Truncation of dg categories and connective resolutions

Abstract

We study the truncation $(-)^{\leq0}$ of dg categories. We first show that given a dg category $\mathscr{C}$ with shifts, for example a pretriangulated dg category, the canonical functor $\mathscr{C}^{\leq0}\to\mathscr{C}$ is a localization whose kernel is compactly generated by the $0$-th cohomology. Next we demonstrate that the derived category of the truncation serves as a triangulated analogue of the Auslander's category of coherent functors over abelian categories. We give a description of the Auslander-Reiten-Serre duality in terms of the inverse dualizing bimodule of the truncation. Building on truncations, we introduce the notion of connective resolutions of dg categories, defined as a localization functor from a connective dg category. This notion encompasses non-commutative resolutions of algebras in module categories, cluster tilting objects in triangulated categories, and also the truncation as the universal connective resolution. We give a sufficient condition for an object in a triangulated category to give a connective resolution of its enhancement in terms of finiteness of resolution dimension. Furthermore, we show that every proper dg module over a connective dg algebra is a direct summand of a dg module giving a connective resolution. Consequently, for every proper connective dg algebra, its bounded dg derived category has a connective resolution. Finally, we give some explicit description of the truncation for some categories including the derived and cluster categories of a Dynkin quiver, and the Yoneda category of a finite dimensional algebra.