Finitistic Dimension Conjectures via Gorenstein Projective Dimension (original) (raw)

It is a well-known result of Auslander and Reiten that contravariant finiteness of the class P^fin_∞ (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 GP^fin_∞ (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 GP^fin_∞ 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 ...

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