 Ah, OK, all right, fantastic. Yeah, so I will talk about the homology of the quote scheme, which I called last time QDGRN. So this was the space parametrizing quotients of CN of rank N minus R in degree D. So then the kernel as rank R in degree minus D. And this is on P1. And I'll point out today when this P1 assumption is or is not necessary. So in any case, what we discussed, we talked a bit about this space. It compactifies. So this compactifies the space of maps of degree D from P1 to the Grassmanian. In case, this is when R is less than N. OK, so when R is less than N, the generic quotient here is actually locally free. OK, so it's a vector bundle. Obviously, when R is equal to N, then we're parametrizing just torsion quotients. So it depends. Right, so I'll comment on this as we go on. So we know that there is a universal sequence on the quote scheme. And so we've established various things that this is a smooth projective variety. In this case, over P1 is a smooth projective variety. There's this universal sequence. And we saw it be our diagonal arguments that we talked about, I guess, last time that, in fact, the churn classes of the universal bundle generate the cohomology multiplicatively. So if we write, well, maybe I'll take the dual here. This is the universal sub-sheep. So the churn classes live in the product space. So if we use a quenet decomposition fix a basis for the cohomology of P1, a point, and the fundamental class, then we can decompose this. And so this is, if you want, point on P1. And so then what we know from our previous discussion is that the class is ai and fi. And this is generate the cohomology. So yeah, we also, so this is one thing that we understand. And last time I had written the Poincaré polynomial. So let me actually write it again. So here, if we also see the Poincaré polynomial. So if we, so this is here, we sum over the Poincaré polynomials of over degrees, basically. So this is, OK, and then we saw that this is a beautiful expression, the following formula. So here we have the, this is the Poincaré polynomial of the grass man in itself. I'm not going to write it. And then you have a product of two types of series. So these are, OK, so this i goes from 1 to r. And so this was calculated by, I guess, by Linda Chen about 20 years ago. And the calculation involves localizing the code scheme. So and just it's using the cellular decomposition that gives, that this localization yield. So in fact, you can do it with isolated fixed points in this case. And so it's, yeah, so it's a calculation one can do. I should say that Linda actually calculated for every variety of flags. So the so-called general, the so-called hyperclose schemes parameterizing not just the not just the subchef of the grass man, but then a flag of a certain type. The most interesting examples are the full flag or the grass man, kind of the more one for representation theory, maybe, and the other for geometry more. But in any case, there's a formula for any type of flag. OK, so what we saw is that so we're going to focus a little bit on explaining this Poincaré polynomial today. So one thing that we noticed last time was that what we have seen is that the homology stabilizes. So we're interested in this. So in other words, we saw that the betting numbers start to stabilize for d greater than or equal to k. OK. And so also then this was basically you read it off from the formula, essentially. We went over this last time. And I also explained, or rather, let me mention briefly at the end of the lecture, what it stabilizes to. How do you calculate it? And in fact, we have that the stable homology. So this is the stable value. You put them in a series. In fact, given, and one can check this easily by multiplying the Poincaré series by 1 minus t and evaluated at 1. There's a simple polar one. And so then this is just of the theory. So this is in fact given by, let me write this. So is this pre-factor, the Poincaré polynomial of the grassmine in itself. And then it's just this expression. And so we commented where we exactly left off. So it's been a few weeks, I guess, but where we actually stopped last time was in commenting that this is actually the polynomial algebra. So we, right. So let me just, so we specialize. So in fact, I will do this today. And I'll explain why and what the general difficulties are. But we saw very quickly. So what's very, very easy to observe is that in the kind of extreme case when r is equal to n. So when r is equal to n. So in this case, so this is this QD, is this QDG rr, if you want. So this just parameterizes sub-sheeps of rank r of CR. And then the quotient is just purely torsion, yeah? So this is a zero-dimensional sheaf. So it's a rank, it's a, so it's a rank zero sheaf one. It's just a torsion sheaf on the curve, you know? So in this case, in fact, we commented that this polynomial is, so the stable polynomial is actually the Poincare polynomial of just the polynomial algebra in the ATIA bot generators in these A and F classes. And so moreover, also, the Poincare polynomial has a very nice form in this case. This pre-factor disappears. So this guy disappears. So there's just one. And in fact, the full Poincare polynomial. So this is the, so there's the stable polynomial and the full polynomial is P of z, I'm sorry, z and t is equal to this. So I will, for the rest of the lecture, I will actually focus on understanding the homology of this particular space. So this quote scheme of torsion quotients of CR, OK? It's just, it's very accessible. So we can say a number of things about it. And I also comment on the general case as well, how, you know, what the difficulties might be. So where the analogies kind of break down. OK, so I should also say that, you know, in, so this is obviously, so the Poincare, these are the Poincare polynomials, but I should say for this punctual, sorry. OK, so for, so from now on, so assume, so we're going to assume from now on, assume, we have always this R, R is equal to n, OK? So this is, this is, and QD will always mean this now. Yeah, so QD, QD denotes just the scheme of quotient, of quotients of length D of CN, OK? So, well, OK, so let's, so I should say that, yeah, so there's a, there's the Poincare polynomial here, of course, and in fact, you know, one can, one can, one can generalize this and there are, recently, there are calculations of the, of the motif of this quote scheme. Yeah, so, so, so sort of kind of along the same line. So the motif, so in this case, the motif of QD is calculated, OK? So, so these are recent papers of my recall fee. Juan Jaro Fanteci, Peroni, so, so, so, so it's a, yeah, so in some, in some ways, this exploits the, the Taurus action and the Biennese-Kibirula decomposition and also a few, a few other sort of a classic multi-dig techniques, but in any case, this is, this is understood. And what I want to kind of, to do today, my point is to go a bit beyond, I mean, to understand what these formulas actually mean. So I'm not going to, why do we, why do we, why do we have this Poincare polynomial? And here, you know, I wrote both the stable polynomial and the, the, just the total polynomial and we can go in two directions, right? I mean, we can, we know that you can approach the study of the homology in, in, in two, in two ways. So one, one thing is that, you know, we have the idea about generators and to, we, we could try to understand something about the, the shape of the ideal of relations, yeah? So we know that these generators become relation free as D, as D is large, gets large. So it's kind of a, it's, it's an interesting, it's an interesting direction. So I can tell you right, and we expect these relations to be natural, geometric. So I'll just say one thing about this direction because I'll actually focus in on a different one. So let's, let's observe the following, yeah? So, so on QD now, because we have the universal sequence, so on QD, so we have this universal sequence, which in this case, actually I will denote by, I will denote the torsion byte, just to the, the quotient byte to E to indicate it's a torsion sheet. So this is on QD cross P1. So if we project here, we can, so we have the complexes. I'll just write something obvious in some sense. So I can consider the complexes, which I, here, I'm gonna call E sub n, which are the push forwards of the universal sub sheaf twisted by just the, just ample line bundle from P1, coming from P1. So for all n, yeah, you can do this. And so a feature of this is that if I look at the Segre class, so you notice that if I look at, I'm looking at the, you can look at the trend classes, but I'm actually interested in looking at the Segre class of the m, this is zero whenever k is greater than D. And the reason is that this Segre class is actually calculates the trend class of the associated sheaf, which comes from the torsion. So you obtain in the same way by push forward. And this is just locally free of degree D. So then the trend class is vanished past D. I'm sorry, locally free of rank D. So the trend class, the trend class is just vanished past D. And so one thing that one can ask is, do these vanishings, do the Segre vanishing generate the ideal relation, the ideal of relations, RQD. And when we take the generators to be these at the abogen, obviously they can be expressed. These are polynomial expressions in the abogen. There's by the Riemann-Roch formula. So you can say, well, so you can ask this question. And then it's somewhat supported by the fact that indeed there are no relations. We know that there are no relations in a homological degrees lower than D. So these relations are all, these would be relations in higher homological degree as they should. So we expect this. So I'll leave this a bit open, because I want to say a bit more about the other direction. But this is a question that sort of, it's natural and it's kind of forced on us by the fact that we know what the stable, what the stable cohomology looks like. So we understand when we know the stable cohomology want an explanation for it. So this is something that one might pursue and I'll defer answering this question, okay, for the moment. But this is a type of geometric relation that you might expect to give you all relations, essentially. This is kind of a Mumford type relation in the homology of this code scheme. Okay, so now in fact, let me go in a different direction now, which is, you see, which is looking at the Poincaré polynomial here. Let me highlight it and I wrote it here. Yeah, which is to sort of understand, produce not a, instead of starting so to speak the general shape of the relation ideal, you may attempt to produce some sort of canonical additive basis for the homology, since the Poincaré polynomial looks very nice. And this can be done just by, I guess, a system of Hecke correspondences. So this is what I would like to explain next. So our goal for the remainder of the lecture is to produce, so we leave this a bit open and here we want to produce additive basis for the homology of the code schemes, okay? So this would be an Akajima type basis, okay? So let's see what we can do. So it's actually straightforward in this case and Poincaré polynomial kind of tells you what to do. So we're gonna consider the nested code scheme. So we can consider for any given D. So here this sits, so Etilda is a locally free sheaf of degree by one less. So you study the sort of nested phenomenon. So the quotient is certainly just a skyscraper sheaf at a point on the curve. So yeah, so there's a, so what is this then? So how do you constitute this? Well, you know, it just sits as a projective bundle over QD times P1. So in fact, so we have that myomorphism. Yeah, so then, so this, right? So what you need to do in order to generate the nested code scheme starting with, so starting with the basic setup of E and CR. So a point in QD here. Oh, you fix a point where you want to have this Hecke modification. And then you're looking at just a surjective morphism from E to E to just the skyscraper sheaf at this point. Okay, so this is essentially, this is the fiber of E dual, okay? So, and of course you look projectively inside the code scheme. So therefore, overall one can show this easily that this is, I mean, this is completely straightforward. This is just a projectivization of the universal sub bundle, okay? So then there are these natural maps. So you can, sorry, you can map to QD here and well, you can also forget about E and just map to the deeper sub sheaf, so to speak. So that maps you to QD plus one in here. And this side you can check is a generically finite of degree D plus one. So it's a D plus one, one map, okay? All right, so it's just kind of a basic geometric setup we have and then on, so this comes of course, there are two basic exact sequences. One is as the exact sequence, the universal sequence. So this would be what is a torsion and this is precisely, I'll explain in a second. Or maybe I, oh no, it's actually okay. Okay, so now this, I'll explain immediately what this is. I mean, here of course this maps to QD cross P one and this diagonal, you see there's a divisor here which is just a pullback of the diagonal on P one times P one. So this figures here, so this parameterizes the elementary modification at the point, yeah? And then there's this L which is just a hyperplane of the, so this is a line bundle which is the hyperplane bundle of this projective bundle, okay? So this is a fundamental universal sequence which connects the two sides on QDD plus one. And so then of course, B is just the exact sequence of this projective bundle. So how, so this is on, so there's a quotient Q but this is just on QDD plus one, okay? And this is pullback from QD via this, yeah, this is pullback from the base from here. From, yeah, it's pullback from here. This is just the exact sequence of the projective point. This is the universal. So then, so this analysis, it's easier, I mean, okay, this setup has been much studied, for example, in the context of just the Hilbert scheme of points on a surface, no? So in fact, by Lotar, so it's nothing, it's nothing very surprising, you know? I mean, it's kind of a standard way to move between, inductively between understanding the Comolge of QD, understanding of Comolge of QD plus one. So also, so therefore, you know, but these exact sequences allow us to relate easily the trend classes, which is something that we want to do. Well, let me just, yeah, so here I have to be, right, so I will call the hyperplane in this P1, I'll call this H0, okay? So then, whereas, so we have two P1s, yeah? I mean, on this, so we have to be a bit so, and on the basic, oh no, I'm sorry, I'm sorry, I'm saying it exactly the way I don't want it, okay? I apologize, so I'm gonna call this H, okay? And when I write a universal sequence in the context when this would be H0, this H0 will disappear, so this one is H0, yeah? So we have two P1s, they each has a, in the picture here, each has a hyperplane, so we have to keep them apart. There's H in the basic one that comes into the projective bundle and H0. So then, if we, so from the universal exact sequence gives, gives the following, well, if you, if we write the trun classes, we sort of, we do this kinetic composition, so we can, now we have two sets of classes for E and for this E tilde. So then you just, well, I'm not gonna go into the details of the calculation, but it's completely standard, but gives, so setting these, so with this notation, okay, we have, we have that, and now I'm just gonna write a full series, so to speak, so this is absolutely standard calculations, okay? So you might not even, it is kind of important, but as one, right, so it's something that you definitely want to understand. We also have that, so these are both classes, so this is on QD, D plus one, yeah? So we're looking, so we decompose, we use universal sequence and the kinetic composition on the second P1, so to speak, in the problem, and this is what we have. So here, just to, right, so I should also say that for the, I'm interested in the trun class of the quotient on the projective bundle, and this is in fact, well, again, this is immediate. I remind you in a second of the classes if this is a bit confusing. So in any case, we have, and this is something that is useful and it's a beautiful formula, in fact, that when everything is said and done, the two universal classes differ by just the trun class of the quotient bundle here, okay? And again, I don't write it by, I mean, this is a multi-degree formula, but okay, okay. So, but let me also quickly remind you how, what this class is, okay, so I didn't say that this, okay, so this lambda is actually this hyperplane class, yeah? So it's one of the, okay? So in other words, if we think of this being just a, so this has a structure of a projective bundle over QD, right, as we, so let me remind quickly here what the classes are. So this H is the hyperplane on P1, lambda is, well, there's a hyperplane line bundle here, so this lambda is C1 of L, and in fact, the entire cohomology of this nested Hilbert scheme then is written over the cohomology of QD, where you have these two generators, you have H and you have lambda, and then, you know, of course the relations are that way squared is zero. And also there's a basic relation on, just on any, on the projective bundle, and this is the top churn class of E dual L is equal to zero. Now it's just a standard relation on any projective bundle. So that's why it's no mystery that this A tilde and F tilde then can be written in terms of precisely these classes up here, AF, which generate, which live here, they generate the cohomology of QD, and then, you know, you have these extra two classes, H and lambda to contend with, one coming from P1, the other one is just the hyperplane class. Okay, so any questions so far? Yeah, so then, so then we're in a position to define these Hecke and Akajima type operators in, right, so we have this, so our setup allows us to define, it's the collection of two R operators. So, right, so we, let's, let's remind ourselves as the basic notation here, this maps to QD by phi, and to UD plus one by psi, and while psi is a finite map, is a generically finite map, this phi is not, it has, the fiber has dimension R, because this sits as a projective bundle over QD cross P1 here. Okay, so it factors like this. Okay, so let's, so we have these operators. They will be indexed by, yeah, by, we have two indices here. So what, so this goes from the homology of QD, the homology of QD plus one, and let's say you take a class alpha, or you pull it back to the nested quote scheme, and then you intersect with this canonical class, which is the class with the term classes of the canonical quotient of the projective bundle, and then also with this hyperplane class on P1. So here I ranges from zero to R minus one, this Q is a vector bundle of rank R minus one, and of course A is either zero or one. Yeah, so, right, so if you, if we're looking at what this does, so basically, for example, A zero zero is just the nested Hilbert scheme itself. Yeah, you don't, and for, to generate a rest of the operators, you don't use just the correspondence, which is the obvious one, the incidence correspondence, this nested quote scheme, but you also use these natural cycles on it, yeah, which are these classes. I found it more convenient to work with the term classes of Q as opposed to powers of the hyperplane class, but essentially it's the same thing, no, I mean, so you can say, well, I could use the lambda to the I here, but I think this is better to see some, to calculate with it, in fact. Yeah, so this operator, then how does, what does it do to us? So if to a cycle of a certain dimension, well, it raises by, oh, I'm sorry, minus, I'm sorry, you can, you can see that this is obvious, you can just, okay. So essentially they raise dimension, no, in the case when, well, when the maximal case, when I look at i is equal to r minus one and h, then it's actually, it preserves dimension, but in general, it will raise dimension. But I'm also, of course, interested in looking at the transpose operator, the one that goes from, so if I look at the transpose operator, so given by the same cycle, but I view this as going from qd plus one, qd, okay, then this, the dimension goes down in this way, okay? So this lowers dimension, okay? So one raises the other one, lowers dimension. Okay, so then, well, so the, there are a couple of basic propositions here, the first one is that these operators commute, okay? So, of course, to see this, you look into, so you have to compose, no, you have to compose the operators and they're raising, they're both raising, so you, so you, I'm not gonna show the argument, but I will just, you have to look into, I'm sorry, you have to look at the double nested quote scheme, and then, of course, there's a projection to the nested quote scheme where the length of the quotient is actually one scheme sits in the other with the quotient of length two, okay? So this is, so this here, you look at, I'm using F for the one that will disappear that you project out of, yeah? So then this, so this is where the cycle, the correspondence, the cycle will be supported that the composition of these two operators will be supported on this quote scheme. And then, you know, you just, this, and if I'm just looking at this inclusion, then the torsion here has length two, I'm sorry, the quotient has length two, it's a torsion shift of length two. So then, you know, you just use the basic geometric cell, this is generically two-to-one map, so you can, yeah, so you can argue the commutation, okay? But so, so basically looking at the Poincare polynomial, you would like to say, well, you know, you just act with these operators on the vacuum, on Q zero, so on, there's a, and then they just sort of freely generate the, the, the, the comology of the, of the, of these torsion quote schemes. And that is indeed the case. So, but you do have to, you're interested also in establishing the commutations with the, with the transposed operators, right? The lowering operator, so to speak. So, so we also wish to calculate, right? This type of commutator. So, yeah. So then again, you know, a calculation is needed here. So, you, so, so I'll just say a few words in the remaining time about what this is and how, so maybe I should kind of be a bit organized here and say that, let's say that you want to look at this composition, of course, as a correspondence. So this is a cycle supported on the following scheme. So this is, I'm going to call this Z minus. So this proceeds not, you know, not unlike the calculation of the Nakajima commutation relations for the Heisenberg algebra on the Hilbert scheme. No, I mean, it's, it's this type of, it's kind of this type of analysis. So this Z minus is just, you're looking at, well, so, so three subsheves here, yeah? So, and the main point is that they're contained in the same, I call this F, I'll explain in a second. Okay, so it's a cycle supported like this. So you have three subsheves of CR, but in this relation, so two of them are actually contained in this F minus. And of course, they're contained with quotients, which is a skyscraper sheaf, no? So then of course, then you want to project this, yeah? So you take, so you, so then you take the, well, well, so in fact, you're, how do I, yeah, so let me, maybe, let me see how I organize this best and it's very little time left or so. Yeah, so let me say also about the other one, right? So if I'm looking at A, maybe plus composed with A, I, A, this is a cycle supported on, let me call this Z plus. And the Z plus is the, why it goes. So again, you have three subsheves and the ones you care, so you have E at this E and D tilde. This I think in this course, Kim, this is E tilde and this is F plus, yeah? And they're in this relation. So these E and D tilde are related by the fact that they actually contain a large subshift is F plus. They both contain it. Yeah, so then these correspondences, both this Z minus and Z plus both have dimension, well, the dimension of the quote scheme QD plus R. Now, they play differently, which is why the commutator in general will not be, will not be zero, yeah? So in fact, the Z minus is smooth irreducible. So in fact, you can say what it is. I mean, it's simpler to describe in some sense. So this is over QD minus one is a product of two projective bundles. So here, you have two, you have a universal E one on the first copy of P one and P two. So this is simpler, but this is not the case about Z plus. Z plus is not irreducible. So the plus is not irreducible. And in fact, it has one component which is smooth, one component. Well, if you look at this, I can look at a component when E and E tilde are the same is given by E in CR being equal as a point in the quote scheme to the tilde in CR. And then of course, you have a, you pick a subchief, okay? So this is one component of it. And in fact, so this component, let's call it C, it's kind of a distinguished component plays a major role here. So C is just isomorphic to the projective bundle over. So you see it has the right dimension, it's a big, it's a big locus in fact, in this Z plus. But in any case, so what you want to do is ultimately project to, so you have this, let me, okay, it's okay, I won't discuss too much. So let me just, because let me write this to comparing. So of course you want to project to QD cross QD where you want to compare these two compositions, no? So here, so let me write them side by side. So this is C plus here. And again, you want to project to QD cross QD. So our operators are cycles supported. So what these compositions are cycles supported on Z minus and Z plus respectively, which you then project down to QD times QD. And you hope then you calculate and calculate the difference, no? So, but largely these calculations will overlap. So these, so basically, yeah, let's think a little bit about this setup, I'll just explain because geometrically it's very simple what goes on. So if I'm on this side, where in fact, I'm looking at sub sheaves, which have a common. I'm looking at pairs, E and D tilde, which have a common. Okay, which are, yeah, I'm sorry, I'm sorry. Just a second, I want to do this right. Which are both contained in a common sub sheaf F minus with some quotients. So unless, you see, unless these maps are the same, so this quotient, unless P is equal to Q, and in fact these two maps are exactly the same, this phi and phi tilde. In other words, if E and D tilde coincide as sub sheaves, unless that happens, there's a pullback here, which maps you to the other side. No, so which is basically will give you a point in the correspondency plus. And conversely here, if I start with a sub sheaf of both, unless again, E and D tilde are identical, there is a push out, no? Which they will be sub sheafs of a common sub sheaf minus of the correct degree, degree by one, higher by one, okay? So, and so this is a, so basically this establishes so there's a, there's an isomorphism between an open sub scheme of Z minus and then open sub scheme of Z plus, okay? So that under which, under this isomorphism, I don't have time to explain this, I'm just kind of giving a flavor of this, of this calculation, the classes, the cycles that are supported precisely match on both sides, okay? So then you have to, you have to see where you don't have, where you don't have this picture, where you don't have this isomorphism, and that is exactly over the diagonal in QD times QD. So you have, you always this diagram works perfectly unlike unless the two pairs E and D tilde are actually the same. So then, yeah, so let me write this as a lemma. So there is an isomorphism. So as I said, this is not at all unlike the calculation of the Nakajima commutators on the Hilbert scheme where this type of analysis is also done, okay? Which basically interchanges, which matches the cycle supported on the, on these two sub schemes on Z minus and Z plus, okay? So then you have to deal with what's left essentially and on one side, so you have to examine and this plays differently on the two sides somehow, which is why you get a non-trivial commutator sometimes, which I'll tell you what it is, okay? And so, yeah, so in here there's a big, there's this, this is actually this component C that I described, which sits as a, so this is just a, so in, okay? So this C sitting inside QD cross QD plus one cross QD is in fact isomorphic to, this is just the, let me write it here, write it here. So this is just a projectivization of the universal bundle on the, over the diagonal, yeah? So over the diagonal in QD times QD cross P1, okay? That's how you should view it. So, so then this leads to the following conclusion in some sense, so we have the following proposition which sort of sums up if one does the calculation in detail, so it's, the point is that this commutator, viewed as a correspondence in QD times QD in the homology of QD times QD is always supported on the diagonal. And so, so in particular, concretely we have, well, A, I mean, maybe I should say one here, that this commutator is zero whenever I plus J plus A plus B is less than R, just for dimension reasons, it cannot be supported on the diagonal, it is for dimension reasons. And two, if you have a match, then the commutator is the identity for complementary pairs of operators. So whenever I plus J is equal to R minus one and A plus B is equal to one, okay? And so it's zero otherwise. Now, it does appear that this algebra does not close in the sense that, so you would say, well, you know, this would be, what happens if you exceed this number? So, so more precisely if I plus, well, then you put it like this, it's okay. So it seems that the commutator is calculated by a cycle which can be determined precisely by a tautological cycle on the diagonal. But in any case with this, so the conclusion in, I'm sorry, I went over it, I just, I'll be done in a couple of minutes, but it's kind of hard to interrupt at this particular moment. So, yeah, so what this means though, is that, let's write this maybe as a theorem, is that the homology of the code schemes in this case has a canonical basis given by applying these operators, the operators IA to a basic state zero, say to the zero where, where this just spans is just the homology of q zero, q zero is just a point. Yeah, so then, you know, you write everything basically, you know, as a, and so, okay, you can always, once you know they can, they commute, you can just propagate the vacuum through them, but this commutation, the fact that you can go back, shows you that in fact, it's the, shows you that in fact, the classes you generate in this manner are linearly independent, okay? And then the Poincare polynomial does the rest, the Poincare polynomial basically strongly suggests this picture, and so, then in fact, they, you know, they, they span the homology as well, just because the Poincare polynomial tells you so, but this computing the commutator with the, with the, with the lowering operator, with the transpose is important for understanding that they're actually linearly independent, so you generate linearly independent classes. Yeah, so you, so you have this picture, and yeah, so, you know, so basically you can say, well, you have these, then these basis vectors are labeled by a, by a multi-plat, NIA, you know, which corresponds to applying each operator a suitable number of times, you know, so, and zero one and so on, A i minus one, zero, these are the basis vectors, and these are sort of canonical, there are no choices involved in, in come, in producing these basis. So it's, it's kind of an Akajima basis, I don't know. So, yeah, so I should say also as a remark that it does not matter at all for this, that C is P one can take C an arbitrary smooth projective curve, and also it does not matter at all that you start with a trivial sheath, it could be, it could be anything, so, so it could be quotients of a fixed sheath is zero, but of course, torsion quotients, yeah, so we're in zero quotients on some arbitrary curve, see nothing changes in the analysis that of course on a curvy of odd homology. So the zero and one that I have here, this second index, of course, is there to model the homology of P one, yeah, so you have the class of a point and the fundamental class for an arbitrary curve, you have to include the odd generators, yeah, so then, so then, so then you have a larger family of operators, but in some sense, you know, the analysis is not, it's not, or difficult, and of course, let me maybe stop sharing because I'm here. Yeah, so, so the analysis basically, I would say, I should say that I knew about this, you know, of course I knew about this Poincaré polymer, but his operators for a long time, but giving this lecture is an opportunity for me to sort of understand how to, you know, present this best and yeah, so, and somehow calculate carefully, and you could say, well, what does this have to do with anything, so I just wanted to, because we started with these lecture SL2s, and of course, the idea is that, you know, how do you calculate, you know, we wrote this in the Chao of an abelian variety, and you know, if say, you want to express this on the code scheme, then having such a basis might help, so you can, either of the two approaches I outlined, one is to, you know, I would produce this canonical basis, so then it might be easy to write the lecture SL2 in terms of these operators, and the other approach too, if you have a good system of generators, when you control a bit the ideal of relations, you understand the ideal of relations, you might also be able to just define the operators on the generators, so on just, and then see that they're compatible with the relations, if the relations are geometric, so basically, it's kind of sets the stage for this type of, any calculation on the homology, for example, of the lecture SL2s, but other, you know, other calculations too, so it's kind of, it's good to have, in some sense, this type of description, so yeah, I apologize, you know, I don't know how I always run over, so I run over a lot of time, so if, I don't know if there are any questions here. Thank you very much. I'm actually gonna approve the statements you gave us today, because I've never seen them before, so I was a bit lost. Sorry? I'm gonna approve the things you discussed today, in the end, like the revision or the theorem. Am I gonna approve the- The last part, like when you give the description of Z minus Z plus? To the commentators in, I try to sketch the, how you approach the, it's a calculation, right? So it's actually kind of, yeah, so I try to sketch a bit how you calculate this commentator, basically there's a large overlapping part in the two sides, when you, in the two compositions, and then you have to somehow come to terms with the difference, and so, yeah, this is, I guess, as I said, I'm sort of working them, I worked them out, I mean, I knew for a long time they were there, but I worked them out basically for the purpose of this lecture, and then, yeah, so there isn't, they should be written carefully, but in some sense, I did indicate a little bit, how you know, how you go about it, but I mean, you mean to have this in all, in full detail, and just the calculation of the commentators. Yeah, yeah, maybe, I don't know, maybe I have to go through the details, please. No, no, I mean, it's, for example, I didn't think, you know, for example, the fact that they commute, I, you know, yes, there are arguments to be completely, to be written carefully, you know, to be given carefully, and of course I could not in one hour do it, you know, but yeah, it's just an intersection theoretic calculations now in successive nested quote schemes that give you these commentators, you know. So, but of course, maybe there is an optimal way to, which still maybe I haven't, you know, to present this, because as I said, if you look at, it's not, so maybe you would hope for a kind of a finite Heisenberg algebra, and this doesn't seem to be that because you do have these, some of the commentators will not, yeah, I mean, you would be properly supported on the diagonal for dimensional reasons, so you know, so you'll have some cycle on the diagonal, but I, so I'll have to think a little bit about that. So the algebra doesn't seem to close completely, you see, but it's enough what already the propositions that I wrote, I mean, the fact that they commute and the fact that in there is, so to speak a, for each creation operator is a matching annihilation operator that, so the committer is the identity is enough to provide this basis, so this basis is given, it's enough to show that these are inject, this is, they're linearly independent. Now, this, if you keep applying the creation operator, the vacuum you generate linearly independent vector, so what I, you know what, the propositions I, the commentators I actually calculate are enough for this purpose, but yes, for an optimal presentation of the algebra, I agree there's still, one has to, one has to work maybe a bit more, so thanks for the question, yeah. Actually, I have one comment to say, but just to, which is that this doesn't, you know, this picture, which is fairly simple, you know, it doesn't, doesn't work for, unless for the case of a non-degenerative grass mania, so when R is less than N, so you look at a code scheme in that case, it's more subtle, I mean, you know, the Pankarebo-Lehmann looks also beautiful, but these hecke type correspondences do not tell the whole story in that case, okay, so one has to work harder. Just as I would say it's, you know, in the modular theory of sheaves on surfaces, it's the same situation for rank one, for the Hilbert scheme, the hecke correspondences capture the cohomology, but it's not the case for higher end sheaves, yeah, so for, so it's kind of an interesting one, yeah, so we've been interesting thing to study basically, the case when R is less than N, so here I only focused on the case of torsion quotients, but yeah, anyway, so just, yeah, so I, yeah, I apologize, I ran over time and yeah, and so. Is there a senior infrastructure for all of the coasting? Was there any general theory about this point is how to work with the crazy as an algebra? But here the coasting is for sheaves on Q1, for general understanding, you know, I'm sorry, I have to say that somehow the acoustic, there's something about, I can't hear, I heard a little bit, but not quite completely. So, I mean, here the algebra structure is for cold scheme of sheaves over Q1, is that a similar structure for other cold scheme like sheaves on hygienist curves? Yeah, you know, so for this particular cold scheme, I ended up discussing of torsion when the quotients are torsion, this is smooth for any curve, but in general the cold scheme on a curve is can be quite singular. So then, you know, you have to write, so the structure is much messier. So you can, yeah, so you have to sort of work virtually, but then, you know, how, yeah, so it's a more difficult picture to make sense of. I'm not saying it's impossible, but in principle, lots of things do not go as you, it's not smooth, it can be quite badly behaved, you know, for higher, for higher genus curve, but and an arbitrary gross mind, you know, I mean, in this case, in this kind of, why I won't say the general, but kind of limit case when you're looking at torsion quotients, it's okay. So that this goes through, as I remarked, this is basically, it's the same picture, but in general, no. So another thing that you could, one could look at is the code scheme of zero-dimensional quotients on a surface. Yeah, so sort of generalizing the Hilbert scheme. So you look at quotients, not of C, but of some higher rank trivial sheaf on a surface. So that's also not smooth. But it's, I don't know, it's a little thing. Thank you. Okay. Thank you.