Global dimension of dg algebras via compact silting objects

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.