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:

${\displaystyle \operatorname {Ext} _{R}^{i}(M,{\overline {\Omega }})\cong \operatorname {Hom} _{R}(H_{m}^{d-i}(M),E(k))}$

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