Cohen–Macaulay Noetherian algebras

Abstract

Let (R,m) be a d-dimensional commutative complete Noetherian local ring and Λ be a Noetherian R-algebra. Motivated by the notion of Cohen–Macaulay Artin algebras of Auslander and Reiten, we say that Λ is Cohen–Macaulay if there is a finitely generated Λ-bimodule ω that is maximal Cohen–Macaulay over R such that the adjoint pair of functors (ω⊗Λ′−,HomΛ′(ω,−)) induces quasi-inverse equivalences between the full subcategories of finitely generated Λ′-modules consisting of modules of finite projective dimension, P∞(Λ′), and the modules of finite injective dimension, I∞(Λ′), whenever Λ′=Λ,Λop. It is proved that such a module ω is unique, up to isomorphism, as a Λ′-module. It is also shown that Λ is a Cohen–Macaulay algebra if and only if there is a semidualizing Λ-bimodule ω of finite injective dimension and P∞(Λ′) and I∞(Λ′) 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 Λ is a Cohen–Macaulay algebra, then for any system of parameters x=x1,…,xd of R, the Artin algebra Λ∕xΛ is Cohen–Macaulay as well. Assume that ω is a semidualizing Λ-bimodule of finite injective dimension that is maximal Cohen–Macaulay as an R-module. It will turn out that Λ being a Cohen–Macaulay algebra is equivalent to saying that the pair (CM(Λ′),I∞(Λ′)) forms a hereditary complete cotorsion theory and the pair (CM(Λ′op),P∞(Λ′)) forms a Tor-torsion theory, where CM(Λ′) is the class of all finitely generated Λ′-modules admitting a right resolution by modules in addω. Finally, it is shown that Cohen–Macaulayness ascends from R to RΓ and RQ, where Γ is a finite group and Q is a finite acyclic quiver.