12 Finitistic Dimension Through Infinite Projective Dimension (original) (raw)

Abstract. We show that an artin algebra Λ having at most three radical layers of infinite projective dimension has finite finitistic dimension, generalizing the known result for algebras with vanishing radical cube. We also give an equivalence between the finiteness of fin.dim.Λ and the finiteness of a given class of Λ-modules of infinite projective dimension. 1. Introduction. Let Λ be an artin algebra, and consider mod Λ the class of finitely generated left Λ-modules. The finitistic dimension of Λ is then defined to be fin.dim. Λ = sup{pdM: M ∈ mod Λ and pdM < ∞}, where pd M denotes the projective dimension of M. It was conjectured by Bass in