# Cohen-Macaulay ring

Macaulay ring

A commutative local Noetherian ring $A$, the depth $\mathop{\rm prof} A$ of which is equal to its dimension $\mathop{\rm dim} A$. In homological terms, a Cohen–Macaulay ring $A$ is characterized as follows: The groups $\mathop{\rm Ext} _ {A} ^ {i} ( k, A)$, or the local cohomology groups $H _ {\mathfrak m} ^ {i} ( A)$, vanish for all $i < \mathop{\rm dim} A$, where $\mathfrak m$ is the maximal ideal in $A$ and $k$ is the residue field of $A$. An alternative definition utilizes the concept of a regular sequence. A regular sequence is a sequence $a _ {1} \dots a _ {k}$ of elements of $\mathfrak m$ such that, for all $i$, the element $a _ {i}$ is not a zero divisor in $A/( a _ {1} \dots a _ {i - 1 } )$. A local ring $A$ is a Cohen–Macaulay ring if there exists a regular sequence $a _ {1} \dots a _ {k}$ such that the quotient ring $A/( a _ {1} \dots a _ {k} )$ is Artinian. In that case $k = \mathop{\rm prof} A = \mathop{\rm dim} A$.

If $\mathfrak p$ is a prime ideal in a Cohen–Macaulay ring $A$, then its height $\mathop{\rm ht} ( \mathfrak p )$( see Height of an ideal) satisfies the relation

$$\mathop{\rm ht} ( \mathfrak p ) + \mathop{\rm dim} ( A/ \mathfrak p ) = \ \mathop{\rm dim} A.$$

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 $A$ be a $d$- dimensional Cohen–Macaulay ring and $a _ {1} \dots a _ {k}$ a sequence of elements of $A$ such that $\mathop{\rm dim} ( A/( a _ {1} \dots a _ {k} )) = d - k$. Then $a _ {1} \dots a _ {k}$ is a regular sequence and the ideal $\mathfrak A = ( a _ {1} \dots a _ {k} )$ is unmixed, i.e. any prime ideal associated with $\mathfrak A$ has height $k$ and co-height $d - k$. 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 $A$ is a local Cohen–Macaulay ring, then the same is true of its completion, of the ring of formal power series over $A$ and of any finite flat extension. A complete intersection of a Cohen–Macaulay ring $A$, i.e. a quotient ring $A/( a _ {1} \dots a _ {k} )$, where $a _ {1} \dots a _ {k}$ 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 $A$( a scheme $X$) is called a Cohen–Macaulay ring (a Cohen–Macaulay scheme) if for any prime ideal $\mathfrak p \subset A$( respectively, for any point $x \in X$) the local ring $A _ {\mathfrak p}$( respectively, ${\mathcal O} _ {X,x}$) is a Cohen–Macaulay ring; for example, this is true of any semi-group ring $K [ G \cap \mathbf Z ^ {n} ]$, where $G$ is a convex polyhedral cone in $\mathbf R ^ {n}$( see ).

Cohen–Macaulay rings are also stable under passage to rings of invariants. If $G$ is a finite group acting on a Cohen–Macaulay ring $A$, and if moreover its order is invertible in $A$, then the ring of invariants $A ^ {G}$ is also a Cohen–Macaulay ring.

If $A$ is a graded ring, the property of being a Cohen–Macaulay ring appears in the cohomology of the invertible sheaves over the projective scheme $\mathop{\rm Proj} ( A)$( see ). If the homogeneous ring $A$ of a cone in $A ^ {n + 1 }$ associated with a projective variety $X \subset P ^ {n}$ is a Cohen–Macaulay ring, then $X$ is called an arithmetical Cohen–Macaulay variety. In that case the ring $A$ is isomorphic to $\oplus _ {\nu \in \mathbf Z } H ^ {0} ( X, {\mathcal O} _ {X} ( \nu ))$, and $H ^ {i} ( X, {\mathcal O} _ {X} ( \nu )) = 0$ for all $\nu \in \mathbf Z$ and $0 < i < \mathop{\rm dim} X$, where ${\mathcal O} _ {X} ( \nu )$ is the $\nu$- th tensor power of the polarized invertible sheaf ${\mathcal O} _ {X} ( 1)$ on $X$. This property holds for projective spaces and their products, complete intersections, Grassmann manifolds and Schubert subvarieties , flag manifolds and generalized flag manifolds .

A module $M$ over a local ring $A$ 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 $A$ has a Cohen–Macaulay module $M$ such that $\mathop{\rm dim} M = \mathop{\rm dim} A$.

How to Cite This Entry:
Cohen-Macaulay ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohen-Macaulay_ring&oldid=46382
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article