CohenMacaulayness

Cohen-Macaulayness is a homological regularity condition in commutative algebra and combinatorics, applicable to rings, modules, and simplicial complexes. In the algebraic setting, a finitely generated module M over a Noetherian local ring (R, m) is called Cohen-Macaulay if the depth of M equals its Krull dimension, written depth M = dim M. Depth is the length of a maximal M-regular sequence contained in m. For a nonlocal ring, R is Cohen-Macaulay if every localization R_p at a prime p is Cohen-Macaulay; equivalently, depth R_p = dim R_p for all p. The property extends to graded rings using the homogeneous maximal ideal.

Typical examples include regular local rings and polynomial rings over a field, both of which are Cohen-Macaulay. If I is generated by a regular sequence, then the quotient R/I is Cohen-Macaulay. Complete intersections are Cohen-Macaulay as well. These examples illustrate that Cohen-Macaulayness often arises from favorable regularity conditions.

In combinatorics and topology, Cohen-Macaulayness has a parallel notion for simplicial complexes. A simplicial complex Delta is Cohen-Macaulay over a field k (when Delta is considered with coefficients in k) if, for every face F, the reduced homology of the link of F vanishes in all degrees i less than the dimension of the link; equivalently, the Reisner criterion states H̃_i(link_Δ(F); k) = 0 for i < dim(link_Δ(F)) and all faces F. Shellable pure complexes are Cohen-Macaulay, and many naturally occurring complexes have this property.

Applications of Cohen-Macaulayness include depth and dimension theory, duality results in local cohomology, and influential uses in algebraic geometry, invariant theory, and combinatorial commutative algebra.