 So this lecture is part of an online algebraic geometry course about schemes, and we'll be about how you move quasi-coherent schemes from one quasi-coherent sheaves from one scheme to another. So suppose you've got a map between schemes X and Y, and we've got a sheaf F on Y. We might want to get a sheaf F on X, which we will denote by F star of F. And if we've got a sheaf G on X, then we will get a sheaf lower star G on Y. And furthermore, these maps will be sort of adjoints in the sense that the set of morphisms from this sheaf to this sheaf and the set of morphisms from this sheaf to this sheaf are more or less the same. So we'll say that F opposite star is going to be left adjoint to F lower star. Well, earlier on we defined some maps F to the minus one and F star for sheaves of Abelian groups. And the F lower star that we're about to define for quasi-coherent sheaves will be more or less the same as this one, but the F opposite star has to be modified from this one for reasons we will see in a moment. So we first look at the affine case. So here we've got a morphism A to B of rings. And we can ask, suppose we've got a module M over A, how do we get a module M over B? So a module over B. Well, that's obvious because we can just make M into a B module by just using the map from B to A. So turning an A module into a B module is kind of easy. So M is also a B module. On the other hand, if we've got a B module N, there's no obvious way in which it's an A module. So B module N is not an A module. But there's a way to turn it into an A module. We can take N tensor over B of A. And this now becomes an A module. And this is a sort of left adjoint. If we look at morphisms from N tensor A to M as A modules and morphisms like this. So these are going to be B module homomorphisms. And these are going to be A module homomorphisms. Then the B module homomorphisms here are more or less the same as the A module homomorphisms here. There's a sort of canonical isomorphism between them. So the map taking N to N tensor over B A is left adjoint to the map taking M to M, where this one is an A module and this one is as a B module. Well, now let's examine whether or not these maps preserve exactness. Because obviously if we're transferring sheets from one ring or one scheme to another, then knowing whether exactness is preserved is going to be really useful. Well, the map taking M to M. So here we're taking M as an A module to M as a B module is obviously exact. More or less because we haven't really done anything to M except pretend we're working over a different ring. On the other hand, the map taking N to N tensor over B A, well, the exactness is a bit more subtle. Map taking N to N is obviously exact, but tensuring with something doesn't always preserve exactness. You remember, tensuring with something is only right exact. So you remember if you've got an exact sequence say nought goes to Z goes to Z goes to Z over 2Z goes to zero. Let's just tensor with Z over 2Z. Well, we get Z over 2Z goes to Z over 2Z goes to Z over 2Z goes to nought and this bit is exact. But this map here is not injective. So it's a really important point that tensuring with something is only right exact, it's not exact. So there's one case in which it is exact. It's exact if A is a flat B module. And that's true because this is more or less the definition of what it means for A to be a flat B module. It just means that tensuring with A preserves exactness. So this is one of the reasons why flatness occurs so much in algebraic geometry. It means that this map here that we're going to define extended schemes is exact. The concept of flatness is very unintuitive. It was sort of introduced by Seher and growth and Dick discovered it had central importance in algebraic geometry. However, it's very hard to explain what it means geometrically and there's probably no really easy explanation because people have been doing algebraic geometry and commutative algebra for about a century before Seher discovered flatness was important. So there probably isn't any obvious reason for it. So let's translate all this into the language of affine schemes. So what we do is we have a morphism from the spectrum of A to the spectrum of B. I hope I've got all these arrows the right way around. So let's call this X. Let's call this scheme Y. And if we've got a sheaf over F, we can pull it back to a sheaf F to the minus one of F over A. So this isn't necessarily a sheaf over A for the same reason that if we've got a module over the ring B, there's no reason why it should be a module over the ring A. And what we remember what we had to do with rings was tensor it with something. So we need to tensor this with something as well. So we tensor it with over F minus one O Y of O X. And this is really just what we did with modules. We just took n tensor over B of A. And if you translate everything into scheme theoretic language, it looks like this rather slightly complicated mess. On the other hand, if we've got a module over A, in other words, a sheaf over spectrum of A, then it's very easy to convert it into a module over B. And this is just the map G goes to F star G where F lower star F to the minus one of the other maps we defined earlier for sheaves of abelian groups. So and as we saw the maps from this sheaf to this sheaf are essentially the same as the morphisms from this sheaf to this sheaf. So these are the same. And we put F upper star F equals F to the minus one F tensor over F minus one O of Y O of X. And we put F lower star G equals the F lower star G we defined several lectures before. And we see that F star and F lower star are adjoints. More precisely F upper star is left adjoint to F lower star. So these are going between categories of quasi coherent sheaves on these two schemes. So all of this is just exactly the same as what we wrote down here except we're writing it in geometric language using sheaves and schemes instead of talking about modules and rings. So although this this only looks a bit of a mess but it's just saying you tensor a module with another module. Now we need to study. Now we just recall exactness properties of these. So for affine schemes. We see that F lower star is exact. And F upper star is right exact. That's because we showed these for modules over a ring and affine schemes are more or less the same as rings. And we also saw that this is exact. If something is flat, some flatness condition holds and I want to really emphasize the word affine here. Because when we look at none affine schemes we will see in a moment that F lower star is no longer exact. So now look at general schemes, not necessarily affine. So we've got a map F from X to Y. And we've got a sheaf F on Y and we've got a map F star of F on X. And we've got a map taking G to F star of G. And you can check for general schemes of jointness still holds. F upper star is defined by exactly the same formula as before. And what F upper star looks like is F upper star F looks locally like N tensor over B of A. We're going to take an open affine subset whose coordinate ring is A and an open affine subset whose coordinate ring is B and so on. And this map here is really just locally taking a tensor product. But we know that F upper star is left adjoint to F lower star. And left adjoints are always right exact. And we saw earlier that F upper star became exact if some sort of flatness condition holds. And the same thing holds for general schemes. F star is exact if some flatness condition holds. It's not difficult to write out this flatness condition explicitly, but it's a bit of a mouthful. We have to sort of cover Y with open affine subsets and then cover X with open affine subsets mapping to them. And then for each coordinate ring here, the corresponding ring here has to be flat over it and so on. But it's easier just to remember this as some vague flatness condition. And similarly F star is always left exact because right adjoints are always left exact. F lower star is no longer exact in general. And we've seen an example of this before when we looked at invertible modules over P1. Let's just take F to be the map from P1 of K, one-dimensional projective line to a point, the spectrum of K. And then we notice that F star for this map is just the same as taking global sections of a sheaf. And now we look at this map O minus one plus O minus one that we saw last lecture and it maps on to O zero. And we noted that the global sections of this or F star of this was just zero. Whereas for this, the global sections were just K. So this map here is not on to. So F lower star is left exact for, sorry, it's exact for morphisms of affine schemes, but it's no longer exact for general morphisms of schemes. What happens in general is over a scheme, if we've got an exact sequence, nought goes to A, goes to B, goes to C, goes to nought, we will later see that in fact we get a long exact sequence, nought goes to F star of A, goes to F star of B, goes to F star of C. And then instead of getting nought, we get something called the right derived functor of F applied to A and goes to the right derived functor of F applied to B and that goes to the right derived functor of F applied to C and that goes to the second right derived functor of F applied to A and so on. Anyway, that's all coming later. There's one more topic I would just want to finish this lecture with. There's one other way, well, actually there are several other ways to get between sheaves on schemes, but one other way in particular that sometimes turns up is if you've got an open immersion from U into X. So informally U is an open subset of X or at least isomorphic to one. And now we've got a map F lower star taking sheaves on U to sheaves on X, but there's another one who sometimes encounter F with a sort of exclamation mark. This is called extension by zero. So I'll just say a little bit about this. The way it works is as follows, if we've got a sheaf on U, we can think of it as being an etal map from an etal space to U. Remember etal means it's a local homomorphism. Well, as U is an open immersion in X, the map from U to X is also etal. So we can just compose these and get a map from E to X. And this is the etal space representing F exclamation mark of whatever sheaf we started with. So if we're taking a sheaf A on U, then F lower star of F will be here. So this is the etal space of F. And if we just compose it, this will be the etal space of F shrink of F. So what is happening is easy to see at the level of stalks. So the stalk of this at each point of X is zero if the point isn't in the image of U. And it's just the same as the stalk of F if it isn't the image of U. So that's what's meant by extension of zero. All the stalks outside U are just zero. Well, next we should note that F star of a quasi-coherent sheaf is not quasi-coherent in general. Well, I mean, sometimes quasi-coherent, but usually not. And let's see an example of this. So let's just take F to be the map from the spectrum of K to the spectrum of R where R is a discrete valuation ring. So you remember the spectrum of R looks something like an open point and a closed point. And the spectrum of K just looks like the open point being included into it. And let's take the sheaf corresponding to the module K. So you remember we indicate the corresponding sheaf by just put a twiddle on it. And let's work out what F star of K on this is. Well, F lower star of K is just the sheaf K. So that's a perfectly good quasi-coherent sheaf on R. On the other hand, F lower sheet of K twiddle is kind of funny sheaf because the stalk at the closed point is just zero. So it has no global sections other than zero. I mean, it's a tall space kind of looks like this. Here we've got the spectrum of R. And then above it, you've got the zero section and you've got all these other sections which just can't be, which just don't, which can't be extended to this point. So the only global section is the zero section. So it's not quasi-coherent because any quasi-coherent sheaf is determined by its module of global sections. So it is a module over the ring, over the sheaf of rings of the spectrum of R, but it's not a quasi-coherent module. So this is one natural way in which modules over sheaves of rings arise which aren't quasi-coherent. The exact relation between F lower star and F lower shriek is F lower shriek is left adjoint to F star, which in this case is F minus one, and F lower star is right adjoint to it. For most morphisms, F opposite star doesn't actually have a left adjoint. However, the special case when the morphism is an open immersion, it does happen to have a left adjoint and it's given by this funny sort of extension by zero. Anyway, because of this, the extension by zero isn't often used for quasi-coherent sheaves, although it's used a bit more for non-quasi-coherent ones. In the next lecture, we're going to discuss coherent sheaves and try and explain why we ended up with such a horrible name quasi-coherent for quasi-coherent sheaves.