Suppose $X$ is an integral scheme. I would like to show that the restriction maps
$res_{U,V} : O_X(U) \rightarrow O_X(V)$ is an inclusion so long as $V$ is not empty. I was wondering if someone could give me some assistance with this. This is exercise 5.2 I on Ravi Vakil's notes. Thank you!
Algebraic Geometry – Injectivity of Restriction Maps in an Integral Scheme
algebraic-geometryschemes
Best Answer
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.
I've left some details out. Hopefully you'll be able to fill them in.