Maximal ideals of coordinate ring of the circle I am currently a master student taking algebraic geometry and a teaching assistent for undergraduate ring theory.
The students had to make the following easy exercise in ring theory:
Consider the ring $R=\mathbf{R}[x,y]/(x^2+y^2-1)$.
(a) Prove that $(x-a,y-b)\subset R$ is maximal if and only if $a^2+b^2=1$. (b) For which $b\in\mathbf{R}$ is $(y-b)\subset R$ maximal?
From algebraic geometry, I know that $R$ is the coordinate ring of the circle. I was wondering if this exercise (maybe replacing $\mathbf{R}$ by $\mathbf{C}$ since algebraic geometers work over algebraically closed fields) has a more advanced interpretation from an algebraic geometry point of view.
I am very interested in any answer.
 A: The interpretation of a) is very simple: if $k$ is a field and $f_1, \dots f_m \in k[x_1, \dots x_n]$ is a system of polynomial equations, then the collection of solutions to those equations over $k$ can be identified with the set of $k$-algebra homomorphisms
$$\text{Hom}_k(k[x_1, \dots x_n]/(f_1, \dots f_m), k)$$
and since any such homomorphism must be surjective the kernels of such homomorphisms are exactly the maximal ideals with residue field $k$. Furthermore if $k$ is algebraically closed the Nullstellensatz implies that every maximal ideal arises in this way. This is the basis of the correspondence between maximal ideals and points on varieties; the quotient $k[x_1, \dots x_n]/(f_1, \dots f_m)$ is the algebra of functions on the affine scheme cut out by the polynomials $f_i$ (I really need to say "affine scheme" here and not just "affine variety" if the ideal $I = (f_1, \dots)$ is not reduced).
b) is a little trickier to understand geometrically, and we actually pass out of the realm of varieties. Quotienting by $(y - b)$ corresponds geometrically to taking the scheme-theoretic intersection of the circle with the line $y = b$. Geometrically there are three possibilities: the intersection can be

*

*two real points (e.g. $y = 0$, corresponding to the quotient $\mathbb{R}[x]/(x^2 - 1) \cong \mathbb{R}^2$),

*one real point with multiplicity two (e.g. $y = 1$, corresponding to the quotient $\mathbb{R}[x]/x^2$), or

*two complex points (e.g. $y = 2$, correspnding to the quotient $\mathbb{R}[x]/(x^2 + 3) \cong \mathbb{C}$).

We only get a field in the third case so those are the maximal ideals. Now it's necessary to have a grip on how to think geometrically about a variety over a field that isn't algebraically closed, and scheme-theoretic intersections of such a variety with other varieties that aren't varieties (there's a nilpotent in the second case). To my mind the cleanest way to do it, and the point of view I implicitly adopt in the description above, is that a variety (or scheme) over $k$ is a variety over $\bar{k}$ together with a Galois action. Maximal ideals (in the affine case) correspond not to $k$-points but to Galois orbits of $\bar{k}$-points; above in the "two complex points" case there are two points but they lie in a single Galois orbit which is why we get a single maximal ideal. And then sometimes intersections have multiplicity which turns out to correspond to nilpotents, at least in this simple case.
