Lemma 57.12.5. Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$. Assume $X$ is regular. Then a $k$-linear exact functor $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ is fully faithful if and only if for any closed points $x, x' \in X$ the maps

are isomorphisms for all $i \in \mathbf{Z}$. Here $\mathcal{O}_ x$ is the skyscraper sheaf at $x$ with value $\kappa (x)$.

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