Homological Invariants of Left and Right Serial Path Algebras

TL;DR

This paper explores the relationship between delooping level and finitistic dimension in serial path algebras, introducing new invariants like subddell and ddell that improve upon the delooping level.

Contribution

It establishes the connection between delooping level and finitistic dimension in serial path algebras and introduces subddell and ddell as refined invariants.

Findings

Delooping level can be computed with a finite algorithm.

Right serial algebras have equal right finitistic dimension and left delooping level.

Subddell and ddell often outperform delooping level in certain contexts.

Abstract

We investigate the relationship between the delooping level (dell) and the finitistic dimension of left and right serial path algebras. These 2-syzygy finite algebras have finite delooping level, and it can be calculated with an easy and finite algorithm. When the algebra is right serial, its right finitistic dimension is equal to its left delooping level. When the algebra is left serial, the above equality only holds under certain conditions. We provide examples to demonstrate this and include discussions on the sub-derived (subddell) and derived delooping level (ddell). Both subddell and ddell are improvements of the delooping level. We motivate their definitions and showcase how they can behave better than the delooping level in certain situations throughout the paper.