Lemma 37.69.3 (0CSK)—The Stacks project

Lemma 37.69.3. Let $A$, $x \in X$, and $U, n, V, Z, z, E$ be as in Lemma 37.69.2. For any $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have

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

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 $U \to V$ is the finitely presented closed immersion $c \circ (b|_ U)^{-1}$.

Proof. Since $b^{-1}(U) = c^{-1}(V)$ and since $c$ is a closed immersion over $V$, we see that $c \circ (b|_ U)^{-1}$ is a closed immersion. It is of finite presentation because $U$ and $V$ are of finite presentation over $A$, see Morphisms, Lemma 29.21.11. 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. Set $W = b^{-1}(U) = c^{-1}(V)$ and denote $i : W \to X \times _ A V$ the closed immersion $i = (b, c)|_ W$. 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 Schemes, Lemma 36.22.1 we can rewrite this as

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

which is what we want. $\square$

The Stacks project

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