Rigidity dimension - a homological dimension measuring resolutions of algebras by algebras of finite global dimension

TL;DR

This paper introduces a new homological dimension called rigidity dimension to evaluate how well singular finite-dimensional algebras can be resolved by regular ones, with invariance properties under various algebra equivalences.

Contribution

It defines rigidity dimension, establishes bounds using Hochschild cohomology, and proves invariance under stable, Morita, and derived equivalences for certain classes of algebras.

Findings

Rigidity dimension measures resolution quality of singular algebras.

Upper bounds are established via Hochschild cohomology.

Rigidity dimension is invariant under several algebra equivalences.

Abstract

A new homological dimension is introduced to measure the quality of resolutions of `singular' finite dimensional algebras (of infinite global dimension) by `regular' ones (of finite global dimension). Upper bounds are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. Then invariance under stable equivalences is shown to hold, with some exceptions when there are nodes in case of additive equivalences, and without exceptions in case of triangulated equivalences. Stable equivalences of Morita type and derived equivalences, both between self-injective algebras, are shown to preserve rigidity dimension as well.