 This lecture is part of an online algebraic geometry course about schemes and we will be giving a couple of examples of affine schemes. So the first one is going to be the spectrum of field with polynomials and two variables with coefficients in the field. You may as well take k to be the complex numbers, just for simplicity. And if we take the field to be algebraically closed, then you can see there are three different sorts of prime ideal. First of all, for any alpha beta in c squared, we can look at the ideal x minus alpha x y minus beta. So this is a maximal ideal and this is just the usual correspondence between the maximal ideals of c x y and the affine plane c squared that we had when we were looking at affine varieties. Second example of a prime ideal is we can just take all multiples of f x y where f is irreducible. And you can see these prime ideals are not closed in the Zariski topology because the closure contains the point or rather the ideal corresponding to the point alpha beta whenever f vanishes at the point alpha beta. And thirdly, we have the rather trivial example of prime ideal, which is just zero. So these are the prime ideals of the polynomial ring in two variables over the complex numbers. And we can sort of draw a picture of it as follows. So the prime ideals of this form can be thought of as points in the complex plane. The prime ideals of the second form can be pictured as irreducible curves. So we might have a curve f x y equals zero. And this point here, so if this is some point alpha beta, it will actually be in the closure of this point whenever f of alpha beta equals naught. So you may be getting a bit confused by whether or not this is a point or a curve. Well, this is actually a point, but it's a really, really big point that sort of looks one dimensional and contains all these other points in its closure. So the problem is the spectrum is not in a non-house store, but it's not got non-closed points. So when you draw it, it starts looking really weird. And some of the points are so big that they contain other points in their closure. And finally, we've got this third type of point just corresponding to the ideal zero. And this contains all the other points in its closure. So you can sort of picture it as being a really huge two dimensional point whose closure contains all other points. It's called the generic point corresponding to zero. And it's sort of you can think of it as somehow being a point the size of the entire complex plane, whatever that means. Next we can try and figure out what the local rings of all these points are. So what we have to do is to localize this ideal. It means we invert everything that's not in this ideal. So here the local ring, well, we invert everything that's non-zero. So we just get the field of all rational functions in two variables. So these functions are defined sort of almost everywhere on this yellow point. For these purple points, the local ring is going to consist of all rational functions g over h with h alpha beta is not zero. So it will be rational functions that are regular at the point alpha beta. So these are just rational functions defined in a small neighborhood of the point alpha beta without being singular. The local ring of this generic point is going to be all functions g over h with f not dividing h. So it must be regular at most points of the curve. So you see that an element of the local ring of this point is actually allowed to have singularities on this blue curve because h might have a zero that sort of cuts across the blue curve. However, it must be regular at almost all points of the blue curve. In fact, this is an example of a discrete valuation ring. So every element here can be written as a power of f times a unit where f is the function vanishing on this curve. So you can actually say if you've got an element of this local ring, you can actually say it vanishes to give an order along this blue point. I mean, if it's f to the n times unit, then it's going to vanish to order n. We should also take a look at what the spectrum of a discrete valuation ring looks like. So the next example will be let's take r to be a discrete valuation ring. And if you can't remember what a discrete valuation ring is, let's just take for example the ring z2, which is all into rational numbers m over n with n odd. Or you can take something like a ring of formal power series over a field. That's another example of a discrete valuation ring. And the discrete valuation ring, well, let's just do this case, for example. It just has a maximal ideal m, which in this case will be all multiples of 2, generated by some element f, which in this case will be the prime 2. The other ideals are just generated by powers of f, and we've also got the ideal 0. And you can see that this ideal is maximal. These ideals are neither maximal nor prime, but this ideal is prime, not maximal. So we see that the spectrum of the discrete valuation ring has exactly two points in it. So if I indicate this one as being the blue point, then it's got a closed blue point. So maximal ideals correspond to closed points of the spectrum. And if I think of this one as being red, it's got a sort of fuzzy one-dimensional point whose closure contains the blue point. And so the spectrum of the ring has two points, a closed one and a non-closed one. So in general topology, this is called the Sierpinski two-point space. It's a topological space with just two points, one of which is closed. And we can also figure out what the stalk at each point is. In other words, what the local ring is. So here the local ring is R, the discrete valuation ring we started with. And here the stalk at this point is the field of quotients of R. So if we were taking a discrete valuation ring to be z-localized at 2, this field of quotients would just be the rational numbers. You can sort of see this spectrum, if you look inside the spectrum of CXY, here if we take the yellow point and the blue point and throw everything away, we would just have a generic yellow point which seems to have changed color to red and a single blue point. The next example we're going to look at is the spectrum of ring of polynomials in one variable over Z, which is going to be a bit more complicated. So first of all, we have a map of rings from Z to ZX. So we get a map from the spectrum of ZX to the spectrum of Z. And we know what the spectrum of Z looks like because we figured it out last lecture. So let's draw the spectrum of Z and I'm going to color its points. So first of all, we've got the point 2 and then we've got a point 3 and then we might have a point 5 and so on. And these will go all the way up to this generic point 0. And the generic point 0 contains all these other points in its closure. So it can be pictured as a sort of one-dimensional point. So what we've got here is spectrum of Z. And above each point of the spectrum of Z, we're going to have some sort of fiber. So I want to figure out what this fiber is. Well, first of all, what are the points of spectrum of Z of X that lie above this point here? Well, they're just the points whose inverse image in Z is the prime 2. So they're really just primes of Z of X containing the prime 2. In other words, they're really just primes of Z mod 2 of X. So what we get here is the spectrum of F2 of X. I guess I should have drawn this in blue, but anyway. So the spectrum of F2 of X just contains a generic point of F2, which is just all multiples of 2 and Z of X. And then it contains all irreducible polynomials in F2 of X. So it contains 2X, 2X plus 1, 2X squared plus X plus 1. And we can think of this as corresponding to the point 0 in the finite field F2, this to the point 1. And these two points in various algebraic extensions of the finite field with two elements. So you can think of this as being a sort of affine line over the field with two elements, except we have to include all points of algebraic extensions. So this fiber here is the closure of this generic point here. So what we've got here is a generic point. And the generic point is really this whole thing here. And we get the same thing over 3. So we get a generic point 3, which contains various points like 3X and 3X plus 1 and so on. And we get another fiber over 5, which does much the same. So we get the point 5X and so on. So what happens over this point here? Well, this time we're looking at ideals whose intersection with Z is 0. So what we can do is we can just tensor with Q and we're getting a map from Z of X to Z of X modulo some prime ideal. I guess I shouldn't use P for a prime ideal. Let's call this some prime ideal Q. And if we tensor that with Q, then we're going to get an algebraic number field where the image of X will be some algebraic integer. And from this, you can see that the corresponding prime ideals are all going to be of the form F of X, where F is some irreducible polynomial in Z of X. And this will be essentially the same as the spectrum of Q of X, because irreducible polynomials over Z are more or less the same as irreducible polynomials over Q. So we've got a generic point. And what this generic point contains is all irreducible polynomials, the ideals of all irreducible polynomials over Q. So we get things like X plus 5 and X squared plus 1. And what you see is that these correspond to algebraic numbers up to the action of the absolute Galois group of Q bar over Q. So for instance, this point contains the algebraic number, can be thought of as consisting of the algebraic number i together with its conjugate minus i. So this is really a sort of pair of algebraic numbers. And just the same way that the points over these fibers kind of correspond to various different elements of algebraic extensions of a finite field. Well, so you might think that's what the spectrum of Z looks like. But that's not really what it looks like, because each of these points here is really a one dimensional point. And so this point here is really two dimensional and really includes the whole space here. Only I can't draw it because then the diagram will get too confusing. So for example, let's look at the point X squared plus 1. Well, this is really a one dimensional point, so it has closures. Its closure meets several of the points here. And so for example, it meets this point because this ideal contains this ideal that 1 squared plus 1 is equal to 0 mod 2. So it kind of goes through here. And similarly, so over here, it will pass through the point 3 X squared plus 1. And over the point 5, it splits into two points. It has the point 5 X plus 2, and it also meets in the point 5 X plus 3. So you notice we're getting a sort of curve which meets these fibers in one or two points depending on whether X squared plus 1 has a root in the finite field. So if it has a root, this thing kind of splits into two, meets this fiber in two points. So the whole of quadratic, the theory of quadratic residues can be sort of seen by looking at some of these horizontal curves in the spectrum of Z of X. Well, if we look at another point, say if we look at the point X squared plus 5, we notice it meets this point here. And I guess it meets this point here and looks like that. So what is going on is we can think of these, the closures of these two points as being two curves, and they meet at this point here. Well, there's also another point they meet at, because if X plus 5 is equal to 0 and X squared plus 1 is equal to 0, then well, that means minus 5 squared plus 1 is 0, which means 26 is equal to 0. So there's another point here lying above the point 13. So I should really pull this coming down here. We're really meeting. And what they're meeting, they're meeting at the point 30, it's the right color, 13 X plus 5. So what's going on is you should think of the spectrum of Z of X as being a sort of surface in some weird way. And this surface has points and it's got some vertical lines in it, which are these fibers. And it's got some sort of horizontal lines. And these horizontal lines sometimes intersect each other. And when they intersect, there's a certain amount of algebraic number theory going on, for instance, where this orange line meets these vertical lines is telling you whether or not minus 1 is a quadratic residue mod p or not. And we can take a quick look at what the local rings are. In other words, what the stalks are. So I'm only going to do a few cases. So let's look at this one. The local ring is going to consist of all polynomials g over h, where the constant term of h is odd, because these are going to be the polynomials that aren't contained in the ideal 2x. If you want to know the local ring of this generic point, it will be of the form f over g, where g has some odd coefficient. The local ring of this generic point here will just be the field of rational functions in x, because you can invert anything that's non-zero. And the local ring of this point here will be all rational functions of the form f over g, with g of nought not equal to 0. So it will be all rational functions with rational coefficients that don't have a singularity at x equals 0. And OK, well, next lecture will contain another example of the strange spectrum of a ring.