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 be a field. We want to draw , so we’ll need to know the prime ideals of to do this. The only prime ideal of any field is the zero ideal . Hence 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 :

Note though that if are two different fields then the schemes and 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 be a discrete valuation ring (DVR) and write denote its fraction field. What are the prime ideals of ? Firstly, since DVRs are integral domains, the zero ideal is prime. I’ll write for the point in corresponding to the zero ideal. The only other prime ideal is the unique maximal ideal of the DVR, and I’ll write for this point in the affine scheme . So has two points. But these two points are *very* different!

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

That’s not the case with ; the ideal is not maximal, so is not closed in the Zariski topology – in fact, the closure of is the whole of ! So you can picture as a point “smeared out” over the whole scheme:

Before we look at the sheaf structure, recall there’s a natural ring homomorphism ; therefore we know from this post that there’s a morphism of schemes . Since occurs as a localisation of , this must be an open immersion and hence must send the unique point of to an open point in . Therefore this morphism has to send the unique point to . If instead it was sent to , 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 ; we can use this to compute the stalk . The stalk is the colimit

where the limit ranges over the open sets containing . The only such open set containing is itself, and hence the stalk is equal to .

We can use the fact that the inclusion is an open immersion to compute the stalk at ; since this open immersion of ringed spaces has to induce isomorphisms on the structure sheaves, we see that the stalk .

**The affine line**

Now for a more “geometric example”, which really shows how schemes generalise varieties. Let be an algebraically closed field and set . To understand , we need to know what the prime ideals of are. In fact, there are only two distinct kinds – the zero ideal since is an integral domain, and maximal ideals of the form where . So we can picture as a **line**, which we’ll denote , called the affine line. The maximal ideals correspond to the **closed** point on the line, and **the line itself** corresponds to the “generic point” (i.e. nonclosed point) :

In a sense, the nonmaximal ideal 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 . 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 as being a zero-dimensional point of , while the generic point 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 the quotient ring is isomorphic to , and so we have closed immersions where the unique point is sent to the ideal . We also know that the stalk at is equal to the localisation

The stalk at the generic point is , the field of fractions of .

**The affine plane**

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

Let be an algebraically closed field and let ; we call the **affine plane**. It has prime ideals (now a two-dimensional generic point, delocalised over the whole plane), maximal ideals corresponding to the closed point in the plane, and a third kind of prime ideal corresponding to one-dimensional **curves** in the plane; these are the ideals where is an irreducible polynomial. These ideals are contained in many of the maximal ideals , which is reflected by the fact that the “curve” in passes through the points in the plane (as inclusion is reversed when we go between ideals in the ring and subsets of the scheme). So the curve is a point of the scheme, but it can be thought of as living everywhere along a curve of points :

The stalks pictured tell us the functions we’re allowed to invert on each part of ; for example, let be the point in corresponding to the maximal ideal ; then the “function” is zero on , so is the unique maximal ideal in the stalk . Any “function” which isn’t in is nonzero near , so in it becomes a unit, meaning we can divide by it.

**Spectrum of the integers**

Now for a more arithmetic example: . The prime ideals of are and for all prime numbers . The second type of prime ideal is maximal, corresponding to closed points in , while is not, and so it corresponds to a “one-dimensional” point. In fact, looks suspiciously similar to when is an algebraically closed field!

Each closed point is isomorphic to , and the stalks at are the local rings consisting of all fractions whose denominators are not divisible by . The stalk at is , the field of fractions of ; in general, when is an integral domain, the stalk at the generic point of will be the field of fractions of .

**An “arithmetic surface”**

The last example for today, with some **real** arithmetic content, is the affine line over , namely . It’s often written. 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 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 is compared to when 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!

Very nice! These pictures are excellently drawn. It makes me wish I was better at drawing myself. 😦

Some comments:

1) It’s helpful, at least in the context of pictures, to think about things geometrically (similar to my answer in your post about ramification). Namely if you think about as being a finite extension then they both look like points. But, geometrically they are very different. Namely, if one base changes the picture to then we get something very different. Namely, decomposes into a bunch of points. Some points are pure, literal points, some are fuzzy points. In fact, if one decomposes as then the picture is essentially many fuzzy points, with fuzz coming from the inseparable part.

One can recover your picture by recalling that everything over has a Galois action telling you how to ‘glue back down’ to . This perspective is also useful for picturing things like . Namely, what is ? It’s , where (this is actually a rigorously correct statement, it is a quotient in the correct sense). So, one pictures as being the plane but for which one folds the upper half plane $\latex \mathfrak{h}=\{z:\text{Im}(z)>0\}$ up and identifies it, under the conjugation action, with the lower half-plane. So, looks like the line and attached to it are two half-planes glued together.

This picture works for , but things get MUCH more complicated. Now, we have infinitely many sheets being glued together, but any point actually sits in a space where finitely many of the sheets are glued together. I have a picture in my head, but it would be nice to see an artist such as yourself draw such a beast. 🙂

2) It’s a good exercise to picture how the morphism can be pictured! Hint: it’s a formally close zoom-in on the point in , where formally close roughly means ‘as close as the Taylor series of a function can discern’, or ‘as close as derivatives can see’.

3) You seem to really like Mumford. Probably my favorite algebraic geometry text is by him. And, no, it’s not the Red Book. It’s the, for lack of a better phrase, sequel. It’s absolutely stupendous and not really well-known. So, on the chance you haven’t seen it, I thought I’d share: http://www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf

Thanks again for the nice post!

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I’m glad you liked my drawings – I think I’ll carry on trying to draw things and post them on here even if they do start to get ridiculously complicated. It seems to be good for certain intuition and also seeing where other things fail in an arithmetic sense – like in my MSE question that you answered.

I did actually try to draw – and I have a picture of this thing in my head – but it’s messy 🙂

Here’s a quick question: you said that the “quotient” is actually a quotient in the correct sense i.e. a quotient in the category of schemes. Assuming that this holds for more general field extensions with Galois group , does this mean that Galois groups – i.e. étale fundamental groups of spectra of fields – are also group schemes in their own right? Or is it just in this special case with ?

I’m going to do a post on formal schemes when I know more about them, so I’ll have a pic of sitting inside up on here then 🙂

Finally, thanks for the link to the Mumford book. I had no idea there was a follow-up to the Red Book and it looks absolutely great (I especially like how Mumford writes and puts in loads of pictures into his books too)

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Dear Alex,

I should have remarked precisely what I meant by the statement . There is a naive notion in which one might interpret this, and it’s not that one. Namely, it is a ‘geometric quotient’ in the sense of Mumford (see here: https://en.wikipedia.org/wiki/Geometric_quotient). In particular, it *co-represents* a functor, and not *represents* it. Specifically, there is an action of on where, here, $G$ is interpreted as finite constant group scheme. One can see that this map is -equivariant. Then, in general one has that correspond to the -equvariant maps . It satisfies more properties though (see that Wiki link I sent you).

In general, the étale fundamental group is *not* a group scheme. One might be tempted to note that since it’s a profinite group which, of course, one can make into a scheme by considering it as the limit , where is the finite constant group scheme, and . Of course, this is actually doomed to failure since, as shocking as this is (if you’ve never seen it before), profinite groups are not their own profinite completion. In particular, is not its own profinite completion!

That said, Nori has constructed a group scheme (called, imaginatively, Nori’s fundamental group scheme) with the property that it’s -points are, in fact, the usual étale fundamental group. It’s essentially defined to be the fundamental group of a certain Tannakian category. See here for more details: https://en.wikipedia.org/wiki/Fundamental_group_scheme.

Also, I hate to be that guy, but note that the formal scheme of (i.e. ) is not the same thing as . The latter cares about all primes, and the former only about open primes. I know what you mean though. 😛 You don’t have to wait until then to draw it. Here is a relevant post of mine that I’m fairly fond of: https://ayoucis.wordpress.com/2014/07/22/what-information-is-contained-in-an-infinitesimal-neighborhood-of-a-point/

No problem! It really is excellently written. Hopefully it’s as informative to you as it was to me.

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I apologize again for the clutter—I was discussing something with a friend today, and felt compelled to add to this after our discussion.

I feel like I should emphasize, that while the assignment is not a stack (you can’t glue schemes in the étale topology) it *IS* true that (the category of algebraic spaces over , with morphisms isomorphisms) is a stack. Not an algebraic stack, but a stack. Note that, essentially by the 4 equivalent definitions I mentioned above, it doesn’t really matter which of the usual topologies I say they’re a stack with respect to—they’re one with respect to all of them! This is, essentially, the reason that one might want to pass from schemes to algebraic spaces.

One other nice thing to mention, is that they play nicely with the complex topology. Namely, as you most likely know, there is an ‘analytification functor’ which takes a finite type scheme to a complex analytic space (where, here is just the complex analytic space consisting of a point, with sheaf assigning to this point). In the case when is smooth, then is a complex manifolds.

Now, it’s classically a difficult problem to determine the essential image of the analytification functor—even in the case of connected smooth varieties—even in the case of smooth proper connected varieties! As an easy example, the complex disk is not in the essential image of the analytification functor.

One natural impediment is the following. Call a connected complex manifold *Moishezon* if where is the field of meromorphic functions. Note that if then and so it follows from basic algebraic geometry that is Moishezon. So, Moishezon is a necessary condition to be in the essential image of the analytification functor.

(NB: Moishezon seems like a super obvious condition that most things satisfy. It’s true that all smooth projective curves are Moishezon—in fact, they’re all algebraic! That said, a complex torus , for a full lattice, can fail to be Moishezon. In fact, it’s true that as long as then there exists a complex torus of dimension such that latex \mathbb{C}$ is a lot like gluing in the complex topology. So, it’s not shocking that we can get ‘almost everything’ if we’re willing to use algebraic spaces.

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Boo! Like half of the post got cut off by WordPress’s archaic TeXing system.

It said:

(NB: Moishezon seems like a super obvious condition that most things satisfy. It’s true that all smooth projective curves are Moishezon—in fact, they’re all algebraic! That said, a complex torus , for a full lattice, can fail to be Moishezon if . In fact, as long as then there exists a complex torus of dimension such that !).

It is not true that Moishezon is sufficient though unless you restrict to some special classes of manifolds—a complex torus is Moishezon if and only if it’s algebraic. That said, there is a notion of analytification of algebraic spaces, and there it’s true that Moishezon is a necessary *and* sufficient condition! This makes sense since, as we imagine, gluing in the étale topology over is a lot like gluing in the classical topology. So, it’s not shocking that we get a lot more if we allow (mostly) arbitrary gluing in the étale site (i.e. algebraic spaces)

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Not a problem – I’m really pleased you’re happy to write so many detailed and helpful comments! I’ll reply to all of the new ones here if that’s OK.

Thanks for clearing up the thing about the quotient for me. I’ve seen “geometric quotients” before in topology and I don’t know why it didn’t immediately ring with me that this is what you meant. I guess it’s probably a good thing that the fundamental group – in general – isn’t a group scheme though. I was starting to visualise towers of fundamental groups of fundamental group schemes and I think this would all end up being really trivially impossible or otherwise horrendously complicated. The fact that the Nori fundamental group scheme exists is really nice though. I would like to learn more about that – all the more reason to get back into Szamuely’s book and learn more about these things 🙂

Ah, good then – maybe I’ll give drawing a go then. I also enjoyed reading your post about infinitesimal neighbourhoods earlier today – it was helpful for bringing together some of the ideas I’ve recently been reading about in other books.

The fact that algebraic spaces interact in a nicer way with the complex topology than schemes over is very cool. I guess being able to essentially glue spaces as if they were in the classical topology provides a lot more freedom. I am a little bit surprised that there are some non-Moishezon complex spaces out there but your counterexample is a good one. I’ll have to have a read about these! I’m also continually impressed with how all of your examples are well-chosen to highlight how when you pass from schemes to algebraic spaces to stacks, things actually become simpler. It’s given me a lot of motivation to understand these objects more.

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