# Cohen-Macaulay ring

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Macaulay ring

A commutative local Noetherian ring , the depth of which is equal to its dimension . In homological terms, a Cohen–Macaulay ring is characterized as follows: The groups , or the local cohomology groups , vanish for all , where is the maximal ideal in and is the residue field of . An alternative definition utilizes the concept of a regular sequence. A regular sequence is a sequence of elements of such that, for all , the element is not a zero divisor in . A local ring is a Cohen–Macaulay ring if there exists a regular sequence such that the quotient ring is Artinian. 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.e. any prime ideal associated with has height and co-height . The unmixedness theorem was proved by F.S. Macaulay  for a polynomial ring and by I.S. Cohen  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.e. a quotient ring , where is a regular sequence, is a Cohen–Macaulay ring. 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 ).

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.

If is a graded ring, the property of being a Cohen–Macaulay ring appears in the cohomology of the invertible sheaves over the projective scheme (see ). If the homogeneous ring of a cone in associated with a projective variety is a Cohen–Macaulay ring, then is called an arithmetical Cohen–Macaulay variety. In that case the ring is isomorphic to , and for all and , where is the -th tensor power of the polarized invertible sheaf on . This property holds for projective spaces and their products, complete intersections, Grassmann manifolds and Schubert subvarieties , flag manifolds and generalized flag manifolds .

A module over a local ring is called a Cohen–Macaulay module if its depth equals its dimension. Many results for Cohen–Macaulay rings carry over to Cohen–Macaulay modules; for example, the support of such a module is equi-dimensional. It has been conjectured that any local complete ring has a Cohen–Macaulay module such that .

How to Cite This Entry:
Cohen-Macaulay ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohen-Macaulay_ring&oldid=16394
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article