 So this lecture is part of an online algebraic geometry course about schemes and in it we will construct the etal space of a sheaf. So we introduced sheaves and pre-sheaves last lecture and most people's initial impression on encountering sheaves as they seem to be an unnecessarily complicated and abstract way of talking about functions on a space. So I'll start by giving an example to show why we might want to work with sheaves rather than functions on a space. So on the left hand side, let's just look at the space of regular functions on the affine line. So we have polynomials over the complex numbers and the affine line C just corresponds to the maximal ideals of this ring. So we recall that if you've got a point alpha in C, it just corresponds to the maximal ideal of all multiples of X minus alpha. And we have the Zariski topology which says the open sets have a basis of sets Df which is the set of points where f is not equal to zero, f is some polynomial. And we also have the regular functions on each open set. So we have f of Df is just all polynomials of the form g over f to the n. That's all rational functions of this form. So these are the rational functions that are regular on the set Df. So f will have a finite number of zeros. The open set will be the complement of the zeros and these are functions which are regular except possibly for poles at those zeros. Now we can do the same for the ring of integers Z. So here we're just going to sort of copy this. So instead of the affine line, well we're going to take the set of maximal ideals. So we have the maximal ideals of two, three, five and so on. So this is called the maximal spectrum of the integers. And as we will see later it turns out to be the wrong thing to do. You should really take all prime ideals rather than maximal ideals but that will be coming later. Anyway we take the open sets Df and here f is going to be an integer and the open set is going to be the set of primes such that f is not in this prime ideal. In other words, p does not divide f. So this is exactly what happened here. The open set just corresponds to the point such that f isn't in the maximal ideal. And we're going to define f Df to be the set of rational numbers of the form g over f to the n where g is some integer. So this is exactly analogous to this. So we're sort of pretending that an integer is a function on this space here. There's a bit of a difference because here a polynomial is a function on c taking f as values in c at all points of c. Well here it's a little bit more complicated because you might think of an integer as being a function on spec z and what's it taking values in? Well here this is just the quotient c of x over the x minus a. So f naturally takes values in this field. Well here the field is going to be z over p. So an integer is taking values in fields but the field varies with the point. So it's a sort of funny sort of function. It's not taking values in a fixed space. It's taking values in a variable field. Actually more generally it would be better to think of as taking values in a local ring rather than a field but we weren't worried about that point now. And this also satisfies the sheaf property. So in other words it really integers really are behaving as if they were functions. So for example let's look at the points 2, 3, 5, 7 and so on. And let's take a couple of open sets. So I might take an open set u1 here and I might take an open set u2 here. And then f of u1 is equal to all functions or all integers of the form a over. Well we're allowed a power of 2 or 3 in it. So 2 to the something times 3 to the something. So we're only allowing 2s and 3s in the denominator. And similarly f of u2 is all functions where we allow 2s and 5s in the denominator. Sorry all integers not all functions. And now if we set u that's u2. If we set u equals u1 union u2. Well this will be all things of the form c over 2 to the something. And now the sheaf property says that if you've got a number in over this open set. So I suppose b over 2 to the something times 5 to the something has the same restriction to u into section u1 as f of u as something in f of u1. Then it has to come from some function on this set u. In other words an integer with only 2s in the denominators and that's kind of obvious. If you've got an inter rational number with only 2s and 5s in the denominator and it's also has only 2s and 3s in the denominator then obviously it only has a 2 in the denominator. So this is going to be if we've got this property then it's of the form c over 2 to the something. So the sheaf property on this rather funny sheaf turns out to be this basic property coming from I guess the fundamental theorem of arithmetic. And the point of this is you can now study the ring Z by thinking geometrically you think of Z as being functions on a space. Well obviously for the ring Z and this isn't going to tell you anything very new about the integers because it's so well known for more complicated rings this way of looking at the ring geometrically can be can be very powerful. So a basic theme of sheaves is that sheaves of sets are sort of very similar to behave in the same way as sets and similarly sheaves of abelian groups are very similar to abelian groups. So we've only defined sheaves of abelian groups that if you want to define sheaves of sets you need to modify the definition very slightly which we won't worry about because we're only going to use sheaves of sets as informal examples. In particular we can form a category of sheaves of sets in the category of sheaves of abelian groups for these we want you to find morphisms from a sheaf u to a sheaf g so sheaf f to a sheaf g. So what's a morphism from a sheaf f to a sheaf g was pretty obvious. What we should do is for each open set u we should give a morphism from f of u to f of g and moreover these should be compatible with restriction maps so if v is contained in u then we have maps f of v to g of v. And we have these restriction maps rho and I'm not going to write rho u of v because I will get u and v the wrong way around. So a morphism consists of morphisms like this for each u such that this diagram commutes whenever v is contained in u. So this makes sheaves into a category. We've got a set of objects and morphisms between them. So the category of sheaves of sets over a space is behaves very much like the category of sets. It's a sort of weak model of set theory. It's not quite a model of set theory because things like the action of choice of a bit of a tendency to fail in it. This is called a topos and there's a whole theory of toposes where the idea is you sort of do set theory except instead of working with sets you work with sheaves over a topological space or something like that. And similarly sheaves of abelian groups that the category of sheaves of abelian groups is very similar to the category of abelian groups. And this is a sort of theme of sheaf theory that when you're studying sheaves of abelian groups what you do is you try and think of some construction or theorem about abelian groups and extend it to sheaves of abelian groups. So the first question is what about exact sequences. So we might have an exact sequence A goes to B goes to C goes to naught of abelian groups, which you remember means A is more or less a subgroup of B, and the quotient of B by A is more or less equal to C. So we want to do the same thing for sheaves. So what what does it mean for a sequence naught goes to A goes to B goes to C goes to naught of sheaves to be exact. It's completely obvious. I mean any idiot can figure out what the definition of this should be. All you do is you say naught goes to A of U goes to B of U goes to C of U goes to naught is exact for all you. I mean this is a very natural definition. And there's no other obvious thing you could have so this must be the correct definition right. Well actually this is completely wrong. I mean it's it's it's it's just it's not completely wrong it's half wrong because that's okay for saying A goes to be as injective. But this gives you the wrong definition for saying the map from B to C is surjective which is really quite surprising. I mean it's very hard if this isn't the right definition it's not at all obvious what the right definition is. And understanding why this is wrong is really one of the fundamental things in sheave theory so we're going to look at an example to see why this is so wrong. So I'm going to take a space. I'm going to take my space X to be a circle. And I'm going to define two spaces X one is going to be the same as X and X two is going to kind of be a circle winding around it twice. And now I'm going to define a sheave f of X two is the sections. Sorry, not f of X two f two of you is the sections from you to X two in other words if I've got an open set you f two of you is just going to be the the maps from you to X two that that sections. And similarly f one of you is the sections you goes to X one. And now we can see X two goes to X one is obviously onto so we've got natural maps from X two to X one and X one goes to X. So we get a natural map from the sheaf f two to the sheaf f one in the obvious way and we've also got a map from X one X two goes to X one and this map here is obviously surjective. And if we look at the maps of sheaves well f two of X goes to f one of X is not surjective. In other words if you think about it, this is just a point. And this is empty. Because there are no continuous maps from X to X two that are sections of it to mean if you if you stop by mapping this point up to him and go around you find you, you get back to the other point of X two so so so f two of X is actually empty. So, well, we're talking about sheaves of sets rather than, rather than sheaves of groups here but we still have this problem that we've got two different possible definitions of surjective we can either talk about X two to X one being surjective or we can talk about two of X goes to f two one of X being surjective. And these are different. So, this is a sort of local definition of being surjective. And this is a sort of global way to define things being surjective so so here. So, we're looking at the point of X the fiber of this point of X two to X one is is surjective. So we're sort of looking locally. If we just look locally to point then the corresponding maps of open sets would would be surjective and this is a global definition and we need to know whether we should be using a local or a global definition of things being surjective. And it turns out that the right thing to do is to is to work with a sort of local definition. So, how do we do that. Well, suppose we've got a any map from a to X where this is continuous, then we can form a sheaf by just letting f of you be this continuous sections from you to a in other words maps from you that such that if you take take the map here and then project to get back to where you started. And this is this is always a sheaf. So it's a sheaf of sets of a is that is just a set and it's a sheaf of groups of a has some reasonable group like structure on its fibers. So we can construct a sheaf from any map from a set A to X and similarly, if we've got a map from A to X and a map from B to X and F is the sheaf of a and G is the sheaf of B, then if we've got a map from A to B making this continuous, we get an induced map from the sheaf F to the sheaf G again in a fairly obvious way. And we now have this concept. We're going to obviously talk about A to B being surjective. So A to B might be surjective, but F of X goes to G of X might not be. So this is just what we had in the previous example. And what we want to do is sort of define the map from F to G to be surjective, if and only if the map from A to B is surjective. Well, there's a bit of a problem with this because if we're given a sheaf F, you know, it doesn't necessarily come from a map A. So we have the following problem. Does a sheaf F come from a map from A goes to X for some A. And it turns out there's a very nice way to construct A from it. In fact, this works for a pre-sheaf. So we're now going to construct the etal space of a pre-sheaf F. And what we want to do is get given some space X, construct a space A mapping to it. This is something to do with the pre-sheaf F. And first of all, suppose we're given a point X in X. We want to know what is the fibre of A over this point X. And we're going to construct it as follows. Point of the fibre is given by a section F of U for some neighbourhood U of P. And we need to say when these are the same. So F and G are the same point of the fibre if F and G are the same near X. So what this means is that the image of F and G in the image of F and G in V are the same for some small V with P contained in V. So you can think of F as being... So we have the point P here. And we've got a set U1 and F is in F of U1. And G is going to be in F of U2. And somewhere inside U1 and U2, we're going to take a small set V and F and G have to have the same restriction to the set V. And if so, they're considered to be the same point of the fibre. So if you think of this pre-sheaf as being continuous sections of something, then the fibre is roughly equivalence classes of sections where two things are considered equivalent if they're the same near P. Next, we have to put a topology on this. So a base of open sets is given as follows. So suppose we're given any set F in F of U. Then we can form an open set for each P in U. We take the image of F in the fibre over P. And the union of these all images will be an open set. And these open sets will form a base of the topology. So this gives us a set A to X. And this is called the etal space of the sheaf F. And the word etal, a map from A to X is etal if for all A in A, there is a neighbourhood or V of A such that the map from V to the image in X is a homeomorphism. So I'm not going to prove this because it's rather easy. In fact, in sheaf there's a whole lot of statements that are rather easy to prove just by unwinding the definitions. And I'm going to leave most such statements just as exercises. So the definition of the fibre sort of says that each fibre is a direct limit over all the open sets of U if you're used to doing direct limits. Now the etal space can have a rather strange looking topology. You see it's locally isomorphic to X and that each point has a neighbourhood that looks like X. However, this is a bit misleading. For example, if X is a manifold, say a real manifold, then this says that each point of A is also locally a manifold. So A is a manifold. Well, that's really misleading because it turns out that the set A can be wildly non-house dwarf. So in the next lecture we'll give some examples of what etal spaces look like and just how strange they can be.