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On modules whose dual is of finite Gorenstein dimension

Victor D. Mendoza-Rubio, Victor H. Jorge-Pérez

TL;DR

This work investigates when the dual of a finitely generated module over a Noetherian ring has finite Gorenstein dimension and derives strong homological criteria under this assumption. A central result connects $k$-torsionfreeness, $k$-syzygies, and the $(\widetilde{S}_k)$ condition via Ext vanishing, with precise equivalences involving total reflexivity and the finiteness of $\operatorname{G-dim}_R(M^*)$. The authors then establish freeness and Gorenstein criteria, including AR-type conclusions, under dual-finiteness hypotheses, and apply these findings to Kähler differentials and derivations to address generalized Herzog–Vasconcelos conjectures on the freeness of differential modules. The paper also develops global and local machinery for analyzing $\Omega^{(n)}_{X/Y}$ and $\operatorname{Der}^n_Y(X)$ on schemes, yielding partial results toward the Lipman–Zariski and GHVC questions. Overall, the results bridge dual-Gorenstein-dimension conditions with module structure (free/totally reflexive) and regularity questions in both algebraic and geometric contexts.

Abstract

In this paper, we aim to obtain some results under the condition that the dual of a module over a commutative Noetherian ring has finite Gorenstein dimension. In this direction, we derive results involving vanishing of Ext as well as the freeness or totally reflexivity of modules. For instance, we provide a generalization of a celebrated theorem by Auslander and Bridger, obtain criteria for the totally reflexivity of modules over Cohen-Macaulay rings as well as of locally totally reflexive modules on the punctured spectrum, and recover a result by Araya. Moreover, we prove that the Auslander-Reiten conjecture holds true for all finitely generated modules $M$ over a commutative Noetherian ring $R$ such that $\operatorname{G-dim}_R(\operatorname{Hom}_R(M,R))<\infty$ and $\operatorname{pd}_R(\operatorname{Hom}_R(M,M))<\infty$. Additionally, we derive Gorenstein criteria under the condition that the dual of certain modules is of finite Gorenstein dimension. Furthermore, we explore some applications in the theory of the modules of Kähler differentials of order $n\geq 1$, specifically concerning the $k$-torsionfreeness of these modules and the Herzog-Vasconcelos conjecture.

On modules whose dual is of finite Gorenstein dimension

TL;DR

This work investigates when the dual of a finitely generated module over a Noetherian ring has finite Gorenstein dimension and derives strong homological criteria under this assumption. A central result connects -torsionfreeness, -syzygies, and the condition via Ext vanishing, with precise equivalences involving total reflexivity and the finiteness of . The authors then establish freeness and Gorenstein criteria, including AR-type conclusions, under dual-finiteness hypotheses, and apply these findings to Kähler differentials and derivations to address generalized Herzog–Vasconcelos conjectures on the freeness of differential modules. The paper also develops global and local machinery for analyzing and on schemes, yielding partial results toward the Lipman–Zariski and GHVC questions. Overall, the results bridge dual-Gorenstein-dimension conditions with module structure (free/totally reflexive) and regularity questions in both algebraic and geometric contexts.

Abstract

In this paper, we aim to obtain some results under the condition that the dual of a module over a commutative Noetherian ring has finite Gorenstein dimension. In this direction, we derive results involving vanishing of Ext as well as the freeness or totally reflexivity of modules. For instance, we provide a generalization of a celebrated theorem by Auslander and Bridger, obtain criteria for the totally reflexivity of modules over Cohen-Macaulay rings as well as of locally totally reflexive modules on the punctured spectrum, and recover a result by Araya. Moreover, we prove that the Auslander-Reiten conjecture holds true for all finitely generated modules over a commutative Noetherian ring such that and . Additionally, we derive Gorenstein criteria under the condition that the dual of certain modules is of finite Gorenstein dimension. Furthermore, we explore some applications in the theory of the modules of Kähler differentials of order , specifically concerning the -torsionfreeness of these modules and the Herzog-Vasconcelos conjecture.
Paper Structure (11 sections, 42 theorems, 31 equations)

This paper contains 11 sections, 42 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Setup and preliminaries
  3. A generalization of a theorem of Auslander and Bridger
  4. Characterization of totally reflexives
  5. Freeness Criteria
  6. Gorenstein criteria and related questions
  7. Preliminary facts on Kähler differentials
  8. Global Kähler differentials
  9. Applications
  10. Applications on the $k$-torsion of the modules of differentials
  11. Applications on Herzog-Vasconcelos's conjecture

Key Result

Theorem 1.1

Let $k$ be a non-negative integer. Let $M$ be a finitely generated $R$-module such that $M^\ast$ has locally finite Gorenstein dimension on $\widetilde{X}^{k-1}(R)$. Then:

Theorems & Definitions (81)

  • Theorem 1.1: See Theorem \ref{['GteosPa']}
  • Corollary 1.2: HomologicalDimensionsOfRigidModules
  • Theorem 1.3: See Theorems \ref{['pos5']}, \ref{['genCM']} and \ref{['seaa1']}
  • Conjecture 1.4: Auslander-Reiten
  • Theorem 1.5: TheAuslander-ReitenconjectureforGorensteinrings
  • Theorem 1.6: See Corollary \ref{['432c']}
  • Theorem 1.7: See Theorem \ref{['rpsCM']}
  • Theorem 1.8: See Theorem \ref{['fjk']}
  • Theorem 1.9: See Propositions \ref{['Gorentein2']}, \ref{['sq884']} and \ref{['Gdual']}, and Theorem \ref{['GG']}
  • Definition 2.1
  • ...and 71 more