Proposition 56.16.4. Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$. If $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ is a $k$-linear exact equivalence of triangulated categories then there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ which is an equivalence and a sibling of $F$.

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