Finitistic dimension conjecture and relative hereditary algebras

Abstract

We introduce the notion of relative hereditary Artin algebras, as a generalization of algebras with representation dimension at most 3. We prove the following results. (1) The relative hereditariness of an Artin algebra is left–right symmetric and is inherited by endomorphism algebras of projective modules. (2) The finitistic dimensions of a relative hereditary algebra and its opposite algebra are finite. As a consequence, the finitistic projective dimension conjecture, the finitistic injective dimension conjecture, the Gorenstein symmetry conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for relative hereditary Artin algebras and endomorphism algebras of projective modules over them (in particular, over algebras with representation dimension at most 3). We also show that the torsionless-finiteness of an Artin algebra is inherited by endomorphism algebras of projective modules, and consequently give a partial answer to the question if the representation dimension of the endomorphism algebra of any projective module over an Artin algebra A is bounded by the representation dimension of A.