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

Comments

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.