The Stacks project

Lemma 30.5.9. Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$ be a point which is not in the image of $f$. Then $s$ is not weakly associated to $f_*\mathcal{F}$.

Proof. The question is local so we may assume $X = \mathop{\mathrm{Spec}}(A)$. By Schemes, Lemma 25.24.1 the sheaf $f_*\mathcal{F}$ is quasi-coherent, say corresponding to the $A$-module $M$. Say $s$ corresponds to $\mathfrak p \subset A$. As $s$ is not in the image of $f$ we see that $X = \bigcup _{a \in \mathfrak p} f^{-1}D(a)$ is an open covering. Since $X$ is quasi-compact we can find $a_1, \ldots , a_ n \in \mathfrak p$ such that $X = f^{-1}D(a_1) \cup \ldots \cup f^{-1}D(a_ n)$. It follows that

\[ M \to M_{a_1} \oplus \ldots \oplus M_{a_ r} \]

is injective. Hence for any nonzero element $m$ of the stalk $M_\mathfrak p$ there exists an $i$ such that $a_ i^ n m$ is nonzero for all $n \geq 0$. Thus $\mathfrak pA_\mathfrak p$ is not weakly associated to $M_\mathfrak p$. $\square$

Comments (2)

Comment #4177 by Zhiyu Zhang on

"The question is local so we may assume ", maybe you mean ? Moreover, I think in order to have , we may need be a closed point, hence it's better to base chagne to at first (as the question is local).

Comment #4178 by on

Good catch and thanks for explaining the fix! I will update the text later. Thanks again.

There are also:

  • 2 comment(s) on Section 30.5: Weakly associated points

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AVN. Beware of the difference between the letter 'O' and the digit '0'.