Lemma 37.69.4. Let A be a ring. Let X be a scheme separated and of finite presentation over A. Let K \in D_\mathit{QCoh}(\mathcal{O}_ X). If R\Gamma (X, E \otimes ^\mathbf {L} K) is pseudo-coherent in D(A) for every pseudo-coherent E in D(\mathcal{O}_ X), then K is pseudo-coherent relative to A.
Proof. Assume K \in D_\mathit{QCoh}(\mathcal{O}_ X) and R\Gamma (X, E \otimes ^\mathbf {L} K) is pseudo-coherent in D(A) for every pseudo-coherent E in D(\mathcal{O}_ X). Let x \in X. We will show that K is pseudo-coherent relative to A in a neighbourhood of x and this will prove the lemma.
Choose U, n, V, Z, z, E as in Lemma 37.69.2. Denote p : X \times \mathbf{P}^ n \to X and q : X \times \mathbf{P}^ n \to \mathbf{P}^ n_ A the projections. Then for any i \in \mathbf{Z} we have
\begin{align*} & R\Gamma (\mathbf{P}^ n_ A, Rq_*(Lp^*K \otimes ^\mathbf {L} E) \otimes ^\mathbf {L} \mathcal{O}_{\mathbf{P}^ n_ A}(i)) \\ & = R\Gamma (X \times \mathbf{P}^ n, Lp^*K \otimes ^\mathbf {L} E \otimes ^\mathbf {L} Lq^*\mathcal{O}_{\mathbf{P}^ n_ A}(i)) \\ & = R\Gamma (X, K \otimes ^\mathbf {L} Rq_*(E \otimes ^\mathbf {L} Lq^*\mathcal{O}_{\mathbf{P}^ n_ A}(i))) \end{align*}
by Derived Categories of Schemes, Lemma 36.22.1. By Derived Categories of Schemes, Lemma 36.30.5 the complex Rq_*(E \otimes ^\mathbf {L} Lq^*\mathcal{O}_{\mathbf{P}^ n_ A}(i)) is pseudo-coherent on X. Hence the assumption tells us the expression in the displayed formula is a pseudo-coherent object of D(A). By Derived Categories of Schemes, Lemma 36.34.2 we conclude that Rq_*(Lp^*K \otimes ^\mathbf {L} E) is pseudo-coherent on \mathbf{P}^ n_ A. By Lemma 37.69.3 we have
Rq_*(Lp^*K \otimes ^\mathbf {L} E)|_{X \times _ A V} = R(U \to V)_*K|_ U
Since U \to V is a closed immersion into an open subscheme of \mathbf{P}^ n_ A this means K|_ U is pseudo-coherent relative to A by Lemma 37.59.18. \square
Comments (0)