 This lecture is part of an online algebraic geometry course on schemes, and we'll be about the definition of V and Cartier divisors so We saw last lecture that if we've got a curve C over the complex number then a divisor Is a formal sum Ni Pi points pi in C in other words you just take the points of C as the basis for a free Abelian group and if we've got a meromorphic function F Then we have a divisor Which is given by? sum of Ni Pi where Ni is the order of the zero of F at Pi and What we want to do is to generalize this to all varieties We want to define Devises and we want to have a map from meromorphic functions to devices whatever meromorphic functions mean and There are two sorts of divisors We can have V divisors Or we can have Cartier divisors So the names they can't device and Cartier device were actually introduced probably introduced by David Mumford in his book lectures on Curves on an algebraic surface so here I think lecture nine and see the Original introduction of the term Cartier divisor here He didn't introduce the notion of V and Cartier divisors these have been introduced earlier by possibly V and Cartier I guess anyway So let me first give a sort of overall summary of What's going on? What we're going to do is we're going to introduce a group of Cartier divisors and this will map to the group of V divisors and The group of V divisors is often denoted by Div for divisor and this will map to a group of V divisor classes and It's the quotient of V divisors by Well, it's quite often by a Field case star that sort of just assume the scheme X is integral Simplicity and Cartier divisors also have Map from rational functions on the scheme to them again assuming the scheme is integral and the quotient is going to be a group called the Cartier divisor class which I always Get confused about it being calcium chloride so Moreover Cartier device that the group of Cartier divisor classes is very close to the group pick of Invertible shears so these are the groups we're going to be considering and Quite often these maps here are often Isomorphisms that isomorphism whenever the scheme is reasonably well behaved in particular It should have if it doesn't have singularities and has some of the mild properties then then all these three groups actually naturally Isomorphic and similarly this map here is often but not always an isomorphism And the difference in philosophy is that Cartier divisors are somehow given by They sort of look like locally like the zeros of a function or poles of a function, whereas very divisors sort of look locally like co-dimension one Irreducible subsets and you may think that the zeros of a function are just the same as a union of irreducible Substets and you're mostly right, which is why Cartier divisors and very divisors are really quite similar So let's first define very divisors on Scheme X and let's assume X is notarian Which covers almost all cases and effects isn't notarian things really get rather tiresome and technical So the very divisor Is just the free Abelian group? generated by co-dimension one irreducible Irreducible subsets of the underlying topological space of X. So this corresponds to points Or closed points on a curve Just what we had before And We're sort of going to let's assume X is integral which is not necessary but is Covers again covers most cases and avoids a certain amount of Complication then it has a function field k consisting I mean for any open set you can take the take the Field of quotients of it and these are all isomorphic of X's integral And then we get a map from k star to the group of divisors and All this does is it takes a function f to sum over all Co-dimension one irreducible subsets of the order of the zero of F at P Times P So this is essentially what we did for curves and the problem is is this well defined So first of all We need this sum to be finite So this sum is finite essentially because F is equal to zero or infinity on a proper closed subset and Since the scheme is notary in a proper closed subset is going to be the union of a finite number of irreducibles Soon you get a finite number of of varieties P turning up here a slightly more subtle point is We sort of casually said the order of the zero of F at P Well, what on earth is the order of the zero of F at a point P So we can ask what is the order of zero of F at P Where P is a co-dimension one point. Well, if P is reg if if P is if the If the variety is non-singular at P So if the local ring at P is a discrete valuation ring Which is what heart shorn assumes. This is easy We just take the valuation of F so you remember for a discrete valuation ring It has a valuation which can informally be thought of as the order of the zero of a function at that point in general You have to be a little bit more careful and what we can do is we can define The order of the zero of F at P to be the length of R over F where R is equal to the local ring at P. So R is a one-dimensional local Notarian ring and um, if That there's no problem doing this because F is a non-zero divisor because we've assumed the scheme is integral If the scheme isn't integral, you have to fuss around about being careful to take F to be a non-zero divisor but we weren't worried too much about that and Now we have a following problem is the order of Fg equal to the order of F plus the order of G I mean by the order I mean the order of the zero at P And this follows by looking at the following exact sequence We just take nought goes to R over G and then we Multiply it by F and we get R over Fg and then we just map this to R over F is zero. So the length Of R over G plus the length R over F is the length of R over FG Which is what we wanted. So so this shows that the map from K star to The divisor class group is a homomorphism of groups So can we work it out? Well in general, it's a bit tricky But there are some cases when it's quite easy to work out so suppose A is a unique factorization domain Then Code I mentioned one subsets. So code I mentioned what irreducible subsets Of the spectrum of a just correspond to primes Of a In the naive sense that it's a every element of a can be written as a product of primes and And for any divisor So so the the the group of divisors is just the free Abelian group on the primes And now we can work out the group of Divisor classes the group of divisor classes is just zero because any divisor is principal So I mean if you've got a prime then more or less by definition of a unique factorization domain, there's some prime in a Corresponding to it. I mean sorry the Code I mentioned one prime ideals Correspond to prime elements of the ring a because it's a unique factorization domain And if you've got any divisor sum of n i p i then this is Principle because you just take the product of p i to the n i where here p i is the prime ideal and here p i is the Element generation of prime ideal So For an example Where it's not trivial suppose a is a dedicated domain Then you can see that the ideal class group which is the divisors Modulo principle divisors Is exactly what number theorists call the ideal class group and there are plenty of examples of dedicated domains Where the ideal class group isn't zero for instance z root minus five so for z root minus five the group of Very divisor classes is non-zero. In fact, it happens to be of order two as we've Well, I think we showed there were two different elements of it earlier We didn't quite show it was of order exactly two So there are plenty of examples where the group of very divisor classes is trivial and plenty of examples where it's non-trivial Now let's look at Cartier divisors and The definition of Cartier divisor looks a bit technical at first sight But I'll point out after giving this technical definition that's in practice. It's usually much easier. So what we do is we define a sheaf K on x which is the sheaf of Total quotient rings. So what you do is for any set you You first of all map it to the total quotient ring of O of you which is equal to O of you Where you invert All non-zero divisors And this is a pre-sheaf and we take the corresponding sheaf Well, what on earth does this look like? I mean, it sounds like a rather complicated construction Well in practice, it's nearly always can be given a much easier description. Suppose x is integral Then all we're doing is we're taking each open set u to k where k is the function field of Of x and this is for you not equal to the empty set for the empty set. We just take u to zero so This apparently rather complicated sheaf is in practice Just a rather complicated way of specifying the function field of x at least when x is integral You need this more complicated construction if you want to look at Cartier divisors of non-interval schemes, but we won't worry about that too much and then This sheaf has a sub sheaf O star There's the K star where this is just the units of O and This gives us a quotient sheaf K star over O star Zero and a Cartier divisor It's just a section Global section of K star over O star Well, what does this mean for integral sheaves? So sorry for integral schemes well, all this means is You cover Your scheme by a finite number of open sets say ui uj and UK and on each of these open sets You choose an element K i of the function field K K and K i over K j has to be a unit of O on ui intersection uj so So Furthermore if you multiply these elements K i by units that doesn't change the the Cartier divisor So you can think of the Cartier divisor as being something like The zeros of K i at least if the scheme is non-singular as we will see a bit later So for example Let's find the Cartier divisors Of an affine line just to see what's going on. Well, we have a map from Cartier divisors to Vade divisors because on On any open set ui We can just we've got a we've We've got some set K i and on ui you can just map K i to the corresponding Vade divisor On ui and since K i And K j have the same zeros on ui intersection uj This gives a map from Cartier divisors to Vade divisors So for the affine line, let's see what happens Well The map from Cartier divisors To Vade divisors Is obviously surjective In this case because a Vade divisor just consists of the point. I mean Vade divisors are linear combinations of points on the affine line And for each point on the affine line You can just take a function vanishing at that point and that will give you a Cartier divisor mapping to the corresponding Vade divisor So this is obviously surjective. It's also injective because If you've got this Cartier divisor it means you take a cover and you choose functions on here and if it's injective It means that each function K i Has image Has no zeros or poles on the open set ui And This means it must actually be a unit of The coordinate ring of the open set ui Here we're using the fact that if something on one of these open sets has no zeros or poles then it must actually be a unit In the case of the affine line um so So the Cartier divisor is Um Must then be trivial. So in other words Cartier divisors Are really the same as Vade divisors in this case and you may think that this sort of argument Applies much more generally than the affine line and you're sort of right in many cases Cartier divisors are the same as Vade divisors um So what we'll do next lecture is we'll give a couple of examples Where the map from Cartier divisors to Vade divisors is neither injective nor surjective