 So this lecture is part of an online algebraic geometry course on schemes, and will be a review of valuation rings, which we will be using when we describe various criteria for maps to be separable or proper later on. So we'll just recall the definition of a valuation ring. So R is a valuation ring if it is an integral domain, and so that with quotient field K, and it has the property that if X is in K and X is not zero, then X or X minus one is in the ring R. A typical example, we could take the ring of formal power series over a field K, and then a formal power series. So the quotient field consists of all Laurent series, and Laurent series starts with A and X to the N for some N, and if N is positive it's in R, and if N is negative, then its inverse starts with X to the minus N and so is in R. Another example is just the ring ZP, that's the ring localized at P, which consists of all integers M over N, with P does not divide N. And again it's sort of obvious that the quotient field is the rational, isn't any non-zero rational, either rational or its inverse is in here. So valuation rings have the property that if A and B are not equal to zero, then A divides B, or B divides A. In particular R is a local ring, meaning the non-units form an ideal. You have to think for a second or two to see the sum of two non-units as a non-unit, but if A and B are not units, then say A divides B, so B equals AC, and then A plus B equals A by one plus C, so this is also a non-unit. So if you've got a valuation ring with quotient field K, then we can form a group, if you take K star modulo the units of R, this is a group, and it's totally ordered by whether something is R times something else. And it's called the valuation group. And we have a map from the non-zero elements of K to this group G, let's call it G. This map is denoted either by V or by V of K or by the absolute value of K. If G equals Z, then R is called a discrete valuation ring, and both examples I gave earlier were discrete valuation rings. Discrete valuation rings, which are kind of often abbreviated DVR because discrete valuation ring is rather a lot to write out. So discrete valuation rings are principal ideal domains. The ideals are just the elements A where the valuation of A is 0, 1, 2, 3, and so on, or we can take the ideal zero. So in the ring K of X, if F is the power series, then the valuation V of F is just the order of zero of F at the point zero. And in general, discrete valuation rings are sort of similar to this. The valuation is often the order of the zero of something somewhere. So the following equivalent for a valuation ring. First of all, it can be a discrete valuation ring. I guess you might want to count fields as discrete valuations as well and allow the valuation group to be trivial. So it could be notarian, or it could be a principal ideal domain. So proving the equivalence of these for a valuation ring is very easy, so I'm not going to bother with it. The spectrum of a discrete valuation ring is very easy because the only prime ideals are 0 and P where the valuation of P is equal to 1. So its spectrum looks like this. It's got one closed point, which is the ideal P, and it's got one generic point, which is the ideal 0. And you should think of the spectrum as being somewhat analogous to a short, smooth curve where we've taken it so short that we can only see one point on it, but whatever. And valuation rings to some extent play the same role in algebraic geometry that short, smooth curves sometimes play an analysis. We'll see one or two examples of this fairly soon. So that's discrete valuation rings, and everybody likes discrete valuations rings and agrees they're good things. I now want to talk a little bit about non-discrete valuation rings. And these have had a rather strange history. Everybody is a bit wary of discrete valuation rings. It's like some sort of relative of yours who seems to have lots and lots of money, but there are sort of suspicions that the relative runs an organized crime gang or something. So non-discrete valuation rings are kind of useful, but everybody is a bit wary of them. The first obvious problem is they're non-notarian, and this caused some delay in people using them. They were then very fashionable a few decades ago, and Zariski in particular made heavy use of them in his work on resolution of singularities. Anyway, let's see some examples of them. The first example are Pruise series. I can't remember how many vowels this has in it. So the Pruise series are the union over n greater than zero of the formal power series in x to the one over n. And you can see this is a valuation ring. And its quotient field is going to be where you allow negative powers of x as well. And the valuation group is now the rationals Q, because a formal power series can start with x to the r for r, any rational number. And it's an example of a rank one valuation ring. So what does rank one mean? Well, what we do is we take the value group G and we tensor it over the integers with the rational numbers Q. And the rank is then the dimension of this as a vector space over the rationals. So discrete valuation rings are certainly rank one valuation ring. And this gives an example of a rank one valuation ring that isn't a discrete valuation ring. Well, what about the valuation rings of higher rank? Well, we can find lots of examples as follows. Let G be any ordered group. I guess I mean here a totally ordered group. Then we can form a valuation ring with value group G as follows. What we do is we take all power, all formal power series of the form sum over n in G for n positive. So we're taking just non-negative elements of G of a n times x to the n. Well, you will immediately notice there's a problem with this. These don't form a ring because you can't really multiply them together. I mean, you just get infinite sums. So we have to add a condition such that the set of n in G with a n non-zero is well ordered. That means every set of them has a minimal element. And then if you've done a bit of set theory and discussed well ordering things, you see that these sorts of power series can be multiplied together and form a valuation ring with value group G. So we can get examples of valuation rings that have ranked bigger than one just by taking G to be some totally ordered group of rank bigger than one. For example, we could take G to be the real numbers. And this is uncountable dimension over the rational numbers. So there's a valuation ring with uncountable dimension. So what use are these in algebraic geometry? Well, here are some examples of where they turn up. First of all, suppose that we take P0 to be a point on the surface. Let's call this surface X. And let's take Y to be the blow-up of X at the point P0. So Y is obtained by replacing P0 by a copy of the projective line P1. Let's assume X is a non-singular surface. And we can look at this P1 contained in Y as it's a co-dimension 1 variety. So its local ring is a discrete valuation ring. Because if you take any co-dimension 1 variety in some variety, the local ring at that point is a discrete valuation ring. And in fact, that's a common way to produce discrete valuation rings. Well, this isn't an example of a non-discrete valuation ring yet. So let's continue. What we're going to do is pick P1 in the blow-up of P0. So this is another point here. And then we're going to blow up P1. And then we're going to pick P2 above P1 and blow up P2. And we sort of continue like this. So we get a whole series of points P0, P1, P2, and so on. And then we can look at the local ring at P0. Let's say RP0. And this is a map to the local ring RP1 at P1. And this is a map to the local ring RP2 at P2. And if we take R to be the union of these, then R gives an example of a non-discrete valuation ring. Well, as you see from this construction, non-discrete valuation rings are kind of hairy. You tend to get these rather bizarre infinite constructions in order to get them. So these were sometimes used by Zariski as follows. First of all, let me do the case of curves. So suppose C is a curve over a field k. Then let's put big k to be the function field of the curve. And the problem is, how do we reconstruct C from k? Well, the answer is you can't, because there are lots of different curves that have the same function field. For instance, you could remove a few points from the curve or you could have a curve with a singularity. And this is the same function field as its singularity. So instead of reconstructing C, we should try and reconstruct, say, a complete non-singular model of C. Or we could start with C being complete and non-singular. And the answer is we can take the points of C to be the valuation rings of k containing little k rather than big k itself. And this construction was given in Hartshawn's book on Algebraic Geometry, Chapter 1, Section 6, I think, and I didn't go into that in detail, but whatever. So this gives a sort of correspondence between curves and function fields where if you've got a function field, you just take its valuation rings. In the case of curves, these are all actually discrete for curves. And you can ask what happens if you do this for a high-dimensional variety. So suppose C is a variety of dimension greater than 1 with function field k. And what Zariski did was he defined something well, he rather confusionally called it a Riemann surface, but it's not a surface at all in general and it was very little to do with Riemann. So it's sometimes now called a Zariski Riemann surface. This is an example of the important principle that if you name something, you should give it a bad name so that people have to rename it and with any luck they'll rename it after you. So the Zariski Riemann surface is a set of valuation rings of big K containing little K and sometimes you miss out the valuation ring K but it doesn't really make much difference. And you can think of this as a sort of model of C. The trouble is it's very bizarre as we saw a little bit earlier, the valuations of K can be very weird indeed. You can sort of get valuation rings where you blow up an infinite sequence of points in C and that was only for surfaces and high dimensions it gets even weirder. Anyway, this Zariski Riemann surface is a sort of non-singular model of C. So it is non-singular in some sense. There's a theorem called Zariski's local uniformization theorem which says that in some sense this Zariski Riemann surface can be thought of as being a non-singular object. The trouble is it is locally very complicated. So Zariski tried to use this in order to resolve singularities. The idea is you first take this Zariski Riemann surface which is non-singular but kind of a bit weird and then you try and contract it down a bit so it actually becomes a variety. And Zariski was able to do this in dimensions two and three but beyond that things just got a bit too complicated. Zariski Riemann surfaces were sort of precursor to schemes. So the points of a scheme correspond to prime ideals of a ring. Whereas the points of a Zariski Riemann surface correspond to certain valuations of a field. And there are a lot of similarities between schemes and Zariski Riemann surfaces. First of all, you can put a topology on both of them which in both cases is called the Zariski topology and is kind of similar. Next you can put the structure of a ring space on both of them. So these things can be thought of as geometric objects. They're kind of locally ringed spaces. Zariski Riemann surface may have actually inspired growth index definition of a scheme. There were several, in the 1950s on growth index defined schemes there were several ideas floating around that were kind of a little bit similar to schemes and Zariski's concept of the Zariski Riemann surface was one of them. So you can think of Zariski Riemann surfaces as being sort of parents of schemes. There are two problems with Zariski Riemann surfaces. First of all, they're a bit too special. You see this works for any rings so you can do it over the integers. Whereas this only works if you've got a field so it only works for integral domains. The other problem is schemes are locally quite nice. That schemes are locally affine schemes and you know what an affine scheme looks like. What is the Zariski Riemann surface looks locally is not really very clear. As I said, you've got all these weird infinite constructions going on. Anyway, growth index pretty much eliminated nondiscrete valuations from algebraic geometry. He really had an intense dislike of them. I even found a quote by him. So this is a letter he wrote to Seher. And if I magnify it a bit, you can see his opinion of discrete valuations. So here it says, I've proposed several times in the vein that chapter six on valuations, he's talking about Bourbaki's book on commutative algebra, should be purely and simply thrown out. And then he goes on to say, it would be premature to predict whether and in what form valuations will be needed. I suspect not at all and so on. And in his EGA, he goes out of his way to eliminate nondiscrete valuations whenever possible. So growth index judgment is that nondiscrete valuations are probably not very useful and you should just stick to discrete ones. So next lecture we'll be discussing how to use valuation rings to give a criterion for a morphism to be separable and later on we'll use it to give a criterion for a morphism to be proper.