Fourier–Mukai functors in the supported case

Abstract

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 ...

Fourier–Mukai functors: a survey

“…The kernel turns out to be unique in perfect analogy with Theorem 3.1. This is observed in [17], following a suggestion by Orlov.…”

“…Remark 5.14. The techniques used to get the extension θ 3 were improved in [17] (see, in particular, Sections 3.2 and 3.3 of that paper). Indeed, we consider a wider class of triangulated categories and we deal with extensions of natural transformations rather than isomorphisms of functors.…”

A tour about existence and uniqueness of dg enhancements and lifts

“…(3) The supported case: In [13], the techniques introduced in [42] are enforced and extended to deal with exact functors (with some special assumptions) between the categories of perfect complexes with cohomologies supported on projective schemes. Remark 6.4.…”

Uniqueness of dg enhancements for the derived category of a Grothendieck category

“…Notice that, beyond the geometric situations studied in [24] and described in the introduction, the uniqueness of enhancements has been investigated in other cases, e.g. for the derived categories of supported quasi-coherent sheaves in special situations (see [8,Lemma 4.6]).…”