Unmixedness theorem

In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well-understood in many ways.

They are named for Francis Sowerby Macaulay (1916), who proved the unmixedness theorem for polynomial rings, and for Irvin Cohen (1946), who proved the unmixedness theorem for formal power series rings. All Cohen–Macaulay rings have the unmixedness property.

For Noetherian local rings, there is the following chain of inclusions.

For a commutative Noetherian local ring R, the depth of R (the maximum length of a regular sequence in the maximal ideal of R) is at most the Krull dimension of R. The ring R is called Cohen–Macaulay if its depth is equal to its dimension.

More generally, a commutative ring is called Cohen–Macaulay if is Noetherian and all of its localizations at prime ideals are Cohen–Macaulay. In geometric terms, a scheme is called Cohen–Macaulay if it is locally Noetherian and its local ring at every point is Cohen–Macaulay.

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