Local ring in mathematics
In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring formula_1 of Krull dimension "d" ; 0 that satisfies any of the following equivalent conditions:
The last condition implies that the localization formula_18 is Cohen–Macaulay for each prime ideal formula_19.
A standard example is the local ring at the vertex of an affine cone over a smooth projective variety. Historically, the notion grew up out of the study of a Buchsbaum ring, a Noetherian local ring "A" in which formula_20 is constant for formula_8-primary ideals formula_5; see the introduction of.
Notes.
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