Grothendieck local duality

Suppose that R is a Cohen–Macaulay local ring of dimension d with maximal ideal m and residue field k = R/m. Let E(k) be a Matlis module, an injective hull of k, and let {{overline|Ω}} be the completion of its dualizing module. Then for any R-module M there is an isomorphism of modules over the completion of R:

: $\backslash operatorname\{Ext\}\_R^i(M,\backslash overline\backslash Omega)\; \backslash cong\; \backslash operatorname\{Hom\}\_R(H\_m^\{d-i\}(M),E(k))$

where H_m is a local cohomology group.

There is a generalization to Noetherian local rings that are not Cohen–Macaulay, that replaces the dualizing module with a dualizing complex.

• Matlis duality

• {{Citation | last1=Bruns | first1=Winfried | last2=Herzog | first2=Jürgen | title=Cohen–Macaulay rings | url=https://books.google.com/books?id=LF6CbQk9uScC | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-41068-7 |mr=1251956 | year=1993 | volume=39}}

Category:Commutative algebra

Category:Duality theories

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