Kyoto J. Math. 63 (1), 1-22, (February 2023) DOI: 10.1215/21562261-2022-0029
Abdolnaser Bahlekeh, Shokrollah Salarian, Zahra Toosi
KEYWORDS: Cohen–Macaulay Noetherian algebra, dualizing module, Cotorsion theory, Tor-torsion theory, Auslander class, Bass class, 20J05, 20J06, 13H10, 18G20
Let 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 induces quasi-inverse equivalences between the full subcategories of finitely generated -modules consisting of modules of finite projective dimension, , and the modules of finite injective dimension, , whenever . It is proved that such a module ω is unique, up to isomorphism, as a -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 and 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 of R, the Artin algebra 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 forms a hereditary complete cotorsion theory and the pair forms a -torsion theory, where is the class of all finitely generated -modules admitting a right resolution by modules in . Finally, it is shown that Cohen–Macaulayness ascends from R to and , where Γ is a finite group and is a finite acyclic quiver.