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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.

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.

We prove that exact functors between the categories of perfect complexes supported on projective schemes are of Fourier-Mukai type if the functor satisfies a condition weaker than being fully faithful. We also get generalizations of the results in the literature in the case without support conditions. Some applications are discussed and, along the way, we prove that the category of perfect supported complexes has a strongly unique enhancement.F : Perf (X 1 ) → Perf (X 2 ) between the categories of perfect complexes on the projective schemes X 1 and X 2 are of Fourier-Mukai type. To show this, Lunts and Orlov prove that such fully faithful functors admit dg lifts. At that point, they can invoke the representability result in [Toë07]. Indeed, Toën proved that, in the dg setting, all morphisms in the localization of the category of dg categories by quasi-equivalences are of Fourier-Mukai type (in an appropriate dg sense). Notice that the strategy in [LO10] allows the authors to improve the results in [Bal09].To make clear the categorical setting we will work with, let X 1 be a quasi-projective scheme containing a projective subscheme Z 1 such that the structure sheaf O iZ 1 of the ith infinitesimal neighbourhood of Z 1 in X 1 is in Perf (X 1 ), for every i > 0. This last condition is satisfied, for instance, when either Z 1 = X 1 or X 1 is smooth. Moreover, let X 2 be a separated scheme of finite type over the base field k with a closed subscheme Z 2 . One can then consider the categories Perf Z i (X i ) of perfect complexes on X i with cohomology sheaves supported on Z i . The definition of Fourier-Mukai functor makes perfect sense also in this context (see Definition 2.5).A rewriting of (1.1) in the supported setting which weakens the full-faithfulness condition in [LO10, Orl97] requires a bit of care. Indeed, assuming X 1 , X 2 , Z 1 and Z 2 to be as above, one can consider exact functors F : Perf Z 1 (X 1 ) → Perf Z 2 (X 2 ) such that the following condition holds.Property ( * ).(1) Hom(F(A), F(B)[k]) = 0, for any A, B ∈ Coh Z 1 (X 1 ) ∩ Perf Z 1 (X 1 ) and any integer k < 0;(2) for all A ∈ Perf Z 1 (X 1 ) with trivial cohomologies of positive degrees, there is N ∈ Z such that Hom(F(A), F(O |i|Z 1 (jH 1 ))) = 0, for any i < N and any j i, where H 1 is an ample (Cartier) divisor on X 1 .At first sight this condition may look a bit involved, but if Z 1 = X 1 is smooth with dim(X 1 ) > 0, then part (2) of ( * ) is redundant and thus ( * ) turns out to be equivalent to (1.1) (see Proposition 3.13). In general, full functors always satisfy ( * ), if we assume further that the maximal zero-dimensional torsion subsheaf T 0 (O Z 1 ) of O Z 1 is trivial. Actually, due to [COS13], a non-trivial full functor is automatically faithful if Z 1 is connected. We will discuss in § 3.4 the existence of non-full functors with property ( * ).We are now ready to state our first main result.Theorem 1.1. Let X 1 be a quasi-projective scheme containing a projective subscheme Z 1 such that O iZ 1 ∈ Perf (X 1 ), for all i > 0, and let X ...

This paper surveys some recent results about Fourier-Mukai functors. In particular, given an exact functor between the bounded derived categories of coherent sheaves on two smooth projective varieties, we deal with the question whether this functor is of Fourier-Mukai type. Several related questions are answered and many open problems are stated.