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.

单词	CohenMacaulayModule
释义	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.
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