[Theorem 2.2, Orlov-K3]; this is shown in [Noah] without the assumption that $X$ be projective

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 $D_{perf}(\mathcal{O}_{X \times Y})$.

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

$F_ i : D_{perf}(\mathcal{O}_{X_ i}) \to D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$

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

1. there is a surjection $\mathcal{N} \to \mathcal{F}$ and

2. $\mathop{\mathrm{Hom}}\nolimits (\mathcal{F}, \mathcal{N}) = 0$.

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

$\mathop{\mathrm{Hom}}\nolimits (\mathcal{F}, \mathcal{L}^{\otimes n}) = H^0(X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{L}^{\otimes n})) = H^0(X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{O}_ X) \otimes \mathcal{L}^{\otimes n})$

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. $\square$

Comment #8732 by Hao on

The original version in Orlov's paper concerns the coherent category, but the statement here is about perfect objects. Does this implies the coherent version? Or Does every fullly faithful embedding preserves perfect objects?

Comment #8733 by Hao on

Sorry I just found the answer that by \tag{0FDC} they are the same.

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