# Equivalent definition of maximal Cohen-Macaulay modules over a Gorenstein local ring

$$\newcommand{\Ext}{\mathop{\rm Ext}\nolimits} \newcommand{\depth}{\mathop{\rm depth}\nolimits} \newcommand{\dim}{\mathop{\rm dim}\nolimits}$$

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.

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.