 So this lecture is part of an online algebraic geometry course on schemes and will be about divisors on dedicined domains, or at least their spectra. So we recall that in previous lectures, we've defined a map from Cartier divisors or Cartier divisor classes to Ray divisors or Ray divisor classes. And we've also defined a map from Cartier divisors to the Picard group, which is usually an isomorphism. So this map here is called the churn class. And I guess actually before going on a bit I better explain why this is called the churn class. Suppose X is a complex manifold, might be say a Riemann surface. Then on this complex manifold, we have the following exact sequence of sheaves. 2 pi i z goes to 0 star. I'm not quite sure whether to put a 0 or 1 here. It depends whether you write this additively or not. Here, this is the sheaf of meromorphic functions. This is the sheaf of non-zero meromorphic functions. That's ones that are not 0 anywhere. And this here is the exponential map. And the kernel of the exponential map is just multiple integer multiples of 2 pi i. And from this exact sequence of sheaves, we can get an exact sequence of cohomology groups. And yeah, I know I haven't defined cohomology groups yet, but this is just for background information. In particular, we get a map from h1 of 0 star to h2 of 2 pi i z as part of the long cohomology exact sequence. And when we were discussing line bundles, I said the line bundles sort of correspond to the first check cohomology group. And this is more or less the first check cohomology group, so it's more or less the Picard group. So we get a natural boundary map from line bundles to the second cohomology group of a manifold. And this is the map that is the churned class in differential geometry. More generally, you can define churned classes not just for line bundles, but for arbitrary vector bundles. And they take values in the even dimensional cohomology classes. But we're just going to be looking at the simplest case of line bundles. Now, what we would like to do is do an analog of this in algebraic geometry. And we run into a serious problem with this because we've got this exponential map is not algebraic. And we've got another problem in that, well, you can define the sheaf Z in algebraic geometry over a scheme. However, the cohomology of the sheaf Z turns out to be rather badly behaved. And it doesn't give you the cohomology classes you expect. And it's a really rather difficult problem to find the correct analog of integral cohomology that this was solved by growth that using etal cohomology, but we're not going to go into that. It's not yet. And so what we do well, a second cohomology group is sort of dual to a second homology group. It should be the homology of X with coefficients in Z. And by point current duality, a second cohomology group is sort of dual to a homology group of a homology group of dimension n minus two. And homology of dimension n minus two is something to do with co-dimension to subsets. Well, if you're working over the complex numbers, then there's something a real co-dimension to is something complex co-dimension one. So that's something to do with divisors. That's very divisors rather than Cartier divisors. So in other words, very divisors are something to do with the second cohomology group. They're sort of analogous to it. And we saw that Cartier divisor classes, at least in nice cases, correspond exactly to the Picard group. So this group here is sort of analogous to the Cartier divisors. And this group here is sort of vaguely analogous to V divisors. And the map from the Cartier divisors to V divisors that we defined turns out to be the analog of the churn class map in differential geometry. So more generally, in high dimensions, the Picard group would be replaced by some sort of K theory where K theory describes vector bundles over a manifold. The V group would be represented by some high dimensional generalization of co-homology, either using intersection theory or etal co-homology. So that's why the map from Cartier divisors to V divisors is sometimes called a churn class. Well, now we get back to Dedekind domains. So if we've got a Dedekind domain R, we're going to look at the scheme spectrum of R and try and figure out what the V divisors and the Cartier divisors look like on that. So let's just recall what a Dedekind domain is. So a Dedekind domain is the following properties. First of all, it's an integral domain. Secondly, it's notarian. And thirdly, every prime, it's not zero, is maximal. And fourthly, it's integrally closed in the quotient field. So this is the definition in terms of algebra. Now let's translate this to geometry. So geometry, well, if you've got an integral domain, this just means you've got an affine scheme, which is integral. Notarian translates to notarian. Every prime being maximal, every non-zero prime being maximal just says the dimension or the crawl dimension is at most one. And integrally closed in the quotient field means that the scheme is normal. Now for one-dimensional schemes, that's more or less equivalent to saying it's regular. There's no singular points. So Dedekind domains correspond to one-dimensional affine things with no singular points. So there's an obvious example of this. First of all, any non-singular affine curve over a field. So for example, we could take the curve y squared equals x cubed minus x, and the coordinate ring of this would be a Dedekind domain. The other really important example are integers of algebraic number fields. So a typical example of this would be the ring of Gaussian integers, z of i. And back in the 19th century, when people were studying curves and integers of algebraic number fields, people noticed that there was a strong analogy between them. And these days we say this is because they're both basically one-dimensional affine schemes. So now there are two classical approaches to the ideal class group. So in algebraic number theory, there's something called the ideal class group which measures how far the ring of integers is from being a unique factorization domain. And there are traditionally two opposing ways to define this, and people used to sort of argue a bit about which the best way was. So the first method is using divisors. So what's a divisor? Well, here we take the free abelian group on the maximal ideals of our Dedekind domain. So for instance, for z of i, if you took z of i, the maximal ideals would be 1 plus i, 3, 2 plus i, 2 minus i, and so on. So you take the free abelian group generated by these elements and call that the ring of divisors. Principle divisor, well, if f is an element of r, then you can write the ideal f as a product of prime ideals because it's a Dedekind domain. So it would be p1 to the n1, p2 to the n2 and so on. And n1, p1 plus n2, p2 and so on is called a principle divisor. And then you can define the divisor class group to be the group of divisors modulo the principle divisors. And this is trivial if and only if the Dedekind domain is a unique factorization domain. And what you can do is you can think of the elements of the Dedekind domain as sort of being like functions. For instance, if we take r equals z, so the divisors are 2, 3, 5 and so on. Then you think of, say, number 12 over 5 would have a zero of order 2 at 2 and of order 1 at 3 and a pole of order 1 at 5. This isn't the same way that if you take, say, x squared times x minus 1 over x minus 2, this is a zero of order 2 at the ideal x and a zero of order 1 at the ideal x minus 1 and a pole of order 1 at the ideal x minus 2. So this is part of the analogy between algebraic number fields and curves. Anyway, if you notice, the divisors that we've defined for a Dedekind domain are exactly the vague divisors of a scheme. So divisors in algebraic number theory are the same as what geometers would call vague divisors. And the divisor class group is just the vague divisors modulo principle vague divisors. So it's just the group CL of vague divisor classes. So this group was sort of discovered more or less independently by people doing number theory and people doing geometry. Well, because it wasn't really independent since the people doing it knew perfectly well about the analog. Okay, so that's one way of defining the ideal class group. I guess I better just do a quick example where the divisor class group isn't trivial. So the simplest case is zero minus five that we've had a few times before. And here we can see that, for instance, the principle divisor two factorizes is two one plus root minus five squared and three factorizes as three one plus root minus five three one minus root minus five and one plus root minus five factorizes as two one plus root minus five three one plus root minus five and so on. So, so these are principle. And, however, the ideals to one plus root minus five three one plus minus root minus five, not principle. So the divisor class group is definitely non trivial because these are non trivial elements in it that aren't principle. On the other hand, there was a second approach to the, to the ideal class group, which uses ideals, rather than, rather than divisors. Here we recall that a fractional ideal is a non zero finitely generated sub module of K, which is the quotient field of quotients of the data kind domain. And the, for a data kind domain, the fractional ideals actually form a group and multiplication of, well, multiplication of ideals extended to fractional ideals in the obvious way. And you can quotient out this by the group of principle fractional ideals, which are just all multiples of, of some particular element of the quotient field. And this is called the ideal class group. So, so for example, for C, root minus five. There are two elements of the ideal class group, you can either have the ideal one or the ideal to one plus root minus five. And this is a non principle ideal. And it's fairly straightforward to calculate that any fractional ideal is equal to a principle ideal times one of these two. So here the ideal class group is order two. And we notice that fractional ideals are essentially the same as invertible sub sheaves of the field K or rather of the sheaf corresponding to the field K. And these are essentially the same as Cartier divisors. So the two classical approaches to the class group of a number field, the divisor class group or the ideal class group, turn out to be these two, they correspond to the two different ways we've had a defining divisors, either Cartier divisors, or Cartier divisors and the classical equivalents between the ideal class group and the divisor class group in number theory turns out to be just the isomorphism we've given between the Cartier divisors and the very divisors. I guess you could if you want quotient is out by principle divisors. I'm arguing about which of these is the correct approaches a bit silly they're too perfectly good approaches which just happened to be the same for data kind rings. So I think it probably helps a bit to see what's going on if I give you an example where these two groups are not the same. So here I'm going to look at the spectrum of C root minus three. So Z root minus three is the classic example of an order in an algebraic number field that is not a data kind domain. And the reason it's not a data kind domain is it's not integrally closed. Because the element root minus three plus one over two is a root of x squared plus x plus one equals zero so it's integral but it's not in here so so we actually Z root minus three is contained in Z one plus root minus three over two. And this one is a data kind domain. And so in particular we get maps from the spectrum of Z root minus three and spectrum of Z one plus root minus three over two. This scheme is non-singular whereas this one has a sort of singularity in it. If you want to sort of visualize a picture of them you can think of the spectrum of this has been kind of like a curve with a node and this is kind of like curve without a node maps on to it or something like that but of course these aren't really algebraic curves this is just a sort of picture of what's going on. Anyway, this scheme has a singularity at the point two so this is a prime ideal of it and it's actually got a got a singularity there. So let's look at the Cartier divisors and the Baye divisors. First of all, the Baye divisor class group is non-trivial. In particular, there's the prime ideal to one plus root minus three, which is not principal. Sorry, I just realized I made an error here. This isn't the point to ideal to one plus root minus three of course. So this prime ideal is not principal so the Baye divisor class group is not one. Now let's look at the Cartier divisors. So here we have to look at the fractional ideals and it's not difficult to work up what they are. There are two sorts of fractional ideal. There's a principal one and there's the ideal two plus root minus three. Two one plus root minus three and this is, well, so it looks as if the group of Cartier divisors isomorphic to the group of Baye divisors. Here we've got two candidates for Cartier divisors. However, we actually made an error because this one is not a Cartier divisor, or at least there's no Cartier divisor correspondence. The point is, this is a fractional ideal but is not invertible. So in a Dedekind domain, all non-zero fractional ideals are automatically invertible but this actually fails here. In fact, we can work out what its square is. If we take two one plus root minus three and square it we get the ideal four two plus two root minus three minus two plus two root minus three, which is two times two one plus root minus three, which is the ideal we started with times two. So if this idea were invertible it would be the same as the ideal two, which it isn't. So, in fact, the Cartier divisor group only has one element, which is just generated by the principal ideal. In fact, we can see other things going wrong. If we look at the local ring, if we kind of localize it two and add root minus three, then the maximal ideal is now two one plus root minus three. And we see M over M squared is two dimensional over the field, quotient field, which is two elements. So M is not principal in the local ring. So in particular, it's not locally free at this point and it doesn't give us an invertible sheaf or an element of the Picard group or anything else. So in other words, there's a whole cascade of things that go wrong, this example. So here are things that go wrong. So first of all, so in spec of Z root minus three, we have a local ring is not regular or normal. Secondly, the ideal two one plus root minus three is not principal in this local ring. So it's not invertible. So for a data kind domain, all local rings are discrete valuation rings. So every ideal becomes principal in the local ring. So, so this fails for this example. Similarly, the map from the Picard group to the vague group, or this would be corresponds to fractional ideals, I should say invertible fractional ideals. This corresponds to divisors is not an isomorphism. So that's enough about data kind domains. The next lecture will be more examples of Cartier and Wade divisor class groups.