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Corresponding author. E-mail: takahashi math.

## Cohen-Macaulay modules over Cohen-Macaulay rings - Semantic Scholar

Cite Citation. Permissions Icon Permissions. Abstract Let R be a Cohen—Macaulay local ring. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals. Issue Section:.

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Sign in via your Institution Sign in. All Cohen—Macaulay rings have the unmixedness property. 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 it 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.

## Cohen-Macaulay ring

Rational singularities over a field of characteristic zero are Cohen—Macaulay. Toric varieties over any field are Cohen—Macaulay. Then the section ring of L. See also Generalized Cohen—Macaulay ring. Cohen-Macaulay schemes have a special relation with intersection theory.

Precisely, let X be a smooth variety [6] and V , W closed subschemes of pure dimension. In general, that multiplicity is given as a length essentially characterizes Cohen—Macaulay ring; see Properties.

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Multiplicity one criterion , on the other hand, roughly characterizes a regular local ring as a local ring of multiplicity one. For a simple example, if we take the intersection of a parabola with a line tangent to it, the local ring at the intersection point is isomorphic to. In that case. If is a prime ideal in a Cohen—Macaulay ring , then its height see Height of an ideal satisfies the relation. In particular, a Cohen—Macaulay ring is equi-dimensional and it is a catenary ring. A fundamental result on Cohen—Macaulay rings is the following unmixedness theorem. Let be a -dimensional Cohen—Macaulay ring and a sequence of elements of such that.

Then is a regular sequence and the ideal is unmixed, i. The unmixedness theorem was proved by F. Macaulay [1] for a polynomial ring and by I. Cohen [2] for a ring of formal power series. Examples of Cohen—Macaulay rings. A regular local ring and, in general, any Gorenstein ring is a Cohen—Macaulay ring; any Artinian ring, any one-dimensional reduced ring, any two-dimensional normal ring — all these are Cohen—Macaulay rings. If is a local Cohen—Macaulay ring, then the same is true of its completion, of the ring of formal power series over and of any finite flat extension.

A complete intersection of a Cohen—Macaulay ring , i.

Finally, the localization of a Cohen—Macaulay ring in a prime ideal is again a Cohen—Macaulay ring. This makes it possible to extend the definition of a Cohen—Macaulay ring to arbitrary rings and schemes. Indeed, a Noetherian ring a scheme is called a Cohen—Macaulay ring a Cohen—Macaulay scheme if for any prime ideal respectively, for any point the local ring respectively, is a Cohen—Macaulay ring; for example, this is true of any semi-group ring , where is a convex polyhedral cone in see [6].

Cohen—Macaulay rings are also stable under passage to rings of invariants. If is a finite group acting on a Cohen—Macaulay ring , and if moreover its order is invertible in , then the ring of invariants is also a Cohen—Macaulay ring.