 This talk will be an introductory overview of the Chao ring in algebraic geometry. So what the Chao ring is, is the following. We take a non-singular variety v and we form a ring, which is denoted by A of v or sometimes by C H of v and the idea is the elements of the ring are something to do with subvarieties of v and the product of two elements is going to be something to do with the intersection of two subvarieties. Well there are several problems in making sense of this and I'll try to explain what the problems are and how you fix them. So first of all, if we've got this ring whose elements are subvarieties, it ought to be graded A of v where this is something to do with subvarieties of co-dimension i and it's graded by co-dimension because if you take the intersection of two things of co-dimension i and co-dimension j the intersection should usually have co-dimension i plus j although it sometimes doesn't and this is one of the complications we have to deal with. So the first attempt might be to define z i of v to be the cycles of v, the co-dimension i cycles. So the cycles of co-dimension i are just the free abelian group generated by closed co-dimension i subvarieties and the idea is to define, try and define a product on z i of v by defining y intersection z to be the sum over w of some intersection number y z w times w. So this is a sort of provisional attempt at defining the product. So this is a sum over co-dimension i plus j components of y intersection z and this is going to be some sort of intersection number telling you the multiplicity of the intersection of y and z at the component w. So the first problem is how do we define this intersection number? So let's ask how do we define i y z w. So the picture you might have is something like this. So here's v and y might look like this and z might look like this and you see in this particular example the intersection is going to have two components I'd have a component w1 and here a component w2 and at the component w1 it looks plausible that the intersection number should be one and here the two y and z are sort of touching to order two so the intersection number ought to be two and y intersection z should be something like one times w1 plus two times w2. So how do we define this intersection number? Well there's an obvious way to do it. Let's take a to be the local ring of v at the irreducible component w. Then y and z just correspond to ideals a and b of the local ring and w which corresponds to y intersection v should correspond to a modulo the ideal generated by a and b. So means y is sort of correspond to a modulo the ideal a and z to a modulo the ideal b. So this corresponds to the scheme theoretic intersection. So this might have nilpotent elements. So if it didn't have nilpotent elements which happens in a case like this then then it's just the coordinate ring of the variety w. But in general in a case like this element hit like this intersection here this ring will generally have nilpotent elements so but you can still think of it as the coordinate ring of a scheme. And we can just look at the length of a modulo ab and ask is this equal to the intersection number of y and z along w. And in many cases this works quite well. So suppose we just take y to be say a parabola y equals x squared and we take z to be the x-axis y equals zero. Then the ring we want to work in the local ring kxy sort of localized at zero. I should have said we're taking v to be the affine plane here of course. So the ideal a is going to be y minus x squared and the ideal b is going to be the ideal y. So we look at kxy zero modulo the ideal generated by y minus x squared and y. And this is just isomorphic to k of x modulo x squared. And the length is obviously equal to two and two is a very reasonable number for the intersection multiplicity of these two sub varieties. So that works just fine. However it sometimes fails. So here's an example where it where it fails. Here we take y to be the union of two planes in four dimensional affine space. So you can picture y as being a two plane there and another two plane here. And they just meet at a point because we're working in four dimensional space not three dimensional space. They don't intersect along a line. And we're just going to take z to be another plane through this point zero zero zero zero. So they're going to intersect at one particular point. So for example y that the two planes might be x equals y equals zero and z equals w equals zero. And this plane might be say x equals z y equals w just taking the simplest way you can get three planes through there. And here we're taking the point w to be zero zero zero zero and v to be eight four of course. And the intersection multiplicity of y and z along w really ought to be two because z intersects each of these two planes. Of y in one point so it ought to intersect y in two points. So we should really have this. On the other hand if we try and work out the length of this local ring. Well we look at the ideal of y which is given by x z equals x w equals y z equals y w equals zero. And the ideal of z is obviously given by these. So we should look at the ring k x y z w modulo x c x w y z y w x minus c y minus w. So here we have the ideal of giving us the algebraic subset y and this comes from algebraic subset z. And this is isomorphic to k x y localised at zero modulo x squared x y y squared. And the length of this is equal to three and you notice that three is not equal to two. So our definition of intersection multiplicity just gives the wrong answer for this case. And you notice that absolutely nothing pathological or weird about this case. So we're just looking at intersections of hyperplanes in affine space. You couldn't get anything simpler and it's already failing for this case. Well there are several ways of fixing it and the neatest one is due to Sarah. Who found the following definition of the intersection multiplicity of y z along w. All you do is you take a to be the local ring of v along w as before. And you just take this to be the sum over all i of minus one to the i of the length of tor i of a over a over b. So these are both modules over the local ring a and you can just take take the torsion. Okay well so how's this relation to the previous definition? Well if we take i equals zero we're just getting tor zero of a a over a a over b. Which is just a over a tensor over a with a over b. Which is just a modulo a b. Which is our previous definition of the intersection multiplicity. So what is going on is that the first term of Sarah's formula or maybe the zeroth term is just our old definition of the intersection multiplicity. And then you have to add correction terms coming from these various higher tors which happen to be zero in simple cases but a non-zero in more complicated cases. And this turns out to give a really nice definition of intersection multiplicity. For instance it's always non-negative which isn't entirely obvious because some of the terms in this formula are actually negative. Well there's another problem that we've been a bit quiet about that we'd better comment on now. Now which is what if y intersection z has components of co-dimension less than co-dimension y plus co-dimension z. So generically the intersection of y and z its components will have co-dimension equal to the sum of these co-dimensions. And in that case Sarah's formula works just fine and we have no problems. However sometimes it has the wrong co-dimension. For example we could just take y equal to z and then it definitely has the wrong co-dimension. So let's look at the simple example of this. So let's just take v to be the project of plane and we can just take y to be a line and z is also a line. So z looks like that and the intersection ought to be a point but it isn't a point it's an entire line. So what can we do? Well what we can do is we can deform z slightly to form a new line z dash. So we just perturb z to get z dash and we take z dash intersection y. And there's a problem with this obvious problem. Y intersection z dash depends on the choice of z. You see the intersection is going to be some point on y but the point on y will depend on how we move z off y. And there's no way to choose a canonical point because the automorphism group of p2 acts transitively on all points of y. So we're just stuck with the fact that the intersection, even if we define it, is going to be a bit ambiguous. So the problem is that y intersection z is only defined up to some equivalence relation. So we want to say for example that all points on y are going to be equivalent to represent the same element of the chow ring. So what is this equivalence relation? The equivalence relation turns out to be rational equivalence of cycles. So what do we do for this? Well we say two cycles of dimension j are equivalent if, let's call these cycles c1, c2. So c1 minus c2 is equal to f, the zeroes of f for f is a rational function on some sub-variety of dimension j plus one. Well this isn't actually an equivalence relation so we have to take the equivalence relation generated by these. So, and what chow showed is that given cycles y and z we can find z dash equivalent to z so that y intersection z dash is well defined. In other words its components have the right co-dimension. And furthermore y intersection z dash is equivalent to y intersection z double dash whenever z dash is equivalent to z double dash. So the result of this is that we get a well defined ring. So we get a well defined ring called the chow ring on equivalence classes of cycles. So it's these equivalence classes that are going to be the modules a, i, of v. Actually this is actually a little bit subtle so let's illustrate chow's theorem or lemma by taking s to be say some surface p2 blown up at a point. So we have a little copy of p1 on this blown up surface which is the exceptional curve usually denoted by e. And suppose you try working out the intersection number of e with itself. Well first of all we have to start by moving e, we have to move e slightly to some other curve e prime. And the problem is you can't do this. If you remember from the discussion of exceptional curves you can't actually deform exceptional curves. They're sort of rigid so this seems to be a counter example to chow's moving lemma. Well it's not because you can actually do something a little bit subtle. Suppose we take a line a in p2 and if we move it so that it passes through the point we blew up then it becomes the union of two curves b and e. And this is a rational equivalent so we find a is equivalent to b union e. And now you see this means that e is going to be equivalent to a minus b. And now a minus b has a well defined intersection with e because a doesn't even meet e and b meets e transversely. So a minus b intersection e is well defined. And you see what has happened is that we've deformed e not to a curve but to a cycle with negative coefficients. And in fact whenever we deform e to something that has a well behaved intersection number with e we have to have something with negative coefficients. And that's because the intersection number of e with itself is equal to minus one times a point in some sense. And the fact that the self intersection number of e is negative means that you have to have negative signs turning up somewhere when you deform e into a well behaved cycle. So Chow's moving lemma doesn't say you can deform sub varieties it says you can deform cycles. And even if you start with a cycle with positive coefficients its deformation might pick up negative coefficients. So there's one minor point that I haven't emphasised which is that v should be non-singular. And what happens if v is singular? Well then you definitely get some extra complications and I'll just illustrate what can go wrong. Or not really wrong but illustrate one of these extra complications by just taking v to be a cone with a singular point at the origin. And now suppose I take some sort of conic section on this so it's going to be some sort of hyperbola and let's call this a. And let's take a line b and you notice that a intersection b is just a point. That's just this point here there's no problem at all. And now let's slide a down or move a around so that it becomes a line here except that if you slide a around so it becomes a line it doesn't actually become a line. It becomes a double line so it might be 2 times c where c is a single copy of this line here. So 2c intersection b should also be a point up to equivalence but this means c intersection b should be half a point. Well that's not really that much of a problem it does mean that instead of working with cycles with integer coefficients you need to work with cycles with rational coefficients on this particular non-singular variety. So you can get a sort of intersection on varieties with singularities but you definitely do get extra complications turning up. Well we should give some examples what chow rings are. So the problem with the chow ring a of v is in general very hard to calculate. Well we can sometimes calculate a few bits of it so this is going to be the sum of a i of v and a nought of v is easy. It's just equal to z and it's generated by the whole variety v which is the identity element of the chow ring because if you intersect anything with the whole space v you just get the thing you started with. A1 of v which is the co-dimension 1 is also not too difficult to describe this turns out to be it's just hyper surfaces, modulo linear equivalents and this thing generated by hyper surfaces are more or less divisors. So what we've got are divisors opto linear equivalents and this is more or less the Picard group of v. And already you notice this can be rather big for instance if v is an elliptic curve then the Picard group of v is actually an uncountable Abelian group. So except in very special cases the chow groups are uncountable Abelian groups so they're very large. There are some cases when the Picard group is much simpler. If we take v to be a projective space for example we just find that a i of v is equal to z from nought less than or equal to i less than or equal to n. And the reason for this is that any co-dimension sub-variety v turns out to be linearly equivalent to a finite union of linear subspaces. Number of linear subspaces are just the degree of the variety. The reason why projective space has a chow group that's so easy is that p to the n is paved by affine spaces in some sense. What this means is we can write p to the n as a very nice well behaved union of affine spaces. So remember p to the n can be just a point union, a line, a one union, a plane, a two and so on. And it turns out that the point and the line and the plane or rather their closures actually form a basis for the chow ring. And the same thing happens for any variety that you can write as a very nice union of affine spaces. For example, the same thing works for grass manians. Again, grass manians can be written as a somewhat more complicated union of affine spaces. And so we get that the chow ring of grass manians is also easier to describe and has a basis of these affine spaces. The intersection numbers of these affine spaces are quite complicated to work out. This is the famous Schubert calculus and the coefficients are given by things called little wood richards and coefficients and the sort of a semi infinite number of papers written about these. Okay, that will end the introduction to the chow ring. The next lecture will be a few basic properties of the chow ring and a few applications of it. For example, we'll show how you can define churned classes of vector bundles taking values in the chow ring.