Grothendieck local duality

In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves.
Statement

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 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:

\( \operatorname{Ext}_R^i(M,\overline\Omega) \cong \operatorname{Hom}_R(H_m^{d-i}(M),E(k)) \)

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.
See also

Matlis duality

References

Bruns, Winfried; Herzog, Jürgen (1993), Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, ISBN 978-0-521-41068-7, MR 1251956

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