 This lecture is part of an online course on algebraic geometry about schemes, and we will be talking about what happens to sheaves if you've got a map between two topological spaces. So suppose f is a continuous map from a topological space to a topological space to another topological space. So so far in the first three lectures we were discussing sheaves on just one topological space. Now we want to talk about moving sheaves from one space to another. So we've got these two categories. We've got sheaves on x and we've got sheaves on y. So for the moment I think we'll mostly talk about sheaves of a billion groups, although there are modifications that work for sheaves of sets. And we're going to define two functors between these categories. One will take a sheaf on x to a sheaf on y and would be called f subscript star. And the other will take sheaves on y to sheaves on x and will be called f to the minus one. And if you've come across category three you may be wondering why this is called f to the minus one and not f superscript star, which would be a more obvious notation. And the answer is that f superscript star is used for a different map later on. Roughly speaking the difference is that f to the minus one is used for sheaves of a billion groups. However the important thing turns out to be sheaves of modules over a certain sheaf of rings. And for that one you would use the map, a different map f of a star is used. So that's why we use this slightly non-symmetric notation. So we remember there are two different ways of viewing sheaves. So a sheaf can either be a map f from open sets u to an abelian group f of u. Or we can think of it as being sort of an etal space over x. So we have a map from a to x and f of u for u contained in x is now sections of a to x. And these two ways of thinking about a sheaf are rather useful for these two different ways of moving sheaves from one space to another. In particular if we think of f as being a map from open spaces to abelian groups then it's easier to define this map. Whereas if we think of f as corresponding to an etal space it's easier to define this map and it's kind of rather harder to define these two maps using the other interpretations. So let's first define f star, f lower star. So here f is a sheaf on x, f is a map from x to y and f star f will be a sheaf on y. Now to define it we need to say what f star of f is on an open set u. So we just say f star of f of u, so u is a subset of y, is equal to f of f minus one u. So this looks a bit confusing when you first come across it. So let's try and draw a picture. So we've got a set u here contained in y and we can form the inverse image f to the minus one u here. So this is now an open set of x and all we're doing is saying that the value of this sheaf on u is the value of this sheaf here on x which is a reasonably natural thing to do if someone is confusing. So that sheaf on u is that sheaf on x and that's all this formula says. So let's have an example. Suppose we take y to be a point. So we've got a map from x to a point and suppose we've got a sheaf on x. What is f star of f? Well a point only has one non-empty open set. So a sheaf on a point is really just a group. We just take the value of the sheaf at that point. So and its value at the point is just the value of f on the whole of x. So f star f of the point is just f of x. So this is global sections. By the way we see from this that f star does not preserve surjections. In other words if f1 to f2 is a map of sheaves that are onto, f star of f1 mapping to f star of f2 need not be onto. We saw several examples of this in the previous lecture where a map of sheaves could be onto but the set of sections over some open set wasn't necessarily onto. And it's useful to know when this map f star preserves exactness. So suppose we've got a map of sheaves naught to a goes to b goes to c goes to naught on x. And we have this map f x to y. So we have naught goes to f star a goes to f star b goes to f star c goes to naught. And as we said this need not be exact here. So we should really cross off this zero. So it's not exact. However the rest of it is exact even if we omit this bit here. So this is exact and recall this exactness for sheaves can be defined as just saying it's exact from all the fibres over these points. So we can show that this bit is exact in two ways. We're first going to do it just straight from the definitions and later we'll have a slightly neater way using a jointness. So here we're looking at f x to y and we need to know what the fibres look like. So suppose y looks like this and here is a point p of y. And x might look something like this mapping onto y and it's got a fibre f minus one of p. And now we want to know what is the fibre of f star a. Let's put it f star b at p. Well its elements are represented by taking a small open set u around p. And so that these elements are given by elements of f star of u where p is contained in u. And remember there's this equivalence relation on elements that they're considered to be the same if they're the same on a smaller open set of u and so on. So we should do something about taking direct limits but let's not worry about that too much. We can specify an element of the fibre just by giving an open set u and an element of this. So what is this? Well this is just an element of f of, so big f of, sorry that should be f star f of u. So these are given by elements of f of u, so f of f minus one u. There are too many f's here. So what that means is we take this open set u and we take its inverse image here and this is going to be f minus one u and we want to take the values of the sheaf big f on this. So if we've got an exact sequence nought goes to a goes to b goes to c goes to nought of sheaves on x then and we take an element of f star f, f star of b and we can look at its fibre over p and the fibre is just going to be some, an element of the fibre can be represented by section of the sheaf b over this set here. Now if it is image zero here then that means all the fibres must have image equal to zero and the fibres of c which must mean that in the image of the fibres of a which means that this section here is actually the image of a section in a which means that if something in f star b has image zero in f star c then it's actually the image of something in f star a. So if this sequence of sheaves on x is exact then nought goes to f star a goes to f star b goes to f star c is also exact. So next we're going to look at maps that go in the other direction. So this time we have a map from x to y and we've got a sheaf g on y and we're going to try and find the sheaf f minus one of g which will be a sheaf on x. And how do we do this? Well we remember the sheaf g on y corresponds to an etal space over y. Never get the L's and the T's right around an etal so this is etal over y in other words it's a local homeomorphism. And now we can pull it back to get an etal space over x just by taking b times over y of x. So this is the pullback and it consists of the subset of b times x of points bx which have the same image in y. So another way of thinking about this is suppose we take a point p of x and then we can look at the point fp as a point in y and all we're saying is the fibre over p is the same of this space here is the same as the fibre of fp. So it's the same as the fibre of b over fp so we're sort of pulling back the fibres and you can easily check that if b is etal over y then this pullback is etal over x so it gives a sheaf and f to the minus one of g is the sheaf of the etal space from b times over y x goes to x. So from the point of view of etal spaces this operation f to the minus one is a very natural operation it's rather more complicated to define if you think of a sheaf as being a map from open-sets to abelian groups. Cartchon gives this alternative definition. Well you remember the map f lower star was not always exact, f to the minus one preserves exactness and this is almost obvious because we're defining f to the minus one by just pulling back fibres of etal spaces and exactness of sheaves can be tested by looking at whether it's exact on all the stalks of the sheaf. So if you're just pulling back the stalk then a set of stalks that are exact will still be exact if you pull them back here. So we should give an example. Suppose that x is actually a subset of y so f is just the identity map on x is a subset of y and if g is a sheaf on y we can ask what is f minus one of the sheaf g. So you can think of this as being restriction of g to x and the fibres of this are very easy. The fibre of f minus one g at p in x is the same as the fibre of g at p in y so it's just almost the most trivial possible thing you could do to get a sheaf on x. Next there is a relation between f star and f minus one or precisely f minus one is left adjoint to f lower star. What this means is as follows this is always a bit confusing that we hope I try and get it the right way round. So suppose you've got the map from x to y and f is a sheaf on x and g is a sheaf on y then you can look at f minus one g which will be a sheaf on x and you can also look at f star of f which will be a sheaf on y and the relation between these is as follows on x we have a map from f minus one g you can consider maps from f the minus one g to x and over y you can consider maps from g to f star of f and informally we say that these two are adjoint if these two sorts of maps are really the same in other words a map from g to f star f lower star of f is sort of the same as a map from f minus one g to f that there's a natural there's a natural bijection between these sets in order to define natural bijection you can go and look a book of category theory because whenever I give this definition I always get completely muddled. One way of seeing this is that is that both maps maps from g to f star of f and maps from f minus one g of f are the same as collections of maps from g of v to f of u whenever we have open sets u contained in x v contained in y with f of u contained in v and furthermore these maps have to be compatible with the restriction homomorphisms in other words if you've got a map v prime contained in v and u prime contained in u then you get all the usual maps you get g v prime most f u prime I should have said f of u prime should be contained in v prime and then we have restriction maps going one way or the other I think they go this way so there's a restriction map there and there's a restriction map there so for each u in x and v in y such that f of u is in v you are given a map from this space to this space which are commute with all these restriction maps if you enjoy this sort of abstract definition you should really go and study the theory of stacks which is even more complicated and abstract than this anyway left adjoints and right adjoints of the following nice property that if a functor a is left adjoint to b then a preserves right exactness and b preserves left exactness you notice that this is yes another case when terminology has been messed up because left adjoints preserve right exactness and right adjoints preserved left exactness so what this means when preserving right exactness means that if a goes to b goes to c goes to nought is exact then a of a goes to a of b goes to a of c goes to nought is exact and left exactness is the same except we put the zero on the left so to apply this we know that f to the minus one is left adjoint to f lower star so f lower star is right exact so so sorry f lower star is a right adjoint so it is left exact which is what we showed earlier and we see that this is automatically right exact um so the right exactness of f to the minus one on the left exactness of f star follow for sort of trivial category theoretic reasons that there are joints to each other in fact we get a bit more because f to the minus one is not just right exact it is in fact exact so um we get a surprise extra left exactness of this which we couldn't guess from this this all this categorical nonsense um the failure of f star to be right exact is of major importance um there's the entire subject of sheaf co-homology is entirely for the purpose of trying to fix the fact that this functor here is not right exact um for example if we have a sequence of sheaves over some space and we apply f star to it then um homology theory tells us what group we have to put here in order to extend this exact sequence further and um then higher co-homology groups give more and more complicated um sets there so that's the topic of chapter three in heart shorn which we won't be getting to in this particular series of lectures okay that's quite enough abstract nonsense about sheaves so next lecture we're going to define um we're going to start the definition of a scheme