Lemma 57.6.4. Let X be a proper scheme over a field k which is regular. Let G : D_{perf}(\mathcal{O}_ X) \to \text{Vect}_ k be a k-linear homological functor such that
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).
Proof. Consider the contravariant functor E \mapsto E^\vee on D_{perf}(\mathcal{O}_ X), see Cohomology, Lemma 20.50.5. This functor is an exact anti-self-equivalence of D_{perf}(\mathcal{O}_ X). Hence we may apply Theorem 57.6.3 to the functor F(E) = G(E^\vee ) to find K \in D_{perf}(\mathcal{O}_ X) such that G(E^\vee ) = \mathop{\mathrm{Hom}}\nolimits _ X(E, K). It follows that G(E) = \mathop{\mathrm{Hom}}\nolimits _ X(E^\vee , K) = \mathop{\mathrm{Hom}}\nolimits _ X(K^\vee , E) and we conclude that taking K^\vee works. \square
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