I'm working on problem 3.7 from Qing Liu's text on algebraic geometry, and I'm having a bit of difficulty formalizing my intuition. I searched around, and was unable to find the answer elsewhere on the site. The problem is as follows:
"Let $X$ be a scheme over a field $k$. Let $\varphi: k[T_{1},..., T_{n}] \to \mathcal{O}_{X}(X)$ be a homomorphism of $k$-algebras, and $f: X \to \mathbb{A}^{n}_{k}$ the morphism induced by $\varphi$. Show that for any rational point $x \in X(k)$, via the identification $\mathbb{A}^{n}_{k}(k) = k^{n}$, we have $f(x) = (f_{1}(x),..., f_{n}(x))$, where $f_{i} = \varphi(T_{1})$ and $f_{i}(x)$ is the image of $f_{1}$ in $k(x) = k$ (the residue field)."
Let us fix $x \in X$. I know that the identification $\mathbb{A}^{n}_{k}(k) = k^{n}$ is obtained by assigning each prime ideal $p \in spec(k[T_{1},...,T_{n}])$ the point $\alpha = (\alpha_{1},...,\alpha_{n}) \in k^{n}$ where $\alpha_{i}$ is the image of $T_{i}$ in the residue field. Hence, it should suffice to show that the following diagram commutes:
\begin{center}
\end{center}
where $p = f(x)$. By definition of the induced map, the top square commutes, and thus my question regards the bottom bottom part of the diagram.
Question 1: How should it be shown that the bottom part of the diagram commutes?
I strongly suspect that this has to do with the fact that all the maps are $k$-algebra morphisms. Since $f^{\#}_{x}$ is the local morphism on stalks induced by $\varphi$, the image of a coset under $f^{\#}_{x}$ should remain constant (since cosets are elements of $k$) - but I'm not sure how to formalize this.
I also do not have much geometric intuition about schemes at this point, and thus wish to ask a potentially superficial, ambiguous question.
Question 2: Are rational points of schemes a generalization of rational solutions to equations? This seems strange to me, because in past experiences the term 'rational points' has strictly meant solutions to some polynomial in $\mathbb{Q}$ - whereas here we're allowed any field. Is there some tidbit of motivation one could offer on why rational points of a scheme are a noteworthy concept?