Let $A$ be a local CM ring, and $M$ a maximal CM $A$-module. Is it true that $\operatorname{Supp}M=\operatorname{Spec}A$ ? This suspicion stems from such statements as:


* If $\omega$ is a canonical module of $A$, then $\omega\_{\mathfrak{p}}$ is a canonical module of $A\_{\mathfrak{p}}$ for every $\mathfrak{p}\in\operatorname{Spec}A$.
* If $\omega$ is a canonical module of $A$, then $\mu\_i(\mathfrak{p},\omega)=\delta\_{i}^{\operatorname{ht}\mathfrak{p}}$ for every $\mathfrak{p}\in\operatorname{Spec}A$, where $\mu$ denotes the Bass number.


And am I correct in understanding that maximal CM module is by definition nonzero?



Are maximal Cohen-Macaulay modules supported everywhere?
No. $R = k[x,y]/(xy)$, $M = R/(x)$.


The zero module has infinite depth and support of dimension $-\infty$, so should not be considered MCM.


