The strong global dimension of piecewise hereditary algebras

TL;DR

This paper explores the strong global dimension of endomorphism algebras of tilting objects in derived categories of hereditary abelian categories, linking it to tilting mutations and hereditary subcategories.

Contribution

It introduces a new way to compute the strong global dimension using tilting mutations and hereditary subcategories, extending previous understanding in the field.

Findings

Expresses the strong global dimension via sequences of tilting mutations.

Relates the invariant to hereditary abelian generating subcategories.

Provides formulas connecting the dimension to tilting theory.

Abstract

Let T be a tilting object in a triangulated category equivalent to the bounded derived category of a hereditary abelian category with finite dimensional homomorphism spaces and split idempotents. This text investigates the strong global dimension, in the sense of Ringel, of the endomorphism algebra of T. This invariant is expressed using the infimum of the lengths of the sequences of tilting objects successively related by tilting mutations and where the last term is T and the endomorphism algebra of the first term is quasi-tilted. It is also expressed in terms of the hereditary abelian generating subcategories of the triangulated category.