Uniqueness of dg enhancements for the derived category of a Grothendieck category

TL;DR

This paper proves the uniqueness of differential graded (dg) enhancements for the derived categories of Grothendieck abelian categories and certain subcategories, with implications for algebraic stacks and schemes.

Contribution

It establishes the uniqueness of dg enhancements for derived categories of Grothendieck categories and their subcategories under specific conditions, extending to algebraic stacks and schemes.

Findings

Unique dg enhancement for derived category of Grothendieck abelian categories.

Extension of uniqueness to subcategories of compact objects under additional assumptions.

Application to derived categories of quasi-coherent sheaves and perfect complexes on algebraic stacks and schemes.

Abstract

We prove that the derived category of a Grothendieck abelian category has a unique dg enhancement. Under some additional assumptions, we show that the same result holds true for its subcategory of compact objects. As a consequence, we deduce that the unbounded derived category of quasi-coherent sheaves on an algebraic stack and the category of perfect complexes on a noetherian concentrated algebraic stack with quasi-finite affine diagonal and enough perfect coherent sheaves have a unique dg enhancement. In particular, the category of perfect complexes on a noetherian scheme with enough locally free sheaves has a unique dg enhancement.