# 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 [1] for a polynomial ring and by I.S. 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 $A$ 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 [6]).

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 [4]). 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 [7], flag manifolds and generalized flag manifolds [8].

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 $\mathop{\rm dim} M = \mathop{\rm dim} A$.

#### References

 [1] F.S. Macaulay, "The algebraic theory of modular systems" , Cambridge Univ. Press (1916) [2] I.S. Cohen, "On the structure and ideal theory of complete local rings" Trans. Amer. Math. Soc. , 59 (1946) pp. 54–106 [3] O. Zariski, P. Samuel, "Commutative algebra" , 2 , v. Nostrand (1960) [4] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) [5] J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965) [6] M. Hochster, "Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes" Ann. of Math. , 96 (1972) pp. 318–337 [7] M. Hochster, "Grassmannians and their Schubert subvarieties are arithmetically Cohen–Macaulay" J. of Algebra , 25 (1973) pp. 40–57 [8] G.R. Kempf, "Linear systems on homogeneous spaces" Ann. of Math. , 103 (1976) pp. 557–591

For the concepts of depth, dimension, regular local ring, normal ring, Gorenstein ring, cf., respectively, Depth of a module; Dimension; Local ring; Normal ring; Gorenstein ring. For a description of the invertible sheaf ${\mathcal O} _ {X} ( 1)$ cf. also Projective spectrum of a ring, and for a discussion of the local cohomology groups $H _ {m} ^ {i} ( A)$ cf. Local cohomology and Koszul complex.