Let $A$ be a local CM ring, and $\omega$ a canonical module of $A$. Here are two properties of $\omega$ from Bruns & Herzog:


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


These properties seem to imply that $\operatorname{Supp}\omega=\operatorname{Spec}A$. As [Graham Leuschke](https://mathoverflow.net/questions/84685/are-maximal-cohen-macaulay-modules-supported-everywhere/84688#84688) pointed out, this is not a property of maximal CM modules. Why, then, are canonical modules supported everywhere?



Why are canonical modules supported everywhere?
See (1.7) on page 87 of [*Some basic results on canonical modules*](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-23/issue-1/Some-basic-results-on-canonical-modules/10.1215/kjm/1250521612.full). For a local CM ring condition (b) there holds.


