Given how technical my last few posts have been, I decided that I’d ditch any more technicalities for a couple of posts and instead show you some pictures of schemes. The ones today will all be affine (which already become pretty complex to draw!) but perhaps I’ll do some general pictures too some other time.

Spectrum of a field

As with all things, let’s start on pretty much the simplest example possible. The simplest rings – at least in the context of ideals – are fields. Let $K$ be a field. We want to draw $\text{Spec}(K)$, so we’ll need to know the prime ideals of $K$ to do this. The only prime ideal of any field is the zero ideal $(0)$. Hence $\text{Spec}(K) = \left\{\bullet\right\}$ is just one point. The stalk of the structure sheaf at this point – which, as a one-point space, coincides with the ring of global sections – is just $K$:

Note though that if $K, L$ are two different fields then the schemes $\text{Spec}(K)$ and $\text{Spec}(L)$ are NOT THE SAME! Even though they are homeomorphic as topological spaces, the schemes carry different information in their structure sheaves. It is important to remember that schemes have the structure of their “functions” built in as the structure sheaf.

In all my pictures, I’ve drawn the points of the scheme in blue, and the various sections of the structure sheaf in red.

Spectrum of a DVR

So that was pretty boring. What about another slightly more interesting example? Let $R$ be a discrete valuation ring (DVR) and write $\text{Frac}(R)$ denote its fraction field. What are the prime ideals of $R$? Firstly, since DVRs are integral domains, the zero ideal $(0)$ is prime. I’ll write $t_0$ for the point in $X=\text{Spec}(R)$ corresponding to the zero ideal. The only other prime ideal is the unique maximal ideal $M$ of the DVR, and I’ll write $t_1$ for this point in the affine scheme $\text{Spec}(R)$. So $\text{Spec}(R) = \left\{t_0, t_1\right\}$ has two points. But these two points are very different!

The point $t_1$ corresponds to the maximal ideal $M$; recall from this post that the closed points of an affine scheme correspond exactly to the maximal ideals of the ring. Therefore, $t_1$ is a closed point, meaning the subset $\left\{t_1\right\}$ is closed in the Zariski topology.

That’s not the case with $t_0$; the ideal $(0)$ is not maximal, so $\left\{t_0\right\}$ is not closed in the Zariski topology – in fact, the closure of $\left\{t_0\right\}$ is the whole of $\text{Spec}(R)$! So you can picture $t_0$ as a point “smeared out” over the whole scheme:

Before we look at the sheaf structure, recall there’s a natural ring homomorphism $R\rightarrow \text{Frac}(R)$; therefore we know from this post that there’s a morphism of schemes $\text{Spec}(\text{Frac}(R))\rightarrow \text{Spec}(R)$. Since $\text{Frac}(R)$ occurs as a localisation of $R$, this must be an open immersion and hence must send the unique point of $\text{Spec}(\text{Frac}(R))$ to an open point in $\text{Spec}(R)$. Therefore this morphism has to send the unique point $\text{Spec}(\text{Frac}(R))$ to $t_0$. If instead it was sent to $t_1$, this would still be a morphism of ringed spaces, but not of locally ringed spaces and hence not a morphism of schemes. So this is a first example of subtleties arising from schemes and their morphisms.

Now we know that the ring of global sections is equal to $R$; we can use this to compute the stalk $\mathcal{O}_{X,t_1}$. The stalk is the colimit

$\mathcal{O}_{X,t_1} = \text{colim}_{U\ni t_1} \mathcal{O}_X (U)$

where the limit ranges over the open sets $U\subseteq X$ containing $t_1$. The only such open set containing $t_1$ is $X$ itself, and hence the stalk $\mathcal{O}_{X,t_1}$ is equal to $\mathcal{O}_X (X) = R$.

We can use the fact that the inclusion $\text{Spec}(\text{Frac}(R)) \rightarrow X$ is an open immersion to compute the stalk at $t_0$; since this open immersion of ringed spaces has to induce isomorphisms on the structure sheaves, we see that the stalk $\mathcal{O}_{X,t_0} = \text{Frac}(R)$.

The affine line

Now for a more “geometric example”, which really shows how schemes generalise varieties. Let $K$ be an algebraically closed field and set $X = \text{Spec}(K[x])$. To understand $X$, we need to know what the prime ideals of $K[x]$ are. In fact, there are only two distinct kinds – the zero ideal $(0)$ since $K[x]$ is an integral domain, and maximal ideals of the form $(x-a)$ where $a\in K$. So we can picture $X$ as a line, which we’ll denote $\mathbb A^1_K$, called the affine line. The maximal ideals $(x-a)$ correspond to the closed point $a$ on the line, and the line itself corresponds to the “generic point” (i.e. nonclosed point) $(0)$:

In a sense, the nonmaximal ideal $(0)$ is a point that isn’t anywhere in particular on the line, but its closure is the whole line, so it’s helpful to think of it almost as a “subvariety” of $X$. Thinking of things this way (which we will do lots later) is nice, because it forces us to make the notion of “point” more general, allowing points of schemes to “parameterise” subschemes in a vague hand-wavey sense. We can think of each maximal ideal $(x-a)$ as being a zero-dimensional point of $X$, while the generic point $(0)$ is a one-dimensional point. In fact, just as divisors are formal linear combinations of traditional points in a variety, the notion of algebraic cycles can be thought of as formal linear combinations of subvarieties of higher dimension, or in this sense points of higher dimension. Anyway, I digress – let’s get back to looking at our picture above:

For each $a\in K$ the quotient ring $K[x]/(x-a)$ is isomorphic to $K$, and so we have closed immersions $\text{Spec}(K)\rightarrow \mathbb A^1_K$ where the unique point is sent to the ideal $(x-a)$. We also know that the stalk at $(x-a)$ is equal to the localisation

$K[x]_{(x-a)} = \left\{f/g: f,g\in K[x], g(a)\neq 0\right\}$

The stalk at the generic point $(0)$ is $\mathcal{O}_{X,(0)} = K(x)$, the field of fractions of $K[x]$.

The affine plane

Now for an even more geometric example, to see how these things generalise to higher dimensions:

Let $K$ be an algebraically closed field and let $X = \text{Spec}(K[x,y])$; we call $X = \mathbb A^2_K$ the affine plane. It has prime ideals $(0)$ (now a two-dimensional generic point, delocalised over the whole plane), maximal ideals $(x-a, y-b)$ corresponding to the closed point $(a,b)$ in the plane, and a third kind of prime ideal corresponding to one-dimensional curves in the plane; these are the ideals $(f)$ where $f\in K[x,y]$ is an irreducible polynomial. These ideals are contained in many of the maximal ideals $(x-a,y-b)$, which is reflected by the fact that the “curve” $(f)$ in $\mathbb A^2_K$ passes through the points $(a,b)$ in the plane (as inclusion is reversed when we go between ideals in the ring and subsets of the scheme). So the curve $(f)$ is a point of the scheme, but it can be thought of as living everywhere along a curve of points $(a,b)$:

The stalks pictured tell us the functions we’re allowed to invert on each part of $\mathbb A^2_K$; for example, let $(a,b)$ be the point in $\mathbb A^2_K$ corresponding to the maximal ideal $(x-a,y-b)$; then the “function” $(x-a,y-b)$ is zero on $(a,b)$, so $(x-a, y-b)$ is the unique maximal ideal in the stalk $\mathcal{O}_{X,(a,b)}$. Any “function” which isn’t in $(x-a,y-b)$ is nonzero near $(a,b)$, so in $\mathcal{O}_{X,(a,b)}$ it becomes a unit, meaning we can divide by it.

Spectrum of the integers

Now for a more arithmetic example: $\text{Spec}(\mathbb Z)$. The prime ideals of $\mathbb Z$ are $(0)$ and $(p)$ for all prime numbers $p$. The second type of prime ideal is maximal, corresponding to closed points in $\text{Spec}(\mathbb Z)$, while $(0)$ is not, and so it corresponds to a “one-dimensional” point. In fact, $\text{Spec}(\mathbb Z)$ looks suspiciously similar to $\text{Spec}(K[x])$ when $K$ is an algebraically closed field!

Each closed point $(p)$ is isomorphic to $\text{Spec}(\mathbb F_p)$, and the stalks at $(p)$ are the local rings $\mathbb Z_(p)$ consisting of all fractions whose denominators are not divisible by $p$. The stalk at $(0)$ is $\mathbb Q$, the field of fractions of $\mathbb Z$; in general, when $R$ is an integral domain, the stalk at the generic point of $\text{Spec}(R)$ will be the field of fractions of $R$.

An “arithmetic surface”

The last example for today, with some real arithmetic content, is the affine line over $\mathbb Z$, namely $\text{Spec}(\mathbb{Z} [x])$. It’s often written$\mathbb A^1_{\mathbb Z}$. Here’s a picture of it taken from Mumford’s Red Book:

It’s called a “line” because it’s the spectrum of a polynomial ring in one variable, but it’s really a two-dimensional object, with two different axes – an “arithmetic axis” along the bottom, and a “geometric axis” going upwards. Even though $\mathbb Z$ is a very simple ring – it’s a Euclidean domain, and thus “nearly” a field in terms of nice structure – it’s startling how much more complicated the affine line $\mathbb A^1_{\mathbb Z}$ is compared to $\mathbb A^1_K$ when $K$ is an algebraically closed field. This “arithmetic surface” provides the first glimpse that schemes encode arithmetic information well, and that they’re the right objects to use to get number-theoretical data out of geometry.

In fact, I’m not going to talk about this scheme today – I will probably devote an entire post to it in the future when we’ve got some more tools – because I can’t do a better job than Lieven le Bruyn did on his own blog in his analysis of “Mumford’s Treasure Map“. Please have a look at it!