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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 pointed out, this is not a property of maximal CM modules. Why, then, are canonical modules supported everywhere?

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2 Answers 2

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See (1.7) on page 87 of Some basic results on canonical modules. For a local CM ring condition (b) there holds.

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    $\begingroup$ I just realized that $\operatorname{Supp}\omega=\operatorname{Spec}A$ can be deduced from the isomorphism $\operatorname{Hom}(\omega,\omega)\simeq A$! $\endgroup$
    – ashpool
    Commented Jan 1, 2012 at 23:09
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    $\begingroup$ In general, $\text{Hom}(\omega, \omega)$ is the S2-ification of $A$ if I recall correctly (see the linked paper of Aoyama, or a different paper by the same author). $\endgroup$
    – Karl Schwede
    Commented Jan 1, 2012 at 23:27
  • $\begingroup$ @ Mahdi Majidi-Zolbanin $\operatorname{Ass}\omega=\operatorname{Ass}A$ from (1.7) also follows from the isomorphism $\operatorname{Hom}(\omega,\omega)=A$. Thanks for the reference! $\endgroup$
    – ashpool
    Commented Jan 1, 2012 at 23:38
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Just a little more information with regards to your first question.

Canonical modules make sense for any local ring with a dualizing complex (for example, a complete ring). In that case, I would define the canonical module to be the first nonzero cohomology of the dualizing complex. If the ring is not Cohen-Macaulay they need not always localize well however (things are fine in a domain regardless). For example, the canonical module of the ring

$$R = k[[x,y,z]]/\langle x \rangle \cap \langle y, z \rangle$$

is only be supported at one of the minimal primes of $R$. The dualizing complex behaves better though.

In the non-local setting, even weirder things can happen (even in domains).

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