A tour about existence and uniqueness of dg enhancements and lifts

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.

Paper appeared in J. Geom. Phys. The present version (38 pages) differs from the published one only in the statement of Thm 4.2 (the old Thm. 4.1). This is due to a gap in the proof of Thm C in the published version of arXiv:1507.05509 which has been corrected in the new version arXiv:1507.05509v5

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Cited by in corpus (14)

• Derived Categories
• Gerstenhaber algebra and Deligne's conjecture on Tate-Hochschild cohomology
• A DG-enhancement of D(QCoh(X)) with applications in deformation theory
• An informal introduction to dg categories
• A derived Gabriel-Popescu theorem for t-structures via derived injectives
• A refined Derived Torelli Theorem for Enriques surfaces
• A categorical characterization of quantum projective spaces
• On the uniqueness of infinity-categorical enhancements of triangulated categories
• Moduli of Bridgeland semistable holomorphic triples
• Liftable derived equivalences and objective categories
• Formality and strongly unique enhancements
• A note on non-unique enhancements
• A constructive approach to Fourier-Mukai transforms for projective spaces via $A_{\infty }$-functors between pretriangulated dg categories
• An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves