Extension dimensions: derived equivalences and stable equivalences

TL;DR

This paper investigates how extension dimensions behave under derived and stable equivalences of algebras, establishing bounds, invariance properties, and conditions for invariance.

Contribution

It introduces bounds on extension dimension differences for derived equivalent algebras and proves invariance under stable equivalences, with conditions for invariance under specific derived equivalences.

Findings

Extension dimension difference is bounded by the minimal length of a tilting complex.

Extension dimension is invariant under stable equivalence.

Provides conditions ensuring invariance under certain derived equivalences.

Abstract

We show that the difference of the extension dimensions of two derived equivalent algebras is bounded above by the minimal length of a tilting complex associated with a derived equivalence, and that the extension dimension is an invariant under the stable equivalence. In addition, we provide two sufficient conditions such that the extension dimension is an invariant under particular derived equivalences.