Lemma 71.2.10. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $y \in |Y|$ be a point which is not in the image of $|f|$. Then $y$ is not weakly associated to $f_*\mathcal{F}$.
Proof. By Morphisms of Spaces, Lemma 67.11.2 the $\mathcal{O}_ Y$-module $f_*\mathcal{F}$ is quasi-coherent hence the lemma makes sense. Choose an affine scheme $V$, a point $v \in V$, and an étale morphism $V \to Y$ mapping $v$ to $y$. We may replace $f : X \to Y$, $\mathcal{F}$, $y$ by $X \times _ Y V \to V$, $\mathcal{F}|_{X \times _ Y V}$, $v$. Thus we may assume $Y$ is an affine scheme. In this case $X$ is quasi-compact, hence we can choose an affine scheme $U$ and a surjective étale morphism $U \to X$. Denote $g : U \to Y$ the composition. Then $f_*\mathcal{F} \subset g_*(\mathcal{F}|_ U)$. By Lemma 71.2.4 we reduce to the case of schemes which is Divisors, Lemma 31.5.9. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)