Igusa-Todorov distances

TL;DR

The paper introduces the Igusa-Todorov distance, a new homological measure for Artin algebras, establishing its finiteness, invariance under various equivalences, and its relation to algebraic properties like the Loewy length.

Contribution

It defines the Igusa-Todorov distance, proves its finiteness for all Artin algebras, and shows its invariance under several algebraic equivalences, connecting it to existing homological invariants.

Findings

Every Artin algebra has finite Igusa-Todorov distance.

The distance is preserved under stable and singular equivalences.

Bounds for the distance are established in recollement situations.

Abstract

A new homological dimension, called the Igusa-Todorov distance, is introduced to measure how far an Artin algebra is from being an Igusa-Todorov algebra. An upper bound for the dimension is established in terms of the Loewy length, leading to the conclusion that every Artin algebra has a finite Igusa-Todorov distance.Using this dimension, we derive an upper bound for the dimension of the singularity category. Furthermore, we investigate how the Igusa-Todorov distance behaves under various relationships between algebras. Specifically, we demonstrate that stable equivalences preserve the Igusa-Todorov distances for algebras without nodes, prove that it is an invariant under singular equivalence of Morita type with level, and establish bounds for the distances of algebras involved in a recollement of derived module categories. Consequently, the Igusa-Todorov distance is an invariant under derived equivalences of algebras.