 Okay, this is the first part of lecture two on algebraic geometry in which we'll be discussing affine space and the Zariski topology on it. So first of all, what is affine space? Well, if you take any field, common choices will be the real numbers or the complex numbers or a finite field. So we take a field k and affine space is then just k to the n, a vector space of n dimensions over k, except it isn't quite, as we'll see in a moment, and it's often denoted by a to the m for affine space. So what's the difference between affine space and vector space? Well, the difference is quite subtle. They have slightly different automorphism groups. So let's first look at the automorphism group of a vector space k to the n. So we have automorphisms of a vector space. Well, these are just invertible linear transformations. The group of automorphisms is denoted by the general linear group over k, which is just n by n matrices of determinant, none zero. So we can picture this if n is 2 as little 2 by 2 matrices of non-zero determinant. On the other hand, if you look at affine space a to the n, then the automorphisms include all linear transformations gln of k, but they also include translations where you map any point x to x plus v for some fixed vector v's. So these are the translations. So the full group, so this group has dimension n squared and this group has dimension n. So the full group of symmetries of affine space is dimension n times n plus one, and it can be pictured as the group of matrices of the following shape where this bit here in the top left corner are just the automorphisms of the corresponding vector space, and this bit here corresponds to translations. So the difference is roughly speaking that a vector space has an origin and an affine space is kind of like a vector space except you've forgotten what the origin is. In other words, if you've got a vector space, then from any vector space we get an affine space and we get the affine space just by forgetting which point 0 is, roughly speaking, and to get from an affine space back to a vector space, all you have to do is choose any point of the affine space as your origin and that makes it into a vector space. For example, take the three-dimensional space we live in and pretend we're in the days before Einstein so it's not curved or anything like that. Then the three-dimensional space we live in is really an affine space, not a vector space because there's no really natural way to choose the origin. But if you choose the origin, for instance, you might choose the origin to be the centre of the Earth or if you're in the ancient astronomer you might choose it to be the centre of the Sun if you're trying to study the solar system or you might choose it to be the centre of the galaxy or whatever. You can choose any point you like as the centre of coordinates and then three-dimensional space becomes a perfectly good vector space but there's no canonical way to choose the identity. So we're actually living in a, well, I guess it's not really an affine space because it's got a metric but never mind. So affine geometry can be sort of as a study of the properties of affine space that are invariant under affine symmetry. So any property that is invariant under translations and linear transformations. So let's list some properties of affine geometry. Well, perfectly good concepts of things like points or lines or parallel lines because if two lines are parallel and you do any sort of linear transformation or translation they remain parallel so that's fine. Connex are also well-defined and even polynomials. So polynomial functions on affine space. Next we should list some things that are not affine geometry. So first of all, circles are not affine. This is because if you take a circle and apply some linear transformation to it there's no reason why the result should be a circle. It might end up as some sort of ellipse. However, if you take all connex including ellipses and circles then that is a perfectly well-defined concept in affine geometry. Similarly, angles make no sense. For example, if you take a rectangle and apply some linear transformation form 1101 to it, the skew transformation becomes some sort of parallelogram and this angle here doesn't get preserved. Similarly, lengths are not well-defined in affine geometry because they're changed by linear transformations. Now, algebraic geometry tends to use the coordinate ring of affine space. So the coordinate ring is just the space of all polynomial functions or all polynomials on A to the N. Notice that whether or not something is a polynomial doesn't depend on the choice of origin so that's okay. I should mention here we're taking our field K to have an infinite number of elements because if it's only got a finite number you've got to be a little bit more careful about how you define the polynomial ring. So we take the polynomial ring Kx1 up to xN or polynomials in the N coordinates. So a point here is going to be x1, x2 up to xN. So if we've got affine space we can reconstruct the ring of polynomials on it's just as polynomial functions. Conversely, if we've given the polynomial ring over K we can reconstruct affine space as the A to the N as the set of homomorphisms from this polynomial ring to the field K. This is homomorphisms as a K algebra. So a homomorphism taking Kx1 up to xN to K just takes x1 to A1 for some number, A1x2 to A2 and so on. So it's uniquely determined by these numbers A. So this corresponds to the point A1, A2 up to AN of A to the N. And you notice this map here is just the value of the polynomial at the point A1, A2 up to AN. So we can go from affine space to the coordinate ring by taking all polynomials and we can go from the polynomial ring to affine space just by taking various homomorphisms. So because of this, the study of affine space is more or less equivalent to the study of this polynomial ring. So anything you can do for a polynomial ring has an analog for affine space and anything you can do for affine space has an analog for polynomial rings. In particular, for example, the automorphism groups of these two things are the same. So we had the affine group acts on affine space and you can also easily check that it's the group of automorphisms of the polynomial ring over K. Next, we're going to discuss the Zariski topology on affine space. So if we do this, we define algebraic sets. So an algebraic set is a set of zeros of some set of polynomials KX1 up to XN. So this is going to be an algebraic set in the n-dimensional affine space. For example, we could take our polynomial F to be the polynomial X squared plus Y squared minus one and then our algebraic set would just be a circle or we could take a set of two polynomials F equals X minus A and G equals Y minus B. The set of common zeros of X minus A and Y minus B is obviously just the point AB. So a single point AB is also just an algebraic set. And now algebraic sets are closed under the following two operations. So they're closed under, first of all, they're closed under intersections. This is because if we have an algebraic set C1, C2, C3, and so on, which zeros of sets P1, P2, and so on of polynomials, then C1 intersection C2 intersection C3 and so on is the set of zeros of P1 union P2 union P3 and so on. So taking intersections of algebraic sets just corresponds to the operation of taking unions of sets of polynomials. They're also closed under finite unions. And you've got to be a little bit careful here because if we take the union of two algebraic sets is not given by the set of zeros of the intersection of their polynomials. Instead, if C1 and C2 are zeros of sets F1, F2, and so on, and G1, G2, and so on of polynomials, then C1 intersection C2 is the set of zeros of the polynomials FI, GJ. So we have to take all products of pairs of these polynomials. And you notice this doesn't work for infinite unions because to do for that, we'd have to take the product of an infinite number of polynomials which doesn't really make sense. Now, we notice that if a collection of sets is closed under all intersections and under finite unions, then they form the closed sets for a topology. And this is called the Zariski topology. So the algebraic sets are the closed sets of the Zariski topology. Of course, to check something's the closed sets of a topology, all you have to do is to check it's closed under arbitrary intersections and finite unions. Arbitrary intersection means the intersection of any collection of subsets. I've sort of written as if it's a countable collection, but that's just because it's easier to write. So let's see some examples of Zariski topologies. So let's first do one-dimensional affine space A1. So this is just a line. And we can ask, what are the closed sets of it? Well, first of all, the whole line is the set of zeros of, you can either take the empty polynomial or just the polynomial zero. Any finite set is going to be the zeros of a polynomial x minus A1 times x minus A2 and so on. So any finite set of points is going to be the set of zeros of some polynomial. And in fact, these, the whole of A1 and finite sets, you can easily check, are the only possible closed sets. If you've got any collection of polynomials, their common zeros are either a finite set or they're the whole space. In particular, this topology is a bit weird. I noticed that it does not house dwarf. And this is a bit of a problem because if you've done a course in topology, probably most of the topological spaces you were seeing were house dwarf. And your intuition gets used to the fact that all topological spaces are house dwarf. And this simply isn't true in algebraic geometry. For instance, if you take a line and take two points on it, if it were house dwarf, we would have to find open sets. We're going to be able to find open sets containing each of these points that are disjoint. But the trouble is the only open sets contain all but a finite number of points. So any two open sets have an infinite number of points in common unless they're empty. That's again assuming that we're working over a finite over an infinite field, over a finite field, but this happens to be house dwarf, but that's a bit misleading. Let's look at two-dimensional affine space A2. So let's write down some closed sets. Well, first of all, we've got points A, B, which as we saw earlier is the set of common zeros of X minus A and Y minus B. And we've also got any curve, any algebraic curve, which are the set of zeros of FXY equals zero. So any irreducible curve will do. And as we said there, closed sets are closed under finite unions. So a typical closed set will look like a few algebraic curves and a finite number of points. Here is a typical open, typical closed set in two-dimensional affine space. Notice by the way that the Zariski topology on A2 is definitely not the product topology on A1 times A1. So if we take the product topology on A1 times A1 and look at its closed sets, it's not very difficult to work out what these are. We get a finite number of vertical lines and a finite number of horizontal lines and a finite number of points. So the product topology on A1 times A1 has far fewer open sets than the Zariski topology on A2. So in higher dimensions, algebraic sets look like this only more so. So we get points and we get one-dimensional curves and we get two-dimensional varieties and we get hypersurfaces and so on. I just finished by giving you one slightly more exotic example of an algebraic set. So this is an example of an algebraic set. It's called a determinant variety for a reason that will be pretty obvious in a moment. And what we do is we take affine space of dimension m times n for m and n integers. And we think of this as being linear maps from k to the m goes to k to the n. And these obviously form a vector space of dimension mn because you can think of them as being m by n matrices. And now an algebraic set is going to be all linear maps of rank less than or equal to i for some constant i form an algebraic set. So this is going to be the determinant or variety. And we want to check that it is actually an algebraic set. So this is given by the vanishing of all i plus 1 minus of an m by n matrix because this is just the condition for a linear map to rank at most i. So this is some set of polynomials. You notice it's actually quite a large set of polynomials in general because we have to choose i plus 1 of the m rows and i plus 1 of the n columns and in general this would be a really quite large number of polynomials. So this determinant or variety is given by the set of common zeros of polynomials but it's a large set. In particular, for example, the subset of maps from k to the m to k to the n that are onto is open in the Zariski topology.