# 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*:

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

## 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

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