I have a question related to Proposition 1.49 of the book Algebraic Group of Milne. Here is the excerpt:
My question is the last line. I don't really understand the reasoning in that line. From what I understand, $H_j$ is a closed subscheme of $G$, which is affine, so if $G = Spec(R)$, then $H_j \subset G$, and $I(H_j) = \bigcap_{p \in H_j}p$, which is an ideal of $R$. So each $f \in I(H_j)$ is an element of $R$, which can be considered as a function on $Spec(R) = G$. How does this function $f$ act on some $g \in G(R) = Hom(SpecR, G)$? And why is that line true eventually?
If you don't understand any notion/symbol from the excerpt, please tell me. Thank you
This notation is explained in 3.2 of the book. Let $G$ be an affine algebraic group.
For a $k$-algebra $R$, $G(R)\simeq\mathrm{Hom}_{k\text{-algebra}}(\mathcal{O}{}(G),R)\simeq \mathrm{Hom}_{R\text{-algebra}}(\mathcal{O}{}(G)_{R},R)$.
An $f\in\mathcal{O}(G)$ defines an evaluation map $f_{R}\colon G(R)\rightarrow R,\quad g\mapsto g(f)=f_{R}(g),$ which is natural in $R$.
If $H$ is the subgroup of an affine group scheme $G$ corresponding to an ideal $I$, then $H(R)=Hom(\mathcal{O}/I,R) =\{g\in G(R)=Hom(\mathcal{O},R)|g(f)=0\text{ all }f\in I\}$