Lemma 76.51.3. Let $X/A$, $x \in |X|$, and $n, Z, z, V, E$ be as in Lemma 76.51.2. For any $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have
where $p : X \times _ A \mathbf{P}^ n_ A \to X$ and $q : X \times _ A \mathbf{P}^ n_ A \to \mathbf{P}^ n_ A$ are the projections and where the morphism $W \to V$ is the finitely presented closed immersion $c|_ W : W \to V$.
Proof. Since $W = c^{-1}(V)$ and since $c$ is a closed immersion over $V$, we see that $c|_ W$ is a closed immersion. It is of finite presentation because $W$ and $V$ are of finite presentation over $A$, see Morphisms of Spaces, Lemma 67.28.9. First we have
where $q' : X \times _ A V \to V$ is the projection because formation of total direct image commutes with localization. Denote $i = (b, c)|_ W : W \to X \times _ A V$ the given closed immersion. Then
by property (5). Since $i$ is a closed immersion we have $i_*\mathcal{O}_ W = Ri_*\mathcal{O}_ W$. Using Derived Categories of Spaces, Lemma 75.20.1 we can rewrite this as
which is what we want. (Note that restricting to $W$ and derived pulling back via $W \to X$ is the same thing as $W$ is étale over $X$.) $\square$
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