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It is a well-known result of Auslander and Reiten that contravariantfiniteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ (of finitelygenerated modules of finite projective dimension) over an Artin algebra is asufficient condition for validity of finitistic dimension conjectures.Motivated by the fact that finitistic dimensions of an algebra canalternatively be computed by Gorenstein projective dimension, in this work weexamine the Gorenstein counterpart of Auslander--Reiten condition, namelycontravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ (offinitely generated modules of finite Gorenstein projective dimension), and itsrelation to validity of finitistic dimension conjectures. It is proved thatcontravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$implies validity of the second finitistic dimension conjecture over leftartinian rings. In the more special setting of Artin algebras, however, it isproved that the Auslander--Reiten sufficient condition and its Gorensteincounterpart are virtually equivalent in the sense that contravariant finitenessof the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ implies contravariantfiniteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ over any Artinalgebra, and the converse holds for Artin algebras over which the class$\mathcal{GP}^{\mathrm{fin}}_0$ (of finitely generated Gorenstein projectivemodules) is contravariantly finite.

Finitistic Dimension Conjectures via Gorenstein Projective Dimension