution of $M$ is a diagram $\mathbf{T}\overset{\vartheta}{\longrightarrow}\mathbf{P}\overset{\pi}{\longrightarrow}M$, where $\mathbf{P}$ is a projective resolution, $\mathbf{T}$ is a totally acyclic complex and $\vartheta$ is a morphism of complexes such that $\vartheta_{i}$ is an isomorphism for all $i\gg 0$.

Assume $\mathbf{T}\rightarrow\mathbf{P}\rightarrow M$ is a complete resolution of $M$. Then, for an $R$-module $N$ and for $i\in\mathbb{Z}$, the Tate homology $\widehat{\operatorname{Tor}}_{i}^{R}(M,N)$ and the Tate cohomology $\widehat{\operatorname{Ext}}_{R}^{i}(M,N)$ of $M$ and $N$ over $R$ are defined as $\operatorname{H}_{i}(\mathbf{T}\otimes_{R}N)$ and $\operatorname{H}^{i}(\operatorname{Hom}_{R}(\mathbf{T},N))$, respectively.

It is known that $\operatorname{\textnormal{G-dim}}_{R}(M)<\infty$ if and only if $M$ has a complete resolution; see [11, 3.1]. Hence $\widehat{\operatorname{Tor}}_{i}^{R}(M,N)$ and $\widehat{\operatorname{Ext}}_{R}^{i}(M,N)$ are defined for each $R$-module $N$ in case $\operatorname{\textnormal{G-dim}}_{R}(M)<\infty$. ∎

Let $R$ be a local ring, and let $M$ and $N$ be $R$-modules such that $\operatorname{\textnormal{G-dim}}_{R}(M)<\infty$.

1. (i)

   If $i>\operatorname{\textnormal{G-dim}}_{R}(M)$, then it follows $\widehat{\operatorname{Tor}}_{i}^{R}(M,N)\cong\operatorname{Tor}_{i}^{R}(M,N)$ and $\widehat{\operatorname{Ext}}^{i}_{R}(M,N)\cong\operatorname{Ext}^{i}_{R}(M,N)$.

2. (ii)

   If $0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$ is a short exact sequence of $R$-modules, where $\operatorname{\textnormal{G-dim}}_{R}(M^{\prime})<\infty$ or $\operatorname{\textnormal{G-dim}}_{R}(M^{\prime\prime})<\infty$, then, for each $i\in\mathbb{Z}$, there is an exact sequence of cohomology of the form:

   $$\widehat{\operatorname{Ext}}_{R}^{i}(M^{\prime\prime},N)\to\widehat{\operatorname{Ext}}_{R}^{i}(M,N)\to\widehat{\operatorname{Ext}}_{R}^{i}(M^{\prime},N)\to\widehat{\operatorname{Ext}}_{R}^{i+1}(M^{\prime\prime},N),$$

   and also of homology of the form:

   $$\widehat{\operatorname{Tor}}_{i}^{R}(M^{\prime},N)\to\widehat{\operatorname{Tor}}_{i}^{R}(M,N)\to\widehat{\operatorname{Tor}}_{i}^{R}(M^{\prime\prime},N)\to\widehat{\operatorname{Tor}}_{i-1}^{R}(M^{\prime},N).$$

3. (iii)

   $\widehat{\operatorname{Tor}}_{i+n}^{R}(M,N)\cong\widehat{\operatorname{Tor}}_{i}^{R}(\Omega^{n}M,N)$ and $\widehat{\operatorname{Ext}}^{i+n}_{R}(M,N)\cong\widehat{\operatorname{Ext}}^{i}_{R}(\Omega^{n}M,N)$ for all $i,n\in\mathbb{Z}$.

4. (iv)

   If $\operatorname{pd}_{R}(M)<\infty$, then $\widehat{\operatorname{Ext}}^{i}_{R}(M,N)=0=\widehat{\operatorname{Tor}}_{i}^{R}(M,N)$ for all $i\in\mathbb{Z}$.

5. (v)

   If $M$ is totally reflexive, then $\widehat{\operatorname{Tor}}_{i}^{R}(M,N)\cong\widehat{\operatorname{Ext}}^{-i-1}_{R}(M^{*},N)\cong\widehat{\operatorname{Ext}}^{-i+1}_{R}(\mathsf{Tr}\hskip 0.72229ptM,N)$ for all $i\in\mathbb{Z}$.

Let $R$ be a local ring, and let $M$ and $N$ be $R$-modules such that $n\geq 1$.

1. (i)

   If $\operatorname{depth}_{R}(M\otimes_{R}N)\geq n$, $\operatorname{depth}_{R}(N)\geq n-1$ and $\operatorname{\textup{length}}_{R}(\operatorname{Ext}^{i}_{R}(\mathsf{Tr}\hskip 0.72229ptM,N))<\infty$ for all $i=1,\ldots,n$, then it follows that $\operatorname{Ext}^{i}_{R}(\mathsf{Tr}\hskip 0.72229ptM,N)=0$ for all $i=1,\ldots,n$; see [21, 4.2(ii)].

2. (ii)

   If $\operatorname{Ext}^{i}_{R}(\mathsf{Tr}\hskip 0.72229ptM,N)=0$ for all $i=1,\ldots,n$, then it follows that $\operatorname{depth}_{R}(M\otimes_{R}N)\geq\min\{n,\operatorname{depth}_{R}(N)\}$; see [21, 4.2(iii)].

3. (iii)

   If $\operatorname{pd}_{R}(M)\leq\operatorname{depth}_{R}(N)$ and $\operatorname{\textup{length}}_{R}(\operatorname{Tor}_{i}^{R}(M,N))<\infty$ for all $i\geq 1$, then it follows $\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i\geq 1$; see [36, 2.2].

4. (iv)

   If $\operatorname{pd}_{R}(M)<\infty$ and $\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i\geq 1$, then the depth formula for $M$ and $N$ holds, that is, $\operatorname{depth}_{R}(M\otimes_{R}N)=\operatorname{depth}_{R}(N)-\operatorname{pd}_{R}(M)$; see [2, 1.2]. ∎

Let $R$ be a local ring and let $M$ be an $R$-module such that $\operatorname{\textnormal{G-dim}}_{R}(M)=n<\infty$.

1. (i)

   There is a short exact sequence of $R$-modules $0\rightarrow M\rightarrow X\rightarrow G\rightarrow 0$, where $\operatorname{pd}_{R}(X)=n$ and $G$ is totally reflexive. Such an exact sequence is called a finite projective hull of $M$.

2. (ii)

   There is a short exact sequence of $R$-modules $0\rightarrow Y\rightarrow X\rightarrow M\rightarrow 0$, where $\operatorname{pd}_{R}(Y)=n-1$ and $X$ is totally reflexive. Such an exact sequence is called a Cohen-Macaulay approximation of $M$. ∎

Let $R$ be a local ring and let $M$ and $N$ be $R$-modules. Assume $\operatorname{\textnormal{CI-dim}}_{R}(M)<\infty$ or $\operatorname{\textnormal{CI-dim}}_{R}(N)<\infty$. Then the following conditions are equivalent; see [9, 4.9].

1. (i)

   $\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i\gg 0$.

2. (ii)

   $\operatorname{Tor}_{i}^{R}(M,N)=0$ for all $i>\operatorname{\textnormal{CI-dim}}_{R}(M)$.

3. (iii)

   $\widehat{\operatorname{Tor}}_{i}^{R}(M,N)=0$ for all $i\in\mathbb{Z}$.

Moreover, if $\operatorname{\textnormal{CI-dim}}_{R}(M)=0$ and $N=M^{\ast}$, then $M$ is free if and only if one of the above equivalent conditions holds: this is because, if $\operatorname{\textnormal{CI-dim}}_{R}(M)=0$ and $\widehat{\operatorname{Tor}}_{i}^{R}(M,M^{\ast})=0$ for all $i\in\mathbb{Z}$, then, since $M^{\ast}\cong\Omega_{R}^{2}\mathsf{Tr}\hskip 0.72229ptM$, it follows that $\operatorname{Tor}_{1}^{R}(M,\mathsf{Tr}\hskip 0.72229ptM)=0$, which forces $M$ to be free; see 2.3 and 2.5(i).

We should also note, if $\operatorname{\textnormal{CI-dim}}_{R}(M)<\infty$, then the conditions (i), (ii) and (iii) stated above are also equivalent for Ext and Tate cohomology modules; see [9, 4.7].

Let $R$ be a local ring and let $M$ be an $R$-module. Then the complexity $\operatorname{cx}_{R}(M)$ of $M$ is defined as $\inf\{r\in\mathbb{N}\cup\{0\}:A\in\mathbb{R}\text{ such that }\dim_{k}(\operatorname{Ext}^{n}_{R}(M,k))\leq A\cdot n^{r-1}\text{ for all }n\gg 0\}$; see [8].

Note, it follows from the definition that, $\operatorname{cx}_{R}(M)=0$ if and only if $\operatorname{pd}_{R}(M)<\infty$, and $\operatorname{cx}_{R}(M)\leq 1$ if and only if $M$ has bounded Betti numbers. Furthermore, the following properties hold:

1. (i)

   If $\operatorname{\textnormal{CI-dim}}_{R}(M)<\infty$, then it follows that $\operatorname{cx}_{R}(M)\leq\operatorname{embdim}(R)-\operatorname{depth}(R)$; see [10, 5.6].

2. (ii)

   If $\operatorname{\textnormal{CI-dim}}_{R}(M)=0$, then it follows $\operatorname{\textnormal{CI-dim}}_{R}(M^{\ast})=0$ and $\operatorname{cx}_{R}(M)=\operatorname{cx}_{R}(M^{\ast})$; see [12, 4.2]. ∎

Let $R$ be a local ring and let $N$ be an $R$-module. Then $[N]=0$ in $\overline{\operatorname{G}}(R)_{\mathbb{Q}}$ for each one of the following cases:

1. (i)

   $N=\Omega^{n}_{R}(M)$ for some $n\geq 0$ and for some $R$-module $M$ such that $\operatorname{\textup{length}}_{R}(M)<\infty$.

2. (ii)

   $R$ is Artinian.

3. (iii)

   $R$ has dimension one and $N$ has rank.

4. (iv)

   $R$ is a one-dimensional domain.

5. (v)

   $R$ is a two-dimensional normal domain with torsion class group.

6. (vi)

   $R$ is a two-dimensional complete rational singularity such that $k$ is algebraically closed of characteristic zero, or $k$ is finite of positive characteristic, or $k$ is the algebraic closure of a finite field which has positive characteristic.