 Start by recording a few facts from last lecture. So much of what I have said in the last lecture can be summarized in the following diagram. So we have, there is a certain two-funkter which I denoted by Meek Circle Regular Singular Composed with plus. So this is a two-funkter which is defined on a category which is itself a growth and deconstruction of, so well, so over the category of smooth regularly stratified schemes. And this is somehow integrate the functor to such a thing as RS. So it's integration of flags on smooth regularly stratified schemes. And so it's a two-funkter that goes to categories. And in fact it factors like this. So it's by construction it is a composition of a functor with a two-funkter. So this is, yeah. So the functor goes to the category of whose object are again the same here essentially smooth and regularly certified scheme. But the morphism are slightly different. So we take here tangential morphism all over K. So this is this composition. This is here the functor M plus. And this is simply the functor Meek Circle Regular Singular. So here it's just, this is a very simple function. It takes a smooth regularly certified scheme to module with integral connection over the open strata with regular singularities. And this is something which takes a pair consisting of an object here and the flag to something which I denote by M plus XD which is basically the normal cone of the small strata in D with respect to the closure of the larger strata essentially. Okay so this is this diagram that I essentially I constructed in the last lecture. But I keep it for a small moment here but on this blackboard. So and I gave the first use of this thing to define, to write down a complex which will be quasi-isomorphic to the DRAM complex. I also recall this very quickly. So we started, we start with a very simple object. So we start with the unit, so if I call one. So this is, I think about this as, so that you consider, so considered as the flaggy pre-sheaf on smooth stratified schemes with values in this two function. Okay so it's very simple. To a pair like this XD, I just look at the unit object in the category which I get by applying this two function. This is the pre-sheaf and I call it one. So and from this we can do some construction. So for example, we can look at an objective resolution and then apply the global section function of modules with integral connections. So we get this r gamma delta on one. Okay so this and this is now, this is a complex. Okay so it's a unit object in the category of flaggy pre-sheaf on those values in on this category or if you prefer a pre-sheaf on this integral. Okay so whenever I have a two function on some, on some category, I can speak about pre-sheaf with value in this two function. So it's like, okay so. A two function into categories. Yes. Okay so I have a two function into categories and for every object, I pick an object in the image of the. Okay so it's contra variant. Yes. This is contra variant. Yes. Okay so I pick the unit object everywhere but then I can, I resolve this in a coherent way and I get, so this will be another, sorry, the yeah. So if I can take an injective resolution and this would be another object of the same kind. So again it's a pre-sheaf on this integral with value in there and then I apply to this the gamma delta which to a model's integral connection give me the constant sections and this will, will be a complex of flaggy pre-sheafs, pre-sheafs on against most stratified systems and then to this I can, I can somehow get rid of this flaggy by applying this Godman construction. Okay so now this is something more familiar, maybe it's just a pre-sheaf on this category. I can also make this guy h-fibrant. Okay and now it's, I can think about this as a pre-sheaf on smooth varieties. Okay so from this maybe simple but somehow complicated object I can do some construction and I get just a complex of pre-sheafs on, on smooth varieties. So if I want to be precise I would, I would, I should apply here this function c or lower star but if you remember this is, this is this morphism of sight but I'll just to simplify notation I will remove this. Okay so I, I, there is this equivalence of topos between the h-sight on smooth varieties and the h-sight on smooth stratified varieties and we know that this is an equivalence of topos and so I will just to simplify notation I will, I will not write it. Okay and so this, there was this theorem that I, I proved last time that this is quasi-isomorphic to the DRAM complex. Okay so this is what I explained last time. All right so I want to, to use this to illustrate what, what somehow, what, what is the, the goal of this, of this consideration. So I want to use this to prove the following proposition. So let me say the following. Assume that k algebraically closed. Then there exists a morphism of algebras in the effective et al k, k that goes from the DRAM complex to the constant pre-sheave with value k. And so in particular, so as a consequence or in particular. So what did you write in the first line, the letter before the parenthesis? Here. So this was, this was this c o-lower star which is this equivalence of sight for the H topology. And I will not write it anymore. I just, to simplify notation I'll come, what? Course. This is the course. C, c o is for the course stratification. What is what? Complex of pre-sheaves. Okay so now I'm proving the final proposition. So I assume that k is algebraically closed. I am saying that there is a morphism of algebras from this complex, the DRAM complex, to the constant pre-sheave with value k. And so in particular, we can easily compute the homotopy limit of the DRAM complex. So this is, this will be isomorphic to k. So in the effective setting, this is the analogous maybe statement of the main conjecture. It is easy and one can prove it. Okay so I want to illustrate, so I want to use this new model for omega to prove this proposition. There are different ways to do this, but I will just illustrate the use of this new model. I will use it for this proposition. So here's how it works. So I want to do one thing here. So I have been working with modules integral by connection, but in fact one can replace this by any good notion of local system in algebraic geometry. And so in particular, I can use a variation of mixed-horse structure, Vmh. So let me maybe explain this. So for this we need to fix, so I fix an embedding to the complex number. And so for x smooth, I denote by variation of mixed-horse structure of x. This is of course again ind object or ind admissible of mixed structure on x. So the notion of variation of mixed-horse structure, for your structure, is it polarizable? Yes, yes, graded polar. So I want graded polarizable. So you put all the strongest possible condition. So what reference is that for? So I had a hard time to find references for this. There is the paper of Saito on mixed-horse model where these things are done in more generalities and that would apply. So if you want you can also take this to be the least object in the category of mixed-horse modules as a definition for this. But I think that maybe the usual reference is maybe the paper of Zacher and I don't remember. Anyway, it won't be very important. I mean the precise definition won't be very important for us. So I want also- On x, extended to the complex numbers. Yeah, okay, so right. So I would also like to put plus a k-structure. It's not, again, it's not very important. We can just work over the complex number, but I think it's maybe nicer to- Yes, on the connection. So I want the connection to be defined on x, not on xc only. Yes, weight-filtration. The weight-filtration and the connection are defined. And those should be placed. Yes, yes. All right, so this is for x, and again then if x is stratified, as usual I'll denote this to be the same thing but for xc. Okay, and there is also a good factor of nearby, of limiting along a strata. And so one can define in the same manner the two functionalizes. And there is a natural transformation between these two things which is, I call f-doram, like a doram fiber function maybe, or doram component. There is also an analog of the gamma delta. So we have, so I denote by gamma delta and I put mh here for mixed hodge. So the analogous factor is, I think that go from bmh something to mixed hodge structure. Again, this is really ind and plus a k-structure. Okay, and so this is, again, it's just the global section function, or if you want horizontal sections, horizontal global sections. Okay, and yeah, so of course we have a commutative square like this. And here it's really a fiber function. Okay, so with this in hand we can argue as follows, so try to, so the idea is that you can write down this similar object there, but in the context of hodge structure. Okay, so I can consider same object, so the gamma or gamma delta mixed, and now I take one, but really now it's the object in, maybe like this. So the unit object, but in this, in the, with value in this two function here, not that one. Okay, so I can also pass to the fiber replacement. Okay, so this is just really the analog of the previous thing. So what is this really? This is, this is a complex of pre-sheves on smooth varieties with values in mixed structure. Okay, so simply because this guy takes value in mixed structure. And so a claim is that, so if you apply f the ram to that thing, you get back the thing that I was considering previously. So this is maybe again, what is, yeah, what key is, yeah, k vector spaces. Okay, so, so this is compatible with the previous one via the fiber function of the ram. So this is not completely obvious, but in fact this is really, yeah, it's a quasi-isomorphism. So they're not the same really. But somehow, yeah, I don't, I don't want to say more about this. Okay, so, right, so once we have this, it's not difficult to conclude. So here's how, how the proof continues. So, so now, now look, now consider a different, consider another fiber function on mixed structure, namely the one that you obtained by taking f the ram, but first applying the graded, the graduation with respect to the weight filtration. Okay, so, and, and forgetting about the grading afterwards. Okay, so, so you have the weight filtration, you pass through the graded object, you forget the grading, you apply f the ram, and you get a vector space. And so, this is, this is also a fiber functor on mixed structure. And in fact, we know that, so since, so since, since k is algebraically closed, we know that there is an isomorphism like this was, was a, was a previous fiber functor. This is a non-canonical isomorphism, non-canonical, but functorial, which is also, yeah, so compatible with, with the tensor. Okay, and so as a consequence of this, we see that, as a consequence of what I've just done. So we, we see that, that the ram complex which is, which is isomorphic to f the ram of this guy is also isomorphic to f the ram composed with grue. So weight filtration of what comes afterwards, namely, so this fiber interplacement. Okay, and so what I can do then is to project to, sorry, the grue zero part. And in fact, it's not difficult to see that in fact, that I can even put this inside. So let me, okay, so this is the same, but then it's not difficult to see that, that somehow this commute was, was passing to the H fiber specimen and was passing to, to the Godman complex. So I can do it like this. So I can take the Godman complex or f dr grue w zero r gamma delta makes such structure one hodge. Okay, vibrant. And now we are done because so here's a claim. Okay, so f the ram composed with grue w zero of r gamma delta h. This is just complicated symbol for the constant chief k constant. So maybe I just explain quickly what's happening here. So you should think about this r gamma delta. So if you evaluate this, so maybe, yes, this is an equality of flaggy, of flaggy. So yeah, so if you evaluate this on one object, you are looking at the cohomology of some variety, something like a normal cone, open, open strata of a normal cone. But it's a it's a, it's a, it's a cohomology of a smooth variety. And we are taking its grue zero, and it's well known that so it's a smooth connected variety. The grue zero will be essentially k, which will be always k will be the h zero. Okay, so there is maybe a small issue here because our varieties are maybe not defined over k, but I assume that that case algebraically closed, so there should be no issue here. Okay, so case algebraically closed. All my varieties are connected. And so yeah, so it's really an equal isomorphism on the nose. Okay. And then with this, we see that, that if I replace this object by k, this again, this is also k constant, but now as a, as appreciative on smooth varieties. Okay, so this gives, this gives this, this map here. All right, so this was maybe just to give you an idea of what, what am I trying to do here? So somehow the, yeah, the goal of this is to try to, to extend this, this hot, hot theoretic argument to, to the stable situation to apply this to the, to the T spectrum that represents the Rampke module. Okay, so, all right, so, so I said what I wanted to say about the first, the first new model for, for, for omega. Now I want to start discussing the, the construction of the second new model. So I remind you that I will introduce three models for, for the Rampke complex or for the Rampke spectrum. And I did, so the first one was the one I just used in this small proof. And then I will introduce a second one and then a third one maybe in the, in the first lecture. And so, to introduce the second one I need first to, to discuss the notion of P1 delta localization. So I, so this morning I was, I was, I gave a talk about the P1 localization. This is slightly different. There is a delta here. But it's, it's very much related to what I was discussing. But I tried not, not, not to assume that you have been, that, that you have heard me this morning. So I tried quickly to, to, to introduce this. Right. So, so I, so this is, I think the most natural context to, context to speak about this is the context of foliation. So I, I quickly introduce or remind you what is a foliation. So a k-foliation XF, this is a triple. X is a scheme. Omega XF is an OX module. And DFI derivation with, with value in, in this guy. And there are two conditions. So there is the, so derivation would give me a morphism. An O linear morphism like this. And I ask this to be subjective. And the second condition is that I, I want the RAM complex, essentially. So I, I want that, that is a way to go from, so maybe, let me put one here. A bit confusing. So, so I want to go, to be able to go from one, from omega 1 to omega 2 XF. So this is, of course, the wedge of omega 1. So there exists such a thing, compatible, the usual Duran differential. Okay. So simply in the sense that if you put here omega X over k, omega square X over k, and this is the usual Duran complex, this is commutative. Okay. So very, very simple condition. Okay. So that is an obvious notion morphisms. So I don't want to spell this out. There is, so there are two other notions that I will maybe use today is the notion of being deep smooth. For a, for a foliation, this is asking that this sheet is locally free of finite rank. And there is a notion of deep etymorphism. So you can have a non-deep smooth foliation? By definition, yes. So you need this quotient square sheet. Yeah, which is not locally free. I mean, take any singular variety and take its omega, do not change it. This is a foliation, which is singular, which is not local. No, there's a smooth variety. Yeah, yeah, you can, I mean, there is a, yeah, you can put a singular foliation on smooth varieties. For example, yeah, exactly. Yeah. So deep etymorphism, this is morphism of foliation such that the induced map is an etymorphism. And maybe the last thing I want to say here is that there is, of course, a deram complex. Not necessarily, no. It may be, yeah, it's maybe better to do this, but it's not necessary to do this. Okay. So the remark is that, again, maybe in the only in the deep smooth case, it's, no, but this is true in general. So the omega dot, so there is a deram complex, exist, and it is a quotient of the usual one. And in the case where x is deep smooth, it's, I define the deramc homology as a risky homology of x with value in this complex. And another remark is that, so the deram complex that I was considering before, as well as the deram spectrum, both extend naturally to the category SM4, deep smooth foliation. Now the, okay, so the, all right, let me give some examples of foliation that I will need today. So example of foliation, so there are very simple ones. For example, if you take x, any scheme, I denote by x delta, this is the discrete foliation on x. So namely, this is triple x zero zero. If x is smooth, but not necessarily, there is also, I denote also by x, this is the coarse foliation. And in the sense that I do not change the, the sheaf of k-differentials, same, same omega. And then, slightly more interestingly is the following object. So I denote by x, so this is the spectrum kte minus one, with omega x is given by omega, this guy, so kte minus one, okay. Modulo the following relation, tt minus e minus one. So it's a rank one quotient on x and let's define the solution. So yeah, by the way, there are maps here to a one and to a one minus zero. And both are diff eta, okay. And what will be, what I will need today is a compactification of this thing. So one can define a compactification x bar, so contains x, which is, which corresponds, so if this is a one times a one minus zero, and this would be a one times p one. So I tell you quickly how this is obtained. So, so yeah, the idea is that this differential equation can be written in two ways. You can write it either as e dt minus te or e minus one dt plus de minus one, okay. And when you do this, so both, so there's no more denominator in some sense. And so this defines a foliation on kte and, so this one here, and one on k on the spectrum of these rings. And then you can lose them together and obtain this guy, okay. So these are examples, separate example of foliation. So there are also, there are all obvious maps to a one to p one. And this is again this eta, but this is not. And so here the fibers on close point, over close point here, over close point are just, I don't know, to the discrete p one. Okay, so we can think about this as some kind of a local system with fibers p one. Okay, so with this I can now state the following observation, which is somehow, yeah, what is behind this notion of p one delta local. So here's the position, sorry. Here or here? Oh, here, at infinity. It's a copy of a one. So at infinity and zero, you have two copies of a one, a one with the course was with the course for the issue. Yes. Okay, so here's a proposition. So the DRAM, homology of the issue is p one delta local in the following sense. So maybe I give you directly the strong sense. So for all morphism, so given a foliation x and f and given a morphism to a one, the induced math in the DRAM homology is an isomorphism. All right, so in particular also if you, so the commotion of xf is the same as the commotion of xf times p one delta. So in particular case if you take f for example the zero map, we get also p one delta invariance, virtual sense. Okay, so just a word on the proof, it's very simple really. So it's, so the idea is that, is to use the fact that that this x bar to a one is some kind of a local system p one deltas and then, so this reduces the case of showing that the commology of x is the same as the commology of xf times p one. But then there is a CUNIT formula in this context and the DRAM homology of p one is just its coherent homology. Then you use the CUNIT formula and the fact that the DRAM homology of p one delta, this is just coherent homology of p one delta was, of p one was, was, was O, which is just k and degrees. So p one delta is the, the foliation? This is discrete foliation, yes. So the omega is zero and then the DRAM complex has only one term O. And so, so another fact related to this equation is as follows, maybe before I need to, to tell you what is the p one localization. So this is maybe kind of informal, I don't know. I don't want to, at least for the moment to give a precise definition. So, so the, the p one delta localization of a complex for l on smooth foliation is the operation of enforcing the conclusion of the, of the previous proposition. So it's a way of transforming a complex appreciate l into one which has, for which this, this proposition holds. So, and so this, this function is denoted by log p one delta. So the hypercomology for, for fixed topology, the hyper, the hypercomology with value in, in this complex has the same property as the DRAM comology. Okay, so with this I can state the following theorem, which is somehow behind the, the second model that I will introduce. So, okay, so here we work on this site smooth foliation with the topology. So I did not maybe say what was the topology, but I don't think it's, it's more or less clear what it means to have an entire cover in this of foliation. So, okay, so yeah, so yeah, for a time cover is a time cover of the scheme itself. Yes, just, just that, just on the quotient space. Yes, of the total space. So if you forget about this, if you forget about the foliation it's an entire morphism. And the foliation of the target is induced from the foliation on the, on the, all right. So, so we have, so the obvious morphism of t-spectrum from the infinite suspension spectrum on the, on the DRAM complex to, to the DRAM t-spectrum. So exhibits on a gauntlet line as the level-wise P1 delta equalization. Okay, so it's a difference if I, if I try to impose these properties that I, that are there. These omega's are already the extension to foliation. Yes, so I said it's, it's going, it's happening here, so it's, it's already the extension. Okay, so if I, if I, if I try to impose these properties level-wise on this t-spectrum, I just get this, this one here. So at least the first time I saw this, I understood this, I was kind of surprised. Now it's, maybe I'm a bit less surprised, but I still think it's a, it's a nice statement. Okay, so, so once said differently, so the DRAM complex that we care about, this is DRAM, this is DRAM spectrum that we are trying to understand is given by applying this localization function level-wise on the infinite suspension spectrum, which we should consider as being something much, much simpler than, than, than this one for, for some. So, and this is really level-wise quasi-isomorphism. It's really very strong. All right, so the, the new model for omega is, as I said, somehow is a consequence of this formula and some, some computations. So, this, so this is completely false if you would rate for the usual DRAM. For the usual DRAM? Yes, but it is not for this. It doesn't make sense, I mean it's just, it's not false, it doesn't make sense. You do A1 localization. Yeah, no, yeah, if you want, if you replace P1 delta by A1, it's not true, yeah. This is completely false. But, yeah, I mean in some sense there is also something, somewhere hidden here. I didn't, I didn't really tell you what, what was this function. So, but you also need to make an A1 localization here. So, yeah, in fact, you recognize both for A1 and for P1 delta. All right, so, all right, so now I am maybe in good shape in introducing this new model. So, the second, yeah. So, I mean, I did not tell you what was locked P1 very precisely. And for, for this formula to be correct, I really need to, to localize for, also for A1. So, the, I need to take an A1 and P1 five minute replacement. Both should be, sorry, I should impose both properties, A1 invariance and P1 invariance, P1 delta invariance. Okay, for this to be true. Okay, so you have to compose in some order the truth. You have to, yeah, if you want to compose in some order. It does matter a lot, of course, I mean, you have to do it in all orders. I mean, it should be done simultaneously. So, no, not one and the other. It's really both of them. Ah, so it's more than doing one and that. Yeah. So it's not the composition, it's just. Yeah, it's, yeah, I don't know how to call it. So it's not exactly what you have written. No, it's not exactly. Maybe it's better to put an A1 here to say that it's both. Is it like obtained by doing it in an alternate order? You can, yeah, for example, you can do it like this. You can apply it in infinitely many times and one after the other. This would be one way to do this. All right, so the second new model for the drama. So now we are really, we will be really giving model for the drama spectrum, not the complex anymore. So the idea, as I said, this is obtained by, so this is obtained from this formula by a computation, which is somehow similar to, I mean, at least some of the ingredients are similar to the proof I gave last time, showing that this first model I introduced was really a model for the drama complex. So we will see again the same somehow machinery of stratification flags and so on that will appear in the outcome. And of course, so this new order will rely at the concrete or explicit understanding of this p1 delta localization. So I really need a way to ride, to be able to write down this localization, which is the same spirit as somehow the Susan Wojewski complex. And this is in fact quite complicated to write down. So what I will do, I will use this somehow as a black box. I will not give you an explicit complex, which compute this localization. I will just assume that there is such a thing and somehow give you just a good way of thinking about it. Okay, so I try to write some words now. So as a fact or a black box. Okay, so as for the a1 localization and the associated Susan Wojewski complex construction, so that is the p1 delta localization. And to be really honest here, I should say variant of this, variant of the p1 delta localization can be achieved by an explicit construction by an explicit construction that I denote SGP1 delta for the moment because I usually denote the Susan Wojewski construction by SGA1. So just to say that it is a similar construction. Okay, so I can apply it to a complex of pre-ships and get one which is p1 delta localization. But unfortunately this construction is much more involved than the Susan Wojewski construction. So I want to give you just a way of thinking about it without too much really details. So let's see. So this SGP1 delta of F is built by an explicit homotopy collimate construction from the following pieces. Okay, so the thing that contributes to this homotopy limit are, as in the Susan Wojewski case, they are internal homes from some kind of loop space between, so here instead of having just a1, so in the Susan Wojewski, I would just have this algebraic simplest. Here I have something more complicated, so but not so much more complicated a priori. So it's just this affiliation I introduce x bar over a1 and I take some fiber product, so some power over a1. So this is just essentially p1 to zr1 over a1, bundle of this. But then I need to take product of similar things from rn and then there is just an extra copy of a1 minus p1 to some power n. And then this is internal home with my F. So these are what replaces this in the Susan Wojewski setting. So this is analog of this. So these are integers which are varying, so n and vary are 1 up to our n, and varies, all these varying. The gluing maps that are needed to do the homotopy collimate, so the gluing maps induced from again from simple operation, so from simple map are induced, all the maps that you can imagine essentially between these guys. So there are maps going, so you have zero infinity from spec k to p1. So this will induce maps on this factor. That there are also zero infinity from a1 to x, x bar, so x bar has two sections which are isomorphic to a1, zero and infinity. There is also a two morphism from p1 delta to x. They correspond, so if this is a1, these are the fiber at zero and one. And there are also so partial diagonals, partial diagonals, projection. I think that's all. So all these maps somehow are used to build some big diagram, and the homotopy collimate of this diagram will be computing this p1 delta localization. So it's really not useful to tell you what this diagram is. I don't even have this in mind. It's really quite ugly. And in which sense it's a variant of? In the sense that it is, so there are similarities, all right? So the solution for basic construction is a homotopy limit, homotopy collimate. No, that isn't the model for actual what you call p1. Oh, why it's a variant of? Yeah. No, it's not really a model for that. It's not the model for the p1 delta localization. It's something weaker. But it's something which I don't have any good, how to say, which I cannot characterize in a useful way, except that it does, when you apply it to, okay, so I can tell you very quickly what's going on here. So there is this CRM here, that which give you this formula. So it has a proof. And if you look at this proof, you see that you don't need all the properties. You don't need really that you are working with the localization, but something weaker could work enough also. And this weaker, something somehow yell to this construction. It's, I cannot really say more than this. But is it true that if you f is already p1 delta local in this construction, does it change it? It doesn't change, yeah. It will not change. If you start with something which is p1 delta local, it will not change by applying this construction. But the result is not a priority really p1 delta local. But each p1 delta localization is the p1 delta localization of the original. So this is something in between, if you want. It's an approximation of the p1 delta localization. Both this and the construction of the same p1 delta localization. Both this. So if you apply to f and apply this construction, sgp1 delta to f, you have a morphism from f to f. And after applying p1 delta localization, it's an isomorphism. So it's something in between. If you want, it's a good approximation of the p1 delta localization, which has, in the situation I'm interested in, has enough properties to imply this kind of theorem. Is this construction similar to something familiar, like in the same way like such a marvelous case, similar to what we're doing now? So this means that this theorem has a version. And what do you say about this construction relative to the previous theorem? Okay, so if I, maybe I try to write it. So we have this guy, sgp1 delta, which is not really the localization. And it is, somehow it sits in between. It's, in general, there is no reason to expect that these are quasi-isomorphism. But in the case I'm interested in, namely the omega, the sigma. So if this is f, then it turns out that the last map is indeed an equivalence. And this is proven by somehow reproving this theorem there for that guy here, and showing that it's actually just the theorem spectrum. All right, so maybe it's a good time for a break. I just maybe try to finish what I was doing here. Right, so maybe just a few, one notation that we'll be using. So unfortunately, I will have to speak about this construction in the remaining lecture. I mean, most of the remaining lectures. So I will use it as a blackboard. So whenever I want to use this construction, what I will do, I will essentially, okay, I need to say something more, sorry about this. So maybe I take two blackboards before taking the break. All right, so I told you that there is this construction. But in fact, this construction can be even more abstracted as follow. So maybe a remark. So yeah, so the combinatorics behind operation can be abstracted as follow. So whenever we have pieces or objects p, r, m, indexed by a pair where r is itself an n-touple of integers with similar functionalities, more precisely for every map between two such objects induced by one of these guys, you have a map between these pieces and that they should be compatible and so on. So whenever we have an institution, we can build a new complex or a new object by a homotopic collimate procedure. So we can build a whole collimate r, m like this. So I will be using this many times. I will just somehow give you a description of these objects and I just say, okay, I apply then the homotopic collimate and I get something and this something has these properties. I hope this will be okay. I think there is no really no point in giving more details about about this construction. Okay, so I think it's a good place to take a break and then I will now I have I think almost everything to start defining this second model for for the Dirac spectrum. And yeah, I want now to start describing this second new model and now it's really for the t-spectrum omega. So as I said it will be so so this is this new model I denote I will denote this by p2 and so it's a t-spectrum which will be equivalent to omega and it is given by this homotopic collimate that I described of pieces of short not by p2 or m and again this will be t-spectrum on smooth varieties and so what I will do I will simply describe this this object okay so our goal is to tell you what are these. So before doing this I want to introduce a notation which is which I find convenient it's maybe a bit weird but I will so since you see that there was p1 to the m which is which which appeared and it's it's maybe a bit too long to write p1 to the m so I will I will write for p1 I will I will use a different notation I will I will call it e bar and be one and then p1 to the n this is m this is e bar to the m and I will also write e1 for a1 minus 0 and so the reason why why I use this e is because in French there is this word a pointif which means that you have removed a point okay so I will be using this notation hope it's not too confusing all right so the so let us fix these things so remember this is itself it's an r tuple of integers I'll denote by by r this is the sum of these integers r1 plus to rn so I also denote by x bar r underline this is the thing that I appeared previously right so we have we have a map that I will denote by rho rm that goes from x bar r underline times e bar m that goes to e bar r plus m okay so this is just projection on the on the p1 that we have um and so this is my row and I will I call pi rm this map the projection is a point so I'll introduce some notation all right so so with this I can now write down p uh p2 rm except that I will not I haven't yet really defined all the terms so I will spend some time to define the terms later so it will be given by a similar formula than the first model so I will have a good monk complex associated to some r gamma delta applied to some uh flaggy pre-sheaf with values in modulus integrable connection which is now more complicated than just one so but it's still actually not so complicated I will you will see it's rather a simple thing which depends on again the parameters I fixed there's another complication here that remember I am I am trying to define a spectrum this will be only a complex of pre-sheafs and in fact there will be some operation here which uh which is essentially like a suspension infine suspension spectrum but but slightly different so I put a tilde on it so it's essentially that and then of course you need to take an h-fibrant replacement and then you need to restrict to smooth varieties which I will remove from the notation so this this is the except that's not yet and then there is a direct image along this map okay so this guy in fact will live over uh over e bar r plus m and you need to push it forward to the point and yeah this this will be this will be the spectrum okay so I now I try to define the terms of these of this formula so the most important thing is this n and I need I need to I need I need a lemma for that okay so here's a lemma okay so with the same notation here so maybe I put a delta sorry I want to put a delta here okay so let row rm as before okay so this is a this is a morphism of foliation if you want and this is was a coarse structure and these I've told you that is a foliation structure on here and this is a discrete one okay um yeah let so I'll be I'll be considering uh model with integral connection not only regularly singular but even more restrictive I will put log here so this will be uh those of logarithmic type so this is something really very simple it's just essentially um tensor power of of log algebras on so you have a scheme you you pick if an invertible function you can add formally the logarithm of this and this give you some algebra in in mick and yeah taking tensor power of such algebras and some object you get this category of logarithmic of modules of logarithmic types do you need me to say more about this or it's it's quite clear yes yeah there are unique there are extension of trivial of of rank one of trivial module success success of extension sorry no it's not the same it's more more more restrictive you you you might have more extension between one and one which are not logarithmic right so I mean extension of one of one and one there are there are classified by the h1 around and I'm just looking at those which are in the image of the of the delo map in the image of the delo map everywhere or just generically or everywhere on x so you have all star x going to h1 the realm and the image of this classified extension are the global reactions yeah maybe if you can say more I should say more oh okay it's I mean so an object here is a sub quotient of things like like that so you take o x and ln u with u in o star yes and that's so the log log algebras yeah visa to a function of a module with integrable connection of luck so so then there exists unique up to isomorphism object or l r m that lives in this make be the two limit the two sorry there is a unique no the two functor of just I mean just okay I don't know just let this what I whatever okay wanted to make a phrase limit of such when you when you consider as before sometimes you took not only a rank systems but yeah yeah I mean these guys these are already of infinite rank and I take direct some arbitrary direct some so yeah okay any kind all right so there is an object in this guy e or I think I I made I made there so I don't want to take this as before I want to take the open one so not as before but it's the map that goes from the x without a bar and e without a bar okay so I just um if you want the induced map on open strata okay so there is this a unique sorry yeah that one is defeated and in fact no that not not that one yeah that one is defeated exactly uh no I mean not not really sorry because of this delta okay so if m is zero it is defeated but otherwise okay so there is such a thing and a natural isomorphism like this so between so the gamma delta so the functor that that that that takes a model with internal connection of logarithmic type tensor this was guy was this guy and then take the constant sections this this is as a functor it's isomorphic to this thing that we get by taking by first pulling back and then taking global section with respect to this variation okay so the functor that that that takes a model of logarithmic type take its pullback to x r times em delta and take the global section is kind of represented by by this object so this is a really uh maybe very simple proof um so I will not maybe give the proof uh in general but I just I just do two cases for you and now so case case one is uh when we only have uh e1 delta going to delta going to e1 okay so this is the case where where the where n is zero so there's there's no r on the line and m is equal to one well I don't understand the statement there exists a unique such that and a natural isomorphism unique pair if you want up to isomorphism consisting of an object and the natural isomorphism like this so it's like like I'm saying this guy represent this functor here so I have this functor and I want to represent I don't know it's some kind of representability no but you put the same l r m on both sides oh sorry thank you sorry sorry okay I want to discuss so just two cases just to to give you an idea that in fact this is this l r m is essentially uh logarithmic algebra up to some up to constant okay so what is so what happened in this case uh so remember this is spec e e minus one and I can tell you so it's it's really an easy exercise so I just give you a formula for this guy and maybe leave it for you to to check that so this is given like this e so it's an it's a module over this ring and in the connection okay so what you do you add two extra uh element um and a logarithm of a prime okay so with the obvious uh derivation law so uh for example the connection at e prime this is given by uh one tensor d e okay so e prime has the same derivative as as e so somehow uh for example e times e prime minus one will be a constant and that's the log of e prime is e prime minus one d okay so this so this is a module of connection and uh this isomorphism this isomorphism will be induced by uh so we check easily that there is a map module with a table of connection now on on on this uh which uh given by sending e prime to one and log prime to zero okay so so this induces isomorphism here and the the other case I want to discuss is even simpler the case uh case two uh this is the case of x going to e one again this is spec a spec uh k e minus one and here the the l that that that works so it's l 1 0 this is given by just the usual log algebra okay so it's really just a direct proof gives us them but somehow the reason I wanted to introduce these objects in this in this maybe complicated way is because then it is clear that they have some interesting functionality and um for example you see that that this all l r m is really some kind of a of a pre-chief or gets yeah pre-shifted with value in in meek log when I let r and m vary and I take for example these um these maps that are induced by partial diagonals and so okay so it's because of this kind of universal property it makes it clear that that you have that this module has some nice functionality okay so with this I can continue now so um so I want to define remember I want to define this um so I can I try to keep this formula here so my goal is to define my first goal is to define this n here and so n it will be essentially obtained from from this l by by some very simple procedure okay so let me explain this now so uh so we endow r plus m with the obvious certification okay so so p1 is given with certification where the strata r is zero infinity and uh the the complement of these and this induces certification on the on the product of the like this so if you have it if you if you have a strata stratum so every stratum like this is isomorphic is canonically isomorphic to uh to e without the bar for some smaller uh couple r prime and the smaller integer m prime okay this is clear and so on here so over that I have my my my thing I just introduced and I set maybe end of the stratum c to be just this guy okay right so I use this module um and I give it another name but it's really the same it's just again it's a logarithmic algebra um and now it's it lives on on this stratum modulo this isomorphism and this can be extended to uh to flags also so similarly if c underline c0 cn is a flag this thing here then then this m uh m-circ so this thing that I introduced before in this very simple situation uh so this is isomorphic to the largest stratum cn and therefore I can I can think about this um n and c so this is here here I have my mc this guy when I think about it as as a module over that thing I I denote this by okay so this is just just change of notation it's really the same object all the time but when I do this I I get so then then the the association c to this guy is what I call a small fluggy p-sheaf with values okay so it's it's value in this in this uh in this two function but but restricted to to to just one object to to to the to this to the stratified scheme uh e-bar okay so we have this very simple construction uh and in fact this n is essentially that except that you have to put it back to a larger category so I explain this now I'm almost done with this with this guy here so the details are not so important so if you are a bit lost it's not maybe it's not a big deal I just wanted to give it to give this and then we can speak about this result without really knowing what it is we don't really need to keep the definition in mind okay so okay so we can now define if you want the big n that I that I need to make sense of this formula so yeah it's very simple I essentially just need to tell you on which category this is this lives so I denote by regular uh stratification e-bar r plus m this is just category of uh regularly stratified smooth schemes with with a stratified map to e-bar r plus m okay before I was looking at smooth stratified single over a field uh I referring to right to write smooth here because smooth over something mean a smooth morphism and here it's really just the target which is smooth but the morphism is really arbitrary it's just a stratified morphism to that thing so this is this is my category and um see whenever I have an object here and a flag uh of e-bar r plus m is so each so e-bar is just a product of p ones and on each p one you take the stratification given by the point zero infinity and the complement all right so if I take an object there and the and the and the flag I can I can map this flag into a flag see underline in here then I have an induced map on the normal cones here I have my n and c which lives here and like I just uh take its pullback pullback along this map and get what I call n r m d underline okay and so uh fact or the association d is a flag if pretty safe with values in compose with m plus okay and this is exactly the the thing which is here okay but then I can pass to uh to kohmology or or derive double sections apply this thing that I still need to define and then apply godman resolution and then push push forward everything and this will be my p okay so I still need to uh to tell you what is this guy so this is very simple okay so sigma t infinity tilde so is the version of the infinite suspension function but you see um it's it's not on uh complexes of pre-sheaf it should operate on uh complexes of flaggy pre-sheaf okay so that operates complexes okay so more precisely uh if you have a if you have f of flaggy pre-sheaf uh so this guy is given like this so so it's a transfer product of flaggy pre-sheaf and uh with one particular object okay so where is as follow so I just write this into a lemma so um there exists uh pre-sheaf or star tilde which lives on uh smooth stratified scheme with uh tangential map uh so sending x which is stratified and and and smooth to simply o star of of x zero x zero okay so you have a stratified scheme and you you just look at function which but but not on x rather on the open stratum when it turns out that this is a well-defined pre-sheaf on smooth stratified scheme but of course you need to to take tangential map between them not not usual map otherwise it will not work and so so this is an easy exercise and then uh this guy here really is uh the thing that I'm using here is is that thing but composed with maybe sorry no because this is this is so so t is not o star it's equivalent to o star so if you remember okay I mean it depends on on choices but my choice of t is rather something like the free the free uh pre-sheaf on this pre-sheaf of sets so so the multiplication is not function but but rather the formal addition so you want this thing to be at t spectrum right but but yeah so if you are a spectrum for o star you are a t spectrum because you have a map to o star so if you have if you have a multiplication by o star you have a t spectrum also so it's kind of yeah so but uh why is it what about tangential is q how does it affect o star is yeah I mean everything is tensile with q so I'm working up to torsion okay so o star is already o star tensile with q yes yeah but you have not written this you've written o star right but I mean f f is a q is a q valued pre-sheaf so it doesn't f has value with q on q vector space so I can tensor and this would be again okay all right so the yeah pre-sheaf o star yeah okay so this is really yeah my o tilde is really o star can compose with m plus okay this is the o star which is here so this is again it's a it's a flaggy pre-sheaf uh and so it makes sense it makes sense to tense to speak about to tensor it was with that thing up there and then take the godman resolution and then push everything to the point after taking a five meter placement okay so this is this is this is a new this is a second new model for omega let's think I still have a few minutes I would like to um to try to see if it's possible to to repeat this uh this hotch teoretic argument using this this model uh of of omega and see what what what what what are the difficulties in so maybe I can can maybe erase this it's okay all right so so when we when we start to think about this we see one immediate problem is that uh this guy cannot be a priori enriched with with a structure of a variation of mixed mixed hot structure um so here's the first problem a variation of mixed so in the in the in the case of of a complex of pre-sheaf so the effective case I had here one and one was obviously uh endowed with such a structure but here n is uh even if it's very simple um in fact one can show that it doesn't have uh I mean so if you if you fix um I mean if you fix the flag if you if you fix everything each single uh model with internal recognition I mean has actually many many ways of many many structure of uh of vm8 hs but you cannot find one which would be compatible with with all the other by via the the the map between them I hope this is clear I don't know I mean each one is just a logarithmic sheaf the logarithmic module and and it's it's very easy to put such a structure on one of them but if you want to put a structure on all of them in a compatible way then then then this will not be possible I mean is it because the the exponents are defined or you know deleted or some this kind of the exponent uh no I mean it's it's because of okay so it's because of the other okay I can try to say same bit more so you see that there are these maps uh p1 delta so that there are two maps uh zero and one so maybe this is uh okay so this is over over a one so here you have your you have l uh I think it was something like uh one zero and here you have l uh one or or l empty one if you remember so both of them are very simple logarithmic sheaves but this is not canonical so this this this had uh this had a lot of constant in fact um and uh so so yeah and so so these two map will will uh will give you two map from here to here or rather from the pullback to here um and these two map are in fact different and I think it's probably that's it uh one one will um is this and what are the maps from p1 to x so so this is over zero and one so the the fiber so this is delta so the fiber at zero and one there are copies of p1 and and they they map uh so this is this is send so here you have the variable t and it is sent to zero and one um and what is e1 e1 delta is it e1 was was supposed to be at the this is gm a1 minus zero okay and now does it map to zero one uh this is spec k projection to the point projection to the point but you put two of them okay sorry spec k yes so that there are two maps here it's not a competitive diagram but all right so I think yeah if I remember correctly the problem would comes from the fact that the log is sometimes sent to zero and sometimes set to one and um and but but the log has has weight uh has weight two so as an element and so there will be a problem uh because of this x here what is x is this is this spectrum of k t e minus one with this uh d e minus or d t minus so this is this is a problem but one can try to go a bit further around this as as follow so remember n is really is defined over uh over over this e so one can do the following so if you have if I fix x and d uh as before and I look at this make like uh sorry I look at this projection and plus x other line d where c is the the flag which is the image of d so what I did so I had my my uh n and c here and I pull it back here okay so this was the way I constructed my d and then I am doing again a push forward um I'm doing uh yeah I'm taking a global section again um but uh so you see why there is this projection there is a projection formula so if you take something which is which is here put it back and then take the global section it's like first pushing forward tensoring with this object and then and taking a global section again simple projection formula so but by doing this we can rewrite this in a slightly different way so I need to introduce some notation because unfortunately I mean it's very simple but I need to introduce some notation um to write it down so uh I denote uh make uh with a tilde uh singular of uh x uh what yeah I think I want to I want to write like this so I I define this to be make like a singular of of this quotient there so so of this uh okay so for x d as before I I just define a new category which is the model model but over over this this thing which is just the power of g m it's really very stupid um that there is a functor which is uh yeah I call this let's call it p lower star from the thing that I was working with before okay um and with this I can now write this contribution in a slightly different manner again it's the same the same object as before a direct image more complex uh also if it's suspension our gamma delta but now it's a gamma delta over uh over these kind of objects and then of n r m tensored with p lower star or r p lower star okay this is just projection formula applied uh applied here applied yeah was this was this was this thing and now the thing is that this guy has a has a mixed heart structure this has a structure a variation admissible variation in fact a very simple one because we are still working over a product of g m so this is again just essentially a logarithmic sheaf very simple but we have this formula and we could try to so so the idea is that try to get rid of this part here using a variant of the harsh territory now this is a slightly different situation than than before so maybe let me remind you so uh so in the case of of p1 which was which was a model for for the DRAM complex not of the DRAM spectrum so we we we were working so we had to so we used the weight filtration in the category of mix of mix of mixed heart structure that's we we were working somehow over the point and uh it was enough to to construct this this morphism from from omega to the to the constant sheaf uh so here it won't be enough because so I I don't I cannot get um uh mixed heart structure on the point because if I if I do if I if I do this I would have to tensor with uh n r m and then take a global section and and this will destroy uh this this structure here so I really have to to consider this uh this thing here as a as an object and then and work on it before applying this operation and and this forced me to work over with variation of mix of structure over some power of gm so something like okay so yeah so so we need to adapt the argument to variation of mixed heart structure on basis like just the power of of some of some gm so priority very simple but um yeah so so there are many so so there are problems here so for example we don't have any more an isomorphism between uh the the drum uh component composed with was graded for the weight version and the drum component no such isomorphism over a base but somehow the there is there is a way to to get rid of I mean to there is an argument which which would work but it will it will it work somehow only when the base is is fixed so so if if this integers of these data are fixed then there is a way to to adapt somehow what working over finitely many bases one one can still do something the problem comes when you want to to to let the the base or the the dimension of the base being as big as as necessary so then then yeah so problems appears when the dimension of the basis is unbounded so it's sometimes this is a problem of of commuting so you see we my p double is just is a is a homotopy call limit over rm and I'm just telling you that I can work with with each one alone that there is a way to to adopt the argument but there is no way to to do this for for the whole thing so in fact it's it's somehow we are like lacking a commutation between homotopy limit and homotopy co-limit which is essentially which is somehow the essential problem again all right so I could go a bit more in detail but somehow my time is up today so I so in the next lecture what I will try to do I will try to again go back to this argument make it a bit more precise and introduce a third model which is somehow the design somehow or or has some feature which which makes this argument that that that that is here is completely useless make it at least more useful all right so with the third model one can somehow go a step forward using the harsh theoretic argument and and then I can explain to you what is the actual gap in the proof and then then maybe also in the in the next lecture I will propose a conjectural statement which would be somehow enough to somehow enough to to repair the gap or somehow to to completely avoid the gap in some sense okay I think I maybe it's good time to stop here today