Finitistic dimension conjecture and radical-power extensions

TL;DR

This paper explores conditions under which extensions of Artin algebras have finite finitistic dimension, contributing new criteria and generalizations to the longstanding finitistic dimension conjecture.

Contribution

It introduces new sufficient conditions involving radical-power extensions and ideal properties that ensure finiteness of finitistic dimensions in algebra extensions.

Findings

If the extension is right-bounded and certain ideal conditions hold, then fin.dim(B) is finite.

A torsionless-finite algebra with specific ideal properties also has finite fin.dim.

Generalizes previous results and offers new methods to identify algebras with finite finitistic dimension.

Abstract

The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field should have finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this paper, we investigate those extensions of Artin algebras in which some radical-power of smaller algebras is a one-sided ideal in bigger algebras. Our results, however, are formulated more generally for an arbitrary ideal: Let $B\subseteq A$ be an extension of Artin algebras and $I$ an ideal of $B$ such that the full subcategory of $B/I$-modules is $B$-syzygy-finite. Then: (1) If the extension is right-bounded (for example, proj.dim$(A_B)$ is finite), $I A\, rad(B)\subseteq B$ and fin.dim$(A)$ is finite, then fin.dim$(B)$ is finite. (2) If $I\, rad(B)$ is a left ideal of $A$ and $A$ is torsionless-finite, then fin.dim$(B)$ is finite. Particularly, if $I$ is specified to a power of the radical of $B$, then our results not only generalize some ones in the literature (see Corollaries 1.3 and 1.4), but also provide some completely new ways to detect algebras of finite finitistic dimensions.