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Some criteria for Gorensteinness via Gorenstein projective cotorsion pairs

Souvik Dey, Jian Liu, Xue-Song Lu

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.

Some criteria for Gorensteinness via Gorenstein projective cotorsion pairs

Abstract

Let be a noetherian algebra over a Cohen--Macaulay ring admitting a canonical module, and assume that is maximal Cohen--Macaulay over the base ring. We provide a characterization of when is left weakly Gorenstein. We further show that the category of finitely generated Gorenstein projective -modules coincides with the left -orthogonal class of the thick subcategory generated by finitely generated -modules of finite projective or finite injective dimension. As a consequence, finitely generated Gorenstein projective -modules generate a hereditary cotorsion pair. Moreover, we show that a Cohen--Macaulay local ring is Gorenstein if and only if the right -orthogonal class of finitely generated Gorenstein projective modules coincides with the category of finitely generated modules of finite projective dimension.
Paper Structure (5 sections, 28 theorems, 46 equations)

This paper contains 5 sections, 28 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Characterizations of Gorenstein projective modules
  4. Weakly Gorensteinness and virtually Gorensteinness
  5. Gorensteinness via cotorsion pairs

Key Result

Theorem 1.1

(See char-weakly Gor) Let $S$ be a Cohen--Macaulay ring that admits a canonical module $\omega$. Assume that $R$ is a noetherian $S$-algebra and that $R$, viewed as an $S$-module, is maximal Cohen--Macaulay. Choose a short exact sequence in $\mathsf{mod}(R)$\xymatrix{ 0\ar[r] & X\ar[r] & P\ar[r]

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 49 more