Theorem 57.13.3 (Orlov). Let k be a field. Let X and Y be smooth proper schemes over k with X projective over k. Any k-linear fully faithful exact functor F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y) is a Fourier-Mukai functor for some kernel in D_{perf}(\mathcal{O}_{X \times Y}).
[Theorem 2.2, Orlov-K3]; this is shown in [Noah] without the assumption that X be projective
Proof. Let F' be the Fourier-Mukai functor which is a sibling of F as in Lemma 57.13.2. By Proposition 57.10.6 we have F \cong F' provided we can show that \textit{Coh}(\mathcal{O}_ X) has enough negative objects. However, if X = \mathop{\mathrm{Spec}}(k) for example, then this isn't true. Thus we first decompose X = \coprod X_ i into its connected (and irreducible) components and we argue that it suffices to prove the result for each of the (fully faithful) composition functors
Details omitted. Thus we may assume X is irreducible.
The case \dim (X) = 0. Here X is the spectrum of a finite (separable) extension k'/k and hence D_{perf}(\mathcal{O}_ X) is equivalent to the category of graded k'-vector spaces such that \mathcal{O}_ X corresponds to the trivial 1-dimensional vector space in degree 0. It is straightforward to see that any two siblings F, F' : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y) are isomorphic. Namely, we are given an isomorphism F(\mathcal{O}_ X) \cong F'(\mathcal{O}_ X) compatible the action of the k-algebra k' = \text{End}_{D_{perf}(\mathcal{O}_ X)}(\mathcal{O}_ X) which extends canonically to an isomorphism on any graded k'-vector space.
The case \dim (X) > 0. Here X is a projective smooth variety of dimension > 1. Let \mathcal{F} be a coherent \mathcal{O}_ X-module. We have to show there exists a coherent module \mathcal{N} such that
there is a surjection \mathcal{N} \to \mathcal{F} and
\mathop{\mathrm{Hom}}\nolimits (\mathcal{F}, \mathcal{N}) = 0.
Choose an ample invertible \mathcal{O}_ X-module \mathcal{L}. We claim that \mathcal{N} = (\mathcal{L}^{\otimes n})^{\oplus r} will work for n \ll 0 and r large enough. Condition (1) follows from Properties, Proposition 28.26.13. Finally, we have
Since the dual \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{O}_ X) is torsion free, this vanishes for n \ll 0 by Varieties, Lemma 33.48.1. This finishes the proof. \square
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