Let $\phi : X \rightarrow Y$ be a morphism of affine varieties over $\mathbb{R}$. Let $\phi^*$ be the pullback on coordinate rings
$\phi^* : O_Y \rightarrow O_X$. I am trying to understand the relation between this and the scheme theoretic version. Let $\operatorname{Specmax}(X)$ be the set of maximal ideals of $O_X$. This space is bijective with, to my understanding, the points of $X$. Similarly let $\operatorname{Specmax}(Y)$ be the set of maximal ideals of $O_Y$. Is $(\phi^*)^{-1}$ a map from $\operatorname{Specmax}(X)$ to $\operatorname{Specmax}(Y)$?
This question is related to the one in a previous post previous question about affine varieties over $\mathbb{R}$.
Best Answer
Yes, it is true that $\varphi$ induces a map on maximal spectra: the key statement here is that if we have a map of finitely generated algebras over a field, then the pullback of a maximal ideal is maximal. This is proven here on MSE for instance.
On the other hand, it is not true that the points of $\operatorname{Specmax} O_X$ are in bijection with points of the $\Bbb R$-variety $X$, assuming you mean "variety" in the classical sense as a subset of $\Bbb R^n$: for instance, $(x^2+1)$ is a maximal ideal of $\Bbb R[x]=O_{\Bbb A^1_{\Bbb R}}$, but it doesn't correspond to any point in $\Bbb R$. Instead, you should say that the maximal ideals of $O_X$ with residue field $\Bbb R$ are in correspondence with the points of $X$ in the sense of classical varieties. One can get the map of varieties by restricting - the image of any point in the maximal spectrum with residue field $\Bbb R$ is again a point with residue field $\Bbb R$.
All of the extra points we get, first from maximal ideals with residue field $\Bbb C$, and then from generic points, make the theory of varieties over the reals a lot nicer (you can do the same thing with an arbitrary non-algebraically closed field $k$ playing the role of $\Bbb R$ here, just substitute "finite extensions of $k$" for $\Bbb C$ in this sentence). The geometry is a lot better with them in the picture: finite maps are surjective, intersections behave appropriately, you can use stalks at generic points, etc. If you're specifically concerned with the geometry of the real points, you may be helped by investigating some semi-algebraic geometry. Here aresome expository texts on real algebraic geometry to see how all of these ideas play off against each other.