# Cohen-Macaulay module

A module $M$ over a ring $R$ is a Cohen-Macaulay module if its depth is defined and equals its Krull dimension. A ring is said to be Cohen-Macaulay (or just C-M) if it is a Cohen-Macaulay module viewed as a module over itself.

Cohen-Macaulay rings are used extensively in combinatorial geometry and commutative ring theory, and has applications to algebraic geometry as well. For instance, a variety all of whose local rings are Cohen-Macaulay has, in a sense, nicer behaviour than an arbitrary singular variety.

Title Cohen-Macaulay module CohenMacaulayModule 2013-03-22 14:14:58 2013-03-22 14:14:58 mathcam (2727) mathcam (2727) 6 mathcam (2727) Definition msc 13C14 msc 16E65 Cohen-Macaulay ring C-M module C-M ring