Uniqueness of enhancement for triangulated categories

TL;DR

This paper proves the uniqueness of DG enhancements for various triangulated categories, ensuring their structures are canonical and functors between them can be represented by objects on products.

Contribution

It establishes new results on the uniqueness of DG enhancements for unbounded, perfect, and bounded derived categories, especially on projective schemes.

Findings

Uniqueness of DG enhancements for unbounded quasi-coherent sheaves

Uniqueness for categories of perfect complexes

Strong uniqueness results for projective schemes

Abstract

The paper contains general results on the uniqueness of a DG enhancement for triangulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded derived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded derived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.