So this autumn (starting 7 October), I’m teaching a course for first-year PhD students. It’s algebraic geometry, which for me means the theory of schemes. I taught this once before, last spring. There are a few things I want to do different this time – some concepts that I didn’t quite get across terribly well, some examples I didn’t dig into deeply enough.

There are a zillion little choices you have to make when you teach a
course, particularly one that has as many radical new ideas as algebraic
geometry. ^{1} I try to make these choices through
the lens of two questions:

What do I actually know how to communicate well? No one benefits from me stumbling through a discussion of something I don’t understand well or even a story I understand but can’t tell in an interesting way. The moduli space of K3 surfaces seems fascinating, but I know nothing nontrivial about it yet, so teaching a course with that as a centerpiece would be a stupid thing for me to do.

What does this particular crop of students need (or want) to know? Most of the students I’ll be teaching are specializing in other areas of mathematics. If I were to spend a long time on examples over finite fields, I’d probably teach a course that would be of no use to anyone present.

Here I’ll reflect some on these questions in the context of this course.

For whatever reason, I like to organize my thoughts about teaching around aphorisms. Today I have two. The first is about understanding what the word ‘geometry’ actually means in the phrase ‘algebraic geometry’. The second is about understanding the role of the Zariski topology.

This is a coarse statement, but it is broadly reflective of the attitude of the French school around Bourbaki in the late 50s and early 60s.

For example, when we talk about a manifold (smooth, say), what exactly are we talking about? You have a topological space (Hausdorff, second-countable, paracompact), and then what do you have to add? One option is to say that you need a maximal smooth atlas (or an equivalence class of smooth atlases). Using such an atlas, you can now define smooth functions locally on . If , you get a ring (in fact an -algebra) of smooth functions on . On the other hand, once you know what ‘smooth function’ means, you can reverse engineer an atlas.

So the data of a manifold can be expressed like this: it’s a
topological space (satisfying some conditions perhaps)
along with a *sheaf of rings*
.

Well, almost. The stalk of this sheaf at a point
is not just a ring but a *local ring*. For every
open neighborhood
of
, you have the ring map
given by evaluation at
. These are all compatible, so they induce a ring map
. The kernel is a maximal ideal
consisting of the germs of functions that vanish at
. The units of the stalk
are exactly the elements outside the maximal ideal; after
all, if
is a smooth function on a neighborhood of
and
, then
makes sense on a neighborhood of
. Thus the stalk is indeed a local ring.

The fact that the maximal ideal of a local ring is unique misleads
one into thinking that locality is a *property* of a ring. It
should really be considered as *structure*, because the maximal
ideal isn’t preserved by every ring map. ^{2} One
of the ways we try to remind ourselves of that is by writing a local
ring as a triple
, where
is the maximal ideal, and
is the *residue field*. A *local ring
homomorphism*
is a ring homomorphism such that
– or, equivalently, one that induces a field extension
.

So that’s the structure: a topological space
and a sheaf of rings whose stalks are local – a
*locally ringed space*. And the claim is that being a smooth
manifold is a particular *property* that a locally ringed space
might possess. In other words, smooth manifolds and smooth maps between
them form a full subcategory of the category of locally ringed
spaces.

The property that defines this full subcategory is the fact that,
locally, manifolds of the same dimension all look the same: they all
look like euclidean space. In other words, a smooth manifold is a
locally ringed space that is *locally isomorphic* to
.

Once you start thinking like this, it’s hard to stop. You can specify
a style of geometry by just specifying the local models you like. If you
take locally ringed spaces of the form
, then you arrive at the geometry of complex manifolds. And
if you take locally ringed spaces of the form
then you arrive at the category of *schemes*!

This perspective also opens to door to generalizations that provide a much bigger universe of geometries. For example, in analytic geometry, the rings of functions are equipped with something like a topology (or condensed structure). In derived algebraic geometry, the rings of functions are equipped with a derived structure (a cdga or a ring spectrum). With various theories of stacks, one may need to replace the topological space with a site (or a topos).

In a first course on scheme theory, there’s no need to get into these variations of course. Schemes are hard enough. However, I want to communicate the idea that this approach to geometry establishes a theme upon which many variations can be fruitfully created and explored.

But ok. The basic local models are affine schemes . But how do we talk about these locally ringed spaces? The first step is to understand this Zariski topological space. For that, I have another aphorism.

The Zariski topology on
meets the requirements of a topology, but the intuitions
one develops in a general topology course are not terribly useful for
understanding it. It’s (in)famously weird. In particular, the open sets
are comically large, and as a result, it’s rarely Hausdorff, but it’s
always quasicompact (*i.e.*, every open cover contains a finite
subcover). It feels *pathological*.

But our aphorism is an effort to think about this a little differently. The claim is that the Zariski topology can be thought of in more combinatorial terms. I’m not going to give a definition of here today; I’ll do that in a later post. Rather, I want to think about the nature of the Zariski topology, and why it’s so weird.

We can begin by reflecting on the notion of a *stratification*
of a topological space. A stratification of
is sometimes defined as sequences of closed subspaces
whose union is all of
. Some authors require some further conditions, but let’s
be relaxed about this. The
-th *stratum* is the subset
, which is *locally closed*. For instance, you can
consider projective space
, which is stratified via the hyperplanes at infinity:
so that the strata are affine spaces.

In general, if
is stratified in this sense, we have a map
that carries a point of
to the stratum it lives in. Now we can topologize
so that this map is continuous. ^{3} We
say that a subset
is open iff it is an *upper set*: if
, then if
, then
. With that topology – the *Alexandrov topology* –
giving a stratification on
is the same thing as giving a continuous map
.

Of course, there’s nothing terribly special about here. I could do the same thing with any poset : endow it with the Alexandrov topology, define a -stratification as a continuous map . You have a locally closed -th stratum for any , and you also have closed subsets like This lets you think about the combinatorics of more complex kinds of stratifications, like the following stratification on the sphere .

First, let’s identify our poset . Its elements will be the vertices of an -cube: Order this set so that if , then either of is smaller than either of . Now define in the following way: let be the largest index such that , and set .

By the way, this stratification is actually pretty important, and I
want to return to it in another context. ^{4}

Now let’s think about what happens when we contemplate
*algebraically defined stratifications* of
. Algebraic geometry, we are led to understand, is the
study of solutions of polynomial equations. So if I have a collection
of polynomials in
variables, then I’m interested in the sets
. These are closed subsets of
. And these are, like, the only ones we want to think
about.

So an *algebraic stratification* of
consists of a finite poset
and a continuous map
(Alexandrov topology again) such that for every
, the subset
is a subset of the form
for some set
of polynomials. In fact, we might as well choose
, the set of all polynomials that vanish at every point of
.

For example, I could do something pretty simple, and let
, and define
to be the smallest
such that
. But I could also do something far more combinatorially
complicated. When you can write a poset map
, then we think of a stratification
as *finer* than the resulting stratification
.

Now here’s a basic question you can ask yourself: is there a
*finest*, or *universal* algebraic stratification of
? That is, is there an algebraic stratification
such that for every algebraic stratification
, there is a unique poset map
such that
?

The answer, of course, is no. We specifically required the posets in algebraic stratifications be finite, and so any time you have an algebraic stratification, you can always just add a point as a new stratum. Since there are infinitely many points in , we lose.

On the other hand, the answer can be made to be yes if we allow
ourselves to relax the finiteness. But we shouldn’t relax the finiteness
by saying, ‘ok, any size poset will do.’ The way we have to relax the
finiteness is by taking some *limit* of finite posets, because we
want something that will map to every finite poset with which we can
stratify
. What we really want is to define
as the limit of the posets
over all algebraic stratifications
. So an element
of
is a compatible system
of elements
, one for every algebraic stratification
. As the stratifications
get finer and finer, the strata
get smaller and smaller.

If you take a point , then you certainly get such a compatible system: namely, you can take . But there are others!

In , consider the system where is the smallest element such that every point of the form is contained in . Such a smallest element exists because algebraic stratifications can’t break up the locus in a meaningful way: if you have an algebraic stratification , and if is a minimal element such that is contained in , then the irreducibility of implies that all but finitely many points of the form map to .

This gives us a way to think about what’s happening with the Zariski
topological space
more generally. It turns out that this topological space
comes from an *inverse system* of finite posets
. Here,
is itself a poset such that if
, then there exists
such that
and
. (One sometimes calls
a *cofiltered* poset.) The inverse system consists
of the following:

for each you have a finite poset ;

if , you have a monotonic map ; and

these are compatible with composition in the way you’re already imagining.

In our case, the posets
are going to be *algebraic stratifications* of
. We can define these without making explicit reference to
points of
, but let’s postpone that story for now. Instead, I want to
focus on the meaning of the topology.

The point is, once we’ve identified the correct notion of algebraic
stratification, the Zariski topology is just trying to be the universal
algebraic stratification. That isn’t a finite poset itself (usually),
but it is a limit of this inverse system of finite posets, and
*that’s* how the topology arises.

That is, you think of the Alexandrov topology on the posets
, and you form the limit in the category of topological
spaces. The result is the Zariski topology. This is an important way to
use topology that doesn’t get the attention it deserves in undergrad
textbooks: you start with some finite combinatorial gadgets that you
give a ridiculously simple topology, and then you take infinite limits
of these things in topological spaces. This carries us from gadgets to
what are sometimes called *pro*-gadgets.

In fact, you may have encountered this kind of thing before. If you
take an inverse system of finite sets (so posets where the only
comparable elements are equal), then you can endow each of these with
the discrete topology. Then when you take the limit of this system in
topological spaces, you get what’s called a *Stone space* – a
compact, Hausdorff, totally disconnected topological space. Cantor
spaces are a nice example, but there are others, like the Stone–Čech
compactification of an infinite set. ^{5} In brief, profinite sets
are the same thing as Stone spaces.

It’s no big surprise that the topologies that come out of this sort of construction are ‘weird’ in the sense that they don’t have much in common with Euclidean space. The topology is serving a different role than we’re accustomed to: it’s really only there to help us cope with the fact that we took an infinite limit of some combinatorial structures.

What we’re witnessing is the phenomenon that there are really
*two* ways to pass from finite mathematical objects of some kind
to infinite mathematical objects of the same kind. On one hand, you can
take a sequence of maps of finite gadgets
and take the union (or colimit)
. The result is an ‘ind-object’, which is what we tend to
think of first when we think of passing from finite to infinite. It’s
easy to map out of an ind-object: a map out of
is a compatible system of maps out of each
. In other words,
is well approximated by the finite objects
from the left. For instance, if the
are all finite sets, then
is just an infinite set. If the
are all finite groups, then
is an infinite torsion group. If the
are all finite posets, then
is an infinite poset. This is the way we’re used to
passing from finite to infinite structures.

On the other hand, you can take a sequence going the other way:
and take the limit
. This process gives you an ‘pro-object’, which is also
infinite, but in a dual manner. It’s easy to map *into* a
pro-object: a map into
is a compatible system of maps into each
. In other words,
is well approximated by the finite objects
from the right. That means that pro-objects are infinite
in a different way – one that is nicely modelled with topology. The
resulting sets usually have big cardinalities (uncountable, typically),
but as topological spaces they often have good compactness properties
(by Tychonoff). For instance, if the
are all finite sets, then as we have seen,
is a Stone space. If the
are all finite groups, then
is a profinite group (i.e., a compact, Hausdorff, totally
disconnected topological group).

If the
are all finite posets, then
is a *spectral topological space*. This is a
topological space with the following properties:

it is quasicompact;

it is sober –

*i.e.*, every irreducible closed subset contains a unique generic point;finite intersections of quasicompact subsets are quasicompact;

the quasicompact opens form a base for the topology.

In brief, profinite posets are the same thing as spectral spaces.
Melvin Hochster showed in the 1960s that spectral spaces are exactly ^{6} the topological spaces of the form
.

The upshot here is that the Zariski topology is really just a
*profinite poset*, and it has *exactly the same
information* as the collection of all algebraic stratifications of
.

Once you take on this attitude toward the Zariski topology, it becomes a lot easier to accept. You can’t ask too much of it: it merely assembles an infinite family of combinatorial structures.

In a later post, we’ll explain our notion of algebraic stratification, and we’ll describe the structure sheaf.