 So this lecture is part of an online algebraic geometry course, giving an introduction to schemes. This lecture we'll be discussing when maps of sheaves are injective or surjective or exact and so on. So last lecture, when we were given a pre-sheaf F, we constructed an etal space of F, which was a map from, which was a space A mapping to our base space X that F is a sheaf over. And you remember we did this by first constructing the fibres of A by taking a direct limit and then gluing these fibres together. So we're first going to start by looking at a couple of examples of what this etal space looks like. So you remember this map being etal means it's a local homeomorphism in the sense that each point of A has a neighbourhood that's homeomorphic to a set in X. So let's first of all look at the example where we take X to be the reals and we take X to be the sheaf of smooth functions. So F of U is just smooth functions on U. And the fibre at zero is just, you can just think of it as being germs of smooth functions at the point zero. So if we draw a graph of a smooth function, it might look something like this. So we have a smooth function defined in a neighbourhood of zero. And we might have another smooth function that say looks like this. But if these two smooth functions are the same in some neighbourhood of zero then they count as the same point of the fibre. So the points of the fibre are sort of germs of smooth functions where you can only see the smooth function very, very close to zero in some sense. And the point is this example, the space A in this case is not house dwarf. So you sort of might think well it's locally a homeomorphic to the reals so it's just a one-dimensional real manifold. Well it isn't some sense. The problem is it's a non-house dwarf real manifold. So let's see two points at zero that don't have disjoint open subsets. So the first point at zero I'm going to take to be the fibre of zero. So f is just equal to zero. The second point at zero I'm going to take to be the following function. It's going to come down like that and it's going to be zero there and then it's going to have a bump here and it's going to be zero there and then a bump there and zero there and a bump there and zero there and so on. So this is G and you notice that f and G are different points in the fibre at zero because there's no open neighbourhood where they're the same. These bumps go on forever but any open neighbourhood of G is given by taking some neighbourhood of naught and the open neighbourhood of G might sort of consist of the image of this bit of G and similarly an open neighbourhood of f might consist of you take a small open neighbourhood of zero and any function of zero in that open neighbourhood is in the image of f. But now you see any open neighbourhood of G and any open neighbourhood of f have a point in common because you can find some small patch near the origin where both f and G are identically zero in an open set and that will give you a point in the same open neighbourhood. So these etal spaces can be a bit weird because they can sometimes be none house dwarf. Notice that if G is the sheaf of analytic functions then G is in fact house dwarf. You can do that as an easy exercise. You see you can't get this phenomenon because if a function is analytic it can't have an infinite chain of zeros approaching zero unless it is actually zero. So somehow the question of whether this pre-sheaf is house dwarf or not house dwarf is rather like whether your functions have a sort of notion of analytic and analytic continuation. If you look at smooth or continuous functions and the sheaves you get are not house dwarf. If you look at analytic functions or regular functions in algebraic geometry then the sheaves you get often are house dwarf. So from any pre-sheaf we can get the etal space and from the etal space we can get a sheaf of sections. So this gives a way of going from a pre-sheaf to an associated sheaf and I'm just going to list some properties of this construction. I'm not going to bother proving most of them you can take these as exercises. So first of all this is indeed a sheaf and it's universal. If we have a map F goes to G where this is a pre-sheaf and this is a sheaf then it factors through the map from F plus to G. So there's unique morphism of sheaves making this commute. So this is a sort of universal way. It's the best possible way of constructing a sheaf out of F that any other way of constructing a sheaf out of F you sort of start with F plus and then embed it into this other sheaf. So also you can check if F is already a sheaf then the map from F to F plus is an isomorphism. So if you try and turn a sheaf into a sheaf you get back the sheaf you started with not very surprisingly. Also if A goes to X is already etal and F is equal to the sheaf of sections then the etal space of F is isomorphic to A. So in other words what we find is that sheaves are more or less the same. So sheaves over X are sort of equivalent to etal spaces A mapping to X. So incidentally the reason for the name sheaf is you can think of the etal space A as being the union of the fibres or stalks. So in sheaf theory when you talk about the fibre of a map from A to X you quite often call that the stalk of a map from A to X. And a sheaf in agriculture means you get a lot of stalks of corn or something and bundle them all up. So if you think of these as being sheaves of corn bundled together you can sort of think of the etal space A as being a bit like that except that doesn't give a very good picture of the topology of A but never mind. So a sheaf when viewed as an etal space kind of looks like a lot of stalks all stuck together. And in particular if F goes to G is a homomorphism of sheaves then it is an isomorphism if and only if it's an isomorphism of the corresponding etal spaces which is the same as saying it's an isomorphism of all stalks. And this follows easily from the comments we made on the previous sheet that if it's an isomorphism of etal spaces if the corresponding map of etal spaces is an isomorphism then as we said we can just reconstruct F and G from these etal spaces and if the etal spaces are isomorphic then that shows that F and G are isomorphic. Well, we want to define whether, first give another example. So let's have an example of constructing the sheaf of a pre-sheaf. So let's take F to be the constant pre-sheaf where we just put F of any open set U is equal to some set A. So this is going to be a fixed abelian group and we saw earlier that this is usually not a sheaf. Well, so we want to work out what is the etal space of this? Can't spell etal. So the etal space of this, well, what's the fibre at each point? Well, the fibre at each point is kind of trivial to work out because F of U is just A so when we take a direct limit the fibre at each point is just A and if you work out the topology on the fibre you can see the etal space is just X times A mapping to X. So here A has the discrete topology and now we take a section U and want to know what is F of U? Well, F of U is just the sections of this map here. So we want functions from X to X times A such that if you do that then that you get the identity and you can see this is just continuous maps from U to A. Now, since A is the discrete topology this means that F of U equals A if U is connected and I don't want to get into arguments about whether or not the empty set is connected. So let's just say it's connected and non-empty. A good case for saying the empty set is not connected but anyway. And F of the empty set is just the zero and if U has more than one component then for instance if U has two open components then F of U will be A times A sorry this is F plus of sorry this is the sheaf F plus that we can construct from the pre-sheaf F. So if U has two open components then the sheaf F plus will be a product of two copies of A and in general if U is a union of open components then F plus of U will be a product of copies of A and you can get more complicated examples where U where the components of U aren't open in which case slightly more complicated things happen but I'm not going to worry about those. Now we now want to define whether or not a homomorphism of sheaves of Abellion groups is exact or injective or whatever. So we say this is exact if it is exact on the fibres. Similarly F to G is injective or surjective if it is on the fibres. So this is the definition of a map of sheaves being injective or surjective. In case you're worried about whether or not this is the right definition we've got a category of sheaves of Abellion groups and in the category we say something a map from A to B is onto if for any two maps from B to C these are the same if the composed maps from A to C are the same. So this gives a definition for map A to B and any object in the category to be an epimorphism and this definition is in fact equivalent to one defined using category theory and similarly for injective ones. So this just says that maps of sheaves are injective or surjective if the corresponding maps of etal spaces are injective or surjective and you may wonder why don't we just define sheaves to be etal spaces. In fact you can do this if you want so I think it was sometimes done in the early days of sheave theory. The problem is it doesn't really work when you do more general cases of sheaves over a category instead of sheaves over a topological space. So the definition we gave of sheaves as being a map from open sets to Abellion groups is the one that generalizes better. So let's finish let's let's do a couple of examples of exact sequences of sheaves so first of all we have a skyscraper sheave so a skyscraper sheave is defined as follows we pick a point P in X we pick an Abellion group A and we put F of U equals A if P is in U and F of P is not if P is not in U and let's try and draw the etal space of F so here's the base space X and the fiber at any point other than P is going to be zero so we just get the etal space has a zero point here but above the point P you can see the fiber is A and you can now see why it looks like a skyscraper because it looks like a sort of tall building just sticking up with everything empty around it and now we'll see an example of the skyscraper sheave if we take so this is the skyscraper sheave of A so let's put F so this is the etal space of F so this is the skyscraper sheave of F and now we're going to let G be the sheave of smooth functions on the reels and we're going to define a map from G to G which is just multiplication by X where X is pointing the reels and at all points X not equal zero the corresponding map on fibers is an isomorphism so this map is an isomorphism except at zero and at zero turns out we get the skyscraper sheave here so if we look at the fibers over each point if we look at a point X not equal zero here the fiber looks like naught goes to it's going to be some big space of all germs of functions at naught like this where this is an isomorphism because multiplication by X is an isomorphism if X is equal to zero then we don't get this we get if we've got a function at zero then we can write it uniquely as a constant the constant value of the function at zero plus X times another smooth function so X equals zero that the fiber looks like all germs of functions at zero but this time the fiber is actually the real numbers so we get the skyscraper sheave corresponding to the point X equals zero and A being the real numbers so that's a non-trivial example of an exact sequence of sheaves incidentally if you take any vector bundle over a space then you can obviously form the sheave of all sections of a vector bundle so for instance this sheave G is the sheave of sections of a smooth one-dimensional vector bundle now you see if you work with vector bundles you have a bit of a problem you can get quotients of vector bundles in a nice way here we've got two vector bundles but the quotient of the two vector bundles is really a sheave that isn't a vector bundle there's no way to get a vector bundle whose sections are not everywhere except zero where they're the reals so sheaves can be thought of as a sort of generalization of vector bundles that allow you to take quotients so we'll finish with an example that you probably came across in complex analysis so for this example I'm going to look at the following exact sequence nought goes to 2 pi i z goes to holomorphic functions goes to non-zero holomorphic functions so here these are all going to be functions over complex these are all going to be functions over open sets of the complex number so this stands for the constant sheave of the group 2 pi i times z and this is the sheave of holomorphic functions and this is going to be when I say non-zero holomorphic functions I mean functions such that f of x is not equal to zero for all x so I don't just mean functions that aren't zero identically everywhere I mean the function is not allowed to be zero everywhere and this map here is just going to be the exponential map and now we know that x of 2 pi i times any integer is zero so the map from here to here and from here to here is indeed well defined this is an exact sequence of sheaves and the reason is that it's exact on fibers because if you take a point in some fiber here it's represented locally by a little by holomorphic function near that point and if you've got a holomorphic function that's non-zero then you can't always take its logarithm but you can take its logarithm in the neighborhood of a point so the map from here to here is indeed on to on every fiber however it's not exact on sections of open sets for example suppose I take u to be the complex numbers minus the origin and I take the function in here to be the function x now if it's the image of something here then we would have to have a function f such that x of f is equal to x at all points so we would need a global form of logarithm of f but as you know from complex analysis we cannot define the logarithm of x as a holomorphic function for all non-zero complex numbers because when we go around the origin we get this extra factor of 2 pi i turning up so this is an exact sequence of sheaves but it's not exact on some if you take global the global sections of this over open sets are not necessarily exact they might be exact for example if you take u to be the whole of the complex numbers then the global sections are exact the problem is whether or not this u has none trivial loops in it okay next lecture we will be continuing discussion of sheaves