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Module-Theoretic Characterizations of Gorenstein Morphisms

Andrew Soto Levins, Prashanth Sridhar

TL;DR

This work extends the classical, module-theoretic characterizations of Gorenstein rings to the relative and dg-setting by developing a unified framework for non-positive graded commutative dg-algebras. Building on Avramov–Foxby theory and Minamoto–Shaul constructions, it proves a suite of equivalent conditions for Gorenstein dg-algebras, including finiteness of injective and projective dimensions, Auslander bounds, Gorenstein dimension and Gorenstein projective/dg–CM modules, and dualizing-dg-module criteria. The main result provides a comprehensive dg-analogue of Foxby-type characterizations, with derived-category formulations and Koszul-type reductions $A//\underline{x}$ that connect to classical cases via specialization to ordinary rings. These equivalences unify several strands of homological and categorical Gorenstein theory, enabling relative and higher-algebraic perspectives on duality and homological dimensions. The findings have broad implications for understanding Gorenstein morphisms in a dg context and for translating classical results into the language of modern homological algebra.

Abstract

The Gorenstein property in local algebra admits several characterizations via its module category. The goal of this paper is to collect and generalize such characterizations to the relative setting, i.e., to Gorenstein morphisms as defined by [AF92]. We achieve this by proving these characterizations more generally for graded-commutative Gorenstein dg-algebras.

Module-Theoretic Characterizations of Gorenstein Morphisms

TL;DR

This work extends the classical, module-theoretic characterizations of Gorenstein rings to the relative and dg-setting by developing a unified framework for non-positive graded commutative dg-algebras. Building on Avramov–Foxby theory and Minamoto–Shaul constructions, it proves a suite of equivalent conditions for Gorenstein dg-algebras, including finiteness of injective and projective dimensions, Auslander bounds, Gorenstein dimension and Gorenstein projective/dg–CM modules, and dualizing-dg-module criteria. The main result provides a comprehensive dg-analogue of Foxby-type characterizations, with derived-category formulations and Koszul-type reductions that connect to classical cases via specialization to ordinary rings. These equivalences unify several strands of homological and categorical Gorenstein theory, enabling relative and higher-algebraic perspectives on duality and homological dimensions. The findings have broad implications for understanding Gorenstein morphisms in a dg context and for translating classical results into the language of modern homological algebra.

Abstract

The Gorenstein property in local algebra admits several characterizations via its module category. The goal of this paper is to collect and generalize such characterizations to the relative setting, i.e., to Gorenstein morphisms as defined by [AF92]. We achieve this by proving these characterizations more generally for graded-commutative Gorenstein dg-algebras.

Paper Structure

This paper contains 3 sections, 13 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Result

Key Result

Theorem 1.1

Foxby_iso A commutative Noetherian local ring $R$ is Gorenstein if and only if there exists a non-zero finitely generated $R$-module $M$ such that $\operatorname{inj\,dim}_R(M)<\infty$ and $\operatorname{proj\,dim}_R(M)<\infty$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Special case of \ref{['thm:main']}
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: minamoto - Definitions 2.12 and 2.17
  • Remark 2.6
  • Definition 2.7: BSSW
  • Definition 2.8: BSSW
  • ...and 35 more