 Okay, so this is part two of lecture two, where we'll be discussing notarian spaces and notarian rings. We'll start by calling what a notarian ring is. So a notarian ring is a ring satisfying any one of the following three equivalent conditions. First of all, every ideal is finitely generated. And secondly, every non-empty set of ideals has a maximal element. And thirdly, every chain of increasing ideals is eventually constant. In other words, n equals n plus one equals n plus two and so on for some n. So the basic example of a notarian ring is the ring of polynomials pay x one or two x n in n variables. So this was originally proved by Hilbert at the near the end of the 19th century. So Emmy Nertzer found a rather easier proof of this. What Nertzer showed was that if R is notarian, this implies that the ring of polynomials in R is also notarian. And since a field is trivially notarian, and by repeatedly applying this, we see that all polynomial rings over fields are also notarian. So we'll just quickly review the proof that R of x is notarian. So this is a proof of our notarian implying R of x notarian. We'll do a fairly brief sketch of the proof. So we look at the ideals i naught contain an i one, contain an i two and so on, where i n is equal to the leading coefficients of L's of degree n, degree less than or equal to n in i, where i is an ideal of R of x. So we've chosen an ideal i of R of x and we're trying to find a finite set of generators for it. Since R is notarian, the stabilizes, this means that i, bigger n equals i, oops, i n plus one and so on for some n. And now what we do is we take sets of polynomials let's call these sets S naught, S one up to S n, where S naught of degree naught and the leading coefficients generate i naught, S one are polynomials of degree one whose leading coefficients generate i one. And so I'm all the way up to S n whose leading coefficients generate i n. Then you can check that S naught, S one up to S n generate the ideal i. So we take finite sets here, which we can do because the ring R is notarian. So fill in the details, this set of polynomials and variables is notarian. Well, that's notarian rings. We're now going to discuss notarian topological spaces. So a topological space is called notarian if equivalently. So either the closed sets satisfy the descending chain condition. What this means is that any decreasing set, any decreasing sequence and C naught contains C one contains C two and so on of closed sets stabilizes. So C n equals C n plus one equals C n plus two et cetera for some n or equivalently any non-empty collection of closed sets has minimal element. So you see these are sort of duels of the conditions we had for notarian rings. So notarian rings, the ideals satisfied an ascending chain condition and any non-empty collection of ideals had a maximal element. And a consequence of the thing we proved is that A n with this risky topology is notarian. And the reason for that is that closed sets of A n correspond, that should be a capital N, correspond to some ideals of the coordinate ring K x one up to x n because to any closed set is determined by the ideal of functions vanishing on it. So if there's a descending chain of closed sets of A n this gives you an ascending chain of ideals which must be eventually closed. So, so a find over any field is a notarian topological space. I should point out that notarian spaces are weird if you're used to looking at topological spaces in analysis. So first of all, the notarian condition is equivalent to saying every open set is compact. This is a easy exercise which I won't bother writing out. By the way, in algebraic geometry the word compact sometimes instead. So what's the difference between compact sets and quasi-compact sets? Well, there isn't any difference at all. What happened was we have this definition of compact and Burr-Barkey decided that all useful compact spaces were house-storffs, so he changed the definition of compacts that compact in Burr-Barkey means compact and house-storff. And then shortly afterwards they discovered that there actually were some compact spaces that weren't house-storffs, so they had to invent a new word for it, quasi-compact. A rule of thumb is whenever you see the word quasi in mathematics, it's a sort of indication that somebody somewhere really screwed up the terminology and had to put in the word quasi to try and fix it. Anyway, what you notice is that in analysis open sets are almost never compact unless they happen to be finite. In fact, you can easily check that if a space is notarian and house-storff, this implies it is in fact finite. So since most naturally occurring spaces house-storff it's probably going to be interesting. So all your top-to-goo-ish spaces you often get in analysis. So there's one concept that occurs quite a lot for notarian spaces, which is a set is called irreducible if and only if it is non-empty and not a union of two proper closed subsets. Sorry, I should have said not a set of topological space, but never mind. Now this is a completely useless concept for house store spaces because you can easily check that the only irreducible house store spaces are single points. However, it turns out to be very useful in algebraic geometry. So if you remember the picture we had of a two, a typical closed set of a two, something like this. So it's a bunch of curves and a bunch of points and the curves are in fact irreducible. For instance, an affine line is irreducible because any two closed sets, any two non-empty closed sets intersect, which easily implies that an affine line can't be written as a union of two proper closed subsets. And if you look at this space, you can see it seems to be the union of a finite number of irreducible subsets where the irreducible subsets point. And this took very good tea of all notarian spaces. So we have the following theorem, which says any notarian space is a union of irreducible subspaces should have said finite union. So this means you can reduce the study of notarian spaces to the study of irreducible notarian spaces. And this follows quite easily by induction. In fact, it follows by something called notarian induction. So notarian induction means you pick a maximal closed set of some collection of closed sets. So what we're going to show is that every, so we're going to prove by notarian induction, every closed subset is finite union of irreducibles. If not, pick a maximal counter example, sorry, a minimal counter example, C. It's very easy to get maximal and minimal modeled up because for ideals, notarian conditions says there's a maximal element of any set. And for spaces, the notarian conditions says there's a minimal element of every set. So it's these things you constantly get muddled up about. So now there are two cases, C might be irreducible or it might be not irreducible. Well, if it's irreducible, we're done because we get a contradiction. And if it's not irreducible, then we can write C is equal to C1 union C2 where C1 and C2 are smaller closed subsets. Well, then by induction, by notarian induction, C1 and C2 are finite unions of irreducibles. And if C1 and C2 are finite unions of irreducibles, then so is C. And this again gives a contradiction. So by assuming that there was a closed subset that's not a finite union of irreducibles, we obtain a contradiction. Therefore, every closed subset in particular whole space is a finite union of irreducible subsets. So as an application of this, every algebraic set is a finite union of irreducible algebraic sets. Now irreducible algebraic sets are sort of called algebraic varieties. So variety sort of means irreducible. So we can provisionally define an algebraic variety as being an irreducible closed subset of affine space. Well, that's actually not quite right. There's a slight problem we run into. So let's give a few examples of this. Well, suppose you take the variety x, y equals 1. So this is just a nice hyperbola. And suppose we take another example. Let's take the set of points in A1 with, let's take the set of non-zero points in A1. So here we're just taking non-zero points in A1. Now, in some sense, this is not an algebraic variety because it's not a closed subset of the affine line. However, it's sort of more or less the same as the hyperbola because you can map the hyperbola to this just by mapping a point x, y to the x coordinate there. And this is the sort of one-to-one mapping. So in some sense, the set of non-zero points of A1 should be considered an algebraic variety even though it's not quite a closed subset. So this definition of algebraic variety is kind of provisional. We'll give a slightly better definition later on. So let's have some other examples of irreducible components. So a logic defined by two equations, x squared plus y squared minus c squared equals naught and 2x squared minus y squared minus z squared. So that should be x squared plus y squared minus 2z squared and 2x squared minus y squared minus c squared equals naught. And so these are two cones and we're taking the intersection. And the question is, how does this decompose into irreducible subset? So it's the union of four irreducible subsets. So the irreducible subsets x equals y equals z, x equals minus y equals z, x equals minus y equals minus c, and x equals y equals minus c. So it's just four lines. So it's quite common for intersection of irreducible subsets to be non-irreducible. You see, these are both irreducible, but the intersection decomposes as four lines. Another example, you've got to be a bit careful about the relation between being irreducible and being connected. So for example, if we take x, y equals zero, it's just the union of the x-axis and the y-axis. So this is reducible but connected. On the other hand, if we take x, y equals one, it looks like this. And this is irreducible and it certainly looks as if it's disconnected. Well, whether it's connected or not depends on which topology you're using. So it's connected in the Zariski topology. However, it's obviously disconnected in the usual Euclidean topology. So when you talk about things being connected, you've got to be a little bit careful about which topology you're talking about. So that's all for, I want to say, about notary in spaces for the moment. So the next part of the lecture will be on the Hilbert-Nelstellensatz, which explains the relation between algebraic sets and ideals in war.