# Are dualizing modules stable under localization

Let $(R,m,k)$ be a (noetherian) regular local ring of depth=dimension $d$, and let $D$ be a dualizing module for $R$ (say, the injective envelope of $R/m$).

Then is $D_p$ dualizing for $R_p$ for any prime $p$ of $R$ (more generally, if $R$ is Gorenstein and $p$ is a prime such that $R_p$ is also Gorenstein)? If it is true, could I have a reference for a proof?

Bump!

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it may be time to ask your question on Math Overflow. –  Pete L. Clark Mar 22 '11 at 20:31
@Pete L. Clark: I found out that this is actually true. It's an exercise in section 9.5 (9.6 maybe; the section on local duality anyhow) in Enochs Relative homological algebra, but I'm still not sure on how to prove it. –  dtripleez Mar 22 '11 at 22:40
See for example, Hartshorne - Residues and duality, Chapter V, Corollary 2.3 for a proof that being a dualizing complex is a local property. –  the L Apr 27 '11 at 12:59

## 1 Answer

I hope by dualizing module you mean canonical module. Please check page no 110 theorem 3.3.5 of the book Cohen Macaulay ring by Bruns and Herzog. Please do inform me if you have still difficulties.

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