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: : \operatorname{Ext}_R^i(M,\overline\Omega) \cong \operatorname{Hom}_R(H_m^{d-i}(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. ==See also==