For affine schemes, is the value of the function at a prime same as the germ at that point?

Question

Let $X = \operatorname{Spec}A$, let $U \subset X$ be open. Consider $f \in O_X(U)$ and $P \in U$, where $P$ is a prime ideal. Is it then that $f(P) = [(f,U)] \in A_P$, where $[(f,U)]$ is the equivalence class of $(f,U)$ in $A_P = O_{X,P}$? Thank you!

Answer 1

I think this is all a matter of taste and definition, and the expression "$f(P)$" may possibly have a slightly different meaning for various authors.

What has a very fixed meaning is $\mathcal{O}_X(U)$: the structure sheaf of an affine scheme is an object on which everyone agrees. One way to define it is indeed the "étale space" definition, where an element of $\mathcal{O}_X(U)$ is a section of the natural map $\prod_{P\in U}A_P\to U$, with some gluing conditions to make it a sheaf. There are other possible definitions (which are all equivalent).

This does not necessarily mean that this is literaly what people have in mind when they view $f\in \mathcal{O}_X(U)$ as a "function" on $U$. My personal taste (and I think the most common convention) would be to write $f_P\in A_P$ for the germ of $f$, and $f(P)\in A_P/P=\kappa(P)$ for its residue in the residue field of $P$. But I would not be scandalized by an author who would write $f(P)$ for the germ.