 This lecture is part of an online algebraic geometry course about schemes and will be mostly about a couple of ideas due to growth and dick. The first is that you should look at morphisms of schemes rather than schemes and the second is the idea of the functor of points associated to a scheme. So the first idea is that instead of thinking about schemes. You should think more about morphisms of schemes and take the morphisms as being the basic objects. For example, instead of looking at the scheme, a k to the n, say affine n space over the field k, so this is just the spectrum of k x1 to xn. What we should think of is this scheme together with a morphism to the spectrum of k. So this corresponds to thinking of this ring here as being an algebra over k. And instead of asking for morphisms of a scheme to akn, so if we've got a scheme s mapping to akn, we shouldn't think about so much about this. What you should do is take a scheme over say spectrum of k and look at morphisms to s that commute with the map over the base scheme k. So in terms of rings, this would correspond to looking at algebras over k and only looking at morphisms that preserve the algebra structure over k. For example, if we have varieties a and b over k, these correspond to schemes say x and y, because you can convert any variety into a scheme. And morphisms of varieties a to b don't correspond to morphisms of schemes x to y. What they do is they correspond to morphisms of schemes over the base scheme spectrum of k. So if you don't have this condition, then you get all sorts of other weird morphisms coming from things like automorphisms to the field k and so on. And if you're just doing this over a field, this doesn't really make that much of a big difference. But what you can do is you can replace this by any base scheme s whatsoever. So instead of looking at the category of schemes, we can look at schemes over s. What this means is we're looking at schemes together with a fixed morphism to the base scheme s. And a morphism of schemes over s is just going to be a morphism from a to b that makes this diagram commute. This is actually a very general construction you can do in any category whatsoever. It doesn't have to be the category of schemes. You just take an object of your category and you look at you form a new category consisting of objects together with morphisms to this fixed object. For example, suppose we take an elliptic curve y squared equals x cubed plus bx plus c. So this is just an elliptic curve and for fixed b and c is going to mean elliptic curve in the c squared. However, you can think of it as being a subset of c to the four consisting of all things y squared equals x cubed plus bx plus c, where now you think of b and c. So we're taking all solutions x, y, b and c of this and c to the four. This is no longer an elliptic curve. It's a three dimensional variety in c to the four, but you can map this to c squared by just mapping this to the point bc. And you can think of this as being a base scheme where I think of this as being the affine plane considered as a scheme. And you can see that what we have here is a map of schemes. If we fix b and c, then the inverse image is going to be an elliptic curve. Well, most of the time, the special values of b and c, it might not be. So you can think of this as being a family of curves parameterized by the base scheme c squared. In general, if you've got a scheme over a scheme s, you can think of this informally as being a collection of schemes parameterized by s because you can take the fibres of points over s. At least we will be able to take the fibres of points over s when we discuss pullbacks a little bit later. You don't actually lose anything by considering schemes over s because any scheme has a unique map to the spectrum of Z. So talking about schemes is exactly the same as talking about schemes over over spectrum of Z. So in the terminology of category theory, spec Z is a terminal object. There's unique morphism from any object to it. So so objects over this are the same as objects, roughly speaking. The other thing is part of growth next philosophy is you shouldn't really define properties of schemes. What you should do is you should really define properties of morphisms of schemes. So an example of this is the concept of a complete variety. So a complete variety is a slight generalization of a projective variety. Informally, you can think of the variety has all its points at infinity or is a sort of analog of a compact space. And you don't really define complete schemes. What you do find is proper maps of schemes. And if the base scheme happens to be a field, then a proper map from a to the base scheme is more or less the same as saying that a is complete. As a variety in the classical sense. But it turns out to be much more convenient to be able to work with an arbitrary base scheme rather than just a special case of a field spectrum of a field. So that's one theme of scheme theory that you should focus more on morphisms rather than schemes. So we'll see that this a bit later when we spend a lot of time defining properties of morphisms such as being a finite type and so on rather than properties of schemes. Okay, the next idea is that a scheme can be thought of as a functor from schemes to sets. And this is very easy suppose X is a scheme. Then we can define a fun to HX from schemes to sets by putting HX of T to be morphisms from T to X. So it should of course be. As I said in the first part of these lectures should really be schemes over some base scheme S and you should be morphisms over S but I'm a miss out S most of the time just because it's my next or complication. In fact, it's a contra variant functor. You can see if we've got a map from schemes from T one to T two, and we've got a morphism of T two to X, then we get them that then by composing it we can get a morphism from T one to X. So this is a contra variant functor. Meaning if we've got a morphism from T one to T two, we get a map from HX of T two, going back the other way to HX of T one. On the other hand we can just look at the special case of affine schemes, which are essentially just rings and in this case. So if we sort of put H prime X of a ring to be the same as HX of the spectrum of the ring. Then if we've got rings R one goes to R two, then we get a map from H prime X of R one to H prime X of R two because you remember the map from rings to schemes kind of reverses or arrows. This time, H prime would be a covariant functor from rings to sets. So so so schemes can be thought of as covariant functions from rings to sets and although this idea looks a bit abstract, it's actually very natural. For instance, if we just put X to be say the affine line over over Z spectrum of Z of X. And H prime of X of a ring R is just the morphisms of the rings Z of X to R. So it's just equal to the ring R itself. So you can think of this as being a sort of affine line and HX of R can be thought of as points of the affine line with values in in R. So you see the points of the affine line with values in say the complex numbers are just going to be the complex numbers and so on. Think of this functor as being we think of this functor as in form is being points of T with values in X. And in the special case when X is something like the spectrum of a field. This just gives the usual idea of points of a variety with values in the field but we can now do this much more generally we can take points with values in any scheme whatsoever. So this leads to two obvious questions which motivated a lot of growth in this work. Given a scheme what functor so given a scheme X what functor HX does it represent. And the second one is given a functor from schemes or possibly rings to sets. Is it represented by a scheme X. In other words is it of the form H of X for some scheme. This question is usually not too difficult to figure out this one turns out to be very deep and subtle and motivated and a lot of growth index work trying to find a good answer this question for some common functors. So, for example, let's let's look at X being spectrum of Z. So we can ask what functor does it represent. Well for any any ring or any scheme there's a unique map from any scheme S to the spectrum of Z this sort of follows from the fact there's a unique homomorphism Z to any ring. So this represents that the functor HX of T is just a one point set. It's not really a terribly interesting functor but start off with something easy. So for the next example. What happens if we take X equals the spectrum of Z this joint union the spectrum of Z. So what is HX of T going to be well that looks pretty obvious you're just taking a disjoint union of two functors. Two schemes each of which you get by taking a one point set to a scheme so presumably HX of T should be a two point set. And the answer is no it isn't a bit more subtle than that let's cross that out. Show that it's wrong. It's a two point set. Only if T is connected. So if T is connected, then a map from T to this set test the image of T must either be contained in this set or contained in this set in general HX of T kind of corresponds to the subsets. Of T that are open. And closed. So if T is not connected then there can be other open and closed subsets of the other than the obvious ones. So in particular this suggests that there's a certain obstruction given a function to representing it by a scheme because if we try and take the function that just takes any scheme to a two point set with the identity map this is a perfectly good function but we can't represent it by a scheme. Another obstruction to this is that if we're looking at maps from schemes to a scheme X if a scheme is covered by other schemes, then that then the maps from scheme T to X are given by taking the maps from this scheme to X and the maps from this scheme to X and taking the ones that are identical on here. So there's a sort of sheaf covering condition for that a functor has to satisfy before it's represented by X and examining this sort of sheaf covering condition is a very central theme and growth in this work. I won't say much more about just now because so let's look at another example suppose you take X to be spectrum of say Z X Y and we can ask what what is the functor H of X Well the functor H of X on rings is fairly easy that just means you want all the homomorphism from this ring to that ring. So this is just going to be equal to R squared because you look at the image of X and the image of Y and R and that gives you R squared. Similarly, the function H X on a on a scheme T is just pairs of elements X Y that are regular functions on T. So it's so X and Y regular functions. So we start taking X to be the spectrum of Z X Y modulo some polynomials say Y squared minus X cubed minus X. Well the functor this represents will just be pairs X Y in R squared such that Y squared minus X cubed minus X is equal to zero. You can think of this as being a functor that takes rings R to elliptic curves. So this this H prime X takes rings to elliptic curves over R except they and we haven't actually defined what elliptic curve over arbitrary ring is but it's plausible that this is one. Another example might be you take X to be the spectrum of say Z X over X the n minus one and then H X of R is just the nth roots of one in R and you notice that this is a group for each R in a natural way under multiplication and furthermore all them if we have a homomorphism from R one to R two this induces a map between the corresponding groups. So H prime X becomes a functor from rings to not just sets but groups and this actually gives an example of an algebraic group scheme. So an algebraic group scheme can be thought of as some sort of functor from rings to groups that's represented by a scheme. In fact in general if we've got schemes then morphisms X to Y are same as morphisms of functors from H X to H Y where morphism of functors are sometimes called natural transformations. Right the proof of this is completely trivial and I'm not going to give it because although it's completely trivial it always confuses me and I always get it wrong but if you look at the definition of a natural transformation of functor then it's a completely routine piece of book keeping to show that morphisms of X to Y are the same as natural transformations like this. In other words we can think of the category of schemes as being contained in this category of functors making functors into a category is a bit confusing but you can do it. So the question is suppose you've got a functor from schemes to sets is it in the image of a scheme? So we'll discuss this a bit and this turns out to be a very deep and tricky question so I can't do more than just sketch why you might be interested in it. Suppose you want to classify elliptic curves whatever that means. So you might want to classify them by a scheme so we might ask can we find a scheme so that say the morphisms say the points correspond to elliptic curves over the field K. Well the answer is trivially yes because the points corresponding to elliptic curve over a fixed field K is just some sort of set and we can always find the scheme so that it's points at any given set so we need to ask more than that. Instead of classifying elliptic curves over K what we should really do is look at a functor f of s which consists of nice families of elliptic curves over s and there's a big problem here because we don't know what a nice family is and finding out what a nice family means is itself a rather deep question and results by growth in it with the adjective flat. So if you ever wonder why algebraic geometries get so excited about flat things it's because flatness turns out to be the key technical condition that you need to define a nice family. We also haven't defined elliptic curves over an arbitrary scheme either but I think I'll skip that for a moment and just observe that there is a non-trivial problem in defining this. So the problem is, is this represented by a scheme? So is this functor represented by a scheme and if it was you could think of informally as the points of the scheme being different isomorphism classes of elliptic curves. Well, there's a very subtle obstruction to this and the answer is no, it's not represented by a scheme. And to see what the problem is let's try a much simpler problem. Can we represent two point sets by a scheme? So what I want to do is to have a scheme such that points with values so points of the scheme X with values in S is families of two point sets over S. And so you notice that if S is equal to spectrum of K say well there's only one two point set so the functor should the functor of S should just be a point. So, you know, there's only one two point set up to isomorphism so the only reasonable thing you can choose for this functor is it's the functor that takes any scheme S to a single point. However, if S is a scheme looking like this then you can have a very natural family of two point sets over it looking like this there's a sort of double cover. For instance, you could take this to be C star and this to be C star and this the map taking X to X squared. So there's a non trivial family of two point sets such that each of the fibres is the unique two point set if you see what I mean. So this is sort of an example of a my so trivial family that anyway. The point is this means you can't represent two point sets in any reasonable way because there are non trivial families of two point sets over some schemes that still look trivial everywhere locally. The problem is that a two point set has automorphisms and you see the problem is if you sort of go around this circle and try and follow it up here. You're doing a non trivial automorphism on your two point set and what Grossendick discovered is that the existence of automorphisms is always an obstruction to constructing nice families and moreover it is in some sense almost the only obstruction and the trouble is elliptic curves also have automorphisms. For instance, an elliptic curve if you think of it as being a complex space modular lattice it's got the automorphism that just takes X to minus X. So there's a complication in finding a scheme representing any sort of object with automorphisms. And this is a bit of a problem because classifying elliptic curves by finding a scheme representing them is a really fundamental idea in algebraic geometry and I just mentioned there are several good ways of getting around this. The first method is you can just kill off the automorphisms. You can add extra structure to the elliptic curves so that it no longer has automorphisms. For example, you could add the points of order three in this elliptic curve as part of the structure. So you see this group here has nine points of order three and you could color them different colors and that would give you an elliptic curve that didn't have any automorphisms because that would change the color. So that's sort of called rigidifying it to curve or adding some level n structure to it. And the other much scarier way of dealing with this problem is to introduce the notion of stacks which are invented by growth and Dick specifically to deal with this problem. So one motivation for stacks is it was an attempt to define things that could represent objects with automorphisms. But as I mentioned before, the definition of a stack is a bit hair-raising so I'm not going to go in for it. Okay. That more or less concludes the section on schemes and next we're going to define far too many properties of schemes and morphisms. Thank you.