Examples of affine schemes

Question

I have to introduce affine schemes as topological spaces in a small seminar. Could you suggest me three examples of affine schemes in order to put in evidence the correlation among traditional algebraic geometry and the geometry of schemes. For example, how can I see the line or the parabola as affine schemes?

Answer 1

I will give you some ideas but you should probably choose according to your taste and the audience's taste. It may very well happen that nothing of what follows will help you, in that case I am sorry. I will try to give examples where you can (more or less) visualize "points" (as you are interested in the Zariski topology), but in a scheme there is much more structure than a topology. Also, for each example, it would be great to understand what is the generic point of the scheme (when it is irreducible). I include pictures that may help you or your audience to visualize what happens.

I. First examples.

The first example I saw in my life was $$\textrm{Spec }\mathbb Z,$$ and a comparison of $\textrm{Spec }\mathbb Z$ with the affine line $$\mathbb A^1_k=\textrm{Spec }k[x].$$ (I remind that at the very beginning of Eisenbud's book "Commutative algebra with a view towards algebraic geometry", the author introduces the analogies between these rings...). But $\textrm{Spec }\mathbb R[x]$ or $\textrm{Spec }\mathbb C[x]$ could be other good starting points, even from a topological point of view.

II. Examples illustrating connections with number theory.

1. One of these might be $\textrm{Spec }\mathbb Z[i]$, the spectrum of the Gaussian integers, and its structural morphism to $\textrm{Spec }\mathbb Z$. If one remembers what primes $p\in \mathbb Z$ ramify and what are inert, then a nice picture could illustrate this morphism:

2. There is an amazing picture by Mumford:

describing the structural morphism (and much more!) $\textrm{Spec }\mathbb Z[x]\to \textrm{Spec }\mathbb Z$. I think, but I am not sure, the picture could be inside "The red book of varieties and schemes".

3 (Added). The ring of integers $\mathcal O_K$ of a number field $K/\mathbb Q$ is a Dedekind domain. Correction (thanks to Alex's alert eyes): such a ring cannot be a DVR. The ring of integers of a finite extension of $\mathbb Q_p$ (a non-archimedean local field) is a DVR, though. They are, by (a possible) definition, complete with respect to a non-archimedean discrete valuation. [I apologize for my mistake]

The spectrum $\textrm{Spec }R$ of a discrete valuation ring $R$ is an interesting example of a one dimensional scheme whose underlying topological space consists of two points (the closed point corresponds to the unique nonzero prime ideal of $R$). You may also want to look at the morphism $\textrm{Spec }K\to \textrm{Spec }R$ corresponding to the inclusion $R\to K=\textrm{Frac }R$. More generally, you might be interested in Dedekind schemes. (just google the term...)

III. Something you can actually draw

If you want to be more concrete (you were talking about a parabola), you may want to introduce $$\textrm{Spec }\mathbb C[x,y]/(y-x^2),$$ a closed subscheme of $\textrm{Spec }\mathbb C[x,y]=\mathbb A^2_{\mathbb C}$. Or you might introduce the affine (group) scheme $$\mathbb G_m=\mathbb C^\times=\textrm{Spec }\mathbb C[x,y]/(xy-1),$$ whose real points you may draw as an hyperbola. There is the double line $\textrm{Spec }\mathbb C[x,y]/(xy)$, a singular plane curve which can be viewed as a degeneration of the nonsingular curve $\mathbb G_m$. More generally, you might want to introduce "irreducible plane curves" $$C_f=\textrm{Spec }k[x,y]/(f)\subset \mathbb A^2_k,$$ and perhaps show that the only closed subsets of $C_f$ are the finite sets (not containing the generic point). (Here $f$ is any irreducible polynomial and $k$ is a field.) Or maybe surfaces $\textrm{Spec }k[x,y,z]/(f)\subset \mathbb A^3=\textrm{Spec }k[x,y,z]$...