Example 56.9.6. Let $X \to S$ be a separated morphism of schemes. Then the diagonal $\Delta : X \to X \times _ S X$ is a closed immersion and hence $\mathcal{O}_\Delta = \Delta _*\mathcal{O}_ X = R\Delta _*\mathcal{O}_ X$ is a quasi-coherent $\mathcal{O}_{X \times _ S X}$-module of finite type which is flat over $X$ (under either projection). The Fourier-Mukai functor $\Phi _{\mathcal{O}_\Delta }$ is equal to the identity in this case. Namely, for any $M \in D(\mathcal{O}_ X)$ we have

\begin{align*} L\text{pr}_1^*M \otimes _{\mathcal{O}_{X \times _ S X}}^\mathbf {L} \mathcal{O}_\Delta & = L\text{pr}_1^*M \otimes _{\mathcal{O}_{X \times _ S X}}^\mathbf {L} R\Delta _*\mathcal{O}_ X \\ & = R\Delta _*( L\Delta ^*L\text{pr}_1^*M \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X) \\ & = R\Delta _*(M) \end{align*}

The first equality we discussed above. The second equality is Cohomology, Lemma 20.51.4. The third because $\text{pr}_1 \circ \Delta = \text{id}_ X$ and we have Cohomology, Lemma 20.27.2. If we push this to $X$ using $R\text{pr}_{2, *}$ we obtain $M$ by Cohomology, Lemma 20.28.2 and the fact that $\text{pr}_2 \circ \Delta = \text{id}_ X$.

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