Relative Igusa-Todorov functions and relative homological dimensions

TL;DR

This paper extends the Igusa-Todorov functions to a relative setting within exact categories, connecting these to various homological dimensions and generalizing classical results in homological algebra.

Contribution

It introduces the $ ext{E}$-relative Igusa-Todorov functions and dimensions, broadening the scope of homological invariants in exact categories and their relation to other dimensions.

Findings

Defined $ ext{E}$-relative Igusa-Todorov functions and dimensions.

Established relationships with relative global and finitistic dimensions.

Generalized classical Igusa-Todorov results using exact structures.

Abstract

We develope the theory of the $\mathcal{E}$-relative Igusa-Todorov functions in an exact $ IT$-context $(\mathcal{C},\mathcal{E}).$ In the case when $\mathcal{C}=$mod$\, (\Lambda)$ is the category of finitely generated left $\Lambda$-modules, for an artin algebra $\Lambda,$ and $\mathcal{E}$ is the class of all exact sequences in $\mathcal{C},$ we recover the usual Igusa-Todorov functions. We use the setting of the exact structures and the Auslander-Solberg relative homological theory to generalise the original Igusa-Todorov's results. Furthermore, we introduce the $\mathcal{E}$-relative Igusa-Todorov dimension and also we obtain relationships with the relative global and relative finitistic dimensions and the Gorenstein homological dimensions.