Lemma 57.12.2. Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated. Let $F : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ be a $k$-linear exact functor sending $\textit{Coh}(\mathcal{O}_ X) \subset D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ into $\textit{Coh}(\mathcal{O}_ Y) \subset D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$. Then there exists a Fourier-Mukai functor $F' : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ whose kernel is a coherent $\mathcal{O}_{X \times Y}$-module $\mathcal{K}$ flat over $X$ and with support finite over $Y$ which is a sibling of $F$.
Proof. Denote $H : \textit{Coh}(\mathcal{O}_ X) \to \textit{Coh}(\mathcal{O}_ Y)$ the restriction of $F$. Since $F$ is an exact functor of triangulated categories, we see that $H$ is an exact functor of abelian categories. Of course $H$ is $k$-linear as $F$ is. By Functors and Morphisms, Lemma 56.7.5 we obtain a coherent $\mathcal{O}_{X \times Y}$-module $\mathcal{K}$ which is flat over $X$ and has support finite over $Y$. Let $F'$ be the Fourier-Mukai functor defined using $\mathcal{K}$ so that $F'$ restricts to $H$ on $ \textit{Coh}(\mathcal{O}_ X)$. The functor $F'$ sends $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ into $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ by Lemma 57.8.5. Observe that $F$ and $F'$ satisfy the first and second condition of Lemma 57.10.2 and hence are siblings. $\square$
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