Why are canonical modules supported everywhere? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:47:06Z http://mathoverflow.net/feeds/question/84695 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84695/why-are-canonical-modules-supported-everywhere Why are canonical modules supported everywhere? ashpool 2012-01-01T19:24:47Z 2012-01-02T03:01:44Z <p>Let $A$ be a local CM ring, and $\omega$ a canonical module of $A$. Here are two properties of $\omega$ from Bruns &amp; Herzog:</p> <ul> <li>$\omega_{\mathfrak{p}}$ is a canonical module of $A_{\mathfrak{p}}$ for every $\mathfrak{p}\in\operatorname{Spec}A$.</li> <li>$\mu_i(\mathfrak{p},\omega)=\delta_{i}^{\operatorname{ht}\mathfrak{p}}$ for every $\mathfrak{p}\in\operatorname{Spec}A$, where $\mu$ denotes the Bass number.</li> </ul> <p>These properties seem to imply that $\operatorname{Supp}\omega=\operatorname{Spec}A$. As <a href="http://mathoverflow.net/questions/84685/are-maximal-cohen-macaulay-modules-supported-everywhere/84688#84688" rel="nofollow">Graham Leuschke</a> pointed out, this is not a property of maximal CM modules. Why, then, are canonical modules supported everywhere?</p> http://mathoverflow.net/questions/84695/why-are-canonical-modules-supported-everywhere/84700#84700 Answer by Mahdi Majidi-Zolbanin for Why are canonical modules supported everywhere? Mahdi Majidi-Zolbanin 2012-01-01T20:47:40Z 2012-01-01T20:47:40Z <p>See (1.7) on page 87 of <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.kjm/1250521612" rel="nofollow"><em>Some basic results on canonical modules</em></a>. For a local CM ring condition (b) there holds.</p> http://mathoverflow.net/questions/84695/why-are-canonical-modules-supported-everywhere/84710#84710 Answer by Karl Schwede for Why are canonical modules supported everywhere? Karl Schwede 2012-01-01T23:26:23Z 2012-01-02T03:01:44Z <p>Just a little more information with regards to your first question.</p> <p>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</p> <p>$$R = k[[x,y,z]]/\langle x \rangle \cap \langle y, z \rangle$$</p> <p>is only be supported at one of the minimal primes of $R$. The dualizing complex behaves better though. </p> <p>In the non-local setting, even weirder things can happen (even in domains). </p>