Finitistic dimension conjecture and extensions of algebras

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 $f:B\to A$ be an extension of Artin algebras. We denote by $fin\mathrm{.}dim(f)$ 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 $fin\mathrm{.}dim(f)\le 1$, then $A$ has finite finitistic dimension. For the case of $fin\mathrm{.}dim(f)>1$, 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 $K\supseteq rad(A)$. If both $A\mathrm{/}I$ and $A\mathrm{/}J$ are $A$-syzygy-finite, then the finitistic dimension of $A$ is finite.