Some criteria for Gorensteinness via Gorenstein projective cotorsion pairs

TL;DR

This paper characterizes when a noetherian algebra over a Cohen--Macaulay ring is Gorenstein using Gorenstein projective cotorsion pairs, linking module categories and homological properties.

Contribution

It provides new criteria for Gorensteinness via cotorsion pairs and characterizes Gorenstein local rings through Ext-orthogonality conditions.

Findings

Finitely generated Gorenstein projective modules form a hereditary cotorsion pair.

Characterization of left weakly Gorenstein rings in terms of Gorenstein projective modules.

A Cohen--Macaulay local ring is Gorenstein iff certain Ext-orthogonality conditions hold.

Abstract

Let $R$ be a noetherian algebra over a Cohen--Macaulay ring admitting a canonical module, and assume that $R$ is maximal Cohen--Macaulay over the base ring. We provide a characterization of when $R$ is left weakly Gorenstein. We further show that the category of finitely generated Gorenstein projective $R$-modules coincides with the left $\Ext$-orthogonal class of the thick subcategory generated by finitely generated $R$-modules of finite projective or finite injective dimension. As a consequence, finitely generated Gorenstein projective $R$-modules generate a hereditary cotorsion pair. Moreover, we show that a Cohen--Macaulay local ring is Gorenstein if and only if the right $\Ext$-orthogonal class of finitely generated Gorenstein projective modules coincides with the category of finitely generated modules of finite projective dimension.