 So, today I will start by recalling the G-spectrum P second, the second model for the DRAM spectrum. And I want to somehow go with some details through this hot theoretic argument and explain somehow the limitation of this argument in the setting of the second spectrum. So, yeah, let's recall that the last lecture I've introduced a T-spectrum by P2, which was defined as some homotopy polemic over some diagram where the object R somehow pairs. R underline is an n-touple of integers of some T-spectrum like this. So, this is a model of omega. So, I did it maybe say it clearly last time, but so one has really a map between omega and P2, which is in fact even so it's a level-wise quasi-isomorphism if things are done correctly. So, this P2, so it's a T-spectrum which is obtained by, good forward, a T-spectrum on regular or smooth schemes, and though it was a map to E bar R plus n. So, in fact, it's a stratified regular schemes. We maybe remind you that maybe that this E bar, this is E bar 1, this is just P1, so this is just a product of P1's. So, yeah, it's a T-spectrum like this and it has the following shape, so, which is as follow. So, this T-spectrum that we push forward is given. So, can you recall, I will read it a little bit. So, what is this? You define a T-spectrum in the first in your general setup of some pre-shifts of certain things, but then when you say on this, it means what? So, it's a T-spectrum on this, it means that I have, so I have a collection of complexes of pre-shifts on this category. So, it is the same and then the T is the same. T is the same, T is just essentially gm or any, yeah, so T would be just the free pre-shift on gm, pre-shift of groups on gm as a pointed set. Okay, so, and this T-spectrum is given like this, so it's obtained by applying the Godma construction on a flaggy pre-shift, so something which depends on the data of a flag. So, it's precisely like this. Okay, so this is the formula for the T-spectrum, so it's a, so it has to take an H-fibrant replacement before pushing forward to get the spectrum on smooth varieties. This is some kind of an infinite suspension operation which happens on the level of a flaggy pre-shift, so, and then this is some simple model with integrable connection or more precisely a pre-shift of such things. Okay, so, this is a formula, so maybe let me say again, so this nrm is, is a, again, flaggy pre-shift with values, model with integrable connection, so this is some too functor. Okay, and in fact, this is, more precisely, it's obtained by pulling back something which already lives on this scheme. So, in fact, obtain small pre-shifts really on e bar r plus n. So, in fact, so it's just, it's appreciative really on the flags on this very simple stratified scheme. Okay, so it's, yeah, all right. So, and because, so maybe let me be a bit more precise here. So, in fact, if you want to evaluate this guy on x and d, so x is a, is an object in this regular stratified over e r plus m, d is a flag in x. So, the way you do this is by taking a pullback along this, this map here. Okay, so where, where d is maps to c, c is a flag in this product of p1s. And then you, you pull back n, right. So, first one defines this and then this is obtained just by pulling back that thing. And, and these are very simple things. So, these are logarithmic, so sheaf of log type. So, just some up to, up to some constant is just algebra on the, on, on, on logarithm. All right. So, this is more or less the thing that I constructed yes, last time, sorry. All right. So, and then I, I, I went, so I did this, so because of this, of this fact that, that this guy is coming from something below, one can use the projection formula. So, using the projection formula for computing r gamma delta of, of these guys, we can rewrite. So, we can, we can rewrite this as follow. So, this guy, this is r gamma is in fact also r gamma. So, it's the same notation, but now this is really that one. And then I tensor with r p lower star of one. All right. So, this is, so this is computed over m, m-circ x d. This is computed over m-circ e bar r plus m. So, it's the same notation, but, but so here it's a global section over this scheme. This is over that scheme. And, and p is just this projection here. Okay. I hope this is clear. Okay. So, instead of, of, of taking this n here, pulling back and then computing comology, I, what I do, I, I compute the comology of the unit. I compute the direct image of the unit and then tensor with n and then compute comology again. This is a projection formula. Okay. So, when I do this, somehow we, we, we can, yeah. So, then, then I, I can replace in this formula, I can replace this part by, by this part. So, it's projection forward of a map of, of this or a map of? It's a morphism of schemes here. Yeah. But it's a, it's a diagram of scheme, but, but I mean, I can, I, I do the projection formula at each part of the diagram. Yeah, and the projection formula is valid without, let's, let's find it. Okay, here it's everything. It's, it ships forward. So, these are modules, integrable connections. Ah, okay. This is our gamma, this is like, this is the derived function of, of the constant. So, you have a module, you take the constant of, and this is the derived function of this, of this operation. And so, because n is really just an extension of, of, of, of the trivial make, you have such a formula. You can, I mean, you can prove it by induction if you want. You can, I mean, if this is one, this is, this formula is clear. This is just the composition of two functions. So, what do you, you wrote, argama tells? So, I, yeah, so this is argama, but computed over this scheme. This is argama computed over that, that one. So, to compute the argama over that one, you can first, you can apply direct image and then compute argama here. This is the same as computing argama here. So, maybe I, I, I mean, this is here also. This is my P. Yeah, it's a complex of modules with integral, with integral connection. So, you take the, the right factor only for modules with integral connection you'll take away. Yes. This is enough to get a correct answer. Yes, exactly. Locally, it's an, it gives you the, the right answer because of Baylinson's theorem. Yeah, yeah, it's everywhere. I mean, it's, yeah, Baylinson's theorem is somehow used behind this, yeah. Okay, so, yeah. So, let, let me now make the following observation. I think it's maybe better to, because I, I think I need some space for this observation. So, let me erase one of those big blackboards. All right, so I told you that, that omega is, is given by, by this spectrum that I have erased, but, and from this we can construct. So, so from, from this equivalence we can construct morphism, cosimplicial spectra. So, this is really up to stable, the morphism is not, maybe is not really defined, but only up to stable A1 equivalence. Okay, so, but in the derived category I can, I can write it like this. So, this is the, this is the cosimplicial object on the derived complex that I'm interested in, maybe I could go here too. And so, then this will go to the following object. So, essentially what I wrote before, but with a small change. So, okay, so I write it again, I write it because so it's a whole co-limit over these indices. I remember that then I have a push forward of the spectrum which is obtained by taking the Godman complex over some fluggy object. Okay, so if I, if I do this, and this is just the spectrum I, I, I just was discussing about. But so here the new thing is that I, I put a church resolution, but, but only on, on this part today. Okay, so, so yeah. Okay, so this is a cosimplicial object and I'm saying that this, the one that I'm interested in somehow maps to that, to that thing. So, this is really not much, not, not so, so the thing that without the C that you have this map. So, without the church, you have a map. And then you have to show that the church, if you put it here somehow can be moved up to here. And this is possible because somehow all the functors are kind of they, they have the pseudo monoida. So there is, there is, there are monoida in the right sense and, and somehow this is what is happening here. Okay, so this is the observation. And so the, and moreover, so you see it here, so moreover one can show that the right-hand side is degree-wise, stably, degree-wise, somehow this object is good enough for computing the Hamilton P limit. So if I, if I want to compute the Hamilton P limit of this guy, I know that it will go to the Hamilton P limit of this thing. And this can be computed using just the total totalization operation on complexes because of this property. Okay, so, so remember the, the, so I had this weak version of the main conjecture which was about constructing a map from the Hamilton P limit of this object to something rather simple. And so the idea is that if I can do this here, I will, I will get the map I want if, if I can come compute this, this part. Okay, I hope, I hope this is clear. And so the strategy is to, let me see here, so, so the strategy for computing the right-hand side or the Hamilton P limit of the right-hand side is to use Poch theory to get rid of this R P lower star of one. So, and by get rid I mean by E, E2 to replace it by, maybe rather if you put Poch to replace it by just one. Okay, so I, I, I want to, so the, the, the idea is that if you, if you pass to the Hamilton P limit and assuming, so, so of course if you could commute Hamilton P limit with all this operation and then show that the Hamilton P limit of this gadget is just one, this is what I want somehow to do in some sense. Then you would be replacing this guy by one and indeed if you can do this then you can prove the, the weak form of the, of the conjecture. So, so this is somehow the main objective is somehow to, to get rid of this part of this diagram. Okay, so let, let me explain to you what I mean by, by using Poch theory. So there is, there is a statement which, which somehow I would like to use to do this. So I, I maybe now state this as a theorem. Okay, so more precisely I would like to use the following theorem. So precisely I would like to use, start the statement here. It's, it's maybe it's a bit long but I start here. So here's a theorem. So I, I said in an abstract situation because it's simpler to do this. So I, I fix C a small category. I give myself a founder M from C to smooth varieties with stratified smooth variety with tangential morphism factor. And I also give myself a, a complex or sorry an algebra maybe commutative algebra in complexes, pre-shifts with values variation of mixed or structure composed with M. Okay, so remember we have a function V H, V M H 0 that goes from here to categories. I compose a function with M, I get two function from C to categories. And I want A to be an algebra in complexes of pre-shifts with values in this two functions. Okay, so this is, this is the setting. So of course, I have in mind C is probably the category of the, so the, the, the, the growth and the construction on the functor flags from, from stratified schemes over, over E bar R plus M. And so for A I have in mind this guy here. So this is, this is my, my A. So I would like to apply this theorem to, to this A. So of course I didn't tell you what is this theorem is about. So this is just the beginning. So we assume, we will assume two conditions and then there will be a conclusion. So let me tell you what are the assumptions. Okay, so we assume, so this is the continuation of the theorem. Assume that so first I would like that, so the group for the weight filtration on A maybe evaluated at an object X. This is, this should be zero for I negative. And it should be, so this is quasi isomorphic. And quasi isomorphic to, to the unit object for I equals zero. So maybe unit variation of force structure. And second, I want that there exists an integer such that the dimension of, of M of X is smaller than, than N for all X and C. So these are the two conditions. And then I can see what is the conclusion. Okay, so then the, the, the church complex church object on A is strongly, is strongly contractable. In the following sense for all, for all D larger than N, N plus one for all N. The following map church less than M plus D is zero in, in the derived category of pre-sheaves, the values in this VHM Composers. Okay, so in the derived category of such object, this map is zero. Okay, so this is the statement of the theorem that I would like to, to use in, in this situation. Okay, so maybe just a comment. So such, such, such a strong contractibility somehow will be sufficient. So if, if I could show that this, that this object here, this church complex is strongly contractable, then this will be enough to compute the homotopy limit of this co-sympatial gadget in the sense that I could then just replace this guy by one. This would be the answer of the homotopy limit. And this is quite simple because you see, I mean, I told you that it's, it's hard to commute homotopy limit with other functors. I mean, namely, for example, with homotopy co-limits and so on. But there is a good situation where this, when, when this is possible is exactly a situation like, like, like this, like where we have strong contractibility. The complex here is the homological or the homological? It's homological. Homological. And so when you, like, be less than or equal to m, it's homological. Yes, yes, yes. It's the co-sympatial degree, yes. So maybe I should say what, what is so, so c less than two, what is this of a, this is just this complex, a, a tensor, a, maybe a tensor to the three. Okay, so, and then, then, then you put zero. So this is the stupid truncation. Exactly. It's a quotient. Exactly. Yes, exactly. It's a quotient. And somehow the total thing is, is kind of a homotopy limit of these truncations. So it's like a tower of, right? So as a pro-object, this is exactly given by this tower. And I'm saying that the, the, the pro, the tower of object if you want is, is isomorphic to the unit tower. This is what, what is this strong in the prosense here. This is what, what, what is this, what this strong contractibility is about. And so if you have this in this process, then, then you can somehow use this to compute effectively this, this, this thing here. Okay, so somehow this is somehow the, this is in some sense the only tool I, I know about, which would, you know, enables, I want to compute such, such a monster here. But unfortunately it does, it, it will not apply to our situation. And I, so for obvious reason, but let me somehow spell this out. Okay, so I, so I told this, I, so how I would like to apply this, I'd like to apply CRM in the case where I take my A to B, this RP lower star of one, one hodge, of course. Okay, so again, so I, I said also, yes, last time that somehow these things do not have a compatible hodge structure, but, but this guy of course has such a thing, because it's just given by push forward of a longer morphism of variety. So we, we do know that this guy has a hodge structure. And in fact it's essentially given by, by, by this. So the RP lower star of one hodge. And yeah, so I would like to apply the CRM to this A. Now let's see what, where this A somehow lives. So you see, so A, so if I fix, if I fix spare RM, then if you remember this RP lower star hodge lives in PHM composed with mixed hodge structure. Okay, so it lives in one of these guys for some, for some flag. But you see the dimension, I mean this is just again GM to some power and, and this power is bounded by R plus M. Right, so in particular this, this, this, the dimension here is less than R plus M. So for, for a fixed indices like this, the, the theorem applies. Okay, so for a fixed pair applies and give us the, the, the motopy limit lower star, et cetera, each vibrant. This can be computed. This is just the same thing except that you don't have any more one. You just have, maybe I put here N, M, insert. So you replace this just by N, okay. And, and of course you, you, you have, you have guessed. So the problems appear when you, when you want to do this independently of the, of the choice of RM, this will not work anymore. So, but, for, but, but you can, but one, one cannot make this, one cannot make this argument for all RM independently. I mean, okay. And this is, of course, it's, it's essentially the same, same problem. All right, so let me know. Okay, so somehow the, this somehow maybe it's a good place to, to tell you what I'm trying to do now. So the, the next thing will be, so I will introduce a third model, which somehow is, on which I can apply somehow the theorem in a more efficient way. So it will not, as I will explain, so that there is a gap and it will not enables me to somehow to, to finish the proof. But somehow it at least it somehow brings us maybe somehow a bit closer to, to, to, to do this computation. Okay, so, so the third model is somehow designed in a way which way, so that it, one can apply the theorem in a more efficient manner. And one cannot. Sorry. Yeah, so if you fix RM, you can compute the homotopy a little bit. But this will not be enough because, so you can compute, you can compute this if you want. So this can be computed. But what, what we really want is computing the homotopy limit of the collimit. Right, so what, what we really want is to compute this. This is what, this is what you want. But this argument can, does not apply here because then the RM is arbitrary large. Ah, so you have to do first whole limb and then whole collim. Yes, you have to do whole limb and then whole collim. This is what, this is what you want to, to do, but, but if you, but this theorem does, does not help you to do this. It helps you to do something different, which is, which is this. And this is not enough. I mean, this is. You need to kind of a junction between, you need, you need to, yeah, if you, if you can commute homotopy limit and whole collimit, you will be done. But in general, it's not something that you, you, you know how to do. I mean, this is, of course, I mean, there are many examples of where, where, where, where the homotopy limit does not come with the homotopy collimit. So here you are not sure if it commutes. This is the point. Yeah, yeah, I mean, I'm not sure. I'm not even trying to, to, to prove this. It's, I'm not trying to commute the homotopy limit with the whole, with the whole. Ah, what are your, ah. I mean, I will abandon this model. It will not, I mean, this is not, I, I, I'm not, I will not try to, to make this argument work. I will, I will introduce a third model on which, in fact, this homo, this, this particular homotopy collimit and this homotopy limit will not be a problem to commute, but there will be another problem that I will explain in a moment. Okay, so. So, so your goal is to get a non-trivial map to, to some, to what to. I want to, to do this. I want to go from here to here, from the homotopy limit. And you can do it if you can exchange the whole limit. Yes. So I have a map if you want from the homotopy collimit. Of the homotopy limit. You need to do homotopy limit also. Yes. So one can go from the homotopy collimit of the whole limit to that thing. But, but somehow the, the natural map is in, in this direction. Hole in, hole collim. So if you have a map like this, you don't, a priority to do some up from here. Okay, so should I, should I erase this and, and make it? I, do you have any map comparing hole in the whole limit? Yeah, yeah, yeah. So you can go from the whole collimit of the whole limit to the whole limit of the whole collimit. Right? But this is not, not going to help you. Oh, because you don't know that the map will extend. Yeah. Something like this. Yes, yes. So it doesn't go in the direction that, okay. Yeah. All right, so this is again, let's, if you say it again. So the, what, what we will try to do is try to, to introduce some more complication in, in, in the models that I'm working with. So this will lead us to the third model. And in, in the third model we will have somehow more. So we'll have a larger sub diagram on which I can apply this, right? So, so, so I have this, this, I have a large diagram of, of varieties. And so I told you that if I, if I look at a sub diagram, this is the one that corresponds to RM fixed. This theorem apply. But this is too small in some sense. If I, if I bound RM, I only get a very small part of this. Huge diagram. And so the, in the, in the third, in the third model I will have again a huge diagram, but somehow the part on which I can apply this theorem is, is significantly larger than, than, than, than what I have here. So this somehow, this is the idea of this amount behind introducing this third model. And so, yeah. I will now start doing this. So I start introducing some notation to, to describe this third model and try to give you an idea or at least a sense why one can somehow go a bit further using this theorem and the third model. Okay, so the, the third model will, will be denoted by P3. And again, it will be also a homotopy columnate over the same, in fact, same diagram. So, but, but this will be of course different. And so I, I will, I only describe, I will only describe this, this t-spectra. Okay, so that's our t-spectra or, yeah. So I, I, I will start with some preliminaries that might look a bit weird. But then I try to, yeah, maybe later I try to somehow, maybe motivate or at least explain where, where these are coming from. Okay, so I, I start with a, with a simple definition. So let, let C be a category. I call a wedge object in C is, is simply a diagram of the following shape. So it's called X. It's a diagram which has just three object that are denoted like this. X11, X01, and X10 was just maps like this. So it's some kind of a wedge diagram. And morphism of, of wedge object is just more commutative diagram as you imagined. Nothing weird. So the, the category of objects in C is denoted by C was a wedge. And are you interested in the probability of such a diagram? No, no, no, not necessarily. Maybe in some sense, yeah. But I will never really take a co-limit of this diagram. But, yeah. All right, so the, the wedge object they form a category, and so example that we are, of thing that we will be considering are like wedge, wedge schemes or stratified, stratified, possibly regularly stratified. So these are the, the example of object that will be considered. When you have schemes in C, when you have such a wedge diagram in scheme, it's often convenient to consider the probability itself. Yeah, but it, it will not exist a priori, yeah? To steam over a common factor. But somehow, yeah. For example, in schemes, this, there's no co-limit in general. You don't have a co-limit, yeah. But somehow, yeah. I will be somehow kind of approximating the co-limit and, and yeah, you will see it's, it's maybe a good. Maybe by general nonsense there must be a co-limit. Okay, yeah. Maybe there is a co-limit. Of a scheme. You don't include finite time. I say could be probably some. Oh yeah, you can take the limit of all the, but it's, yeah. No, maybe. If it's, if it's affine, huh? Yeah, okay. This is the result of, yeah. But I, okay, yeah. Sometimes it can exist, yeah, of course. Not, not always, but. Okay, so that's a diagram of morphing of stratified, okay, so the main point is that morphing of regularly stratified, like what you said. Yeah. Like what you explained before, okay. Yes. Also showing particular finite time over. Yes, finite time over, over a field. Yeah. So everything would be a finite time over a field, smooth and regularly stratified. In fact, this is the main case of, I'm going to sit in. Okay, so, so I want also to define an H topology in this, in this setting. So, so we define H wedge topology on schemes, which is okay. By, as the one generated by families, families of morphism of which schemes of the following, of the following type. So there will be three type of generators. So, yeah, the first one is. And your category, your category of that, of wedge over, over the schemes. Yes, so just so the first one, you ask that the family F11i, okay, so the one on the top is an H cover. And you ask that the other morphism, so the F10i and F01i are quasi-finant. This is the first type of covers. And the second one is, you ask that F10i is an H cover. And you ask that, sorry? Sorry? What is the H topology? Yeah, so this is a topology generated by a surjective proper morphism and etal morphism, etal covers. Or the risky covers. Or the risky covers. Okay, so this is an, okay, wants us to be an H cover. And I want the square to be Cartesian. Okay, so we start, so we start with an H cover of X10. And we just take the cartesian, the pullback of it, cover of X11. And I want the Fi01 to be again quasi-finant. And third is just as in two, but you exchange zero and one. Okay, so maybe remark. So this is the topology for which the points are as follows. So the points are given diagrams of spectra of valuation. Spec of valuation, yeah. So if you want, I can maybe like this. So valuation, okay. Which are, well, this is dominant and local, this map. And I want, and the three of our absolutely integrally closed, all of them. Yeah, it's, in particular, it's in Cillian, yeah. And the whole thing over the base field, you mean? Yeah, yeah, over the base field. And you can take them to be of finite height. But then it will not be all points. No, okay, enough points. Enough points are given by such things. Yeah. And you can assume that the transcendent degrees are finite. Yes. And therefore, you're finite. Rightly many spectra are finite. Yes, and the ranks are finite. Yes. Relational rank of fact. Exactly, yes. Okay, so this is the wedge H topology. Another thing that you can do in this setting is the following. So if you have X regularly stratified. So I define the category of flags. By definition, this is flag of X11. Okay, so a flag and a wedge scheme is just a flag in the top scheme. Of course, once you have a flag in X11, you get a flag also in the two others by push forward. And so if I have a flag in here, I can define stratified, regularly stratified wedge scheme like this. So M plus X11 of the flag and then M plus X10. Okay, so these of course are the projection of D in X10 and so on. And in fact, I will usually just, for simplicity, just write it like this. Okay, so this is wedge object M plus X10. And yeah, in fact, so we would like to take the push forward of such a thing. And so this is why we introduce the following category. So we set, so this is in latex, it's aleph of X, no, not aleph. What is this? Yeah, I want to do this. So how do I write? I don't know. Okay, like this. I don't know. Let me just, is it okay to write it like this? I don't know how to write aleph on the blackboard. But you don't know how to write aleph 0. Okay, I don't know. Let me write like this, aleph. Okay, aleph of XD. So this is the category of commutative squares. So commutative squares in the category. So smooth certified morphism, but with tangential maps. So you is, so this is, so you complete the square. So in this diagram, these are fixed. This is what is varying. So when you vary this, you get a category. Okay, so all such thing, they form a category which I call aleph of XD. Yeah, so X and D is fixed. This determines these three objects and then U is varying. Of course, if there is a co-limit, then there will be, sorry, if there is a push forward, there will be an initial object. And this category will have a new. Like previous question. Huh? That was like. Exactly. So yeah, I will not, I mean, I don't know in general, that there will be no good, no, no good collimates. So I take, I take all these guys. Of course, there is a final object because it's just the point. Initial, did I say final? Initial, initial is also final. Oh yeah, yeah, okay. Final, there's always a final object, yes. Yeah, so the reason I take squares in the category of tangential map in order to have the following fact. So, so this Aleph becomes a contravariant functor from the growth and deconstruction, in fact, growth and deconstruction of wedge stratified schemes. So she needs the category to do the, to the final factor. Yeah, so a two functor will give you a two functor. So, so yeah. Of the co-contravariant, this is a fact, what category? It's a contravariant. So no, flags, it's co-variant, but the functor is contravariant. The functor, so what I'm trying to write here, so sending. So this is a contravariant functor. So what you do, you have, if you have a morphism from x prime d prime to x d, you just somehow change this three thing and you keep the u and let's give you a new square. Okay? Okay, so this is a two functor. And so therefore I can again, again, apply. Anyway, so the, yes, so the callable structure here is just the, the category of stratified scheme and the flag of the type people say there. Yeah. And the morphisms are morphisms of scheme in the usual direction. And, and then the push forward of the flag goes to the, to the flag. As the push forward of the flag goes in the sense of going to the flag means that the, the target flag is, is a small guys. Exactly, yes, yes, yes. So this is that and then f over star of, so this, this dominates this. Ah, the d is less, less. Yeah, this has less than this. Yeah. So, yeah, so, so because I have this function of that, then I can do a double integral. And so this, this category I will just write it as. Okay, so the object here are clear. So an object in this category is, it's a triple x, d and g. So where x is this regularly stratified, which scheme? This is a flag in the top scheme, x11. This is a completion of the square, like that. All right. So, we also, so by applying regular singular to the square, square here. So, sorry, to the square, we get a square of two functions on this double integral. And that I denote by make 1, 0, 0, 1. And that there are natural transformation between them, which are just pullback functions. Okay, and so I need to give a name to this and this. So this is, I call u upper star. This is s upper star. Okay, so for example, what is make regular singular 0, 0. On x, d u, this is simply make regular singular on the open part of u. Okay, so this is, okay, with this I can now start describing this third spectrum. Third spectrum, u is that thing here. All right, so, okay, so give some notation before. So, I fix, as usual, these two indices, r and m. As usual, r is just the sum of entries in r, r underline. So, I'll denote by, so I told you what was r plus m. This is just a Cartesian product of p1's. But from now, I mean, now I will give it slightly different meaning. So this will be the wedge scheme which has the following shape. So, er plus m as before, equality because this choice might seem arbitrary, but this is what I need. Okay, so this is, this would be now my basis. And I also consider the category of regularly stratified scheme, wedge scheme, and over this guy. Okay, so it's a diagram of this shape and n-maps like this and respecting the certification. And this will be endowed with the h wedge topology. So we have a morphism of sites, okay, denote by pi rm, which is, again, as before, it's essentially the projection of this to the point, but one has to make some choice here. So I'll tell you what choice I have to make. So it goes from stratified with this topology to smooth stratified wedge. Oh, no, I don't want this. So here there's no wedge, just the usual thing. And so it's given on an object by sending a stratified scheme to what I call e bar r plus mx, but which is really the following diagram. So just er plus m times x and projecting in on both factors. So e1 bar, this is p1. Okay, I just introduced this because I don't want to write p1 parenthesis. Some power. All right, so yeah, this is set up. So now let's remark the following. So first observation is that this flaggy appreciative I introduced before, this logarithmic type model was a temporary connection. This is naturally appreciative on this double integral with values, sorry. What did I do here? Yeah, that's okay. So it's not, it's maybe I do it like this. Okay, I think I have to write it. So it's slightly different than the thing before, but okay. So the same thing as before except that I have to work over e bar r plus m. Okay, so it's appreciative with values, singular one thing. So what I'm saying here is simply that what you do, your n is defined somehow on this factor here. So you think about it as defined here. And whenever you have something over it, like xd, you can somehow pull back your n to get something over that thing. I hope this is clear. If not, I can say more about this. Okay, is this fine? Okay, so what is this thing? So maybe I tell you what is n r. So I have to tell you what is this module. So the thing is that what you do, you look at x10. Okay, so let c be the image of d in e bar r plus m. So then you have this thing here. m plus x plus x10 d. And over it, you have the same thing. Okay, just a moment. So the n was living here. So this makes sense. I know what is n rm over c. Yeah, this is essentially a product of gms. And this is essentially the logarithmic sheaf on gms. I put it back via this map here and I get something that lives here and this is my guy. Of course, it does not depend on you, but I want to take you into account. Okay, so this is what is- Did you define already n rm? Yeah, this was defined last time. But we really don't need anything. We don't need to know what it is. It's just something which- It's just a logarithmic sheaf which we need to keep track of. And the only thing that matters is that it does not have a hard structure. Just some random- All right, so we take this guy, we put it back, we get here. And now we think about this as a pre-sheaf with values in this guy. This is one thing. Okay, so on the other hand, we have something very simple. We have the unit object considered as a pre-sheaf with values zero, one. Okay, so I have one which is in- I have n which lives here and I have one which lives here. And so the constant p-sheaf is just the unit object. So the idea is somehow to mix these two things together, in here. So yeah, so we mix and one simply by taking push forward and transfer product. Lower star, so derived, lower star, insert, which derived, lower star. And also let me say it now. This guy somehow has by construction a hard structure. So we will be trying to apply the hard theoretic argument on this object. Okay, so now to tell you what is, so to say what is p3, I do the following. So we consider, now we define, we first define a complex by setting the following. So I denote this by q, maybe q3, and indices are m. Okay, so this is a flaggy pre-sheaf. So I have to evaluate this on an object. So a wedge object here, x, and then a choice of a flag in D. So if I fix x and D, then you see I still have u which is varying. So what I do, I just take a co-limit, a homotopic co-limit of u and aleph of this guy or global section of these guys. So this is global section at u0 of this object. Okay, and so once you have a flaggy pre-sheaf, we know how to get a spectrum. Maybe say it quickly. Okay, so finally I put this by definition the push forward along this morphism of sight I just introduced before of the Godma resolution. Again, there is this sigma in the infinity and then this guy. And this has to be made final. So this is this guy. Okay, so it's maybe a good time to take a break. It's tea time. Yeah, maybe you can take five minutes break and then I continue. All right, so I want to maybe just say a few words about the following theorem. So I'm saying that that p3 as defined is stably a1 et al locally equivalent to omega. I will not really prove this. I'll just maybe tell you somehow what's behind this. So maybe try to make this construction less arbitrary somehow. I'll tell you how one comes to this model. So let me recall that I explained to you that this tea spectrum is equivalent to the p1 delta localization of the infinite suspension spectrum under the RAM complex. This is an isomorphism which is in over smooth foliation. And in fact this is this localization is really computed using the et al topology on smooth foliation. Now on smooth foliation we have other topologies and I will need to introduce two of them. So we have two more topologies. We have plenty of multiple but we have two more to consider. So the first one is the so-called Psi et al topology. And so I tell you how it is generated. So there will be three type of covers. So there will be the et al cover. There will be so-called maybe formal excision covers. So these are the following. So you start with the foliation which is smooth and affine. You choose a closed sub-scheme in x. So it may be constructable sub-scheme closed without any scheme structure. And then there is a notion of completion along z which is. So here the foliation of the schemes of finite type on smooth schemes of finite. No, no, no. So any case scheme x, not necessarily a finite type. Smooth is just the omega f which is so it's this smooth. Exactly. Anybody need schemes which are large? Yeah, schemes can be arbitrary large. So you take closed construct subset. There is a notion of completion and you add. So you look also at the complement. Okay, so you have these two maps and you declare that this is a cover. So what you think about this is some kind of a handsilization of x along z. But in somehow in the in this in this world of foliation. So it's rather somehow like formal completion. And then this is yeah, this is somehow natural cover to consider. And then there are somehow a bit weird covers which are like this. So you start again with the with the foliation and then you can look at the reduced closed sub-scheme on f. And this is you declare that this is a cover. He likes red more than still a full year. Yes, yes, yes. In fact, it's yeah. And this is actually diff et al. And some same for this actually diff et al. So you assume only that you have a quotient bundle of the omega one. Yes, which is locally free. Which is locally free and integrative and with the integrity condition. Yeah, yeah. So you have the complex and then you claim that when you tensor it is x, red you get again and the ram complex. Yes. So so in fact the the omega of this is just the pullback. The omega of x, red f is just the pullback of is just a restriction to the. Okay, so these are those derivations. Yeah, yeah, exactly. Exactly, exactly. It's just that. So this is the psi et al topology. So the more interesting covers are really those ones which are kind of new. And then there is the psi foliated topology which is generated by the psi et al and an extra. So psi et al covers and covers of the following four. So maybe we should not go into the details. Let's say nice, subjective families of diff et al. It won't really matter for what to follow. What are these nice one? Okay, so the the point of this is that. So locally, what is this? It's defined. Okay, but then you still you want to have et al, say which are not the fire et al in this. Yeah, yeah, that's it. So this is the difference between et al cover and if et al is really very, very big. So et al covers are just I mean on the level of the underlying scheme. It's just an etal morphism. But here you could have arbitrary large fibers. Dimension wise. Yeah, I could say it, but it would really not help us to understand more. So yeah, so locally for this for the psi foliated topology. So I denote this by a Pcft. So this is the abbreviation for this Pcft. So the derangue complex is acyclic except in degrees zero. So this is a Poincare lemma. And so it's natural to to denote by o delta the H0 of this derangue complex. So these are the this is the kernel of the first derivation in the derangue complex. Right, so this means that this map inclusion is a quasi isomorphism as a local quasi isomorphism. Right, so now let's let's go back to this formula. So then I can I can write this formula slightly differently by saying that this is localization P1 localization delta of sigma infinity t o delta. And then then make this cft log cfc fiber. Okay, so up to small things maybe. So one can show in fact that omega is like omega has has descent for this topology up to small things maybe but and therefore the the the cft fiber model of o delta is just given by by omega. So this so this is somehow just a direct consequence of previous formula. Now yeah now it comes something which is not completely obvious is that you can in fact put this vibrant replacement after localization. So in fact you can localize you can do a P1 delta localization of sigma t infinity delta and and then afterwards apply cft vibrant take vibrant model. The the price to pay here is that you need to to work with c etal topology instead of the etal topology. Right, so there is a map like this and this map turns out to be a stable a1 etal or maybe c etal a1 c etal local and somehow the it's very important to the stable here it's not a level wise a priori it's not a level wise. Okay so once I said this I can now somehow tell you what is the origin of this model. So P3 is the result of computing or the outcome the restriction this guy here of P1 delta. So it's the restriction of these two smooth varieties. Okay so this this mysterious or unmotivated object is just really the you start from here you you compute and you end up with it. So this is how how one comes naturally to this. Okay so this is and and so as a consequence of this isomorphism there you get that you get this property and again I should stress that it is stably equivalent and it's this is really so if you remember for this for the model P2 I actually I had here something much stronger I had that this was level wise quasi isomorphic to that here here we have something much weaker we have stably yeah it's not not so easy you know yeah so yeah it's I mean you see the the the result is not easy so the computation should be as I mean to to write down this complex it took me some time and yeah so to arrive to such a complicated thing you have to compute a lot so yeah it's long it's longing yeah okay so all right so in the remaining of my time I want to explain now what happens if I try to to use this third model and try to apply to it this the whole theoretic argument and see up to where I want one can get one can get with this and so tell you more precisely at what so where this gap in the proof somehow lies and maybe even state the conjecture or at least my hope which somehow removes this this gap okay so so the homotopy limit the third model okay so let me introduce some notation so I I introduced this flaggy pre-sheaf before q or m 33 let me remind you so it takes x and the flag and it takes and it gets it gives you the whole the whole co-limit over over all possible completion of this of this diagram and that's one when deduced from x and d and then there was this r gamma delta over u-circ yeah of here so of r u lower star some simple module of logarithmic type then tensor with or s lower star of one and on this guy I do have a hard structure and so what I will do I will as I did in the second model I will introduce it is a church construction exactly at this object so I put here a church construction put an index n and do it like this okay so this is a flaggy pre-sheaf and maybe I try to keep this in the blackboard so maybe erase this one here p and also an n so this is now a t-spectrum and it is p r lower star of codmon complex on sigma t infinity tilde of q triple n r s and this is vibrant h so this is I'm just so when n is equal to zero this is just the model for omega p3 and sorry so p3 and n this is the whole co-limit right so if I if I vary n I do get a co-simple shell spectrum so varying n right so as before as in the second model as before one has morphism of co-simple shell spectra and again this is really up to stable a1 block equivalence which goes from this church complex on omega to this yeah p one important difference that will come up come up today was the previous setting is that we do not know that the a1 et al or are stable and so therefore we need to stabilize before computing a motor filament okay so but for the moment let let us forget about this problem and and somehow try to see what what what do we have here so we so yeah we would like to apply so the hodge theoretic argument or ingredient to the following situation um yes actually I think c is this a triple integral sorry double integral uh this is my my c and my a this is r s our star of one hodge maybe and what is my m my m is is a functor my m uh which which takes simply x d u to you so this is goes to uh smooth so this is my m right and so of course if we look at this we see that there is no no way to to apply this you know because uh I have no control on my my u so u could be absolutely of arbitrary dimension but let's let's remember again what what was u what u was was this completion so I have I had a diagram so u was such a thing and in fact of course I can always uh replace u by something which is which is finer because I'm taking a homotopy co-limit right so it's if I have uh so you could be very large but then I might find something here which has smaller dimension which factor this way so I so what really matters is somehow the the dimension in the co-limit but also I could somehow refine d and and x and and so on because I'm working for the for some topology and so on and so the idea is to try to to use the these somehow to to control as as much as we can the dimension of of u and so there is a principle which I which I use for this which is the following so right so we need to control dimensions of u and let me again say so by control I mean I mean there will be no no way really to to get to prove that the dimension of u is is bounded but so what we could hope is we could we could try to hope for to try to look for for a large sub diagram so large subcategory here that such as when I when I restrict m to this subcategory I could show that that dcu are bounded or up to refinement there will be there will have one dimension and then hope that this is large enough so that it will give us something interesting at the end so this is somehow the idea and so I'll try to to make this a bit more precise so yeah so what there is one useful principle is the following so the dimension of u is controlled by inside the wedge scheme so I I will not really be able to to make this statement more precise to take some time I just want to maybe give you one example of such a so the simplest example of such or for this principle somehow applies so yeah but as I said so you can take you can embed you in a large affine space but then you doesn't doesn't really somehow make sense because you can refine it by something smaller that's why if you if you put here a a 100 somehow in the column it this will not play any role it's it's this one which will so you can you can you can replace your your category aleph by a co-final subcategory without changing the and what matter at the end is what are the dimension in this co-finals of category okay or so definitely you can always assume that these are dominant in some sense you can not not okay so yeah that's okay I don't want to make this precise so the idea that you don't you what you care about really is is somehow the the inf the dim inf in some sense of the of the object which are in in this category okay no it's not like dim inf okay so something like this doesn't mean I'm not trying to define anything here so I'm just saying that you always can so if you can refine something an object by something you could somehow remove it and just care about the dimension of this refinement okay so so here is this the simplest example of this principle is the following so let's assume that x11 is x10 times x01 and that the flag is a stratum not not the not the flag but just consisting of one stratum then in this case we really have the following we have that the dimension of u is bounded by the co-dimension of c okay so if if the c is is has has more co-dimension then then this will forces the u to be of small dimension and this follow from the following very simple observation okay so this follow from the following simple lemma is that if you have if you have a square of commutative square of schemes finite type over a field so commutative integral scheme and dominant maps then and let's assume that that z embeds in the product closed embedding maybe then indeed the dimension of u is smaller than the co-dimension of the inside the product and this is really very simple just follow from the fact that d is inside the further product so the count of dimension and and so so somehow a refined version of this principle applied to flags and somehow is what is behind the strategy and so I try to maybe to make it a bit more precise but yeah so let's okay so let me fix an integer so I claim that there is a geometric way to get rid flags of of depths so I did not define depths but let's say depths bigger than p plus one so I'll tell you what is this geometric way and somehow we use a geometric way somehow to to cut up to cut a subcategory of this large one here where we are sure that the u that appears are they have small dimension and this this depends on on a fixed integer p so the way we the way we we we we cut this subcategory is as follow so let's so so let let x be a regularly certified wedge scheme and let's assume that that projection to the first person from x11 to x01 is nice and I will not say what is nice but in particular is is flat on strata okay so every strata here goes by a flat morphism to a stratum downstairs so this condition allows us to to define so we denote by u up x this is a pro pro open object obtained by removing closed sub schemes in x11 x01 meeting strata in co-dimension larger than p plus one okay so we look at sub schemes which meets strata in co-dimension larger than this number and we remove them this is give us a pro open subset of of x um meeting intersecting but what do you mean in co-dimension because so if you take a stratum you intersect with your closed subset if this closed subset is of course I mentioned bigger than p plus one in this stratum you remove the in this stratum yes and then you remove the closed or locally closed up closed closed the river closed I decide this for every strata yeah and then you take the union I mean the intersection of the performance and then you do something slightly more so you to your x what you do then you consider b of x this is the pro object of obtained by blowing up faces so this is in the sense of block maybe okay and then you you you compose these two use these two constructions you look at this guy so then there is this fact is that so again I will not be able to make this precise I'm sorry but um so locally for for the h uh watch topology on this guy so on up b of x all the relevant just precise relevant uh flags have that's smaller than p so by by removing this closed subset but but by doing this in in somehow in this block tower of blowing ups one can ensure that some some in some somehow to compute the homology of this guy you only need to consider flags which have depths small therefore the number of guys no depth is something which I will not define is so you should think about depths as something like a co-dimension like the co-dimension of of your flag in in x 11 but it consists of several things so in fact the most important is the co-dimension of the smallest stratum okay so a good approximation of the of the depths of a flag is the co-dimension of the smallest strata but yeah but this but this but it's somehow a bit bit more complicated because we need we need to control you see we need to control the dimension of u but u is also some kind of a normal cone somehow and it has a basis and the basis will be controlled by by the smallest stratum but the other strata will also contribute to the torus over over u okay so it's a bit more complicated than this but but it's a good approximation to think about the depths as just a co-dimension of the smallest stratum sorry can I can can you say again relevant flags have depths smaller than p I I will I will not be able to define so relevant mean that they contribute to the comology of this guy if you want okay so what I call relevant here is that they contribute to the to the comology of this guy and the depth is because what you exclude close-ups came with a large co-dimension in the stratum so you cannot don't eliminate certain yes exactly yeah so the dimension will still be so the there will be flags for example of there will be strata of arbitrary large co-dimension yes but you say they don't they are not relevant no they are relevant but but they have they have good they are not they have good depths in some sense yeah okay I see what I'm saying is it's contradictory in some sense but so I mean the depth is really is also that somehow has to do with so it's it's uh it has to do with the projection you see we have we have these guys so for example if you take a stratum mapping to a strata here uh what what's really somehow yeah maybe a better a better approximation of what is the depth of a stratum is the co-dimension but not in x but rather in so you take the image here and the image here and you take the product maybe so the co-dimension of this in the product okay so if you have this if you have a stratum here you look you look at their images and you could so a better maybe definition is the co-dimension of this in something like c one times easier okay so you can you can have a very so a strata of very high very high co-dimension but but they are still of small depths okay yeah I really cannot do this my time is is over already so I cannot I cannot go more you claim that's roughly that all the the flags that occur are said because you can blow up somehow you get on the depth it's like a torpedo or that you can somehow homologically ignore those which are so in fact so I look in the sentence there is a h topology so in fact you you uh it's not only so you you will so we will have to put on this guy uh finer stratification in fact so it's not only the original stratification which so we start with something stratified but and then we take this this tower but on this tower we have we will put more certification right so in the tower we will have finer and finer stratification and and then the strata will have a priori finer and larger and larger co-dimension but I'm saying that this does not does not does not does not matter in some sense because at the end the u that that will appear will always have bounded dimension in this tower okay so somehow this this is a geometric way of of of controlling the dimension so I tell you what can we do with this so uh yeah so so if if we fix the level so if we fix at p uh what we can do we we can do the fine we we could we we we can compute the a1 localization for example or a1 a1 localization this is this is the sustain voivosky construction by replacing a n so here it's really a n in this in this sense so I should maybe write it like this so it's really okay so this is this uh this wedge scheme uh by by replacing this guy by the up b of it okay so we we we use the usual stratification for example and and and we apply this construction then somehow then independently of r and and m we we can bound the dimension of the u okay so the u that that appear in in this computation and therefore we can apply the hot theoretic argument and we get to an answer and the the the outcome is computable and it is it is very much related to weight p to weight p a motivic commodity and the reason why this why this argument can work is because of the following fact from motivic commodity so this is purity purity motivic commodity saying that if you uh if you take x smooth u inside x open and x minus u of codimension bigger than t plus one then motivic commodity does not see the difference between between x and and u in weight in weight q less than p okay so so if you if you fix a p and we we do this this construction of removing close subset of codimension p plus one and then you apply this argument and the hot theoretic argument and then we look at what we have we have something that that is still interesting essentially because of this lemma and so if we if we do this in in weight uh bigger than p in level bigger than p we get a problem because we don't have anymore this equality uh and in fact we can show that the result so this this outcome that i'm speaking about here will be zero but so this is zero if uh level higher if level bigger than p okay so at each level we have to somehow we have to work with larger and larger open subset and there is no way to do this in in a coherent way in all levels but remember uh i told you that uh to compute the motor p limit one need to stabilize because we don't know that the spectra that you are working with are omega spectra and this is somehow um this is some somehow what what is missing um to conclude and um so i i i thought i had a way to to solve this difficulty uh but somehow it it's relied on some on some fact which is which turned out to be wrong or or yeah um and maybe i i i just write down what was this fact i mean very quickly so it's so uh just to tell you what is the gap so so the gap the gap uh in the proof so it is in uh so it is exactly in the place where i had in the place where where was trying where we we tried to stabilize so the spectra p p3n um and so i more precisely i used i used the fact the wrong fact that that um stabilization can be computed equally uh for for the h wedge topology or the etal wedge i did not define the etal wedge but you can imagine what it is so it's okay so there is there is an analogous true fact um which say that the stabilization can be computed equally for the h topology or for the etal topology and i i had the impression that this was also true for for the wedge version but it turns out that the wedge version it's this is not true um right so so yeah what one has to remember is that that somehow the the problem is really in the stabilization of the spectrum okay so i i have been to the to to t the spectrum yeah and it's fixed so so each each guy like this it's a t spectrum and we would like to know that it is an omega spectrum yeah this is um we would like to make it an omega spectrum because there is no reason to expect this to be an omega spectrum okay so let me just very quickly finish today's lecture so um uh right so i i i tried for for a very long time uh to to get around this problem and it seems to be no way to to do this and so very recently i started to uh to to shift direction in some sense and and tried and so i i started to to to um i started to to think that in fact this guy is maybe already an omega spectrum so uh this is maybe the conjecture uh maybe too optimistic so i could say it like this so a version maybe not not not this guy but some things closely related a version of of these guys um are omega spectrum so this as i said i i i uh maybe a month ago i would think this would be a completely crazy thing to to to imagine but so yeah somehow i started to think that this is maybe not so crazy and in the last lecture i will in fact give you uh i will i will give you a somehow a collection of of reasons uh to believe that this is that this is true um of course we will not be really working with this model because it's uh it's completely impossible somehow to to prove anything about this uh it's it's so complicated so i will maybe rather be working with uh in the in the context of uh fallation um and and try to yeah some give you some convincing evidence that that the thing that the thing that i wrote today so this uh this guy here the localization of of o delta made um vibrant maybe with with some power tensor here uh is an omega spectrum yes i'd try to to give you some evidence for this okay so i i think i stop here today