A derived Gabriel-Popescu theorem for t-structures via derived injectives

TL;DR

This paper establishes a derived Gabriel-Popescu theorem within dg-categories and t-structures, showing that certain pretriangulated dg-categories can be realized as localizations of derived dg-categories, with applications to dg-enhancements.

Contribution

It introduces a derived version of the Gabriel-Popescu theorem using derived injectives, extending classical results to the dg-category framework.

Findings

Pretriangulated dg-categories with suitable t-structures are localizations of derived dg-categories.

Derived categories of Grothendieck abelian categories have unique dg-enhancements.

New proof techniques involve generalizing Mitchell's argument and derived injectives.

Abstract

We prove a derived version of the Gabriel-Popescu theorem in the framework of dg-categories and t-structures. This exhibits any pretriangulated dg-category with a suitable t-structure (such that its heart is a Grothendieck abelian category) as a t-exact localization of a derived dg-category of dg-modules. We give an original proof based on a generalization of Mitchell's argument in "A quick proof of the Gabriel-Popesco theorem" and involving derived injective objects. As an application, we also give a short proof that derived categories of Grothendieck abelian categories have a unique dg-enhancement.