Cohen-Macaulay Noetherian algebras

Abstract

Let (R, m) be a d-dimensional commutative complete Noetherian local ring and A be a Noetherian R-algebra. Motivated by the notion of Cohen-Macaulay Artin algebras of Auslander and Reiten, we say that A is Cohen-Macaulay if there is a finitely generated A-bimodule w that is maximal Cohen-Macaulay over R such that the adjoint pair of functors (w circle times?' -, Hom?' (w, -)) induces quasi-inverse equivalences between the full subcategories of finitely generated A'-modules consisting of modules of finite projective dimension, P degrees degrees(A'), and the modules of finite injective dimension, I degrees degrees(A'), whenever A' = A, Aop. It is proved that such a module w is unique, up to isomorphism, as a A'-module. It is also shown that A is a Cohen-Macaulay algebra if and only if there is a semidualizing A-bimodule w of finite injective dimension and P degrees degrees(A') and I degrees degrees(A') are contained in the Auslander and Bass classes, respectively. We prove that Cohen-Macaulayness behaves well under reduction modulo system of parameters of R. Indeed, it will be observed that if A is a Cohen-Macaulay algebra, then for any system of parameters x = x1, ... , xd of R, the Artin algebra A/xA is Cohen-Macaulay as well. Assume that w is a semidualizing A-bimodule of finite injective dimension that is maximal Cohen-Macaulay as an R-module. It will turn out that A being a Cohen-Macaulay algebra is equivalent to saying that the pair (CM(A'),I degrees degrees(A')) forms a hereditary complete cotorsion theory and the pair (CM(A'op), P degrees degrees(A')) forms a Tor-torsion theory, where CM(A') is the class of all finitely generated A'-modules admitting a right resolution by modules in addw. Finally, it is shown that Cohen-Macaulayness ascends from R to RF and RQ, where F is a finite group and Q is a finite acyclic quiver.