 So I start by, in fact, quickly recalling a few facts from the last lecture. So let me, first point is, so I explained that the conservativity conjecture somehow would follow from a good understanding of this, of some particular object in the A, which is a homotopy limit, some explicit cosimplicial motive, so this lives in, and naturally was coefficient in K, cofibrant replacement, so it's like taking a projective resolution of something, because this involves a tensor product operation, so we'd better have some cofibrant object instead. Right, so the conservativity would follow from understanding this object, and so the ideal thing would be to completely compute this and hopefully arriving to something, to the unit object here, so this is a conjecture that this should be just the unit object in this category, maybe something potentially less difficult is to be able to map, to find maps from this object to something which is simple enough, and I told you that somehow the weakest thing that we would like to have is at least something which is like the truncation of the RAM complex. So if you can go from here to here by non-zero map, then conservativity would follow. And I also said that it's important really here to be taking the homotopy limit with respect to, so the homotopy limit should be computed, relative to the so-called the stable A1 et al. local model structure. So this is necessary if you want to application to conservativity for example, but there is, so we can somehow restate this problem without speaking about this model structure and somehow in a more concrete way as follow. Or said differently, I can tell you what it means to be computing this homotopy limit in a more concrete way. So in fact, so here's if you want a formula for this. Okay, so this is, we know that this is the same as taking the total complex of some, so maybe I got the formula and then I explain. Yes, yes, this is a total, exactly, this symbol, exactly, this is very important, yeah. So let me maybe just write the formula and then explain the terms. This is something like lambda infinity. There's a composition of three functors that you apply to this omega, but we also have to apply this really on an etal-fibrant replacement. Okay, so this is if you want a concrete formula for what this mean. So here, the etal-fibrant replacement is taking level, so maybe, yeah, I should say that, so this is a consimplitial spectrum. So there is a level and there is also a consimplitial degree. And so we take the etal-fibrant placement level-wise and also degree-wise. So this is what it means, the symbol on this object. And this is the so-called Susslin-Weibersky construction. So this is something where it's a t-spectrum and also you have the check piece. So how many indices do we have? So this object has two indices. There is a level of the spectrum and there is a consimplitial degree. So it's like, it's a consimplitial t-spectrum. So it's a consimplitial object in the category of t-spectrum. Just two directions. Yeah, but there is also another code in every code. Yeah, yeah, I mean each spectrum, yeah, yeah, but let's forget about this. Yeah, there are three degrees, yes. You can also think about it as a t-spectrum in consimplitial complexes, if you want. So it's like a t-spectrum at each level, it's really a consimplitial complex. Consimplitial object in complexes of pre-sheaves. Yeah, yeah, so, yeah. Okay, so this guy is the Susslin-Weibersky construction. It's really given by, so the formula is something, it's again something like, if you want, it's a homotopy collimate, but it's a rather simple one. It's over this category delta. If you prefer, it's just a total complex, doesn't matter, of something like this. Internal home of algebraic simplices applied to our thing. And so this is also applied level-wise and degree-wise, applied and degree-wise, okay? And somehow the most nasty one is this number infinity. This is the stabilization t-spectrum. It is given by, so it's a collimate over the integer of some internal home of a tensor power of the take motive against a suspension of the spectrum. A shift in the level of the spectrum. So we shift the level n time. We replace the level 0 by level n. And so this is applied, this is applied degree-wise again. But not, of course, level-wise, it's somehow, it mixes the level. You want S, what do you write there? Yeah, this is, yeah, this is a, so this is S minus on a spectrum is just the spectrum E n plus 1 and you apply it n time, okay? All right, so this is what we are, what we would like to compute. Now, of course, it's computation up to, up to a motor piece. So we only, we only really care about the A1 et al homotopy type of this, okay? And there is maybe one thing to be said here. So there's also an extra complication because so, yeah, so this guy is a degree, a degree n, it is given by a tensor power, the spectra, so n plus 1 time. And I didn't really tell you what is, what is the tensor power of spectra. So the tensor power of spectra is actually quite complicated. But fortunately, I'll explain this maybe today that there is, we don't really need to understand, so this will not really be causing any serious problem. But just to tell you that this is not simply just in degree n, it's not just a tensor power of n copies of the complex omega. It's more complicated than this, okay? So if I, if I look at level R, this is different from just the complex omega tensor with itself n plus 1 time. So the polynomial to the polynomial should define the wedge of spectrum? Yeah, there is this problem, but even if you define it in the correct way, what you end up is something more complicated than this. Yeah, no, yeah, but I'm saying that in topology first, there is a kind of a, it allows you, I mean, a way where you choose, you start to do something, then you have some other ways to, yeah. So here it's the same kind. Yeah, it's the same kind of, but this will not be really a problem. The tensor power will not be, so I will not, the reason why I don't define the tensor power here is because I don't need it. I will give you another way somehow to represent this object. But I'm just, just want to say that it's not, even if you do it in the right way, it's not given level wise by this, by this object, by this power. It's more complicated than this. All right, so may also be a remark. So this operation that I, that are here, and even that, so the operations for lambda infinity, SCA1 and metal fiber replacement, are only necessary in costimplicial degrees one and above. And the reason is that because in degree zero, what do we have here? It's just the spectrum omega. So in degree zero, it's the spectrum omega that we know already is stable, it's A1 fiber, A1 local, et al. Fibrant, so it has all the good properties. But somehow the problem is that these properties are completely destroyed when we just, when we tensor omega with itself. So we need to apply them again, and this is why somehow it's a hard problem. So the, yeah, so the, so the approach I was trying to do, to develop, to compute this object, relies on some new models of this spectrum omega. So the approach I try to develop, relies on a new model for the spectrum omega. So why those things are, you said are necessary only in costimplicial degree one and above? Because in costimplicial degree zero, the object that we have is already A1 et al. Fibrant, it's omega, omega underline. And it is an omega spectrum? Yeah, so if you're a sick to smooth, I find varieties, it is already an omega spectrum which is even level-wise A1 et al. Fibrant, so it has all the properties that we are trying to impose here. So there's no need to apply this. All right, so, yeah, so the approach relies on a new model for omega. So by a model of omega, it's again a T spectrum which will be isomorphic to omega in DA but somehow looks different in some sense. All right, so it's a T spectrum isomorphic to omega only in DA. So it's a priori not level-wise quasi-isomorphic to omega but after localizing, it's isomorphic to omega. All right, so, but might look, this is what I call a model for omega. And so this new model, with this new model it will be somehow easier to do the sensor product for example and this is why I don't need the sensor of spectra in general. When we see this model, you will see that there is a natural way to also to extend it to get models for the power, for the tensor power of omega. Yeah, so it will be convenient for many things. And actually, yeah, so to introduce this model it's unfortunately quite complicated and I will probably need most of this lecture and the next lecture to do this. And in fact, I will introduce three models and I'm only interested in the last one but I do this because the first one is actually rather simple. It's not so interesting but at least it give you an idea of what kind of objects I'm trying to describe. The second one is complicated but not as complicated as the third one but it is the third one that I really need so somehow I decided to do this step by step so by introducing these three models. You said you really need but you could not prove. I mean, I will explain where you... Yeah, I'll explain where the gap is at some point, yeah. I believe it's still useful, yeah. But this will be maybe the last lecture. I tried to convince you that it's useful. Huh? You'll be away. Okay, so maybe I will be lecturing with an empty room but this is my plan. Okay, so... Yeah, there will be one guy at least. Yeah, yeah, yeah. Okay, so this is what I'm trying to do and... Okay, so... Yeah, so I start by... Yeah, but by... So this is the subject of today's lecture is new models, the Durham spectrum. Okay, so as I said, these are models which are rather complicated and just to write them, in fact I would need to introduce some tools and develop some notation and so on. So I will be starting with this. So there will be a section about general tools. So as you will see, these are... Yeah, I will always be working with Durham Convergy. So now somehow we can forget it for a while about the problem. I will introduce some very elementary things. And then I will use them somehow to, again, to give this new models later. And so the general theme will be about this stratification on schemes and some construction around this. Okay, so... Let's recall what... Recall that stratification on an Italian scheme, let's say X. This is simply a collection, called S, of locally closed subsets, which we call strata. And sometimes if I want to be precise, I would say S strata. With two properties, such that X to be set theoretically is set theoretically the disjoint union of strata. In particular, that would be only... Yeah, yeah, this is implied actually by this. And second is that every strata, the closure of every strata is, again, union of strata. I should say... I want them to be connected. So close connected. Yeah, by definition... Yes, yes, yes. By definition it's connected, non-empty. This is my definition, I mean... Certainly not far from the... There were some books that were given. The empty set was connected and then... Oh, no, okay. No, no, but then, now in... Now it's not connected. Now it's still like in the stacks, but it is explicit that the empty set is... It's not connected. But there were some... Anyway, we couldn't go to this... Yeah, so for me, connected is not empty and this is my definition of a stratification. And I would... So when I say a stratified scheme, this means... So this is just a pair X with a choice of a stratification. And usually, of course, I just say X. I would not write down specifically this other symbol. There is also a notion of morphism, stratified schemes. So f, y to x, this is... It's said to be a morphine scheme if... If for every stratum in y there exists a stratum in X with f t inside, inside X, inside S. This is the definition of stratified morphism. Okay? You mean that the maps send any stratum into another stratum? Yes, exactly. So every stratum would be mapped inside another stratum. I still have some space here. I would also introduce one... So I would say that a subset of a stratified X is constructable if it is a union. So if I need to be more precise, I would say maybe SX constructable. And so in particular morphism of... So a stratified morphism has a property that inverse image preserves constructable subset. This is actually even also another definition of being a stratified morphism. Okay? So in fact I will be mostly interested in a subclass of stratified schemes that I call regularly stratified. These are very simple actually, very simple certifications. So maybe also... So yeah, if X is regular, is a regular scheme, we say that the stratification... Let me say that it's regular with the stratification and stratified. Okay? So I would say that SX is regular. So the stratification is regular. If there exists a strict normal crossing divisor, we are defining the stratification... Defining SX in the sense that the strata are exactly connected component, intersection, reducible component. So it's really a union of divisor, which are smooth, which are regular. So you have to take the intersection of the reducible component minus the smaller intersection. Right, you're right. So it's a section of... Yeah, okay. Minus the other. Your strata should have to be at some level, at some homogenous level. Yeah, yeah. So the code I mentioned should be also the correct one and so on. So it's... Yeah, so... Then of course D is uniquely determined by this stratification. It's equivalent to give D or to give the stratification. Okay, and so such an X, I would call it regularly stratified. Yes, exactly. The same data. All right, so let me now do some notation. So I'll denote by scheme sigma K. This is the category of stratified, finite type pre-schemes and morphism or stratified morphisms. And inside here I look at those which are regularly stratified. Okay, so I have these two categories. Of course, sigma is for stratification. K is my ground field, exactly. And I have... So there are obvious functors in both directions, but let me just consider those one. So that I call CO. And CO is for course. So the function... What it does, it takes a scheme and it gives you a scheme with the course stratification. Namely, this is the strat out, exactly the connected component of X. And similarly for smooth one. Okay, is this clear? So I call this inclusion CO for course. All right, so now continue. So given a topology, tau on schemes or smooth schemes, I get one on the stratified version by declaring that a cover, by declaring that a family is a tau cover if it is so after forgetting the stratification. Okay, so for example, if you take X stratified, then the map from X to its course version. So this is... I forget the stratification and then I look at the... I put the course one. This will be always covered for any choice of tau. Okay, so I don't ask anything about the compatibility of strata. Okay, so... And you have this basic simple lemma. So the first part is very easy. So for any possible choice of tau, so this functor from schemes K to schemes K induces an equivalence. It's an equivalence of sight. I don't know if this is the usual terminology, but it definitely defines an equivalence of topology. And the second one is more interesting. It's actually only two four. So I just assume here that tau is either the CDH topology or the H topology. Sorry? So the functor course, so that sends a scheme X to the scheme with the course stratification, only the connected component. Okay, so this functor is a morphism of sight kind of in the other direction. This is what you mean. Sorry? Right, yeah, yeah. I don't know how I should write it, but okay, if you prefer this, I also prefer this, so... I should write it maybe like this. No, it depends. This is a delicate convention but I already wrote it in the right way. So maybe I put a star. I don't know. Yes, if I look at the sight, maybe I put the topology here. So if I look at this side, so I have this morphism of sight and this is an equivalence of sight. Is it okay? Now, the second part is... Yeah, equivalent topos. So for tau, one of these two topologies, so we have a diagram of equivalence of sight. We have a square as follow. Okay, I try to put it here. Okay, let me maybe erase a blackboard. So the same one as before. For tau, CDH or H. And then there are the smooth variant. So these are induced by the inclusions. Okay, so all these are equivalence of sight. So this is, again, the course. Yeah, so the CDH topology is a topology which is generated by the... The Nisnevich topology and the blow-up square. So whenever you blow up something, it's like a cover under some... H is generated... Okay, so it's generated by finite... Okay, I think it's generated by the Nisnevich topology and anything which is finite and proper and projected. There is also a delicate point of finding a morphism of sight because the category of smooth things doesn't have a fiber product. So you have... But here also there is no fiber product. I was always working with... But it doesn't matter because it's an equivalence. Here and because of the resolution of singularity, anything can be dominated. And so it's... So the inverse image is an equivalence of category, so it's a morphism of topos and it's an equivalence of... Yeah, and it is a morphism of sight, so the one is to verify something. You get to define the topology in general. If you have a topology, not in particular one you took and you want to restrict it to smooth schemes, you say, well, I take families of smooth morphs of smooth schemes which are covered in... Then you have to verify that this is really a topology which involves this delicacy of the very same... So it doesn't quite work, you have to... There are different notions of... So here the only reason is that we have resolution of singularity. We need... It's used to... I'm in characteristic zero, so... I think for the H topology I think you can... You can use the youngest of singularity also, but I'm working in characteristic zero, so there is no issue. So this is really a very easy... Then I will not say much about... There is nothing really to prove, essentially. But it's somehow interesting because then... Remember, my goal is somehow to define some new T-spectrum on smooth varieties and in fact what I will do, I will... In fact, first define something here using a stratification and then use this equivalence to transfer the T-spectrum on something here. So this will play... This is why I'm introducing these categories. Okay, so... Yeah, so now I want to discuss the notion of Godemont complex in this setting. Maybe Godemont construction is a slightly more general context. Okay, so... Of course you know that there is an order on strata. So if I have two strata S and T inside some stratified scheme, when I write this, this means that S is contained in the closure of T. The natural order on strata. Yeah, okay, so then... When I write it like this, it means that S is inside T. And I now introduce a notion of flag. So let's say X is a stratified scheme. A flag or a flag of course with respect to the certification. So it's an n-tuple. So let's say a flag of length n is an n-plus-1-tuple. I wrote the underline, consist of d0, but up to dn. And these are strata. So d0 and dn are strata. Exactly, so they are in the increasing order. Yeah, and this is strict now. I don't want repetition, okay? This is a flag. So I put a category structure on the set of flags. So I'll denote by flag of X. So this is a set of flag. And I would declare that there is a map between one flag to another if d is a refinement. So I think about refining as dominating another flag. This is just a poset, but I think about it as a category. More, more strata. So c is included in d. It's maybe slightly confusing, but I want that d is larger than c, okay? So are c and d the same length? No, no, no, no. One is contained in the other. So all the strata of c appears in d, but there might be more than that. There is identity. Identity would be just equality. There is also one other thing I want to say. So if I'm given a morphism of stratified, then there is a notion of, there is a functor, a flower star, that goes from flags on Y to flags on X. A direct image of flags. And it takes d to something. d, which is the whose, so the strata which makes this flag are exactly those containing an image, containing Yf of a stratum in d, appearing in d. Okay, so what we do, we look at the images of the strata that are in d and then we just remove the repetitions and then this will give me a flag in X. Okay, so this is a simple construction. Give me a two functor. So this defines... So a flag, you can't have an empty flag. No, no, no. The empty flag does not exist and the lengths start from zero. I don't want an empty flag. All right, so this defines a two functor, flags that goes from a category of schemes, 0 by k, to categories. So it takes a stratified scheme to the category of flags. So it's a co-variant two functor. All right, and so for V, I would consider... So for... If I'm given a subcategory of schemes, okay, so example, and this is maybe essentially the main example I'm interested in is... I look at the smooth object. So we can consider... So called the grotendic construction. The grotendic construction. So consider the grotendic. This is how it's called, the grotendic construction. So which is associated to a two-functor category, which is very simple. This is the object, our pairs. So we give ourselves a stratified scheme and a flag, and morphisms consist of a morphism from y to x, plus a morphism from C. This is correct. I'm confused now. In my note, I have the other... Yeah, so I think I want to decide the other one. So I want... What do I want to see? I'm not confused about this, but let me think just for one moment. Yeah, I think this is correct. This is correct, this is correct. This is kind of the core fiber, the total space of the core fiber category. Yeah. And V is a subcategory or a full subcategory? It's a full subcategory. I mean this is in this case essentially, but... This is x in V, x in V also. Which one is in V? X is always in V. Yeah, sorry, I should say. So x is in V. And f will be also a map in V. Okay? Just... Actually, we can take any... Not necessarily full any subcategory. It doesn't matter. A subcategory? Yeah, I choose a subcategory. Sometimes I don't want to look at that one, but maybe at that one or something even smaller. Okay, so... And now I come to this definition, which is this notion. So it sounds a bit weird. If you have a better name, I would be happy to hear any suggestions. So I call the flaggy pre-sheaf on V as before. So V is a subcategory of stratified schemes. So it's just a contravariant functor on this... on this scrutiny construction. So essentially it's a functor which associates to x and a flag, an object, and has a restriction map and so on. Okay? And so if f is such an object, I sometimes just write fd instead fxt. So if it's clear... If d is clear in which... So a flag in which stratified scheme, I just write this notation. Okay, so... And now I can tell you what is this Godma construction. Yes, I'm sure about it. So f of d is more... Yes. More guys than c. Yeah. So like identity... So if x is equal to y, I want d going to c. I want this direction. A map... Is bigger. Yeah, d is bigger. So maybe I can give you an example to keep in mind. So here's maybe an example. So let's assume that... Or maybe, I don't know, motivation for this definition, not another one. So it's the following. So let's assume that everything is affine. That's for simplicity. Everything is affine. So meaning that also the strat are affines... Affine and... So then if you have x and a flag d, you can construct somehow the completion or the successive completion of x along d. Or you can even do it, for example, for the Henselian. So what you do, you look at the first stratum and then you look at the Henselian localization of x along the first stratum. Yeah? Is this...? Yeah, yeah, I said everything is affine. I'm just giving a motivation. So I'm just... Let's assume that everything is affine. I take the Henselization of x at d0, then pull back d1, and again look at the Henselization of what I get at d1 and so on. Okay? Should I write it, or...? So you first Henselize directly with the smaller... Smaller? Then look at the inverse image of the second one. It could be disconnected, but you don't care if everything and you continue. Yeah. I mean, no, the first one will not be disconnected. Okay, let's say I'm in the smooth... Let's say that this is also smooth. Just... Okay. Start about... Yeah, it's just... Okay, so in good cases you don't have even any problems. So you just... It's still connected and so on. And so whenever you have a map like this, it will give you a map like this. So with this definition, you get a functor. This construction becomes factorial. And this is somehow one of the reasons that... Okay. I want to choose this definition, not anything else. Okay, is this... And similarly for the... For the completion also. Yeah, for the completion. And this is why also we took this direction for the flag. Because when you have a refinement of a flag, it gives you something which goes in this direction. Okay. All right. So now I want to describe the construction of the Godmore... So the Godmore construction. So we start with F, a flaggy pre-sheaf, let's say on V, with values in any additive category. So it could be complexes, for example, of vector spaces. And so we define... So we associate to F a usual complex of pre-sheaf. We associate the complex of pre-sheafs by the following formulas. So that I denote by chi F. So this is my complex. chi dot F. And so I tell you how it's given. So a degree M, this is just a product of sequence. Sorry. Chi. Here? Chi. So in that text, it's chi. Chi. I don't know. Capital chi, I suppose. Yeah. And so this is chi maybe, not chi. I don't know. What is the name for this? So in that text, it's chi. Yeah, probably. This is the capital. So yeah. So a degree M is given by... You look at sequence of morphism of flags. So this is in flag of X. So I should say I'm evaluating at X here. So this is evaluated at X. It's given by taking the product over a sequence of lengths M plus 1 in the category of flags of X. And simply you evaluate F at the first one. This is what is this complex in degree M. In fact, this is just... You can maybe set it here. You raise maybe this. And in fact, this has a... You don't really need this explicit form, but just give it. But in fact, if you might also know that this is just... This thing is another name for... It's for the homotopy limit over the category of flags. So this is an explicit complex which computes the homotopy limit along this very simple category. I mean, this is just a poset. And a homotopy limit over this poset or rather nice, it's just a finite poset. And this is an explicit form which computes this object. So I'm just taking the homotopy limit. So I have a flaggy pre-sheaf. I look at the variety X, which is stratified. And I take the limit over the flags of my flaggy pre-sheaf. And I get one object, and this would be the value of my object at X. This defines this complex of pre-sheaf chi F. Is this clear? E0 is a graph. All these guys are flags. And I'm looking at morphisms of flags. So it's a bit complicated, but maybe it's better to think about it like this. It's a homotopy limit along a category where the objects are flags. So this is a pre-sheaf on this category. If I fix my X, I let D vary. This is a pre-sheaf here. I take its global section, homotopy limit. I could take every arrow to be attached to D. But I take them all. So I take the product over all these guys. It's a product over all possible arrows. This is a degree M. So this gives a complex. It's not important, but it's a way. The chi is a way to pass from flaggy pre-sheafs to usual pre-sheafs. And it's just a variant of the Goodmong resolution complex. And one thing also, if F itself is a complex, then when I write chi F, I usually mean that as a simple complex associated to the double complex. I don't want to put the total everywhere, so it's just... The best way to think about it is just as a homotopy limit. So when it is a complex, you take the product... No, yes, sorry, yes, the product, yes. But it's not really a serious product because this is a fact. So it's really a homotopy limit over a finite poset. It's not really a big... it commutes with filtered collimits, for example. Because here you take the un-overlies version. Yes, if you take the normalization, then there will be finitely many... Exactly, exactly. And that cannot be auto-equivalent. Exactly. Okay. All right, so this... Yeah, okay, I... Now continue. Okay, so now I continue with these general tools. But from now on, I will essentially working with regularly certified schemes. So what will follow somehow depends very much on this regularity assumption. All right, so let's see. Here's a remark. So let X be a smooth, regularly stratified. So remember, this means that X is itself a regular scheme. And the stratification that we have on X is induced by a strict normal crossing device. So if I take a stratum, then the normal bundle along the closure of this stratum... So this is... C bar will be a nice regular close subscheme of X. And this is just a normal bundle. Then this is also naturally stratified. So the stratification on X induces one also on the normal bundle. So one way to... A good way to think about this is by using the deformation to the normal cone. So use... So let's assume that X is defined by some normal crossing device or D. So we're equal to D1, Dn. Then what we do, we look at the strict. So take the fiber at zero, transform of D in the deformation to normal cone. So this fiber at zero will be also a strict normal crossing device in this NXC bar and define the stratification that I'm speaking about. So it's a very simple construction. So this kind... So the stratification somehow... Then one can repeat this. And so in fact, we have the following lemma. Let X as before, so regularly stratified C inside X stratum. Maybe by the way, I want to introduce a notation before that. Let me erase this. So I will not be using this notation for the normal cone. So just a slightly different one, which I find more convenient. So from now on, I will denote by N plus XC. This is NXC bar. And I will denote by N zero XC. This is the open stratum. So here, in fact, there is only one stratum which is open. And I call it N zero. And by the way, if given X a stratified scheme, I denote by X zero the union of open stratum. I should have said maybe this before. So here are two notations that I will be using. Sorry, I cannot. Can you... Moving to open stratum, you mean a stratum that's open? Yeah, a stratum which is open. Can you have a stratum which is non-empty interior but is not open? Can you say again? Can you have a stratum which is non-empty interior but is not... But it's not open. No. No, I don't think you can have that. No, no, you cannot have that. So the exil will be always dense in X. This is always dense. No, in general. Why is it dense? Do you want me to give a proof of this? So if you take a generic point and you look at... So you have finally many generic points, right? So you fix one and all the others that are contained in some strata. You take their closure. This is constructable. The complement is also constructable. It's open and contains a generic point. So the complement is even reducible and open. It's not important anyway. Anyway, for a regular thing it's certainly true. Maybe you can do it while I'm erasing. So do you agree that if X is reducible there is no problem? Okay, good. Okay, I still have this black part. So I wanted to give this lemma some kind of transitivity. So again, X regularly stratified. C inside X is stratum. Now I also take another stratum, but now in the normal cone. So how do I call it? E. E also is stratum. Of course we have a map from this guy to C bar, to the closure of C. And we let D be the image of E in C bar by the natural map. So in fact D is also a stratum. Okay, so with this notation we have the normal bundle in here. So in N plus XC of E, this is canonically isomorphic to N plus XT. So we have this kind of permanent condition. And this is a stratified scheme. Okay, so yeah, so using this lemma we can define the category. So here's a construction that will be important for us. So I first maybe define what I call tangential morphism. So let us take two X and Y regularly stratified schemes, which are assumed to be connected. Assumed connected. It's not important, but I think it's natural to do this. I assume them to be connected. So a tangential from X to Y is a pair of C and XY consist of a stratum Y to X. Stratum C and X and morphism from Y to N X N plus to the normal cone of C bar. But I put a plus instead of bar. Such that, so sending the open stratum inside the open stratum there. Okay, so this is what I call a tangential morphism. And the nice thing that they form a category so we can compose them. So given another such guy, so maybe G, we define the composition of this H. Yeah, I'm defining a category. So I define the composition to be a pair H F where, so I tell you what is H and what is F. Okay, so I need first to do some other notation. So first let E. Okay, such that or as follow. As follow. Okay, so maybe try to do something. So yeah, so what do we have here? We have F and we have this G, so maybe I write it here. So G is a map that goes from G to the normal cone of D and Y. Okay. All right, so and so in particular we have D stratum in Y. So what we can do, we can apply to, so we can look at F star D. Okay, so this is a stratum. Let's call it E, the normal cone. Okay, so this is, this is the only stratum that contains the image of D. By F. Yes, thank you. Right, so. You can apply G to F to that. Yes, so I can, so yeah, so since F, so yeah, so because of this. D is a stratum in Y. Maybe I should make a picture. I don't know. So I have X, I have Y, I have Z. I have two stratums to start with C and D. And I have maps that goes to the normal cone. Okay, so what I can do, I can do one thing to start with is to take the image of this. So the image, the push forward, call it E. Okay, now this map here induces a map from the normal cone. Okay, by functionality of this normal cone. But good, this is the same guy as here. And now I can use this lemma which is, which is here to write this as a normal cone in X, but of something else. This is N plus X F, where F is the image of E in C bar. Okay, so F and X. Okay, and so, and this is my H. Okay, so this is how the composition works. Okay, so I want to check that this is a well-defined composition. It's associative. The unit is given by the identity map with the open stratum. Yeah, so it's given the category and I now give a name to this. It's too late for everybody. Should we make a break? Let me finish maybe. All right, I just forgot about this. But it's maybe not a good time. Let me say a few things and then I... Let me finish these two blackboards and then I take a break. Ah, yes, yes, yes. I'm happy about this, so we can continue without noise. Okay, so you want to... Yeah, so notation, I denote by smooth sigma T over K. So this is the category of connected stratified, so naked smooths, regularly stratified schemes, plus a tangential model. And so maybe I just finish by telling you why it is interesting and natural to do this. Just a statement of proposition which I will not prove, of course. It will be too... Is it that there is a way for extending K to the right closure because the scheme can disconnect and so it's not... Yeah, okay, maybe. I'm not using... Let's see. Yeah, but this is my definition, but maybe, yeah. I'm not going to extend my base scheme today at least. So, yeah, I don't see... Yeah, I keep this in mind, that there might be some problem when extending K. Maybe the better notion would be to ask that it is geometrically connected so that there is no... Maybe this is better, indeed, yeah. But somehow the only... Of geometrically connected, maybe it's better, yeah. Yeah. All right, so... Okay, so why do we care about this? Because of the following proposition, so that it exists. Maybe first notation. I still have some place here, so it is another notation. So if you have a scheme X, let's say smooth, or maybe smooth over a field, I'll denote by M, I, C, or S, X. So this is the category of end module. This is the category of modules, integrable connection, which are filtered unions, or just unions. Over X. Yes, over X. For X, which are equal to the union of their sub-objects, which are locally free, a finite rank, over OK. No, no, here, no, no. X just is smooth, yeah. So a finite rank with regular singularities at infinity. Okay, so I... This is quite clear, so module is integrable connection, regular singular at infinity, and on X, they are smooth. But I allow somehow end object, so they can be as big as I want, but they should be, yeah, filtered union of... What is what, sorry? Where? Here? Yeah, yeah, yeah. Integrrable connection over X, integrable connection. So it's a module with a connection, nabla, integrable, and many nice property. It won't play a very important role, but... Sorry? You mean zero? Yeah, it's locally free, so there's no torsion. And at infinity, it has regular singularities. Okay, so essentially these are, if you are overseas, this is like giving yourself a local system. Okay? But maybe an end local system. And... I... It's not the system of finite rank. Sorry? No, no. A module is not the system. No, but it is a union of one which is finite rank. Okay? And so if X is now stratified and regularly stratified, I'll denote by make... So I just somehow interchange the circle. So this will be... And the reason why I do this is because it's because of the finite proposition. Yes, exactly, yes. Propositions. So there exists a contravariant to function. So make circle RS. That goes from this category of tangential map. So smooth sigma tangential. Okay. Categories. And send X to... And what it does on tangential map is as follows. So if you have FC from Y to X, this gold map, this goes to the following functor. So let's say FC upper star. So this is make regular singular X0. You apply the so-called limiting functor. So this is like taking some kind of limit along this stratum C. And then you would go to make regular singular of the normal cone of the open part. And then since your map F... Remember in the condition that I asked my F to preserve the open strata. So you have an induced F upper star here that goes to make regular singular Y. So this is the definition of the pullback map. Okay. And so this somehow explains the introduction of this tangential map. And I think it's a good maybe place to make a break now. Okay. So the next point I want to discuss is somehow a combination of the notion of flags and the notion of tangential morphism. So maybe I'll first start with a notation. New paragraph notation. So take X, maybe that is stratified. And now I take a flag sort of thing. Just taking strata in X. Okay. And so I define a variant if you want of this N plus... And I call it M instead of N. So this is M plus X of the flag. And this is just the N plus, but not now of X, but rather of the closure of the largest strata that appears in my flag. Relate that to the D0. And in particular you see it depends only on the two endpoints of the flag. It doesn't really depends on what's happening in between. Okay. So, and similarly N0. And all right. So here's a lemma. So say that the association was assignment mapping to N plus XD. So extends to... Extends into function N plus X. From flags on X to this tangents. Okay. So whenever I have a morphism of flag I deduce... So these are stratified smooth scheme and they are connected. And so since it is... You see it's connected. And so whenever I have a morphism of flag, I not necessarily have a morphism of stratified scheme, but I have always a tangential morphism like this. There is a small issue here in this lemma, is that this function is not unital. So as I will explain and we see from the construction that it does not take the identity of flags to identity. Okay. So I will very quickly discuss this. So I don't want to get too much into the details, but just give you an idea of how this is constructed. So, and in fact, as I said here, so it's only somehow the two end point of the flag that matters. So in fact, what I will do, I'll just take three strata. And tell you what are the map from N d2 bar plus d0. And the second one is... So there are two maps. So this is the largest flag. And there are two possible maps. So I want to describe this map to you and somehow then the statement is obtained by composing these maps and showing that it doesn't depend on something. But this is somehow the interesting part is to describe these maps. Okay. So this is one, this is two, and I will tell you what are these tangential morphism. Okay. So here it will be a true tangential morphism in the sense that it's not... So the pair consisting of morphism and strata, so the morphism will not be a morphism between these two things. So you really need to do something. So here the stratum will be d0. That I consider as a stratum here by... So here you have d1 bar as a zero section and therefore d0 is also inside. And so you have this stratum here. And if you do the normal cone of this guy, then you get exactly this by the lemma that I explained before. And so this is the map that is defining this tangential morphism. So you take this isomorphism and this stratum here and then this gives you this. I'll tell you. You will see in a moment. I think it's already visible here. You see, if d1 is equal to d0, so you will have identity here always as always. But so in fact the problem will come from the stratum because the identity in the category of tangential map is always given by the open strata. And here we are forced to take an unopened one. So in this case, if you have d0 equal d1, then this is not the open strata. Also the two objects are the same. The two objects are the same. Yes, so I want to... It should be the identity map because there's only... So I'm looking at the flag d0, d2, d1, or d2 if you want, equal to d0, d1. So because these two strata are equal. And this identity map should go to the identity map if my function would be in Italian. But the way I define the image of this arrow is by saying it's given by this isomorphism and the stratum d0. But the stratum d0 is not the open stratum in this guy. So the identity map here would be... Instead of d0, it would be n0. So this is the first one. The second one is as easy. So again you take the stratum would be d0, the zero section in here. And there is an obvious map. From here to... Actually there is a morphism of stratified scheme already from this to this. It's not difficult to see. Yeah. I mean both of them are over d0 bar. And there are normal bundles over d0 bar. So you see we have the situation. We have x. We have d1 bar. We have d0 bar. Sorry. d2 bar, d1 bar, d0 bar. I'm taking the normal cone of d0 inside here and the normal cone of d0 inside d1. And you see that there is a map from one to the other. Yeah. In fact there are two maps. Yeah. Should I say more about this? Anyway, so... Yeah, so the fact that it's not a unital, it's not really a big issue and I will ignore this from... So let's say I ignore this is not unital. And if this is a problem for you, I just maybe tell you in word what can be done. One way to deal with this is just to add a formal unit to the category of flags. So I just add for every object, every flag that I add a unit formally, and then the identity of flags will not be a unit anymore. But there's an easy way to overcome this. It's not important. All right, so we are almost done with the preliminaries. So, yeah, also I have this for my grandma, is that in fact, the functor... Okay, so we even have a functor M plus that goes from the grotendic construction on flags, tangential morphism. So not only we have functorality for morphism of flags, but also for morphism of stratified schemes. And we can put all this together into such a functor. So this is sending X E to M plus E underlined. Okay, so maybe not so important somehow to remember how all this is constructed then, but just at least on objects. So we have to a flag to a scheme and the flag. We have this essentially some kind of a normal cone of the small stratum inside the large. Okay, and so finally we get this nice two functor. We'll play some role from now on. So finally, putting all these things together, we get this nice two functor, which is a composition of the previous one, the mick regular singular in the open stratum composed with this plus. So it goes from the grotendic construction on smooth stratified schemes on the flags to categories and it's sending a pair to make regular singular M0. What I'm saying here is that so whenever you have a flag, you can look at this some kind of open normal cone and the category of modules with regular singularities there is functorial for some morphism that forms this big category. So we have this object. So this will be used in describing these new models for omega. So I still have some time and I will give you the first model for omega, regular singular. So the model with regular singularities on the open stratum. You didn't put it before, but you assumed that it was in the sense of the laners. So the reason why I put regular singularities is because I want to have this limiting construction. I mean, it exists in more generality, but I don't mean more than this. So I can now try to introduce the first and simplest model for new model. And in fact, it won't be really a model for the spectrum omega but for the Dirac complex. So I will define a complex on smooth variety which will be quasi-isomorphic or equivalent to the Dirac complex using this thing here. Okay, so here's the idea. So I need, let me give this one general more definition. Just practical. So let's assume that you have v a small category and you have some two functor. So like something like that one. So v could be this growth and deconstruction. m could be this mic composes m plus. So we can define contravariant. And I want this also to be contravariant. Okay, so we can define what is, so a pre-chief on v with value in m. It's more or less clear what this means. So it's a way to, for every object x, you would associate an object in m of x and you want a restriction map in the usual way. Okay, so I don't know, maybe I should very quickly space this out. So it's an assignment that for an object, you have an object of x. And for a morphism, you have a map from f upper star f of x to, and of course f upper star is inverse image for this two functor. Okay, and of course plus associativity and the obvious notion. And maybe I just say it orally. So in this particular case, I would also speak about flaggy pre-chief on certified smooth schemes with value in this two functor. Okay, so because it's really defined on flags, so it makes sense. And again, just let me say it orally. So in good cases, for example, if m takes value in an abelian categories and if the inverse image functors are right exact, then the resulting category of pre-chief is also an abelian category. And if you have, for example, if this has like also have a right adjoint and the categories are of growth ending, then also the resulting category will be of growth ending. So you can speak about injectives. So if the inverse image functors in these two functors, so you have this f upper star, they should be exact on the right. If they are right exact, then the resulting category is abelian. The category of pre-chief with values in m. When the categories of themselves are of the kind of functors are they exact? Yeah, add it if an exact. And also it's a growth ending category. Yeah, it's also, maybe, I mean, it would suffice to assume that f upper star has a right adjoint. So then it commutes with filtered co-limit. Co-limit. Okay, so for instance, in this case, the resulting category will be abelian of growth ending and we can speak about injectives and so on. We can do homological algebra there. Okay, so, all right. So I'll try to describe this new complex. That will be quite asomorphic to omega. So let's start with the following simple observation. So we do have an example of an object like this in this situation. It simply is a unit object. So this is a map sending xd to o m0 xd. The unit model is integral connection. So this will be called one because it is a unit object. It's a unit object precious with values in mic, like a single or composes m plus. That's two functions. But since this is an abelian category that we have here, so I could take an injective resolution. So I denote by one fibrant, maybe injective is better. Yeah, injective. So this is not very mysterious in fact. So if I evaluate at a pair like this, xd, I simply get an injective resolution of the unit object in the category of models integral connection. The models integral connection you take are usually a local system of fire attack? No, no, no. They are in the object. Yeah, yeah, yeah. So they are union, they are filtered union of what we usually call a model integral connection. So you need it for the injective. Yeah, for example, yeah. So we have this guy. We also have a construction. So let me denote by gamma delta. So this is, for any x, we have a module. So we have this function of some kind of global section which sends M to just the kernel of the connection. So this is, so therefore you can apply this to one. So gamma delta fibrant. So what is this guy now? This is simply, if you think about it for a minute, this is simply a complex, flaggy species on smooth stratified. So you get something which is more familiar. And so in particular, if I apply to this, the Godman construction, then I get just the complex of species, smooth. And now I also have a way to pass from stratified smooth schemes to just smooth schemes. Remember there is this course function. So the function that goes from smooth scheme to stratified smooth schemes sending a scheme to itself was a course stratification. This is the function CO. And this is the direct image. So it's just, yeah. So I can apply CO to this guy. Now this is not a good idea unless you derive this for the H topology. So this is the guy. And so here's a theorem which is actually not difficult, it's more or less obvious, is that this is a quasi-isomorphic to omega, at least on smooth k-affine. There's always this trouble. So this complex computes the same thing as here. Namely, it computes the RAM-commod. So I was planning to give a proof of this because it's so simple, compared maybe to the other one. Anyway, I'm out of time and I did not do more than half what I wanted to do today. But maybe I could just take a few minutes to say a couple of words about this, just to tell you that it's really something simple. All right, so I want to sketch a proof of this. So this is quasi-isomorphic, means there is a chain of maps. Yeah, I will not construct a chain of maps. I just try to convince you that they are the same. It's just a simple chain anyway. So both guys are, so both complexes are h-fibrant. This one by construction, because I applied C low star to a fibrant replacement of this. So it's still h-fibrant and this is because the RAM-commod has h descent. So therefore it's enough to compare these on the points of the h-deport. So the proof really would go first by constructing a chain of maps and then showing that on the points gives you quasi-isomorphism. So why it's simpler on the points? Because this guy can be written simply. So let V be a point of the h-deport. So this is what is called integrally closed, maybe absolutely, yeah, sorry, absolutely integrally closed valuation ring that I can take of finite lengths. It's called V. In fact, it's a spectrum, really. Spectrum of, yeah, finite lengths. It has only finitely many points. And so the argument goes by induction. So we argue by induction on the lens. Maybe before doing the argument. So if we fix such a thing, then the value of this guy can be written somehow slightly simpler. So if I put a V here, this is, I can write it simply as I can remove the first two terms. So it's just a good one complex. But here we have to be a bit more careful. So here V has somehow a slightly different meaning. Here V is endowed with the finest stratification. So if I don't apply this, this function here, I'm still over, so this is still a complex of pre-sheaves on stratified schemes. So if I want to evaluate a V, I need to put the stratification here. And one should put the finest stratification, namely that all the points are strata. And this is in the sense that you can, you write now V as a limit of stratified schemes. So this stratification should be endowed, so it should be induced by this limit. And so if you do this, then you are allowed to write this instead of this. So now the argument goes by induction on the lens. So induction. So the case lens zero, this is simplest one, then you have just a field. There is no really stratification anymore, so this Goldman complex is not really necessary anymore. So we need to compare, on one hand, the Rammcommology, this extension, and the value of, so that's a global section, one field on a spec K, plus the unique flag, the only possible flag. Now if you look at what we have here, this is just the thing that is computing X. So if I put Gommology now, this is giving us the X group between one and one in the category of modulus integral connection, regular singular over K. Now of course K is considered as a limit of smooth varieties. So we want to compare the X group here and the Rammcommology. So here it's, this is the Rammcommology. And then there is a well-known theorem that says exactly that these two things are the same. Yeah and the other, so the general case is, not, I mean it's more involved, but again somehow the crucial, the ingredient is again Bayneson's theorem. So here's just a few ideas for the general case. So as I said, it's by induction on the lens. So let U be inside V, and of lens one less. So the idea is that, yeah, and then let XI be the closed point and let Z be the closure. So then D is a valuation ring of, or the spectrum of valuation ring of height one, of lens one. And then, yeah, then there is, one shows this, that if you look at omega U over, the cone, if you want the quotient or the cofiber, then this is the same as omega Psi over omega Z. And the same for the other, same for this Psi. I have the same formula for this. So if you assume that we know the case of U and so we are interested in the case of V, then it's enough to show that this quotient is the same as this quotient here. And so we are reduced at the end to the case of lens one. And the case of lens one, then it's done by... And does manipulation of valuation ring, does it use the resolution of singularities? I mean it is everywhere, resolution of singularities. I used it in defining the topology, the site. What do you mean by... All right, so yeah, let me just end quickly now. So we have this, so we are reduced to this thing, so we would like to know that it's the same as this, so the Gaudmo complex of gamma delta, fibrant Psi, but we know that this is already... So we have this, we have two things that we want to compare. Now if we look back at this Gaudmo resolution, so this is the Gaudmo complex, so this is a homotopy limit over the category of flags in Z. But Z has two points, so the flags are not so many, there are in fact three possibilities for the flags. Either you have these two points or you have one of them. And so since you are modding this by this, you are somehow... The flag consisting of this single point will not appear and at the end we see that this is the same as the R gamma delta of something like mz plus small z. So let z be the... or sigma, sigma is the closed point. So this is the flag sigma, I'm mod out by gamma delta, so this is one. Now this can be written also as essentially like the quotient of the x group between one and one in the category of module of integral connection, regular singular on this thing. But this thing is actually very simple to write down. It is sigma, it's essentially a torus over sigma, so it's sigma and the value group of the valuations. So this is like a product of gm over sigma. And the same thing is actually true here. So this is also the Durant-Cahonoge sigma. So I hope this is clear. So sigma is an extension of my base field. This is like... I mean this is like... You can think of it as a lattice. It's not true because here we are in this... Maybe q to the r, but let's pretend that this is just a lattice. So this guy here is like gm to the r. So it's gm to the r times sigma. The Durant-Cahonoge of this is very simple. It's just the Durant-Cahonoge of sigma times some exterior algebra on the r. And the same is true here again by the theorem of Bayesian. That's why at the end we end up showing that these are the same. But the valuations here are cold valuations, so they are not this cold. The value group is not z to the... Yeah, I said that it's not z, so it should be qr, but q... It's algebraic, so it's integrally... Absolutely integrally closed, right? Ah, okay. Usually it's a finite rank and good... It's finite rank also. We can assume that it's finite, yeah. So it could be of rank 1 or of q rank r. Yeah, this is the rank, not the height, r is the rank. Rank of the valuations. So the rank of the valuations is the dimension of the value group. Cancel with q or rational rank, yeah, rational rank. Okay, so I think I stop here today.