Lemma 37.59.3. 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 37.59.1 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).
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)