The lemma below is the standard example.
Lemma 57.4.1. Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is Gorenstein. Consider the complex $\omega _ X^\bullet $ of Duality for Schemes, Lemmas 48.27.1. Then the functor
is a Serre functor.
Proof. The statement make sense because $\dim \mathop{\mathrm{Hom}}\nolimits _ X(K, L) < \infty $ for $K, L \in D_{perf}(\mathcal{O}_ X)$ by Derived Categories of Schemes, Lemma 36.11.7. Since $X$ is Gorenstein the dualizing complex $\omega _ X^\bullet $ is an invertible object of $D(\mathcal{O}_ X)$, see Duality for Schemes, Lemma 48.24.4. In particular, locally on $X$ the complex $\omega _ X^\bullet $ has one nonzero cohomology sheaf which is an invertible module, see Cohomology, Lemma 20.52.2. Thus $S(K)$ lies in $D_{perf}(\mathcal{O}_ X)$. On the other hand, the invertibility of $\omega _ X^\bullet $ clearly implies that $S$ is a self-equivalence of $D_{perf}(\mathcal{O}_ X)$. Finally, we have to find an isomorphism
bifunctorially in $K, L$. To do this we use the canonical isomorphisms
and
given in Cohomology, Lemma 20.50.5. Since $(L \otimes _{\mathcal{O}_ X}^\mathbf {L} K^\vee )^\vee = (K^\vee )^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} L^\vee $ and since there is a canonical isomorphism $K \to (K^\vee )^\vee $ we find these $k$-vector spaces are canonically dual by Duality for Schemes, Lemma 48.27.4. This produces the isomorphisms $c_{K, L}$. We omit the proof that these isomorphisms are functorial. $\square$
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