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

$F : \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{O}_ x, \mathcal{O}_{x'}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_ Y(F(\mathcal{O}_ x), F(\mathcal{O}_{x'}))$

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

Proof. By Lemma 57.8.1 the functor $F$ has both a left and a right adjoint. Thus we may apply the criterion of Lemma 57.12.1 because assumptions (2) and (3) of that lemma follow from Lemma 57.12.3. $\square$

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