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

${\mathrm{Ext}}_R^i(M,\bar{\mathrm{\Omega }})\cong {\mathrm{Hom}}_R(H_m^{d-i}(M),E(k))$

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

• Template:Citation

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