Finitistic Dimension Conjectures via Gorenstein Projective Dimension

TL;DR

This paper explores the relationship between Gorenstein projective dimensions and finitistic dimension conjectures, establishing conditions under which these conjectures hold for Artin algebras and left artinian rings.

Contribution

It introduces the Gorenstein counterpart of the Auslander--Reiten condition and proves its implications for finitistic dimension conjectures in algebra.

Findings

Contravariant finiteness of Gorenstein modules implies the second finitistic dimension conjecture.

In Artin algebras, Gorenstein and classical conditions are equivalent under certain finiteness assumptions.

The classical and Gorenstein conditions are virtually equivalent for Artin algebras with finitely generated Gorenstein projective modules.

Abstract

It is a well-known result of Auslander and Reiten that contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander--Reiten condition, namely contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander--Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in the sense that contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ implies contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ over any Artin algebra, and the converse holds for Artin algebras over which the class $\mathcal{GP}^{\mathrm{fin}}_0$ (of finitely generated Gorenstein projective modules) is contravariantly finite.