Delooping levels

TL;DR

This paper introduces a new homological invariant called Dell for Artin algebras, compares it with existing dimensions, and explores its properties and differences, including cases where it diverges significantly from finitistic dimension.

Contribution

The paper defines the Dell invariant, relates it to other homological dimensions, generalizes existing theorems to truncated path algebras, and demonstrates its potential for large divergence from finitistic dimension.

Findings

Dell can be arbitrarily larger than finitistic dimension for monomial algebras.

The new invariant relates to and extends previous homological dimensions.

Generalization of key theorems to broader classes of algebras.

Abstract

In [8] V. G\'elinas introduced a homological invariant, called {\it delooping level} (dell), that bounds the finitistic dimension. In this article, we introduce another homological invariant (Dell) related to the delooping level for an Artin algebra. We compare this new tool with other dimensions as the finitistic dimension or the $\phi$-dimension (where $\phi$ is the first Igusa-Todorov function), and we also generalize Theorem 4.3. from [9] to truncated path algebras (Theorem 4.18). Finally, we show that for a monomial algebra $A$ the difference dell($A$) - Findim($A$) can be arbitrarily large (Example 4.22).