 This lecture is part of an online algebraic geometry course about schemes and will be about immersions of schemes. So for topological spaces, we have the following concepts. We can have open sets or closed sets. Now for schemes, we also have corresponding concepts which are more or less open immersions and closed immersions. There are also corresponding concepts for rings. So open immersions and open sets are sort of related to localizations of rings where you just invert an element and closed immersions of schemes turn out to be closely related to taking a quotient of a ring by an ideal. So we first discuss open immersions. So suppose Y is a scheme and pick an open subset U of Y. Then we can make U into a scheme just by saying that for each open set of U, the functions on that open set are the same as for Y. So we're sort of saying O U of V is equal to O Y of V for V open in U. So we're just making it U into a scheme in the most obvious possible way. And this is an open immersion. So the map from the scheme U to Y is an open immersion. And in general, an open immersion is anything isomorphic to this. So we say X to Y is an open immersion. If it can be factored, if it's of the form X to U goes to Y where this is the inclusion of the scheme of an open subset into Y as above. And this is an isomorphism. So this is just a slightly fussy variation in order that anything isomorphic to a closed immersion is a closed immersion. But for all practical purposes, open immersions correspond exactly to open subsets of the underlying topological space of Y. So let's have some examples. Well, we can just take A1 minus a point to be contained in A1. So this is the spectrum of KX X minus one. And this is the spectrum of KX. And here we're just taking the complement of a closed set in the affine line and that gives you an open immersion. And similarly, we can take the complement of any closed set of a variety and that will give an open immersion. Another example is the inclusion of the affine line in projected space. So this is an example that doesn't quite come directly from localizing a ring. Next, we can ask about finiteness properties. So we had a whole lot of finiteness properties, morphisms could be finite or quasi-finite, finite implies quasi-finite, or they could be a finite type, or they could be locally a finite type, or they could be quasi-compact. And we can ask which of these do open immersion satisfy? So are open immersions finite? Well, no. There's an obvious counter example we've had one up here. The inclusion of A1 minus a point in A1 is certainly an open immersion, but as we saw last lecture, it's not finite. On the other hand, open emotions are definitely locally a finite type for rather completely trivial reasons. Any, if you've got an open immersion from X to Y, then any point in Y obviously has an open neighborhood that's actually isomorphic to an open neighborhood here. So they're a finite type. For the other three open immersions quite often of these three forms, you notice they're actually equivalent. If an open immersion is quasi-compact, then it's automatically a finite type and quasi-finite. Here we're using the quasi-finite definition in EGA2. You remember I said there were three different definitions of quasi-finite in the literature. So we're using the strongest one. So the question is, are open immersions quasi-compact? So let's do this in blue. So does this happen? And so open immersions are often quasi-finite and the finite type and so on. So they certainly have, if you've got an open immersion, this certainly has closed discrete fibres. That's completely trivial because the fiber is either a single point or it's just the empty set. So there's one obvious case in which open immersions quasi-compact. So this is true if X is notarian because then all open subsets of X are quasi-compact. So this is automatically quasi-compact map. So as long as you stick to notarian schemes, then open immersions are quasi-finite, finite type and quasi-compact and so on. As you might suspect for non-notarian schemes, this breaks down. So let's have an example. We're going to take Y to be the spectrum of a ring of polynomials in infinitely many variables. So this is infinite dimensional affine space. And we're going to take X to be Y minus the point, well, the point corresponds to the ideal X1, X2 and so on, which corresponds to the point nought, nought, nought in infinite dimensional affine space. So the point, here's the points in coordinates, it just does all coordinates zero. Here's the point considered as an ideal, it's just the ideal generation by all of these. So this is space minus the origin. And X to Y is an open immersion because the complement of a point is open, but it's not a finite type or quasi-compact. And the point is that the, it's not quasi-compact because the inverse image of Y is X and X is not quasi-compact. So here we have an example of an open subset of the spectrum of a ring that isn't quasi-compact. And to see this, we just know that X is not quasi-compact. Notice that the open sets of the form D, F, so this is the primes P with F not in P. So you can think of this as being the sets where F doesn't vanish. So the open sets D, F does not contain this ideal, not contain this ideal, let's call this ideal P. So it does not contain P if F is in the ideal generation by X1 up to all the others. So if we cover X by open sets D, F, we can't find a finite sub-cover covering X. So X is not covered by a finite number of D, Fs. And the point is any finite number of D, Fs, so for any finite number of D, Fs, all the Fs will vanish on some point of the form nought, nought, nought, nought. We take a lot of noughts and then have a one there because all the Fs will be polynomials and only a finite number of the variables Xi, so they will vanish provided sufficiently many of these coordinates are zero. So here we found a point in X that's not in any of these finite number of open sets. So the open subset X is not quasi-compact and this open immersion is not a finite type. Incidentally, because X is not quasi-compact, you can see immediately that it's not an affine scheme either. So this is a sort of standard counter example to several other things. Finally, we can ask what is the relation between finite morphisms and quasi-finite morphisms? Well, we know that open immersions and finite morphisms are generally both quasi-finite. Well, this is, this implication holds provided everything is notarian. Well, it usually will be notarian in practice. So we've got two completely different sorts of quasi-finite morphisms, open immersions and finite morphisms. And there's a theorem of growth and Dick which he rather confusingly called Zariski's main theorem which says the sort of converse holds. So if X to Y is quasi-finite and some mild conditions hold, then it can be factored as X goes to Z goes to Y where this is finite and this is an open immersion. So a general quasi-finite morphism, you just get by taking a finite morphism and throwing away a closed subset of it. I'm not going to prove it because the proof is rather too difficult and we don't actually need it anyway. You may wonder what mild conditions mean. Well, the mild conditions are completely unmemorable. I think that you can strengthen them a bit to save everything in sight is notarian and separable. But in fact, people tend to give somewhat weaker and much more complicated conditions for this. But anyway, so separable is something we will be discussing in a few lectures time. Next, we should discuss closed immersions. So a closed immersion, well, a typical example of a closed immersion is taking a spectrum of R over I to the spectrum of R. Now, what this will typically look like is you might take R to be, say K X Y, so it will be the affine plane and R over I, well, I might be say the ideal Y squared minus X cubed plus X and the spectrum of R, R over I will then just be this curve where Y squared is X cubed minus X. So here we have the spectrum of R over I as this red thing here. And it's sort of a very reasonable to call this a closed, you know, it sort of looks like a closed subset of that scheme. So closed immersion is very similar to a closed subset of the scheme except we'll see in a moment, it's not quite the same. So in general, the closed immersion looks locally like this. So a closed immersion, suppose we've got a closed immersion X to Y, what it means is Y is covered by open affines Y I and the, the maps from X intersection Y I to Y I look like the spectrum of R I over I I to the spectrum of R I. So the open affine subset Y I is going to be by some spectrum of R I and the intersection with the sub-scheme X will look like this. So first of all, this is local. In other words, something is, it doesn't matter if Y is covered by open affines with this property then any open affine subset of Y will have this property that's intersection with X looks something like this. And furthermore, it's obvious that closed immersions are all finite. So they have a very strong finite in this condition and this is because R over I is a finitely generated R module rather obviously it's generated by just one element. So let's have a look at some examples. Well, as we've just said, the spectrum of R over I to the spectrum of R is a typical example. So we can look at say the spectrum of KX over X goes to the spectrum of KX is a closed immersion. So this is just a point going to a line where you map the point to the origin. That's not very difficult. However, there's a sort of subtlety here because we could also look at the spectrum of KX over X to the N and the spectrum of this is again a point and its image is exactly the same points in the affine line. So we see that a closed immersion is not determined optoisomorphism by the closed subset F of X in Y. So here we've got a closed immersion from X to Y and here I'm being a bit sloppy and using X and Y for the underlying topological spaces of X and Y. So here we have two different closed immersions from a scheme whose spectrum is the underlying spaces of points to the affine line. And yet the images of the same closed set so this is different from open immersions open immersions. All you need to do know is the open subset of the spectrum and that tells you what the open immersion is but closed immersions are more complicated. Closed immersions are also more or less the same as closed subschemes optoisomorphism. So when you talk about a closed subscheme you can either talk about a closed subscheme you can either talk about an either means an isomorphism class of closed immersions or you can talk about a particular closed subscheme where on each open set you take a specific quotient but anyway, so if you've got a closed subscheme so given a closed subset what closed subscheme correspond to it? Well, in general there can be many because a closed subset say suppose we choose a prime ideal in R so we're choosing irreducible closed subset. So we would have a closed subscheme corresponding to the map from R over R to the P but we could also take R over P to the N and consider the map from R to that and we could also take lots and lots of ideals between P and P to the N. So in general there are huge numbers of subschemes corresponding to a given closed subset. However, there is a minimal one. So there is a minimal closed subscheme associated to a closed subset and this is constructed as follows. So for an affine scheme spectrum of R and a closed subset given by you remember closed subsets correspond to ideals I, we just take the radical of I which is the set of R such that R to the N is an I for some N and then we can look at the spectrum of R over the radical of I and this will be a subscheme of the spectrum of R because you've got a homomorphism from R to R over I and this is a reduced scheme which means it has no nil potents. Taking the radical means essentially we're just killing off all nil potents here and these fit together to give a subcheme of R and a reduced subscheme associated to any closed subset of the spectrum of a ring. However, there's something you've got to be a bit careful of. So the intersection of two reduced closed subschemes need not be reduced. So even if you try and stick to reduced subschemes you sort of end up having to use non reduced subschemes if you start taking intersections. So the intersection of two closed subschemes is a reasonably obvious concept. It's the biggest subscheme that's contained in both of the subschemes you're intersecting. And to see an example where the intersection of two reduced subschemes is not reduced let's just take Y to be the affine plane and let's take X1 to be the Y axis. So this is KXY over Y. Sorry, that's the X axis. So here's Y and here's X called the X1 and let's do X2 is going to be a parabola. So let's quotient out by Y minus X squared. So it will look something like this and the intersection of these two subschemes will be KXY and we quotient out by Y and Y minus X. And this is isomorphic to K of X over X squared. And this is not reduced. So it's a sort of scheme which corresponds to there being a double point at the origin. And this corresponds to the fact this parabola is intersecting the line in a double point rather than the single point. So the intersection is a closed subscheme of X2 and it's a closed subscheme of X1 and these are both closed subschemes of Y. So we have intersection of two reduced closed subschemes is not reduced. Okay, that's enough about immersions. Next lecture, we're going to talk about products and fibred products of schemes.