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 be the completion of its dualizing module. Then for any "R"-module "M" there is an isomorphism of modules over the completion of "R":
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.