Perfect Derived Categories of Positively Graded DG Algebras

Abstract

We investigate the perfect derived category \({{\rm dgPer}}(\mathcal{A})\) of a positively graded differential graded (dg) algebra \(\mathcal{A}\) whose degree zero part is a dg subalgebra and semisimple as a ring. We introduce an equivalent subcategory of \({{{\rm dgPer}}}(\mathcal{A})\) whose objects are easy to describe, define a t-structure on \({{{\rm dgPer}}}(\mathcal{A})\) and study its heart. We show that \({{{\rm dgPer}}}(\mathcal{A})\) is a Krull–Remak–Schmidt category. Then we consider the heart in the case that \(\mathcal{A}\) is a Koszul ring with differential zero satisfying some finiteness conditions.