CohenMacaulay

Cohen–Macaulay is a central notion in commutative algebra and algebraic geometry. For a Noetherian local ring (R, m) and a finitely generated R-module M, depth_R M is the length of a maximal regular sequence in m acting on M. The module M is Cohen–Macaulay if depth_R M = dim_R M, where dim_R M is the Krull dimension of Supp(M). When M = R, the ring R is Cohen–Macaulay if depth R = dim R.

Equivalently, a Noetherian local ring R is Cohen–Macaulay if there exists a system of parameters that forms an R-regular sequence; for the module case, M is Cohen–Macaulay if some system of parameters for R/Ann(M) is a regular sequence on M. The property is stable under several standard constructions and has graded analogues for finitely generated graded rings.

Many rings that arise in practice are Cohen–Macaulay. Regular local rings are Cohen–Macaulay; complete intersections are Cohen–Macaulay; polynomial rings over a Cohen–Macaulay ring are Cohen–Macaulay; quotients by nonzerodivisors preserve the property. In algebraic geometry, the Cohen–Macaulay condition often leads to nicer homological and geometric behavior and interacts with local cohomology and intersection theory.

Not all Noetherian rings are Cohen–Macaulay. A classical example of a non-Cohen–Macaulay local ring is k[x,y]/(x^2, xy), viewed at the maximal ideal (x,y); in this ring, depth is 0 while dimension is 1.

Graded and local variants of Cohen–Macaulayness play a role in the study of singularities, Hilbert functions, and the structure of modules, making CM rings and modules a foundational tool in both theory and applications.