Lemma 56.7.3. Let $X$ be a proper scheme over a field $k$ which is regular.

Let $F : D_{perf}(\mathcal{O}_ X)^{opp} \to \text{Vect}_ k$ be a $k$-linear cohomological functor such that

\[ \sum \nolimits _{n \in \mathbf{Z}} \dim _ k F(E[n]) < \infty \]for all $E \in D_{perf}(\mathcal{O}_ X)$. Then $F$ is isomorphic to a functor of the form $E \mapsto \mathop{\mathrm{Hom}}\nolimits _ X(E, K)$ for some $K \in D_{perf}(\mathcal{O}_ X)$.

Let $G : D_{perf}(\mathcal{O}_ X) \to \text{Vect}_ k$ be a $k$-linear homological functor such that

\[ \sum \nolimits _{n \in \mathbf{Z}} \dim _ k G(E[n]) < \infty \]for all $E \in D_{perf}(\mathcal{O}_ X)$. Then $G$ is isomorphic to a functor of the form $E \mapsto \mathop{\mathrm{Hom}}\nolimits _ X(K, E)$ for some $K \in D_{perf}(\mathcal{O}_ X)$.

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