Global dimension of dg algebras via compact silting objects

TL;DR

This paper defines a new notion of global dimension for triangulated categories using compact silting objects, proving its intrinsic nature and analyzing its behavior in dg algebras and quiver contexts.

Contribution

It introduces an intrinsic global dimension for triangulated categories relative to silting objects and explores its properties in connective dg algebras.

Findings

Finiteness of the global dimension is an intrinsic property of the category.

Established bounds for the global dimension of dg quiver algebras.

Linked the regularity of the singularity category to the finiteness of global dimension.

Abstract

We introduce a notion of global dimension for a triangulated category relative to a compact silting object. We prove that the finiteness of this dimension is an intrinsic property of the triangulated category itself and, therefore, independent of the choice of the silting object. Focusing on the setup of connective differential graded (dg) algebras, we analyse the behaviour of global dimension under dg algebra homomorphisms and establish explicit bounds. This allows us to deduce a bound for the global dimension of certain dg quiver algebras. We also relate the regularity of the big singularity category of a proper connective dg algebra to the finiteness of its global dimension.