Lemma 31.6.1. Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then we have
\[ \text{WeakAss}_ S(f_*\mathcal{F}) \subset f(\text{WeakAss}_ X(\mathcal{F})) \]
Proof. We may assume $X$ and $S$ affine, so $X \to S$ comes from a ring map $A \to B$. Then $\mathcal{F} = \widetilde M$ for some $B$-module $M$. By Lemma 31.5.2 the weakly associated points of $\mathcal{F}$ correspond exactly to the weakly associated primes of $M$. Similarly, the weakly associated points of $f_*\mathcal{F}$ correspond exactly to the weakly associated primes of $M$ as an $A$-module. Hence the lemma follows from Algebra, Lemma 10.66.11. $\square$
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Your email address will not be published.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)