View source for Cohen-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" />.

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

<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>

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.

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]]]).

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.

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]]].

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" />.

====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 <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]].