As a consequence of the results in the paper of Bondal and van den Bergh we get the following automatic existence of adjoints.

Lemma 57.7.1. Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$. If $X$ is regular, then any $k$-linear exact functor $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ has an exact right adjoint and an exact left adjoint.

Proof. If an adjoint exists it is an exact functor by the very general Derived Categories, Lemma 13.7.1.

Let us prove the existence of a right adjoint. To see existence, it suffices to show that for $M \in D_{perf}(\mathcal{O}_ Y)$ the contravariant functor $K \mapsto \mathop{\mathrm{Hom}}\nolimits _ Y(F(K), M)$ is representable. This functor is contravariant, $k$-linear, and cohomological. Hence by Theorem 57.6.3 it suffices to show that

$\sum \nolimits _{i \in \mathbf{Z}} \dim _ k \mathop{\mathrm{Ext}}\nolimits ^ i_ Y(F(K), M) < \infty$

This follows from Lemma 57.5.3.

For the existence of the left adjoint we argue in the same manner using Lemma 57.6.4 in stead of Theorem 57.6.3. $\square$

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