Finitistic dimension conjecture and extensions of algebras

Abstract

An extension of algebras is a homomorphism of algebras preserving identities. We use extensions of algebras to study the finitistic dimension conjecture over Artin algebras. Let be an extension of Artin algebras. We denote by the relative finitistic dimension of f, which is defined to be the supremum of relative projective dimensions of finitely generated left A-modules of finite projective dimension. We prove that: (1) If B is representation-finite and , then A has finite finitistic dimension. (2) Suppose that B is representation-finite and . If, for any A-module X with finite projective dimension, has finite projective dimension, then A has finite finitistic dimension. (3) Suppose that the finitistic dimension of B is finite and . If has finite projective dimension and AB is projective, then A has finite finitistic dimension. Also, we prove the following result: Let I, J, K be three ideals of an Artin algebra A such that IJK = 0 and . If both A/I and A/J are A-syzygy-finite, then the finitistic dimension of A is finite.