Cohen–Macaulay Noetherian algebras

Abstract

Let $(R,\mathfrak{m})$ 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 $(\mathit{\omega }{\otimes }_{{\mathrm{\Lambda }}^{\prime }}-,{\mathsf{Hom}}_{{\mathrm{\Lambda }}^{\prime }}(\mathit{\omega },-))$ induces quasi-inverse equivalences between the full subcategories of finitely generated ${\mathrm{\Lambda }}^{\prime }$-modules consisting of modules of finite projective dimension, ${\mathcal{P}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime })$, and the modules of finite injective dimension, ${\mathcal{I}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime })$, whenever ${\mathrm{\Lambda }}^{\prime }=\mathrm{\Lambda },{\mathrm{\Lambda }}^{\mathrm{op}}$. It is proved that such a module ω is unique, up to isomorphism, as a ${\mathrm{\Lambda }}^{\prime }$-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 ${\mathcal{P}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime })$ and ${\mathcal{I}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime })$ 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 $x=x_1,\dots ,x_d$ of R, the Artin algebra $\mathrm{\Lambda }∕x\mathrm{\Lambda }$ 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 $(\mathsf{CM}({\mathrm{\Lambda }}^{\prime }),{\mathcal{I}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime }))$ forms a hereditary complete cotorsion theory and the pair $(\mathsf{CM}({\mathrm{\Lambda }}^{\prime \mathrm{op}}),{\mathcal{P}}^{\mathrm{\infty }}({\mathrm{\Lambda }}^{\prime }))$ forms a $\mathsf{Tor}$-torsion theory, where $\mathsf{CM}({\mathrm{\Lambda }}^{\prime })$ is the class of all finitely generated ${\mathrm{\Lambda }}^{\prime }$-modules admitting a right resolution by modules in $\mathsf{add}\mathit{\omega }$. Finally, it is shown that Cohen–Macaulayness ascends from R to $R\mathrm{\Gamma }$ and $R\mathcal{Q}$, where Γ is a finite group and $\mathcal{Q}$ is a finite acyclic quiver.