Arrow reductions for the finitistic dimension conjecture

TL;DR

This paper introduces new techniques for simplifying bound quiver algebras by removing arrows, reducing the Finitistic Dimension Conjecture to smaller algebras, and characterizing when such reductions preserve finiteness of dimensions.

Contribution

It develops novel arrow removal methods that work even in cases where previous techniques failed, and establishes conditions under which algebraic properties are preserved during reduction.

Findings

Arrow removal can reduce the FDC to smaller algebras.

Successive arrow removals lead to a unique arrow reduced algebra.

New constructions preserve finiteness of dimensions during arrow addition.

Abstract

We present new techniques for removing arrows of bound quiver algebras, reducing thus the Finitistic Dimension Conjecture $\mathsf{(FDC)}$ for a given algebra to a smaller one. Unlike the classic arrow removal operation of Green-Psaroudakis-Solberg, our methods allow for removing arrows even when they occur in every generating set for the defining admissible ideal of the algebra. Our first main result establishes an equivalence for the finiteness of the finitistic (and global) dimensions of a ring $\Lambda$ and its quotient $\Lambda/K$, under specific homological and structural conditions on the ideal $K$, in the broader context of left artinian rings. The application of this result to bound quiver algebras suggests the notion of removability for sets of arrows, and we prove that successive arrow removals of this sort lead to a uniquely defined arrow reduced version of the algebra. Towards the opposite direction, we characterize generalized arrow removal algebras via removable multiplicative bimodules, and introduce trivial one-arrow extensions as a novel combinatorial construction for adding arrows while preserving the finiteness of the finitistic (and global) dimensions. All these new techniques are illustrated through various concrete bound quiver algebras, for which confirming the $\mathsf{(FDC)}$ had previously seemed intractable to the best of our knowledge.