 So this lecture is part of an online algebraic geometry course on schemes and will be about some examples of quasi coherent sheaves. So you remember that a quasi coherent sheaf is something that sort of looks locally like a module over a ring. So let's start with some simple examples. Let's take our scheme to be the spectrum of Z and look at the quasi coherent sheaf corresponding to the module Z over 12 Z. And what I'm going to do is I'm just going to try and draw a picture of it. So here is spec of Z. So it has a generic point zero and on it it has these points two, three, five and so on. And what I want to do is just kind of draw a picture of the stalks of the sheaf corresponding to this at all the points here. Well, over the local ring two. And this module just becomes Z modulo four Z so we can sort of think of it as being a little stalk there with sort of twice the size of a vector space. And similarly over over the prime three, it looks like Z over three Z and everywhere else it just looks like zero. So you can picture this coherent sheaf as having a bit sticking over this point here and a bit sticking over this point here, nothing everywhere else. So let's have a look at a slightly more complicated example. This time let's take the spectrum of K X Y. So this is just essentially the affine plane, which we can draw symbolically like that. And let F be some sort of irreducible polynomial. And we're going to look at the module or modulo F and ask what what does it look like as a sheaf. Well, if the zeros of F look something like this, so the zeros of F will be some sort of curve. And if we take this module and localize it that any maximal ideal that doesn't contain F, we just get zero. Whereas if you sort of localize it as a point on on F, then what you get is essentially looks like something one dimensional. So we can pitch this module is looking kind of like this. So this is the sort of one dimensional something or other associated to each point of this and outside this, it should be thought of as being zero. We can also look at the module of modulo X Y say so, so X Y is the maximum ideal corresponding to point zero. And if we look at this module and localize it at any other maximal ideal we just get zero. And if we localize it at the local ring of zero we just get a one dimensional vector space or something depending on how you do it so we can picture this. The sheaf of this module is just a sort of one dimensional vector space at the origin in some sense and zero elsewhere. Look at the whole of our as a module over our then it just sort of looks one dimensional everywhere so you can sort of picture one dimensional vector spaces at every point of of this. So you can picture these three types of modules or rather the coherent sheaves there associated with them as sort of associating, you know, maybe a one dimensional vector spaced each point of some sub varieties. So you might think of this sheaf as having kind of support along along this sub variety and looking a bit like a vector bundle or something there. In general, a sheaf corresponding to a finitely generated module over ring is a sort of combination of the sheaves you get from this. And you know, every finitely generated module can be broken up into a sequence of extensions of modules of the form R modulo some prime ideal, and our modular prime ideal for this ring looks like one of these three cases. So a general sheaf of a finitely generated module you sort of take these three sorts of sheaves and combine them together somehow. This combined together is quite a complicated operation, but at least this gives you a first approximation to what a sheaf looks like it's sort of built out of pieces and each piece. So it looks like you take some sort of sub variety and assign vector spaces to each point. Um, so now let's look at another example for this I'm, I'm going to take, um, um, the next example is going to be a ring space. Um, not a scheme, at least to start off with, and then I will sort of convert it into a into a scheme. So we're going to consider the sphere S2. So this is just a ordinary two dimensional sphere and let R be the smooth functions on a sphere. And we can think of the sphere as being a ring space where the ring correspond to each open set U is just smooth functions on you. Now I'm I want you to find a module over this. So the module, let's say f, f of U is going to be the smooth tangent vector fields on the open set U here U is some sort of open subset of S2. You can see that this sheaf is a module over this because if we've got a tangent vector field on some open set and a function on the open set, we can just do point wise multiplication of the tangent vectors by the smooth functions. Um, so this sheaf obviously corresponds to the, in some sense, the tangent vector bundle. You know, at each point we're taking a tangent space and taking a tangent vector there. So we've got a vector bundle where a vector bundle just means you assign a vector space to each point of a variety. Um, so, um, um, so this sheaf looks locally like, um, oh, sorry, f, f looks locally like oh, plus, oh, because if you take a small open set U, then you can trivialize all the two dimensional tangent spaces by taking a, by taking a smooth base. It's for them and that gives you an isomorphism between f restricted to U and some of two copies of, of O. However, f is not globally O plus O because if it was, you could find a section of it. In other words, a tangent field that was none zero everywhere because you can do this for O plus O. So, um, there's a well known theorem, the hairy sphere theorem that says you can't comb a hairy sphere. In other words, you can't find a tangent vector field on the sphere that's none zero everywhere, at least not if you insist on doing it in a reasonably continuous fashion. So here we've got a sheaf that looks locally like the sum of two copies of the regular functions, but globally isn't. In fact, this is an example of a vector bundle that we'll describe in a moment. Well, this isn't a scheme, so let's find a scheme version of this. Well, what we do is we replace S2 by the spectrum of, well, we just take, say the reals, the ring of polynomials in three variables and just quotient it out by x. squared plus y squared plus c squared minus one. Okay, this is sort of related to a sphere in that certainly some of the maximal ideals of this will be points of the sphere. However, you remember this spectrum also has lots of complex points. So, so it's not exactly the same as a sphere. And we can form a module over it corresponding to the tangent space. What we do is we just take m is the sub module of r plus r plus r. Here I'm going to call this ring here r of vectors x, y, z with x, x plus y, y plus c, z equals zero. So informally this just says that x, y, z is orthogonal to the vector x, y, z in the sphere. So you can identify vectors that are orthogonal to that vector with the tangent space at each point. So this module kind of corresponds to the tangent bundle of a sphere in some sense. So now m twiddle is going to be a sheaf over the space spec from of r. And again, this sheaf will look locally like 0x plus 0x where let's call this space x, but not globally like it. Well, this is an example of something called a vector bundle. So let's just have a quick review of vector bundles. And what I'm going to do is I'm going to describe several ways of thinking about a vector bundle. So first of all, there's an informal way of vector bundle v. This is going to be over m where m might be might be a manifold or it might be a scheme. I'm going to be a bit vague about this. So informal means you assign a vector space to each point of m in a nice way. I haven't quite said what nice means, but this is a very informal description. So this gives you a mental picture of a vector bundle. If you've got a nice way of assigning a vector space to each point of a manifold, you've probably got a vector bundle. For instance, you could take each point and assign to each point the tangent space at that point or the cotangent space at that point or a one-dimensional vector bundle or the exterior power of a tangent space and so on. You can define it slightly more formally using fiber bundles. So we might think of a vector bundle as being a map from e to m such that the fibers are vector spaces and it's locally trivial. In other words, you can cover m by open sets u such that it looks locally like, say r to the n times u mapping to u where u is open in m. Well, if we were doing real vector bundles, we would take r to the n. If we were doing complex vector bundles, we would take c to the n and so on. So we can also define it in terms of sheaves. So what we would do is we take a sheaf f over m. And again, this looks locally like, it has to look locally like the sum of the ring of regular functions on u, finite sum of the sheaf of regular functions on u. Here u is an open set. O u is the sheaf of regular functions on u, whatever that will be. I mean, this will be slightly different depending on whether you're working with smooth manifolds or schemes or whatever. And finally, we can also think about what it looks like in terms of rings. Well, here, instead of a ring, we can consider vector bundles as vector bundles over the spectrum of r. And what this would correspond to is a locally free module m. So what does locally free mean? It means that mf to the minus one is isomorphic to rf to the minus one, the n, where the sets dfi cover the spectrum of r. In other words, the ideal generated by f1, f2, and so on is just the ring r. So the picture here is we're covering the spectrum of r with these various open sets dfi. And on each of these open sets dfi, the module, if you localize it, looks like a free module in some number of generators. And so if you've got a module like this over a ring, then you turn r into its spectrum and you turn m into a sheaf over that spectrum, then that gives you a sheaf satisfying these properties. So that's four ways to think of a vector bundle. So one example we've had of a vector bundle is just the tangent space of a sphere. Here's another example. This example is going to come from number theory. So we're going to take r to be the ring z root minus five. And our space is going to be the spectrum of r. And now this ring is the basic example of a ring without unique factorization. So we've got two times three equals one plus root minus five times one minus root minus five. And you should remember this formula because we're going to use it a little bit later. And it also has non-principle ideals, I don't remember how to spell principle, ideals, such as two root minus five. And we can see this is non-principle if we quickly draw a picture of it. So let's draw a picture of r in the complex plane. And r looks like this, some sort of nice square lattice. So here's the point zero and here's one and here's root minus five and here's one plus root minus five and so on. So r looks like a sort of rectangular lattice. And if we take any principle ideal of r, the principle ideal a, all we do is multiply all these points by the complex number a. And that has the effect of rescaling everything by the absolute value of a and rotating by the argument of a. So we would still get a rectangular lattice. On the other hand, if we draw a picture of this ideal here, so let's have this ideal, it looks like this. We take all these points here. And you can see this isn't a rectangular lattice, it's a sort of diamond shaped lattice in some sense. So this is a different shape from the ideal one, which is just r. So you can see visually that r has more than one equivalence class of ideals. It's got non-principle ideals. And what we're going to do is to show this non-principle ideal is a non-trivial line bundle. So a principle ideal, so principle ideal i, as long as it's non-zero, is isomorphic to r as an r module. So that's not terribly interesting. And the ideal 2 1 plus root minus 5, sorry, this ideal should have been 2 1 plus root minus 5 there. The ideal 2 1 plus root minus 5 is not isomorphic to r. However, it is isomorphic to r f minus 1 over certain localizations. So we notice that the spectrum of r is covered by the open sets d2 and d3. So this is the primes not containing 2 and this is the primes not containing 3. And that's because 2 and 3 generate the ideal 1. And in d2, the ideal 2 1 plus root minus 5 is equal to 2 because 1 plus root minus 5 is equal to 2 times 1 plus root minus 5 and then we can divide by 2 because d2 means 2 is invertible. In d3, 2 1 plus root minus 5 is equal to 1 plus root minus 5 because 2 is equal to 1 plus root minus 5 times 1 minus root minus 5 divided by 3 using that formula I wrote down earlier that I told you to remember. Now we notice that 3 is invertible. So this ideal becomes principal in over both these open sets. So the sheaf of the module 2 1 plus root minus 5 can be pictured like this. Here we have the spectrum of r and it's covered by 2 open sets. So we've got this open set d2 and d3. And the sheaf looks trivial over these two open sets. When I say it's trivial, I mean isomorphic to a one-dimensional, the sheaf of a one-dimensional free module, but not over the whole spectrum of r. So this is an example of a non-trivial line bundle. So what's a line bundle? Well, it's just a vector bundle of rank 1 where the rank is pretty obvious. It just means the dimension of the vector space you associate to each point. There's one example of a line bundle that everyone's come across. So here's a picture of a line bundle. You just take the moebius band. So here's a moebius band. And the moebius band can be mapped into a circle. And if you look at the fibres, the fibres just sort of look like this. And so we have the moebius band v mapping to a circle s1. And the fibres are all isomorphic to r1. Well, they're all isomorphic to r1 if you have a sort of open moebius band. I mean, you can see you can sort of put one-dimensional vector space together like this. They form a moebius band. And you can see this gives you a vector bundle over s1. It's sort of obvious that it's locally r1 times a small open set of the circle. And it's not the trivial one r1 times s1 because it's got a sort of twist in it. I mean, if you go around it once, you get back to minus what you were before if you see what I mean. So in general, a line bundle can be thought of as a sort of analog of a moebius band. It's kind of like you assign a one-dimensional space to each point of the manifold. But you do it in a funny kind of twisted way globally. So if we go back to this ideal here, it can be visualized as something like a moebius band over the spectrum of r. It sort of assigns a one-dimensional vector space to each point of the spectrum of r. But there's a kind of strange twisting going on. Finally, we can define a product on line bundles as just taking the tensor product of sheaves. So how do we take tensor products of coherent sheaves? Well, it's pretty obvious how to define this. Suppose f and g are quasi-coherent sheaves. We can ask what is f, tense, and g. There's an obvious way to define this. We can just define f, tense, and g of u equals f of u, tense, and g of u. I mean, what could possibly be wrong with that? It's the most obvious thing we could do, and it's wrong. The problem is that it turns out that this is not a sheaf. It's not a sheaf even for line bundles over, say, one-dimensional projective space that we will be looking at later. What you have to do is you have to take a pre-sheaf defined by f, tense, and g of u equals f of u, tense, and g of u, and form the corresponding sheaf. As I said, we'll see why you have to do this one slightly roundabout process later. I should say this formula is OK for open affine sets u, but it does go wrong for more complicated u like u corresponding to projective varieties. Anyway, you can form line bundles into a sheaf that the tense of product of two one-dimensional vector spaces is one-dimensional. The tense of projective line bundles under this is also a line bundle. And we get the famous Picard group equals isomorphism classes of line bundles under taking the tense of product. And this turns out to be really important in variant of schemes and almost everything else that turns up all the time. In fact, we see the Picard group really amalgamate several different things. So the Picard group combines the ideal class group from number theory. You remember it should take an integer of an algebraic number field. It has something called the ideal class group associated to it. And this turns out to be the Picard group of the spectrum of the ring of integers. And also combines things like the Jacobian of a curve. So just the line bundles over some scheme. And it also is very similar to things like the characters, or at least the one-dimensional characters of a group. So you remember if you've got an Abelian group, there's a dual group consisting of the one-dimensional characters of it. And the one-dimensional characters are in some sense line bundles over the group. You can make sense of that by doing something with topics if you really want to. But it's easiest just to think of them informally as being one-dimensional. All three of these are sort of classifying things that kind of look locally like one-dimensional spaces. And they're all really special cases of the Picard group. Okay, next lecture, we will discuss line bundles over the projective line and see why I was making such a fuss about the definition of the tensor product of line bundles.