# Sections are determined by their germs at associated points

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.

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

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 $$s=t\in\Gamma(U,\mathcal{O}_X)$$