 So this lecture is part of an online algebraic geometry course on schemes and we'll be about the relation between proper morphisms and valuations. So what we will do is we will first we'll start by looking at what happens in general topology. So suppose we have a function f from the positive reels r1 to projective space as projective line over the reels which is just the reels together with the pointed infinity. So let's draw a graph of this function and think what it can look like. Well, it might look like that and have a limit or it might look like this and have a limit. So the limit as x tends to naught is infinity and here the limit is finite, but it might not have a limit. It might sort of oscillate like this or it might kind of oscillate between the finite pointed infinity. So these have no limit as x tends to zero. However, it always has cluster points. So and this is due to the fact that the projective line is compact. So it must sort of approach a point infinitely often. And however, these two cases can't happen for rational functions. So rational functions can't have infinitely many oscillations. And if you've got a rational function defined for non zero reels, it always has a limit as you tend to zero. So the limit might be infinity or a real number. Of course, it doesn't necessarily have a limit if you consider maps to the real line that the real line isn't compact. So what this suggests is that if we've got a map from the positive real line to plane. So the picture we've got here is we've got a map from the positive real line to the plane and it might look something like something like this. And if it can be extended to zero on the x-axis like this. So we've got maps like this, then you can always extend it here. In other words, a limit always exists. So this is of course very much like the diagrams we had earlier. And we now want to change this into the language of algebraic geometry rather than topology. So you remember what we do is we replace these by the spectrum of the quotient field of a discrete valuation ring. And here we have the spectrum of a discrete valuation ring R. So R is a discrete valuation ring. And this map here should be proper or possibly might be proper. So this suggests there should be a relation between the map F being proper and the relation that for each choice of R and K and maps between them as above. There is a unique map. There is unique lifting G. So there is a unique lifting G. Well, uniqueness of the lifting is just the separation of this map F. And the existence is closely related to this condition that the map should be proper. So what Grozendick proved is that these two are equivalent if, let me label these, let's call this map X and this map Y. If the map from F from X to Y is of finite type and Y is notarian. As before, this is slightly different from the version in Hart-Shorn's book. Hart-Shorn has a condition not for discrete valuation rings, but for all valuation rings. And as a result is able to state the equivalence for a slightly more general type of map from X to Y. As usual, it's a matter of preference, which version you use. As before, I'm not going to prove this result, but I'm just going to give some examples of how to use it. So what we want to do is to sketch how to use this result on valuations to show that projective morphisms are proper. Well, one problem about proving this is I haven't yet told you what projective means. There's a sort of cheap definition which says that a projective morphism from X to Y is one which factors from X to P to the N over Z times Y to Y. So this we can think of as being projective space over Y, and this is projective space over the integers, where this is a closed immersion, and this is a sort of projection from projective space times Y to Y. Now, this isn't actually the most general form of projective map. So more generally, we should allow things that are locally like this. In other words, why should have a cover by say open affine subsets on each of those it should look like this. So instead of having a map from projective space times Y to Y, we have a map from some sort of twisted bundle over Y to Y. So don't worry about that because this is just to illustrate the criterion for being proper. So first of all, there's some easy bit closed immersions are finite. So are proper. Proving closed immersions are proper is much easier than proving finite morphisms are proper. So we're kind of being a bit lazy here using that. Secondly, it's easy to check the composition of proper morphisms is proper. And thirdly, it's easy to check that the base change of proper morphisms is also proper. So if we put all these together, we see that the essential part of this is to prove that PNZ to a point spec of Z is proper. So this is the this is the sort of core part of the proof and all the rest of it is just sort of routine bookkeeping. And what we're going to do is we're going to use the condition about valuation rings. So what we have that the picture we have is we have the spectrum of K, which is a point, and we have the spectrum of R. So I guess I could draw this as a one-dimensional red point with the spectrum of R, which looks like this red point together with an extra point, which is closed. And here we have the spectrum of P to the NZ. And here we have spectrum of Z, which is kind of boring because for every scheme, there's unique map to spectrum of Z. So these maps kind of give us no information whatsoever. And we want to know, does there exist a unique map here? So, and uniqueness already follows from separability. So we want to know, does this exist? And to do this, we just recall what are the morphisms from the spectrum of K to the spectrum there to projective space. So a morphism of the spectrum of K to projective space, as we saw earlier, can be described as a tuple of points, not all zero, where this is considered to be equivalent to lambda K naught up to lambda KN whenever lambda is not equal to zero. The ring R is a discrete evaluation ring, and in particular is a local ring. So points here are described by tuples R naught up to RN, where the ideal generated by R naught up to RN is the whole of R. And again, this is equivalent to lambda R zero up to lambda RN for lambda a unit. So what we're asking is suppose we're given data K like this, can we find data R like this? And the answer is almost obviously yes. So we just choose I so that the evaluation of K I is minimal. And then we divide all KN, so KJ by KI. So K naught up to KN is then just the same as K naught over KI, K1 over KI. And then we get a one here in the I position up to KN over KI. And because I is minimal, this implies that KI divides KJ for all J. So these are all in R. This is R naught R1 up to one RN for some R I. So this is a point. So this is a morphism from the spectrum of R to n dimensional projective space over the integers. And it's also easy to show this such a point is unique optimal application by units of the ring R. So we have shown that projective space over the integers is proper over spec of Z and this can be extended to show that all projective morphisms are proper. So next lecture, what we're going to do is discuss the relation between projective varieties and complete varieties and give an example of a proper morphism that isn't projective just due to Hiranaka.