A classification of derived-discrete graded algebras

Abstract

A finite-dimensional algebra is derived-discrete, in the sense of Vossieck, precisely when it is a piecewise hereditary algebra of Dynkin type or a gentle one-cycle algebra not satisfying the clock condition. Building on the discrete triangulated categories of Broomhead, Pauksztello, and Ploog, we study derived-discreteness for locally finite non-positively graded algebras, regarded as connective locally finite dg algebras with trivial differential. Our main result extends Vossieck's classification to this setting: such a graded algebra is derived-discrete if and only if it is graded Morita equivalent to a piecewise hereditary algebra of Dynkin type, or it is a graded gentle one-cycle algebra not satisfying the graded clock condition. Along the way, using surface models we prove the conjecture of Kalck and Yang that the graded clock condition is invariant under derived equivalence. We also establish a restriction on semi-orthogonal decompositions of the bounded derived category of a path algebra of Dynkin type $D$.