Finitistic dimension conjecture and extensions of algebras

AbstractAn 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 f:B→A be an extension of Artin algebras. We denote by fin.dim(f) 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 fin.dim(f)≤1, then A has finite finitistic dimension. (2) Suppose that B is representation-finite and 2≤fin.dim(f)<∞. If, for any A-module X with finite projective dimension, AA⊗BX has finite projective dimension, then A has finite finitistic dimension. (3) Suppose that the finitistic dimension of B is finite and fin.dim(f)<∞. If BA 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...