 This lecture is part of an online algebraic geometry course about schemes. And in it, we will be defining notarian schemes and giving a brief discussion of their properties. So suppose we've got a scheme. We say it is notarian if it satisfies the following two conditions. First, it must be quasi-compact and it must also be locally notarian. So what does it mean to be quasi-compact or locally notarian? Well, quasi-compact means the underlying topological space is quasi-compact. Quasi-compact means just the same as compact. It's called quasi-compact for uninteresting historical reasons. Locally notarian means that the scheme S can be covered by open affine sub-schemes of the form spec of S i, spec of R i, but R i is a notarian ring. So this is fairly typical of scheme theory. We have a sort of cascade of definitions where each definition depends on several other definitions and you sometimes feel there's a sort of infinite regress going on, but anyway. So let's first discuss quasi-compactness. We first note that all affine schemes that form spectrum of R are quasi-compact. So this is sort of similar to the proof that all affine varieties are quasi-compact in the Zariski topology. Suppose spectrum of R is covered by open sets and we want to show it's covered by a finite number of them. Well, we may as well assume that it's covered by open sets of the form D F i because these form a basis for the topology. So remember D F i consists of the primes such that F i is not in the prime. Well, if it's covered by these open sets, this means no maximal ideal contains all the elements F i because the D F i just consists of the prime ideals such that F i isn't in here. So if these cover spectrum of R then no prime ideal can contain all of them. So the ideal generated by the F i is equal to R because it's not contained in any maximal ideal. So sum of a i F i equals one for sum a i almost all zero. So a finite number, so we can find a solution of this with, so we can find a solution using a finite number of F i. And this just means that the spectrum of R is covered by a finite number of our sets of the form D F i which shows. So the quasi-compactness of the spectrum of R has something to do with the fact that if you can write one as a linear combination of elements then you can write it as a linear combination of a finite number of them because we're doing algebra not analysis. So scheme S is quasi-compact if and only if S can be covered by the finite number of open sets of the form the spectrum of R i. So in other words, the quasi-compact schemes are just the ones that can be built out of a finite number of affine schemes. Examples, the schemes of projective and affine varieties are all quasi-compact. You know, affine varieties are affine schemes and projective varieties or schemes can be covered by a finite number of affine ones. So they're quasi-compact. An example of one which isn't quasi-compact is just a disjoint union of accountable number of schemes. We'll usually be not quasi-compact. And in practice, the schemes you come across that aren't quasi-compact quite often turn out to be disjoint unions of accountable number of schemes. A fairly typical example is the so-called Hilbert scheme which parameterizes subschemes of projective space and it splits up into accountable number of components because if you've got a subscheme of projective space there are accountable number of degrees it can have, for example, so that splits it up into accountable number of subschemes. Next, we'll talk about notarian spaces. So we recall that a topological space X is called notarian if and only if every non-empty collection of open sets has a maximal element. And this is the same as saying it satisfies the descending chain condition for closed sets. So that means if you've got any descending chain of closed sets it must eventually be closed. If you've got any descending chain of closed sets it must eventually become constant. And you might guess we define a spectrum so we do not define S to be notarian if its topological space is notarian. And the reason for this is the following problem. The problem is I'm supposed to take R to be say K X1 X2 and so on with an infinite number of variables and we quotient it out by the ideal generated by X1 squared X2 squared and so on. Then the spectrum of R is a point. You can see the only prime ideal is X1 X2 and so on. So this is the only prime and the point is notarian as a topological space. However, this ideal is not finitely generated so the ring R itself is not notarian. So if we defined a scheme to be notarian if its underlying space was notarian we would have the embarrassing fact that the spectrum of a non-notarian ring was a notarian scheme which we really don't want. We have to define it in a slightly more roundabout way. We say S is locally notarian if it's covered by open affine sub-schemes with the form spectrum of R i with the ring R i notarian. And we say it's, we remember it's notarian is locally notarian plus quasi-compact. So quasi-compact you remember means that you can cover it by a finite number of open affine sub-schemes. So this is the same as saying if it's covered by a finite number of open schemes of the form spectrum of R i with R i notarian. So for example, projective varieties are all notarian. So they're covered by a finite number of affine varieties which are notarian. The Hilbert scheme, I'm not going to tell you what it is, but exactly, but as I said vaguely that it was a space parameterizing sub-schemes of projective space. And growth and dick show that it was locally notarian. In fact, it's the disjoint union of a countable number of projective schemes. So you do occasionally get examples of schemes that are locally notarian, but not notarian. But in practice, most locally notarian schemes who get a simply a disjoint union of notarian schemes. And now there's a major split in algebraic geometry which is should you restrict to notarian schemes and there are some advantages to this. Most schemes in practice are notarian and it does make it much simpler in many ways. There are a lot of technical complications you have to deal with if you want to deal with the notarian schemes. Also as a disadvantage that some schemes are not notarian and they do sometimes turn up. For example, the spectrum of a nondiscrete valuation ring and nondiscrete valuation rings do turn up. They used to be a lot more popular before growth and dick introduced schemes and they were heavily used in resolution of singularities of varieties, for example. And as I said, algebraic geometry seemed to split into two camps. The ones who work with notarian schemes have the attitude that there is no point spending a lot of effort on this technical complication that you almost never need. And I think they secretly suspect the people who work with non-notarian schemes of being the sort of people who wear hair shirts and so on. On the other hand, the people who work with non-notarian schemes definitely have the attitude that people who work with notarian schemes are a bunch of wimps and as if you were a real small furry creature from Alpha Centaur you handle the tough case of non-notarian schemes. So you have to pick which you do. We're going to mostly just stick with notarian schemes for out of laziness, I guess. As the name suggests, being locally notarian is a local property. So in particular, the spectrum of a ring R is covered by open affine sub-schemes of notarian rings implies R itself is notarian. So in other words, you can check whether a scheme is notarian or not by looking at small open affine subsets covering it and if all those are notarian the ring itself is notary the whole scheme is notarian and if it's an affine scheme then the corresponding affine ring is also notarian. So the key point in showing this is that being in an ideal is a local property. So what does this mean? Well, suppose X is in R and I is an ideal of a ring R. Then the following are equivalent. First of all, X is in I. Secondly, X is in I F to the minus one which is an ideal of R F to the minus one where the open sets DFI cover the spectrum of R. So in other words, if you've got an open cover of the spectrum of R we can check whether X is in the ideal by checking on each of the open sets of this cover. And the third property says that X is in I P which is a subset of R P. This is the local ring at P for all P in the spectrum of R. Or even all maximal P. We don't need to do all prime ideals. You could just test for maximal ideals if you want. And in order to verify this, we notice that one implies two and two implies three are kind of trivial. To show that three implies one what we do is we just look at that's J which is the ideal of all Y such that X, Y is in I. And we want to show that J is equal to R. Well, if M is maximal, if M is a maximal ideal, then X being in I M implies A times B X minus I equals zero for some I in I and A B not in I. So this is just the condition for X to be in this ideal of the local ring. And this implies A B X is in I. So J is not contained in the maximal ideal M because A B is not in, so A B is in J, but not in M. So I shouldn't say A B not in I, I should say A B not in the maximal ideal M there. So J is not contained in any maximal ideal. So J is equal to R. So X is in I because one is in the ideal J. So this shows you can test whether something is an ideal just by testing it at all local rings of prime ideals of the ring. So now suppose the spectrum of R is covered by open sets DFI. Then an ideal of R is determined by its images I F I to the minus one in the rings R F I as above because these open sets cover R and an element is in I if and only if it's in all these ideals I. So if R has an increasing sequence of ideals I one contained in I two and so on. What we can do is we can look at all the sequences I one F I to the minus one contained in I two F I to the minus one and so on. Now if all the rings are F I a notarian then all these sequences must eventually terminate and as R can be covered by a finite number of these this means that the sequence of ideals in R terminates so this sequence here terminates so R is notarian. So we've shown that if the spectrum of R is covered by open subsets of notarian schemes and R is itself notarian which sort of shows that in some sense being notarian being being that locally notarian really is a local property. Finally I just make a few comments about what it means for ring to be notarian being notarian for a scheme or ring is sort of vaguely related to being finite dimensional and it is often true that notarian rings are finite dimension and it is often true that finite dimensional rings are notarian. So this this is usually true but there are counter examples for instance we can take the example we had earlier K X one X two up to we take a count on number of X I's and then just quotient out by the ideal generation by X one squared X two squared and so on. So this is a finite dimensional ring that zero dimension but it's not notarian and it's usually true that if something is notarian then it's finite dimensional so it's true for local rings if you've got a local ring it's a basic theorem of commutative algebra that it's finite dimensional in fact the dimension is bounded by the number of elements needed to generate its maximal ideal and it sort of sounds very plausible that if something is locally finite dimensional and it's compact for heaven's sake then surely it should be globally finite dimensional seems very plausible but there's a weird counter example that Nagata found which instead of writing out I'll just show you Nagata's only own description of it so Nagata himself calls these bad rings let me see if I can zoom in a bit okay so so here's his example of a bad notarian ring so he's saying a notarian ring whose altitude is infinite well the altitude was a sort of old name for the dimension of a ring so here's got an infinite dimensional notarian ring and what he's doing is he's taking a polynomial ring an accountable number of variables and then carefully localizing in a rather cunning way in order to end up with a ring that's still infinite dimensional but it's still possible to prove that this ring is notarian and the proof isn't all that difficult but I'm not going to give it because I'm feeling too lazy so this is from Nagata's book Local Rings and it's kind of rather a famous book because it's got an appendix with lots of horrible examples of notarian rings and I think before Nagata started finding all these examples people were hoping that notarian rings and schemes were the nice collection of rings and schemes but unfortunately it turns out you can get some pretty nasty rings and schemes that are still notarians this leaves the problem of what is a good condition that covers all the rings we want and excludes the nasty counter examples and as far as I know there's no really good answer to this the most promising approaches may be growth index notion of an excellent ring unfortunately the definition of an excellent ring is so complicated no one can remember it so it's still in some sense an open problem I guess okay the next lecture will be on various finiteness conditions like being a finite or locally a finite type