In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves.
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 be the completion of its dualizing module. Then for any R-module M there is an isomorphism of modules over the completion of R:
| i(M,\overline\Omega) | |
| \operatorname{Ext} | |
| R |
\cong\operatorname{Hom}R(H
| d-i | |
| m |
(M),E(k))
where Hm 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.