Restriction maps in an integral scheme are injective (Ayman Hourieh's proof)

Question

I have some questions concerning Ayman Hourieh's proof of the fact that restriction maps in an integral scheme are injective:

Let $X$ be an integral scheme. Let $\xi$ be its generic point. If we show that the canonical map $\mathcal O_X(U) \to \mathcal O_\xi$ is injective, we're done. Since $U$ can be covered by affine open subsets, we can assume that $U = \operatorname{Spec} A$ is affine. Now the map $\mathcal O_X(U) \to \mathcal O_\xi$ corresponds to the canonical map $A \to \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ is the quotient field of $A$. This map is clearly injective as desired.

(1) Since $U$ can be covered by affine open subsets, we can assume that $U = \operatorname{Spec} A$ is affine.

I understand the case when $U$ is affine, but I don't know how to generalize it to an arbitrary open subset.

(2) I don't know how it follows from his proof that $\operatorname{res}_{U,V} : \mathcal O_X(U) \rightarrow \mathcal O_X(V)$ is injective for arbitrary open subsets $U \supset V$.

If you know another proof of this fact, please also feel free to share it in here or the linked post. Thank you!

Answer 1

(1) Follows from the definition of a sheaf - specifically, the locality axiom. Let $s$ be a section over $U$ which has image 0 under restriction to $V$. We want to show $s$ is 0. For an open cover $U_i$ of $U$, it suffices to verify that $s|_{U_i}$ is $0$ on each $U_i$. Since every scheme has a covering by open affines, this shows us that it is enough to verify the statement that restriction is injective in the case that $U$ is affine.

(2) Consider the directed system of rings indexed over all open subsets $Y$ of $U$ with objects $\mathcal{O}_X(Y)$ and morphisms the appropriate restriction maps. By definition, the limit of this system is $\mathcal{O}_\xi$, the generic point of $U$, since $\xi$ is the only point contained in every open subset of $U$. By construction of the limit, the image of a section under some restriction map being zero is equivalent to the image of that same section being zero in the limit. So it suffices to show that the map from $\mathcal{O}_X(U)\to\mathcal{O}_\xi$ is injective, which is what the original author proves.

Answer 2

For (2) say the injective maps $\mathcal{O}_X(U)\to\mathcal{O}_\eta$ are called $f_U$. Then, $f_V\circ\operatorname{res}_{U,V}=f_U$, so the restriction map is injective as well. Thus, all you need to show is that for any open set $U$ the map $f_U$ is injective.

Cover $U$ by open affines $\{U_i\}$ and take $a\in\ker f_U$. Then, by the same argument as above $f_U(a)=f_{U_i}(a|_{U_i})=0$ and since each $f_{U_i}$ is injective you're done.