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 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, if B is representation-finite and fin.dim( f ) ≤ 1, then A has finite finitistic dimension. For the case of fin.dim( f ) > 1, we give a sufficient condition for A with 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 K ⊇ rad(A). If both A/I and A/J are A-syzygy-finite, then the finitistic dimension of A is finite.

Homological properties of extensions of abstract and pseudocompact algebras

“…Remark 6.12. In [XX13] and in [Guo19] (see also [Xi04,Xi06,Xi08]) the authors study the finitistic dimension conjecture for extensions of Artin algebras. In particular, a new formulation of the finitistic dimension conjecture in terms of relative homological dimension is given.…”