The Stacks project

Lemma 56.8.1. 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 56.7.3 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 56.5.3.

For the existence of the left adjoint we argue in the same manner using part (2) of Lemma 56.7.3. $\square$


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