Thread: What is a scheme?

Subtitle: In which I don't actually define schemes but instead tell you what they should be like locally. 1/n

What I'm going to do in this thread is motivate the basic definitions that go into affine schemes. You won't need to know what an affine scheme is.

Notation: all rings are commutative. 2/n

Cast your mind back 0-80 years to your first time learning about polynomials. They looked like c_0 + c_1x + ... + c_nx^n, and you probably came to think of them as instruction sets for drawing graphs. 3/n

As you learned more about polynomials, you found out you could factor them into small pieces, that you could do long division with them, and about rational functions. In other words, without ever being told this, you found out they're a lot like (integer) numbers. 4/n

This probably never seemed weird to you, because you were smallish when you first learned about all this, but also because by the time you'd returned to these topics you'd learned to think of polynomials as formal objects encoding functions, rather than functions themselves. 5/n

For example, x^3 is not the same polynomial over Z/(3) as x. That's a choice we made. We chose to think of polynomials, which are functions on an object, in an abstract algebraic framework that makes them more like numbers. 6/n

Over time a strange reversal began to happen: we started treating numbers like functions, as if putting them in the same framework were causing them to cross pollinate. For one of many examples, consider an integer m and a prime p: we can consider m mod p, mod p^2, mod p^3... 7/n

All this information together acts like a power series expansion of m 'at p'.

At this point a creeping suspicion begins to rise: if we're thinking of functions on objects as abstract ring elements, and some of these abstract ring elements are turning out to be functions... 8/n

... maybe they're all functions, and you can flip back and forth as much as you like.

So this is how you end up at the concept of affine schemes. You start with a ring A, and you want to construct a space Spec(A), the spectrum of A, on which A is the ring of good functions. 9/n

It would be really nice if this construction were functorial into a category of Specs somehow (let's call this category Geo for now), and it would be extremely extremely nice if it produced an equivalence of categories with CRing, the category of rings. 10/n

With the right definitions this aim is achieved (contravariantly). Caveat: our ring elements won't (except in a useless technical sense) end up being literal functions, but they behave so similarly that you won't care. 11/n

Let us answer five questions that will gradually reveal what the objects of Geo ought to look like.

(1) What is the term 'good function' meant to suggest? We can think about what the usable geometric objects that we already have look like. 12/n

They're a topological space, some target field (R or C), and for each open subset U some subring of the continuous functions to the field. Notice that the topological space isn't enough to encode the geometry. 13/n

The usual ring of functions on C^2 is the ring of holomorphic functions to C, while C^2 is homeomorphic to R^4, where the ring of functions is usually the smooth functions to R. So what a good function is is contextual, but they ought to be like functions into fields. 14/n

(2) What should the points of some Spec(A) look like? This is we'll end up diverging from the hope that our ring elements should be literal functions. Well, a point p on Spec(A) is something we can evaluate elements f of A on. 15/n

In particular, the map ev_p given by ev_p(f) = f(p) should be a ring homomorphism. Since our functions are supposed to map into fields, ker ev_p should be prime, and should be maximal iff ev_p is surjective. 16/n

Whatever Spec(A) is, this defines a function p |-> ker ev_p from its points into the set of prime ideals of A. If Geo is going to be part of an equivalence of categories, Spec(A) shouldn't contain any more information than A, so let's just define this to be a bijection. 17/n

That is, we let the points of Spec(A) be the prime ideals of A. When the definition is motivated this way, we see that we should think of f as vanishing at p iff f is contained in p.

A couple things to mention here. 18/n

First, A/p is potentially a very different ring depending on p, so there is no one ambient field our functions are mapping into. Second, the points that are primes that are not maximal don't have an obvious analogue in the topological space/manifold picture. 19/n

You appreciate them more as you get more into AG (for one thing, you need them to make the equivalence of categories work out), but at this level, the intuition revolves around points as maximal ideals.

(3) What should the topology be? 20/n

Certainly level sets of functions should be closed; we can subtract off a constant in our fictional target field to show a level set is the same as a vanishing set. So for any f in A, the vanishing set V(f) of f, (i.e. the set of prime ideals containing f) should be closed. 21/n

We take the minimal topology in which this is true. This is the famous Zariski topology.

(4) What should the good functions on smaller open sets than the whole space be? On U = Spec(A) \\ V(f), we should have all the functions we had before, except now... 22/n

... we include ones that would blow up along V(f). I'll elide the details but it turns out the right ring here is A[f^-1], and that you can use this to uniquely define the good functions on all the open sets.

(5) How does the functoriality work? 23/n

Given F: Spec A -> Spec B in Geo, we should be able to precompose the good functions on Spec B with F and get good functions on Spec A. This precomposition gives a ring homomorphism F*: B -> A, so we ought to have a contravariant functor Geo^op -> CRing. 24/n

The other direction is conceptually trickier: given the information of how functions pull back, we need to recover a geometric map.

I have not and will not tell you the precise definition of the morphisms in Geo, but we can still understand how the recovery works. 25/n

Let's take an example first. Suppose we have a ring of good functions A on real three space R^3, similarly for B and R, and a ring homomorphism F*: B -> A we believe to be induced by pullback along a geometric map F. Think of x,y,z: R^3 -> R as functions. 26/n

For any p in R, we can use x,y and z to determine F(p): we have F(p) = (F*x(p), F*y(p), F*z(p)).

A coordinate free way of using functions to determine our map arises from the following observation: if F*f(p) = 0 for some f in B, then f vanishes at F(p), i.e., f is in F(p). 27/n

Conversely, if f vanishes at F(p), then F*f vanishes at p. Algebraically, this says p ∩ im F* = F*(F(p)), or equivalently that (F*)^-1(p) = F(p). So F(p) is uniquely determined to be the prime ideal (F*)^-1(p). 28/n

This is one reason we need prime ideals: even if p is maximal it may not be that (F*)^-1(p) is maximal.

When constructed rigorously, Geo is called the category of affine schemes, and as promised we'll have an equivalence of categories between affine schemes and CRing^op. 29/n

Thus we've taken the theory of rings and geometerized it. That means any reasonable statement about rings has a geometric analogue, and vice versa. One thing I like to do on my blog ( https://sheafifiedsarah.wordpress.com/ ) is generalize ring theory statements using geometry. 30/n

I think this is pedagogically good because it really clarifies how to use the basic machinery, which can be obscure. One last thing: what is a scheme? It's a space that's locally an affine scheme. 31/n

Probably the best way to think about this is that a smooth manifold is a space on which the local theory is real analysis, a riemann surface is a space on which the local theory is complex analysis, and a scheme is a space on which the local theory is commutative algebra. 32/n

As with real/complex manifolds, looking at the global picture schemes provide lets you ask new kinds of questions, and provides surprising applications. It takes some time soaking in the language to see what the new picture is, but I promise you it's really beautiful. n/n, n=32