Finitistic dimension conjecture and extensions of algebras

TL;DR

This paper investigates the finitistic dimension conjecture for Artin algebras using algebra extensions, providing new conditions under which the finitistic dimension is finite, especially involving representation-finite algebras and syzygy-finite quotients.

Contribution

It establishes new criteria linking algebra extensions and syzygy-finiteness to the finiteness of the finitistic dimension, advancing understanding of the conjecture.

Findings

Finite finitistic dimension when $B$ is representation-finite and $fin.dim(f) \,\leq\; 1$.

Sufficient conditions for finite finitistic dimension involving $fin.dim(f) > 1$.

Finiteness of finitistic dimension when certain ideals satisfy $IJK=0$ and quotients are $A$-syzygy-finite.

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 $f: B \to A$ 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 $fin.dim(f)\leq 1$, 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 $K\supseteq rad(A)$. If both $A/I$ and $A/J$ are $A$-syzygy-finite, then the finitistic dimension of $A$ is finite.