Theorem 57.14.3 (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 57.14.2. By Proposition 57.11.6 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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)