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''Macaulay ring''
 
''Macaulay ring''
  
A commutative local Noetherian ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c0229701.png" />, the depth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c0229702.png" /> of which is equal to its dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c0229703.png" />. In homological terms, a Cohen–Macaulay ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c0229704.png" /> is characterized as follows: The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c0229705.png" />, or the local cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c0229706.png" />, vanish for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c0229707.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c0229708.png" /> is the maximal ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c0229709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297010.png" /> is the residue field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297011.png" />. An alternative definition utilizes the concept of a regular sequence. A regular sequence is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297012.png" /> of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297013.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297014.png" />, the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297015.png" /> is not a zero divisor in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297016.png" />. A local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297017.png" /> is a Cohen–Macaulay ring if there exists a regular sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297018.png" /> such that the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297019.png" /> is Artinian. In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297020.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297021.png" /> is a prime ideal in a Cohen–Macaulay ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297022.png" />, then its height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297023.png" /> (see [[Height of an ideal|Height of an ideal]]) satisfies the relation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297024.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ht} ( \mathfrak p ) +
 +
\mathop{\rm dim} ( A/ \mathfrak p )  = \
 +
\mathop{\rm dim}  A.
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297025.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297026.png" />-dimensional Cohen–Macaulay ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297027.png" /> a sequence of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297029.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297030.png" /> is a regular sequence and the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297031.png" /> is unmixed, i.e. any prime ideal associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297032.png" /> has height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297033.png" /> and co-height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297034.png" />. 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297035.png" /> is a local Cohen–Macaulay ring, then the same is true of its completion, of the ring of formal power series over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297036.png" /> and of any finite flat extension. A complete intersection of a Cohen–Macaulay ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297037.png" />, i.e. a quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297039.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297040.png" /> (a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297041.png" />) is called a Cohen–Macaulay ring (a Cohen–Macaulay scheme) if for any prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297042.png" /> (respectively, for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297043.png" />) the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297044.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297045.png" />) is a Cohen–Macaulay ring; for example, this is true of any semi-group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297047.png" /> is a convex polyhedral cone in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297048.png" /> (see [[#References|[6]]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297049.png" /> is a finite group acting on a Cohen–Macaulay ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297050.png" />, and if moreover its order is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297051.png" />, then the ring of invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297052.png" /> is also a Cohen–Macaulay ring.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297053.png" /> is a graded ring, the property of being a Cohen–Macaulay ring appears in the cohomology of the invertible sheaves over the projective scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297054.png" /> (see [[#References|[4]]]). If the homogeneous ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297055.png" /> of a cone in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297056.png" /> associated with a projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297057.png" /> is a Cohen–Macaulay ring, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297058.png" /> is called an arithmetical Cohen–Macaulay variety. In that case the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297059.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297060.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297061.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297064.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297065.png" />-th tensor power of the polarized invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297066.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297067.png" />. 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]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297068.png" /> over a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297069.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297070.png" /> has a Cohen–Macaulay module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297071.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297072.png" />.
+
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====
 
====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>
 
<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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297073.png" /> cf. also [[Projective spectrum of a ring|Projective spectrum of a ring]], and for a discussion of the local cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022970/c02297074.png" /> cf. [[Local cohomology|Local cohomology]] and [[Koszul complex|Koszul complex]].
+
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]].

Latest revision as of 17:45, 4 June 2020


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.

How to Cite This Entry:
Cohen-Macaulay ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cohen-Macaulay_ring&oldid=22301
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article