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This comes from the exercise 6.6.W of Vakil's FOAG. I guess it is saying that a function (section) is determined by its germs at associated points. And this sort of motivates the introduction of associated points. The problem is stated as follows:

Problem. Assuming $X$ a locally Noetherian scheme and $U\subseteq X$ an open subscheme. How do we show that the natural map $$\Gamma(U,\mathcal{O}_X)\to \Pi_{\mathfrak{p}\in Ass(X)\cap U}\mathcal{O}_{X,\mathfrak{p}} $$ is injective where I guess (since there is no explannation in the book) $Ass(X)$ denotes the set of associated points of $\mathcal{O}_X$ in $X$.

My attempts. I wish to make use of the sheaf property and glue locally identical sections together. Let $s,t\in\Gamma(U,\mathcal{O}_X)$ such that $s_\mathfrak{p}=t_\mathfrak{p}$ for all $\mathfrak{p}\in Ass(X)\cap U$. Now cover $U$ by affine open Noetherian schemes $\operatorname{Spec}A$. Then $\mathfrak{p}\in Ass(X)\cap U$ must be in some of them. The Noetherian property tells us that $$\mathfrak{p}=Ann_A(a)$$ for some $a\in A$. This would imply the existence of some $s'\in A-\mathfrak{p}$ such that $s'((s-t)|_{\operatorname{Spec}A})=0$, hence $(s-t)|_{\operatorname{Spec}A})\in\mathfrak{p}$, i.e. $(s-t)|_{\operatorname{Spec}A})\cdot a=0$. However, I am not sure how to proceed anymore. Also, if I start with a Noetherian open affine neighborhood of $p$ on which $s$ and $t$ agree, I am not sure how to show that they cover $U$.

Update. I think I figured out a solution, based on the reduction of the problem to the following claim: for a module $M$ over Noetherian ring $A$, we have the following map injective $$M\to \Pi_{\mathfrak{p}\in Ass(M)}M_\mathfrak{p}$$ I will try to give a full answer later.

Any help is appreciated! Thank you.

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  • $\begingroup$ You're missing a word or two in your question statement - "How do we show that the natural map ..." is what? Injective? $\endgroup$
    – KReiser
    Mar 29 at 15:16
  • $\begingroup$ @KReiser Sorry, I have edited it. $\endgroup$
    – Mizutsuki
    Mar 30 at 4:22

1 Answer 1

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Let $s,t\in \Gamma(U,\mathcal{O}_X)$ be sections such that $\varphi(s)=\varphi(t)$. Consider the open affine cover $\{\operatorname{Spec}A_i\}$ of $U$ where $A_i$'s are Noetherian. Now for each $\mathfrak{p}\in Ass(X)\cap U$, we have some $\operatorname{Spec}A_i$ containing it. Then we see that under the injection $$A_i\to \Pi_{\mathfrak{p}\in Ass(A_i)}(A_i)_\mathfrak{p}$$ The sections $s|_{ \operatorname{Spec}A_i}$ and $t|_{\operatorname{Spec}A_i}$ are sent to the same object. Hence $s|_{\operatorname{Spec}A_i}=t|_{\operatorname{Spec}A_i}$. Because such $\operatorname{Spec}A_i$ covers $U$, the identity axiom in the definition of sheaf tells us $s=t\in\Gamma(U,\mathcal{O}_X)$

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