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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\operatorname {G-dim}_R(\operatorname {Hom}_R(M,R))<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>G-dim</mo> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mo>Hom</mo> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\operatorname {pd}_R(\operatorname {Hom}_R(M,M))<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>pd</mo> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mo>Hom</mo> <mi>R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. Additionally, we derive Gorenstein criteria under the condition that the dual of certain modules is of finite Gorenstein dimension. Furthermore, we explore some applications in the theory of the modules of Kähler differentials of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, specifically concerning the k-torsionfreeness of these modules and the Herzog-Vasconcelos conjecture.

On modules whose dual is of finite Gorenstein dimension

期刊论文
发表日期：

2025年1月27日

查看全文

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

摘要

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 $ ${G-dim}_{R} ({Hom}_{R} (M, R)) < \infty$ and $\operatorname {pd}_R(\operatorname {Hom}_R(M,M))<\infty $ ${pd}_{R} ({Hom}_{R} (M, M)) < \infty$ . 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 $n\ge 1$ $n \geq 1$ , specifically concerning the k-torsionfreeness of these modules and the Herzog-Vasconcelos conjecture.