Lemma 76.51.3 (0CTS)—The Stacks project

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

$Rq_*(Lp^*K \otimes ^\mathbf {L} E)|_ V = R(W \to V)_*K|_ W$

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

$Rq_*(Lp^*K \otimes ^\mathbf {L} E)|_ V = Rq'_*\left((Lp^*K \otimes ^\mathbf {L} E)|_{X \times _ A V}\right)$

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

$Rq'_*\left((Lp^*K \otimes ^\mathbf {L} E)|_{X \times _ A V}\right) = Rq'_*(Lp^*K|_{X \times _ A V} \otimes ^\mathbf {L} i_*\mathcal{O}_ W)$

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

$Rq'_* Ri_* Li^* Lp^*K|_{X \times _ A V} = R(q' \circ i)_* Lb^*K|_ W = R(W \to V)_* K|_ W$

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 $X$ .) $\square$

The Stacks project

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