A tour about existence and uniqueness of dg enhancements and lifts

TL;DR

This survey reviews recent progress on the existence, uniqueness, and lifting of dg enhancements for triangulated categories in algebraic geometry, highlighting key cases like derived categories of sheaves and perfect complexes.

Contribution

It consolidates recent results on when triangulated categories have unique dg enhancements and discusses conditions for lifting functors to dg functors.

Findings

Unique dg enhancements for derived categories of quasi-coherent sheaves on stacks

Conditions for lifting exact functors to dg functors

Recent advances in understanding dg enhancements in algebraic geometry

Abstract

This paper surveys the recent advances concerning the relations between triangulated (or derived) categories and their dg enhancements. We explain when some interesting triangulated categories arising in algebraic geometry have a unique dg enhancement. This is the case, for example, for the unbounded derived category of quasi-coherent sheaves on an algebraic stack or for its full triangulated subcategory of perfect complexes. Moreover we give an account of the recent results about the possibility to lift exact functors between the bounded derived categories of coherent sheaves on smooth schemes to dg (quasi-)functors.