 This lecture is part of an algebraic geometry course on schemes and will be about morphisms of finite type. The main thing you have to remember to start off with is that these are not the same as finite morphisms. We'll be talking about finite morphisms next lecture and this is a much stronger condition than saying morphisms are of finite type. So it's an analogue of the condition of the condition that if you've got a map of rings, you can say that S is a finitely generated algebra over the ring R. And so you can look at the corresponding map of schemes from the spectrum of S to the spectrum of R and we will see that if S is a finitely generated algebra over R then the spectrum of S, then the map from the spectrum of S to the spectrum of R is a morphism of finite type. Morphisms of finite type are more general than this because you can patch together various schemes. Notice that this is a relative notion. It depends on a morphism, not on a scheme. The reason for this is that if you ask for ring R to be finitely generated as a ring, these are just too restricted. I mean, you can get some examples like an algebraic number field or a finite field but most fields, if you take an infinite field K, this will not be finitely generated over the integers so it's not finitely generated as a ring. So being finitely generated is more useful as a relative term. You want to know whether some ring is finitely generated over another ring, otherwise it's just too restricted. The finitely generated rings are still very important. I mean, algebraic number theory concerns itself mostly with rings like this. So we'll look at varieties over finite fields or rings of integers of algebraic number fields. So you can think of the study of finitely generated rings as basically being algebraic number theory. But if you're doing algebraic geometry, you generally want to work over infinite fields so you can't restrict yourself to finitely generated rings. Anyway, we're going to define three concepts. Amorphisms can be a finite type and a morphism of finite type is defined to be one that is quasi-compact and locally of finite type. So you remember this is rather like what we had for notarian, a scheme was notarian if it was quasi-compact and locally notarian. So we're breaking this up into two conditions here. Well, I've defined quasi-compact schemes but I haven't yet defined quasi-compact morphisms. So what does it mean for morphism to be quasi-compact? We say X to Y is quasi-compact if the inverse image of an open quasi-compact subset of Y is quasi-compact in X. So this is a condition for continuous maps of topological spaces. Very informally, it says the fibres of the map from X to Y are not too big. In fact, there are a whole lot of conditions in algebraic geometry that say in one sense we're another the fibres of a morphism are not too big. So for example, if Y is a point, then the condition of morphism from X to Y is quasi-compact is just the same as the condition that says the space X is quasi-compact in the sense of the previous lecture. So locally of finite type on the other hand means suppose we've got space X and we think of it as mapping to a space Y. So it's got various fibres like this. What it says is that given a point X in here we can find some neighborhood of X and some neighborhood of its image. So this is the point F of X where let's call this morphism F. So for all X in X, we can find an open affine U with X in U and an open affine V with FX in V. And we also want F of U is contained in V. So we can actually map this open set to this open set. And finally, the key condition is that O of U is finitely generated algebra over O of V. So this is the sort of core condition. So it's like the condition that says some ring is a finitely generated algebra over another ring. Only we're sort of doing it for lots and lots of different open sets covering Y. So let's see some pictures of what this looks like. So let's look at a sort of typical example. So let's take Y to be a union of a point on the line and a plane and a copy of A3, A4 and so on. So Y can be pretty huge. It's infinite dimensional. It's got infinitely many pieces. And that doesn't really matter. And let's take X to be the following. It could be a point and a line and a line. And let's take a plane and let's take a plane here and let's take a plane and A3 and let's take A3, A3 and so on. So we're taking X to be a union of lots of points and lines and planes and so on. And we're going to project X to Y like this, sorry. And now this is, this morphism from X to Y is locally of finite type. And the reason is if you pick any point in X, say you pick a point in A3, it's got a neighborhood, say this entire A3 and we can take its image which is say this open subset of Y and the map from there to there is just a map from a three space to a line. And the corresponding map here is certainly map of finitely generation algebras. For instance, this will be a coordinate ring and this will be a polynomial ring in one variable and this will be a polynomial in three variables. So it's certainly finitely generated. So it's locally finite type even though all the fibers are kind of huge and Y is itself quite huge but kind of locally all the fibers, if you pick one fiber then locally it doesn't look too big and that's sort of what locally of finite type means. If you want to see an example of morphism of finite type you could take something like this. So let's take Z to be these things. So we're only taking a finite number of the components of X over each point of Y and then we see Z to Y is a finite type. And the point is it's not just locally a finite type but it's also quasi-compact. If you take any open set in Y, say this one here it's inverse images, the union of a finite number of affine varieties. So it's also quasi-compact. So finite type sort of means all the fibers are not too big in some sense, whereas locally if finite type the fibers can be really huge but are locally reasonably small. Another example, let's just take Y to be the spectrum of a field. So Y is a point, which is an important special case and then we can ask what does finite type and locally of finite type mean? Well, we can have examples that are locally a finite type but not a finite type. Well, one stupid thing we can do is just take X to be a union of a lot of components but we could also take X to be something like the line with an infinite number of origins. So you remember we had a line with two origins where we stuck together two copies of the line where we can perfectly well stick together lots and lots of copies of the line and get a line with infinitely many origins and then X to Y is not locally a finite type. Sorry, it's not a finite type but it is locally of finite type. So each point of X you can find a neighborhood that's just an affine line which maps to Y and that's a finite type but this isn't quasi-compact because the inverse image of a point is this non-quasi-compact space here. Another example, let's take X to be a polynomial ring in infinitely many variables then the map from the spectrum of X to the spectrum of, sorry, X is the spectrum of this space. So the map from the spectrum of X to the spectrum of K is not locally a finite type. See, this is not a finitely generated algebra over K so this is a sort of infinite dimensional affine space and that's not allowed for things that are locally a finite type. So if Y is a point and let's take X to be say some sort of variety over K or strictly speaking I should mean the scheme associated to a variety but that's good enough. Then this is covered by a finite number of things of the form spectrum of RRI where RRI is a finitely generated K algebra. So X is of finite type over the spectrum of K and this is not far from characterizing varieties over K. So in general abstract varieties over K so you remember an abstract variety is something you can get by kind of gluing together affine varieties. So these are the more S correspond to schemes over the spectrum of K. So we have this map from X to the spectrum of K such that first of all the map from X to the spectrum of K is a finite type or maybe locally a finite type. The difference is whether you allow abstract varieties to be obtained by just gluing together a finite number of affine varieties or an infinite number and furthermore we should have X should be reduced. Remember this means no nil potents allowed and we also want X to be irreducible. In other words, it's not the union of various irreducible components. And you remember these two conditions here just say X is integral. So abstract varieties over K are more or less the same as schemes of finite type satisfying these conditions about being reduced and possibly irreducible. You can think of informally that morphisms of finite type are something to do. They're sort of vaguely related to the fibers being finite dimensional. But this correspondence isn't too close. For example, suppose you take the spectrum of K of X to the spectrum of K. So this is a perfectly good morphism of finite type. And now suppose we pick a point here, say we pick a point corresponding to the ideal zero, then you can localize at this point. So we take the spectrum of K X, sorry, that's the ideal X, and we localize it at X and look at this. So this is just the local ring at the point naught in the affine line, but the local ring is not finitely generated. So K X localized at X is not finitely generated as a K algebra. So this map is not a finite type. So maps from local rings to fields are a common example of morphisms that are not a finite type. So this is still finite dimensional. This is a one-dimensional scheme. So the fibers of this map are one-dimensional, but it's not a finite type. So finite type and dimension aren't too closely related. We also have Hilbert's theorem, which says that if S is a finitely generated R algebra, then R being notarian implies S is notarian. And you can translate this into schemes. It says that if you've got a morphism X to Y of finite type, then Y is notarian implies X is notarian. And similarly, Y being locally notarian implies X is locally notarian. I hope I've got these all the right way around. And the proof of this is essentially just Hilbert's theorem with a certain amount of completely routine and easy bookkeeping. You just cover X and Y by affine subsets and then apply Hilbert's theorem to each of these affine subsets. And this result turns out to be a completely routine consequence of Hilbert's theorem. So the hard part of this is the commutative algebra theorem proved by Hilbert. And the extension from that from rings to schemes is just a completely routine piece of bookkeeping. Finally, we should look at properties of morphisms. So we've got several properties like being finite type, et cetera, or locally finite type or being quasi-compact. And we can ask several questions about these properties. Is it preserved under composition? In other words, if you've got X to Y to Z and if F and G have this property, then does the composition G, F have this property? And most properties of morphisms we define turn out to be preserved under composition. And usually it's not too difficult to prove. For instance, quasi-compactness is preserved under composition. If F and G are quasi-compact, it says that an open quasi-compact subset of Z has inverse image that's open and quasi-compact. And the inverse image of that is then open and quasi-compact. So the composition of two quasi-compact morphisms is quasi-compact. Secondly, we can ask, is it local on Y? So what this is saying is that if we cover Y by various open affine subsets and check it on each of these open subsets, is it does the morphism have that property? And this usually reduces to saying that suppose Y is affine covered by affine sets YI. And then we can look at the morphism restricted to each of these YIs. And if all those morphisms have some property, does Y have that property? And the answer is usually. Properties of morphisms we define tend to be local on Y. For instance, morphisms of finite type and morphisms that are locally of finite type are all local on Y. You can just check them for a cover of Y by affine subsets. And it's then true for all affine subsets of Y. Thirdly, we can ask, is it local on fibers? And the answer here is sometimes. So here we're sort of asking, we've got a morphism X to Y. And instead of asking whether the morphism is local on Y, we can ask, is it local on X? So if we look locally at each point, if you look at an open neighborhood of each point of X and the map from this open neighborhood to an open neighborhood of Y has some property, does the whole morphism from X to Y have that property? And here finite type, no, it's not local on X. So for example, if X has an infinite number of components, then locally it might be a finite type, but globally it wouldn't be a finite type. However, morphisms that are locally of finite type are indeed local from the point of view of fibers of X. And generally you say, you put the term locally into a property of morphisms if it's local on fibers, on fibers. So it's easy to get confused about what locally means. It can either mean locally on Y or locally on fibers. And when you say a morphism is locally with some property, you usually mean it's locally on fibers because it's usually automatically true that it's local on Y. Okay, next lecture we're going to talk about finite morphisms which shouldn't be confused with morphisms of finite type.