The Stacks project

Lemma 37.56.3. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. If $E$ in $D(\mathcal{O}_ X)$ is $m$-pseudo-coherent relative to $S$, then $H^ i(E)$ is a quasi-coherent $\mathcal{O}_ X$-module for $i > m$. If $E$ is pseudo-coherent relative to $S$, then $E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Proof. Choose an affine open covering $S = \bigcup V_ i$ and for each $i$ an affine open covering $f^{-1}(V_ i) = \bigcup U_{ij}$ such that the equivalent conditions of Lemma 37.56.1 are satisfied for each of the pairs $(U_{ij} \to V_ i, E|_{U_{ij}})$. Since being quasi-coherent is local on $X$, we may assume that there exists an closed immersion $i : X \to \mathbf{A}^ n_ S$ such that $Ri_*E$ is $m$-pseudo-coherent on $\mathbf{A}^ n_ S$. By Derived Categories of Schemes, Lemma 36.10.1 this means that $H^ q(Ri_*E)$ is quasi-coherent for $q > m$. Since $i_*$ is an exact functor, we have $i_*H^ q(E) = H^ q(Ri_*E)$ is quasi-coherent on $\mathbf{A}^ n_ S$. By Morphisms, Lemma 29.4.1 this implies that $H^ q(E)$ is quasi-coherent as desired (strictly speaking it implies there exists some quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ such that $i_*\mathcal{F} = i_*H^ q(E)$ and then Modules, Lemma 17.13.4 tells us that $\mathcal{F} \cong H^ q(E)$ hence the result). $\square$


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