In mathematics, a Cohen-Macaulay ring is a commutative noetherian local ring with Krull dimension equal to its depth (algebra). The depth is always bounded above by the Krull dimension; equality provides some interesting regularity conditions on the ring, enabling some powerful theorems to be proven in this rather general setting.

A non-local ring is called Cohen-Macaulay if all of its local rings are Cohen-Macaulay.

=Examples=

# Every regular local ring is Cohen-Macaulay. # A field (mathematics) is a particular example of a regular local ring, so is Cohen-Macaulay. # If k is a field, then the formal power series ring in one variable k X is a regular local ring and so is Cohen-Macaulay, but is not a field. # Any Gorenstein ring is Cohen-Macaulay. In particular, complete intersection rings are Cohen-Macaulay. # Rational Singularity are Cohen-Macaulay but not necessarily Gorenstein. # Any 0 -dimensional ring is Cohen-Macaulay. # Following the last idea, if k is a field and X is an indeterminate, the ring k [ X ]/( X2) is a 0 -dimensional local ring and so is Cohen-Macaulay, but it is not regular. # If k is a field, then the formal power series ring k t2, t3 , where t is an indeterminate, is an example of a 1 -dimensional local ring which is not regular but is Gorenstein, so is Cohen-Macaulay. # If k is a field, then the formal power series ring k t3, t4, t5 , where t is an indeterminate, is an example of a 1 -dimensional local ring which is not Gorenstein but is Cohen-Macaulay. # More generally, any 1 -dimensional (Noetherian local) integral domain is Cohen-Macaulay.

The naming here is, in part, for F. S. Macaulay, who worked in elimination theory. The other half is for Irving S. Cohen, one of Zariski s students from his days at Johns Hopkins University.

One meaning of the Cohen-Macaulay condition is seen in coherent duality theory, where it corresponds to the dualizing object , which a priori lies in a derived category, being represented by a single module (coherent sheaf). The finer Gorenstein condition is then expressed by this module being projective (an invertible sheaf). Non-singularity (regularity) is still stronger--it corresponds to the notion of smoothness of a geometric object at a particular point. Thus, in a geometric sense, the notions of Gorenstein and Cohen-Macaulay capture increasingly larger sets of points than the smooth ones, points which are not necessarily smooth but behave in many ways like smooth points anyway.

=References=

Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry (Springer)

=External link=