 So this lecture is part of an online mathematics course on algebraic geometry. It's going to cover the basics of scheme theory loosely following chapter two of the book Algebraic Geometry by Hart-Shorn. For chapter one of Algebraic Geometry, there's another series of lectures, which there might be a link to at the bottom of the screen if I figure out how to put links on. So to motivate the introduction of schemes, we first recall what affine varieties are like. So affine varieties over a field K correspond to commutative rings with the following properties. First of all, the ring is an algebra over K, which more or less means it contains K. Secondly, it's finitely generated as an algebra. And thirdly, it has no nilpotent elements than zero. So nilpotent element is one such that x, the n equals zero, but x is not equal to zero. So the correspondence is as follows. If you've got an affine variety over a field K, you can take its coordinate ring, which is a commutative ring with these properties. And basically it just consists of polynomial functions on this affine variety. Conversely, if you've got a commutative ring with these three properties, you can reconstruct the affine variety. For example, the points of the affine variety correspond to maximal ideals of this commutative ring and so on. Now, there are three, I give three examples to show why you don't really want to have these properties. First of all, we might look at the coordinate ring of a line, which is just the ring of polynomials over a field K. And it's known this behavior of this is very similar to the behavior of the integers or algebraic number fields such as the Gaussian integers C of i. And it would be really nice if algebraic geometry could include these examples as well. It was already noticed in the 19th century that the theory of algebraic number fields was very, very similar to the theory of curves over a finite field. So you really want algebraic geometry to include these as geometric examples. And these are not algebras over a field. So maybe we should just delete this condition. Secondly, if we're looking at the affine line as well as its coordinate ring, we might want to look at the ring or field rather of all rational functions on the line. And another thing that's good to look at is a local ring such as Kx localized at the ideal of the point x, which would just consist of all rational functions of the form p over q with q of naught, not equal naught. So these are rational functions that are defined at zero. And these local rings are very useful. Well, the problem is these rings are not finitely generated over K. So maybe we should delete the second condition too because there are lots of examples we want to apply it to where that doesn't hold. And thirdly, if we are doing Bezut's theorem, we might intersect two varieties. For instance, we might take the intersection of y equals x squared and y equals zero. So we've got a line y equals zero and a parabola y equals x squared. And usually if you want to take the intersection of two varieties, you just question out by the ideals generated by both varieties. So here we question out by the ideal where y is zero and we should question out by the ideal y x squared minus y. And this is just isomorphic to the ring Kx over x squared. Now in classical algebraic geometry, you would just say, well, replace this by the ring Kx over x to get the ring of a variety. So this would just be K. Well, the trouble is you're throwing away information by doing that because this is the coordinate ring of a point and although the intersection is a point that's a bit misleading because it's really a double point and you would rather like to keep track of the fact that this is really two points that have squished together rather than being a single point. And this ring does that. You see, it's two dimensional over K so it sort of remembers there are really two points. The only disadvantage is that it's got nil potent elements. So you can't represent it as functions with taking values in a field, but it does suggest we should drop this condition that our commutative rings should have no nil potent. So what are we left with? Well, we're left with all commutative rings and now we will define affine schemes. They're going to correspond to all commutative rings. So the idea is the relation between affine varieties and these special sorts of rings is very similar to the relation between affine schemes and arbitrary commutative rings. What we will do is we will try and imitate the way you construct affine varieties from these rings. And it doesn't quite work. We need to modify the construction a little bit because rings with nil potents really do take a little bit of extra care to handle. So in order to define affine schemes, we need to use a technical tool called sheaves. So sheaves were introduced by Leray, possibly around 1950, and were very soon used with a very powerful effect by Seher and Cartan to study analytic varieties. And Seher introduced them into algebraic geometry in a very famous paper called Algebraic Coherent Sheaves. If you want to see this paper and don't read French, if you search on the web for the term Algebraic Coherent Sheaves, you'll probably find some English translations of it. So this paper is still possibly the best introduction to sheaves in algebraic geometry. It was written slightly before the invention of schemes, but that doesn't really matter. So before explaining what sheaves are, well, the problem with sheaves is the definition looks kind of abstract and a little bit dry at first. So try and motivate the introduction of sheaves. In algebraic geometry, before the introduction of sheaves, people have found all sorts of invariance. For instance, there's the arithmetic genus. And people studied surfaces embedded into three space and discovered that the arithmetic genus of a surface didn't seem to depend on how it was embedded into three space. Unfortunately, the arithmetic genus wasn't all that easy to define or calculate. For instance, one possible way of defining it would be to say that it's mu naught minus one choose two minus mu naught minus four times epsilon naught plus epsilon one over two plus minus epsilon naught plus two t. Well, what are all these numbers? Well, mu zero, for example, is the degree of the section of the surface by a hyperplane and epsilon naught and epsilon one and t other similarly rather strange looking invariance of the surface. So people sort of discovered by brute force that this combination didn't change if you changed the embedding of a surface. But this is a bit of a problem. I mean, you want to extend the arithmetic genus to three dimensional varieties, for example. And how do you do it? I mean, good luck guessing how this generalizes from two dimensions to three dimensions. Fortunately, if you introduced sheaves, the arithmetic genus turns out to a much simpler definition. For surfaces, the arithmetic genus turns out to be the Euler characteristic minus one where the Euler characteristic is simply the dimension of a zero-sheaf co-homology group minus the dimension of a first-sheaf co-homology group plus the dimension of a second-sheaf co-homology group of your surface. In fact, with coordinates in the sheaf of regular functions which we'll talk about later. So this is very similar to the Euler characteristic of a topological space, which is just the alternating sum of the dimensions of all its co-homology groups. So it's a very natural definition and much simpler to think about than this mess here. Similarly, another invariant of the surface was the irregularity which was originally, well, the arithmetic genus suggests that there should be another sort of genus and there is, there's another one called the geometric genus which were both defined for surfaces and analogy to the genus of a ream and surface. And again, the irregularity was a slightly mysterious thing relating to the difference between the arithmetic and the geometric genus and it wasn't at all clear what was going on or how you'd generalize it to high dimensions. Well, using co-homology of sheaves, it turns out to be just the dimension of H1 of a certain sheaf over a surface. And now it's completely obvious how to generalize this to high dimensions because you can just look at the dimension of high-dimensional co-homology groups of sheaves. And more generally, you can define things called hodge numbers, HPQ, which are the dimension of a certain homology, Q-th homology group of a sheaf of P forms over a surface. So once you have defined sheaves and co-homology, it makes it really easy to define a lot of invariants whose definition looked extremely mysterious before sheaves and co-homology were introduced. So the next question is, what is a sheaf? Well, we'll start off by giving a sort of motivating example. So let's suppose X is a topological space and it has some open sets, U. And we're going to define a sheaf or pre-sheaf, let me explain the difference in a little bit, by putting F of U to be the continuous functions from U to the reals. So F goes from open sets of X to Abelian groups. So it's going to be something called a sheaf or possibly pre-sheaf of Abelian groups. And it has the following properties. First of all, if V is contained in U and V and U are open sets, then we have a restriction map, row UV taking F of U to F of V, which just restricts a function from U to V. And what properties does it have? What has the following really obvious properties? If W is contained in V is contained in U, then row UW equals row VW times row UV. I hope of all these subscripts the right way around, they're really easy to get them muddled up. And also row UU is the identity. And we now define a pre-sheaf of Abelian groups on X to be something similar. So a pre-sheaf of Abelian groups means you're given a map F taking open sets to Abelian groups and you're also given maps row UV from F of U to F of V for each U and V, satisfying these conditions here. Well, this definition looks like a rather abstract and cumbersome way of defining open functions on a topological space. And the reason we're introducing it is we'll see a bit later that there are some very natural examples of sheaves which don't come by taking, well, which don't come directly by taking open functions on by taking functions on the open sets of a topological space in any particularly obvious way. So we can see some, another way of thinking about sheaves is we can think of F is a contra variant functor from the category of open sets of X to the category of Abelian groups. So what on earth does this mean? Well, first of all, what is the category of open sets of X? Well, you remember a category has some objects and it has some morphisms between these objects. So the objects are going to be the open sets. And what are the morphisms from U to V? Well, there is one if U is contained in V and naught if U is not contained in V. So we're just defining the set of morphisms to be a set with either zero or one elements depending whether U is contained in V or not. And now if you unravel the definition of a contra variant functor for this, you'll find that F is really just assigning an Abelian group to each open set together with a map from this Abelian, from the Abelian group of V to the Abelian group of U and F of U is contained in V. So this definition is just a, looks like a really abstract and silly way of making the definition of a sheaf even more abstract than it already is. The reason I'm mentioning it is that it suggests some amazingly powerful and useful generalizations of a sheaf because you can replace the category of open sets of X by any other category and still define pre-sheaves. And in fact, you can replace the category of Abelian groups by another category. So this was used by growth and dick to define things like etal cohomology. You replace the category of open sets of a variety or scheme by a rather bigger category and defined pre-sheaves on that and was able to use that to define all sorts of useful things like etal cohomology and so on. We're not actually going to use this very abstract approach to pre-sheaves in these introductory lectures. I'm just mentioning this as a generalization you'll get later on. Anyway, let's have some more examples of the rather more elementary definition of sheaves we have. So here's some more examples of pre-sheaves and sheaves. We can take X to be a smooth manifold and we can take F of U to be the smooth vector fields on the open set U. So we can think of X as being some maybe a torus or something. So for each open set, we just look at the Abelian group of all vector fields and it's sort of obvious that's a pre-sheaf and we'll see in a moment it's a sheaf. Or you can take X to be an affine variety and we take F of U to be the regular functions on U or we can take X to be a Riemann surface and we can take F of U as you can probably guess to be the holomorphic functions on U. So in all of these cases, we have a manifold with some nice notion of functions that might be smooth functions or continuous functions or rational functions or holomorphic functions and we can form a sheaf by taking that sort of function on every open set. Notice that if you've got a Riemann surface, say the Riemann sphere, you can consider it as a complex manifold or a smooth real manifold or as a topological manifold and corresponding to that, you get different sheaves of functions. You can take the sheaves of holomorphic functions or smooth functions or continuous functions. So the different sheaves you put on this kind of correspond to the different sorts of geometric structures you can put on it like a smooth structure or a complex manifold structure or whatever. Another example of a pre-sheaf is let's just put F of U equals A where A is a fixed abelian group. And again, this is a pre-sheaf for rather trivial reasons. And by the way, this isn't a pre-sheaf according to Hopchon's definition of a pre-sheaf because he adds the condition that F of the empty set should be the zero group. Most authors don't add this condition and in fact, there's, doesn't seem to be any good reason for it. It doesn't really make any difference because we're going to be working almost entirely with sheaves rather than with pre-sheaves and sheaves automatically satisfy Hopchon's extra condition. So just in case you're wondering why this doesn't satisfy Hopchon's definition of a pre-sheaf it's because I'm slightly changing Hopchon's definition. Now what is a sheaf? Well, most of the examples above but not this example, satisfy the following extra conditions. Suppose the set U is a union of sets Ui. So we might have U1, U2 and U3. And all these things like smooth functions or continuous functions or whatever have a sort of local property that a function is smooth if and only if it's smooth locally everywhere. And a sheaf sort of captures this property of being defined locally. So we have two conditions. First of all, suppose F is in F of U. If the image of F in all Ui is zero then F equals zero. By the image, I mean under the maps row from U to Ui unless I mean Ui to Ui in which way around the direction things go. So if a sheaf satisfies this condition we say it is separated. So it says that it's the condition that function is defined by its values locally everywhere. We're not allowed to have a function which is zero on here and zero on here and zero on here but somehow mysteriously non-zero on the whole set. The second is that it says that suppose we're given F i on Ui, we want to glue these together into a function on the whole of U. Obviously, in order to do that the F i's must be the same on the restrictions. So suppose F i and F j in F U j have the same image in F Ui in F Ui intersection Uj for all ij. Then we can find F in F U which has images F i in F Ui. And again, if we're looking at functions or smooth functions or whatever this is just the obvious property that if we define a smooth function on each of these sets and if these smooth functions are the same on the intersections then that defines a smooth function on the whole set and by condition one, this function is unique. So these two properties of pre-shefes capture an extra property of functions. And so a sheaf is a pre-shef satisfying these two conditions here. So this is a pre-shef satisfying one conditions one and two. So this says it kind of behaves like, this says it really does behave like functions. I'll just finish this lecture by giving you an example of a pre-shef that is not a sheaf. Well, we can just take F of U equals A where A is fixed and A is not zero. And this is not a sheaf for all the trivial reasons. So a sheaf has the property that F of the empty set is equal to naught. This is because the empty set can be covered by the empty collection of open sets. And if you think rather hard about definitions involving open sets you see it means that this must actually be zero. Well, okay, this isn't a sheaf for rather stupid reasons. Suppose you just change it so that F of U equals A for U not equal the empty set and F of the empty set equals zero. Well then it's still not a sheaf in general because what you can do is you can take U to be the union of two disjoint open sets U one and U two with U one intersection U two equals zero. And then we find F of U equals A. However, if it were a sheaf then F of U would have to be F of U one times F of U two because for each element of F of U one and each element of F of U two we would have to have a unique element of F. So it'd have to be A times A if F is a sheaf. And for most Abelian groups A is not equal to A times A so this isn't going to be a sheaf. If you like you can write the sheaf theoretic condition as a sort of exact sequence. So this is not goes to F of U goes to product over I of F of U of I and then you can take the product over I and J over F of U I intersection U J. So saying this is exact just means that if you've got anything in this set here in other words, if you choose an element of F U I for all I and if this element is the same image under these two maps, well what are these two maps? Well, there are two ways to map F of U I intersection U J because you can either map you can either use the first component to find the map from here to here or you can use the second component to find the map from here to here. And saying this is exact means that if something has the same image under both of these maps then it's the image of a unique element here. So this is just a rather roundabout way of giving the sheath theoretic condition. And if you look at this particular example you find that you've got a map A goes to A times A goes to get zero because U I intersection U J is zero and A is certainly not the place where these two maps are equal. So this thing fails to satisfy the sheath theoretic condition. Okay, next lecture we will give some more examples and properties of sheaths.