View source for Cohen-Macaulay ring

<!--
c0229701.png
$#A+1 = 74 n = 0
$#C+1 = 74 : ~/encyclopedia/old_files/data/C022/C.0202970 Cohen\ANDMacaulay ring,
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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

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