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A module $M$ is a maximal Cohen-Macaulay module over the ring $R$ if $\depth M = \dim R$, where $\dim R$ denotes the Krull dimension of $R$. I've been told that if $R$ is a Gorenstein local ring, then $M$ is a maximal Cohen-Macaulay module if and only if $\Ext^i(M,R)=0$ for all $i \geq 1$. (This is taken as the definition here.)

How can this be proved? And does anyone have a source where this has been proved? Is it true for Gorenstein local rings only, or also for Gorenstein (not necessarily local) rings too?

I primarily work with commutative rings and finitely generated modules.

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1 Answer 1

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Old question, but here is an answer anyway.

For the first, part, namely the case of a commutative local Cohen-Macaulay ring, one can use local duality: if $R$ is a Cohen-Macaulay local ring admitting a canonical module $\omega$ then for any finitely generated $R$-module there are isomorphisms $$\text{Ext}_{R}^{i}(M,\omega)^{\wedge}\simeq \text{Hom}_{R}(H_{\mathfrak{m}}^{d-i}(M),E(k)).$$ In particular $$ \begin{align} M \text{ is MCM} &\iff H_{\mathfrak{m}}^{d-i}(M)=0 \text{ for all }i>0 \\ &\iff \text{Hom}_{R}(H_{\mathfrak{m}}^{d-i}(M),E(k))=0 \text{ for all }i>0 \text{ (as $E(k)$ is an injective cogenerator}) \\ &\iff \text{Ext}_{R}^{i}(M,\omega)^{\wedge}=0 \text{ for all $i>0$ (by local duality)} \\ &\iff \text{Ext}_{R}^{i}(M,\omega)=0 \text{ for all $i>0$} \end{align} $$ where the last if and only if follows from the fact that $\text{Ext}_{R}^{i}(M,\omega)$ is a finitely generated $R$-module and the map $R\rightarrow \hat{R}$ is faithfully flat.

Since $R$ is Gorenstein if and only if $R\simeq \omega$, and the definition given on the Mathoverflow page is equivalent.

For the more general case, one does not have maximal Cohen-Macaulay modules, so a suitable replacement class is needed. Over any ring, one can define Gorenstein projective modules:

Definition Let $R$ be a ring. An $R$-module $M$ is Gorenstein projective if there is an exact complex $\mathbf{P}^{\bullet}$ of projective $R$-modules such that $M= \text{Ker}(\mathbf{P}^{0}\rightarrow \mathbf{P}^{1})$ and $\text{Hom}_{R}(\mathbf{P}^{\bullet},P)$ is exact for any projective $R$-module $P$

While this definition is not the most accessible, this class of modules has good properties over certain classes of ring:

Lemma Let $R$ be a Cohen-Macaulay local ring. Then every finitely generated Gorenstein projective module is MCM. If $R$ is a Gorenstein local ring, then every MCM is a finitely generated Gorenstein projective module.

Over a not-necessarily commutative Gorenstein ring much more can be said; here is the relevant part for finitely generated modules

Lemma Let $R$ be a Gorenstein ring. Then a finitely generated module $M$ is Gorenstein projective if and only if $M$ is reflexive with respect to $(-)^{*}=\text{Hom}_{R}(-,R)$ and $$\text{Ext}_{R}^{i}(M,R)=\text{Ext}_{R}^{i}(M^{*},R)=0$$ for all $i>0$.

Lemma 4.2.2 of Maximal Cohen-Macaulay Modules and Tate-Cohomology Over Gorenstein Rings, Buchweitz shows that over any Gorenstein ring, a finitely generated maximal Cohen-Macaulay module (using the definition given on mathoverflow) is Gorenstein projective, and clearly any Gorenstein projective module is maximal Cohen-Macaulay.

Therefore the definition given on mathoverflow (due to Buchweitz) is a good one and is equivalent to the usual definition over commutative local Gorenstein rings.

The lemmas I quoted are from section 10.2 of Relative Homological Algebra by Enochs and Jenda. It is a good introduction to Gorenstein modules and their homological algebra.

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