Weight structures and simple dg modules for positive dg algebras

TL;DR

This paper develops weight structures in triangulated categories generated by compact objects and applies these to establish the existence and uniqueness of simple dg modules over certain dg algebras, linking t-structures and simple-minded objects.

Contribution

It introduces a method to construct weight structures in triangulated categories and applies it to relate simple modules, t-structures, and simple-minded objects in dg categories.

Findings

Existence of weight structures in categories generated by compact objects.

Lifting of simple modules over homology to dg modules in derived categories.

Bijection between hearts of t-structures and simple-minded objects for certain dg algebras.

Abstract

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). In three appendices, we elucidate the relation between Milnor colimits and homotopy colimits and clarify the construction of t-structures from sets of compact objects in triangulated categories as well as the construction of a canonical weight structure on the unbonded derived category of a non positive dg category.