Weight structures and simple dg modules for positive dg algebras

Understand

Using techniques due to Dwyer-Greenlees-Iyengar we construct weight structures in triangulated categories generated by compact objects.

• We apply our result to show that, for a dg category whose homology vanishes in negative degrees and is semi-simple in degree 0, each simple module over the homology lifts to a dg module which is unique up to isomorphism in the derived category.
• This allows us, in certain situations, to deduce the existence of a canonical t-structure on the perfect derived category of a dg algebra.
• From this, we can obtain a bijection between hearts of t-structures and sets of so-called simple-minded objects for some dg algebras (including Ginzburg algebras associated to quivers with potentials).