The Stacks project

Lemma 76.51.4. Let $A$ be a ring. Let $X$ be an algebraic space 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 $A$ (Definition 76.45.3).

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 an ├ętale neighbourhood of $x$. This will prove the lemma by our definition of relative pseudo-coherence.

Choose $n, Z, z, V, E$ as in Lemma 76.51.2. 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 Spaces, Lemma 75.20.1. By Derived Categories of Spaces, Lemma 75.25.5 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 76.51.3 we have

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

Since $W \to V$ is a closed immersion into an open subscheme of $\mathbf{P}^ n_ A$ this means $K|_ W$ is pseudo-coherent relative to $A$ for example by More on Morphisms, Lemma 37.59.18. $\square$

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