Uniqueness of enhancement for triangulated categories

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 derived 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.

Bivariant cyclic cohomology and Connes’ bilinear pairings in noncommutative motives

“…Hence, it can now be applied to schemes. Given a (quasi-compact and quasi-separated) k-scheme X, it is well-known that the category of perfect complexes of O X -modules admits a dg-enhancement perf dg (X); see for instance [21] or [4,Example 4.5]. Moreover, whenever X is smooth and proper the dg category perf dg (X) is saturated in the sense of Kontsevich.…”

*-Quantizations of Fourier–Mukai Transforms

“…We can either consider quantizations geometrically in terms of stacks of algebroids as above, or we can look at deformations categorically as deformations of a dg enhancement of the category of coherent sheaves. Similarly a Fourier-Mukai transform for schemes can be studied either geometrically through its kernel or categorically as a functor between dg categories (the equivalence of the two approaches follows from derived Morita theory [Toë07] and the uniqueness of enhancements [LO10]). …”

Relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano fibrations

“…(2) In the case that X is a projective scheme itself, recent results by Ballard [4] (see also [28]), prove that any autoequivalence of D b c (X) is a (absolute) FourierMukai transform, so that we are studying the subgroup of AutD b c (X) consisting of those Fourier-Mukai transforms whose kernel is supported on the fibred product X × T X. △…”