Lemma 27.24.7. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes. Let $Z \subset Y$ be a closed subset such that $Y \setminus Z$ is retrocompact in $Y$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{F}' \subset \mathcal{F}$ be the subsheaf of sections supported in $f^{-1}Z$. Then $f_*\mathcal{F}' \subset f_*\mathcal{F}$ is the subsheaf of sections supported in $Z$.

Proof. Omitted. (Hint: First show that $X \setminus f^{-1}Z$ is retrocompact in $X$ as $Y \setminus Z$ is retrocompact in $Y$. Hence Lemma 27.24.5 applies to $f^{-1}Z$ and $\mathcal{F}$. As $f$ is quasi-compact and quasi-separated we see that $f_*\mathcal{F}$ is quasi-coherent. Hence Lemma 27.24.5 applies to $Z$ and $f_*\mathcal{F}$. Finally, match the sheaves directly.) $\square$

There are also:

• 2 comment(s) on Section 27.24: Sections with support in a closed subset

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).