## 57.8 Existence of adjoints

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

Lemma 57.8.1. Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$. If $X$ is regular, then $k$-linear any 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 Lemma 57.7.3 part (1) 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 part (2) of Lemma 57.7.3. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).