 Thank you very much for the invitation and for the opportunity to give this series of lecture. All right, so today my plan is to do some kind of introduction to this lecture and so one of my goal is also to introduce the language that I will be using during the remaining lectures. And so it will be quite basic in fact. So my goal in this series of lectures is to discuss my last failed attempt to prove this conservative conjecture and to somehow to explain where the gap is and also to give some ideas of how one might hopefully at some point fix this gap. So I start maybe by stating this conjecture, I assume that most of you know the statement but I'll give it anyway. So I'll have a base field k that I will be working over which will be somehow of characteristic zero but the conjecture somehow also of course makes sense in characteristic p but all right. So the conjecture is the following. So let me write chowk k, this will be the category of chow motifs with rational coefficient. So the conjecture says the following, this is the usual category of chow motifs, right. So the conjecture says the following. So if you fix a failed cohomology theory, let's call it H upper star w. So this is a cohomology theory on smooth protective varieties with some axioms. Let me maybe fix one classical so let's take it. So for example, a latic cohomology or vettic cohomology or the round cohomology, okay. So then, then so a correspondence camera between chow motifs is an isomorphism in this category of chow motifs, chow k. If I don't leave, then used morphism in cohomology, H star gamma, so this would be going from the cohomology of n to the cohomology of n, it's a contravariant, isomorphism. So this is maybe the simplest way of setting this, or maybe the simplest case of this conservativity conjecture. Yes, of course. This makes sense for vettic cohomology, but for cure cohomology or when the field of definition of the commercial is larger, you can think of a variant where you take chow motifs coefficient as in ql or in a larger field and ask whether the same statement holds. I see, yeah. Okay. So does it follow somehow formally from, let us say that you know even when you. I think it's full of formally. To be honest, I've never really did the checking, but I'm completely convinced that it's full of formally, but also it's also natural to just put some coefficient here and ask for the same. Your cohomology field, yes, coefficients, where? Okay, let's say in lambda. So let me fix the field lambda. So later I will be working with the dramcology and I will be working k linearly. But I'm also convinced that it follows formally from the case with q coefficient. All right, so this is the statement of the conjecture. And so as I said, from now on I will be working exclusively in the case where the characteristic of k is zero and I will take, for this vague homology, I will take the dramcology. I'm sorry, I only consider this case. And in characteristic zero, this is not really, it doesn't really matter which homology you will be choosing because all the classical vague homologies are somehow comparable. So it's not really a loss of generality by doing it. Do you need the little question that I asked to make the comparison to make the. Yeah, I mean, okay, so in this case I will take, okay? So this is what I'm trying, what I will be focusing on, okay? All right, so this is a kind of elementary statement, maybe. But so the approach I'm trying to do here for this conjecture relies on another conjecture which I can only formulate using the modern language of the theory of motives. And so I want to start by a paragraph which where I will somehow introduce this language and give you a few well-known theorem in this setting that I will be kind of using constantly later on, okay? So this will be collection, okay? So this is the sense, so maybe I should say a la Witzke, sorry? Mix, so yeah, this will be a mixed model from now on, right? So let me fix some notation, so we'll be often working over this side of smooth varieties over K, so K is fixed of characteristic zero. So this is a category of smooth varieties over this field, and I always somehow work locally for the etal topology, so we always work. And I will be doing something which might seem a bit weird at the beginning. So I will refrain from using etal sheaves, and I will rather always work with pre-sheaves, but somehow think about them etal locally. So of course this is completely equivalent but somehow I find it more natural to do it like this, at least in this context. So I will be often considering complexes of pre-sheaves on smooth varieties. And with coefficient, let's say, in Q or any characteristic zero field. And I will always just say complexes of pre-sheaves to when I want to speak out something like this. So these guys, they form a category, right? So complexes, pre-sheaves, smooth varieties, Q. And this category has several model structures. So I don't really need somehow to tell you what is a model structure. I will just say a few things that I will be using. So this has, let's say, two model structures which will be interesting for us. So the first one is the so-called etal local structure. So this is just to introduce some terminologies that I will be using later. So the etal local model structure, so what will be important for us is the notion of weak equivalence and the notion of fibrant object. So the weak equivalence is, this weak equivalence is. In this case, they are called etal local equivalences, etal equivalences. And they are very easy to describe. So these are morphism of complexes of pre-sheaves inducing isomorphism in cohomology or in homology after sheafy, after sheafy, if you're hiding. So after, so differently, this is the same as saying that if you look at the induced morphism on the complex of etal sheaf, this is just a quasi-isomorphism in the usual sense. Then there are the fibrant object. And again, these are very easy to describe. So these are called the etal fibrant, etfibrant complexes of pre-sheaves. And they have the following descriptions. So these are complexes f, admitting etal descent. In the sense that whenever you evaluate this complex at a variety x, there is this natural map that goes to the derived global section for the etal topology. So this will be computing the etal hypercomology of x with value in f. This is a quasi-isomorphism for all x. This is what it means to be etal fibrant. So of course, to define algorithm etal, you have unbounded things to use. For example, the definition of spartanus type is unbounded complexes. So you have to use the notion of k-injective, so you are using. No, no, I mean, I'm just using the fact that there is and there is. Yeah, okay, I mean, it's maybe equivalent to do it like this. But I'm just using the fact that there is a model structure. For which these are the equivalences. So it's just about, I mean, it's probably the same. But you have to define algorithm etal xf. I'm saying this is more general than the classical. So this is the derived functor. So we have the functor f going to fx. And I think it's derived, right, derived functor for the model structure. Yeah, but it is not enough to use the very classical definition. No, no, no, it's not the classical one. Yeah, so you are using some model structure which, and so, you are trying to define a model structure using a model structure. So this is circular. I'm not defining. I'm just telling you the properties, okay? I'm not proving anything here. I'm just telling you some words which, all right. So this is the first, maybe I should also say that. So the homotopy category for this model structure is just or is equivalent just to the derived category of sheaves on this side. So we don't get anything really interesting here. It's just a convenient somehow way for me to speak about this category. The second model structure is more interesting. This is called the A1 etal structure. And again, I'm not trying to define this. I don't want to do it. I'm just telling you a few facts or just to give you some terminologies. So the weak equivalences here, they are called the A1 etal local equivalences. And so contrary to the case of etal local equivalences, these do not have a useful characterization. So we don't have a useful characterization for these. No useful characterization, which is, of course, one of the source of difficulties to work with them. But at least we know that they are generated, but they are generated in some sense by the previous one, by the etal local equivalences, plus things of the form A1x, tensor q, and x, and so on. So here tensor q is, of course, the free sheaf of q vector spaces, which is generated on x or on A1x. So these maps and the previous one, they generate a class of equivalences, and these are the A1 etal local equivalences. And similarly, I will need also to describe the fibrant object here. So fibrant objects, they are called the A1 etal fibrant complexes. And they do admit a nice characterization, so they should satisfy two properties. So there should be, first, etal fibrant in the previous sense. And they should have, should be that if you, so f is A1 etal fibrant, if it's etal fibrant, and if when you evaluate at x and you map to f A1x, this is a quasi-isomorphism. So they satisfy etal descent, and they satisfy also homotopy invariance. Okay, so and so in this case, the homotopy category has denoted by dA effective etal. And so it is known to be equivalent, so known to Wojcicki's category of motifs. So here I'm assuming, so the effective one. I'm assuming that K is characteristic zero and that I'm working with rational coefficients. So yeah, I should maybe say what do I mean by the homotopy category in case you don't know what this means. This is just the category that one obtains by formally inverting the class of weak equivalences. So by inverting these maps, you get a triangulated category, which is that one. And it is also fibrant objects with homotopy priors. Exactly, yes. I'll say this in a moment, but yeah, you're right. Okay, so, all right, so maybe also it means to do some notation. So given here, so I denote by M effective X the object X tensor Q considered in this category. So this is called the effective motive of X. So because of this equivalence, we can use a theorem of Wojcicki to get an embedding. From the category of effective char motifs inside this category. So we have a fully faithful embedding like this. And in fact, so this is a triangulated category and in fact, it's even compactly generated by the image of this. It's compactly generated. So a good way to think about this is as so this is being some kind of a triangulated completion of the category of char motifs. So the next thing I want to discuss now is the notion of spectra, of T spectra. All right, so maybe I don't know if it's a new program, but so I want to discuss T spectra and T spectra. So this is really when I say T spectra, I really mean T spectra of complexes of and so on. So you don't invert the tape motive in the definition. So the object are just the direct summand of motives of proper smooth varieties. All right, so I need to tell you what is a T spectrum. So there is a choice that we should make here at some point is the choice of T. So but somehow T is T should be should be thought of as a model for the tape possibly shifted. And there are many natural choices that one can take. So one of them is P1 pointed at infinity. So this would be the reduced motive of P1 or A1 minus 0 pointed by 1 and so on. So I'll fix one of these and I'll sometimes I will change my my choice and it's it's it's no. I mean we know that this is doesn't this will not really change anything because so these two things are somehow equivalent. Up to A1 equivalent so they are A1 equivalent so it doesn't really matter which one you choose. So if you make this if you make this choice and then then then a T spectrum very simple is just a collection indexed by the integers. Maybe so a collection consisting of ENR complexes, pre-sheaves, multiple varieties and gamma NR just maps from T tensor EN to EN plus 1. So this is what is what is a spectrum it's like in topology there is there are actually other kinds of spectra. So that there is something called symmetric spectra which one should maybe better use. But I will not I will just take this simplest kind of spectra and pretend that they have all the property I need. Yes, we are trying to do this. OK, so so this so the category of category of T spectra also has also many model structure and I will be considering at least three. Mother structure so two of them is just exactly the previous one but somehow considered level wise. So there is a local structure level wise. So here it's everything somehow is detected at each level. So so just give you an example. So if I have a morphism of spectra, I would I would say that this is is a level wise A1 et al local equivalence. This is just if and only if the induced map are like are so for all. And similarly for for fibrin so spectrum is level wise fibrin if whole GCN are fibrin. There is also a notion of fibrations and cofibrations. Yes, so those things are level wise or not exactly. Also also yeah for for for this one is yeah no no sorry not not not for the cofibration but for. So the fibrations and the and the weak equivalences are level wise. But the cofibration are more complicated. But yeah, I will not really use anything of this kind. I mean, I mean, so level wise you define the equivalences and the fibrations level wise and then you deduce cofibration by the lifting property. Right, so the more the more interesting one is the stable stable A1 et al local structure. And again, I will not define it. I will just give you some properties. So tell you what are the equivalences and what are the fibrant object. So the weak equivalences are called stable A1 et al local equivalences. They don't have a nice description. No no useful characterization. Rather, they are generated by they are generated by the level wise A1 et al equivalences. And the class or and morphism like this, which are which are isomorphism which are such that inducing an isomorphism for n large enough. So morphism of spectra, which are which only differ at some at low level and the previous one, they generate the class of stable equivalences. It's a complicated matter. Yeah, it's it's it's quite complicated. Yeah, but I don't want to to go into into this if it's OK with you. It's it's complicated. Yeah, it's you have also to take push push push out of of things and so on. And transform your composition. So it's not really a useful thing. I'm just saying words, which just to give you an idea what's what's going on. And then there is there is a vibrant object in the setting. These are called the stable or the stably fibrant the stably A1 et al fibrant object. And they do have a nice characterization. So if E is this is like this. So if first you at each level, so the E n are A1 et al fibrant, no surprise. And then the other condition is that you want the map that you deduce from the gamma n by adjunction. So the map that goes from E n to the internal home or is a quasi isomorphism. Where the internal home is computed using the model stack. No, no, no, this is not a derived one. It's just the naive one. You don't need because I already assumed these guys to be fibrant. So it doesn't matter. Yeah, the end is already I mean this is already the derived one because this guy is fibrant. All right. So the the homotopy category for this third one. There's some model structure for the last structure is denoted by dA et al k. And again, it's known to be equivalent to Waeworski's dnk. Here we don't even need k to be of characteristic zero. It's true in general. And so again using Waeworski's embedding, we get a fully faithful embedding of non-necessarily effective motives, chow motives into that. And then we have the same properties. So this is compactly generated by the image. Sorry? Effective. Effective, yeah, exactly. Effective you do not invert the take motive and when there is no effective you have inverted the take motive. Right, so but here we get something more. We get an extra property, which is somehow the whole reason why to pass through the non-effective setting. So we get the plus, we get an extra property. The following so every compact object is strongly dualizable. So there is a dual which really behaves like taking duals for vector spaces of finite dimensional vector spaces. So maybe I won't say exactly what this means. Maybe you already know. Okay, so another thing also which is interesting, which is also due to Waeworski modulo these equivalences that I just mentioned. We know that so that there is this infinite suspension function from the A effective sends a complex of pre-sheaf to the infinite suspension of F, which is by definition, this is the spectrum you get by putting in level n just F tensor t to the n and taking identity as gamma n. So this factor is fully feasible and this is a non-trivial statement. It's really a remarkable statement, I should say. Okay, so this has to do with, I mean, I think, yeah, the name of the theorem of Waeworski that is used here is the cancellation theorem of Waeworski, which is behind this. Okay, so yeah, maybe I should maybe give some motivation for introducing these definitions. We'll give you a new course for the Waeworski category. It's new, it's slightly new, it's not really completely new. So you say it's equivalent? Yeah, it's slightly different from the original. But it's not the description of given boundary. It's slightly different from this and it's actually not completely obvious that there are equivalents. It's a theorem and, for example, we don't know this if, in the effective case, if K is a positive characteristic, we don't know this equivalence. So it has to do something. It's not a completely empty statement. So this is true in characteristic zero. Yeah. So that's equivalent? No, no, no, it's something different. It's something different. In Waeworski's setting, because he uses finite correspondences, like inseparable extension somehow are covered. Whereas if you just use a tiled topology, these are not covered. So it has to do with this. And in the stable setting, there is a way to go around this difficulty. I'm sorry. I think that in the definition of your DA, in fact, if you use your finite correspondences. Yeah, there's no finite correspondences. Why? We already use smooth push-ups over smooth over K that you could deduce this embedding. I don't understand. Why? Because it's known that these two categories are equivalent and we have this embedding from here to here. It uses finite correspondences, but you can't use it. Yeah, but I told you, they are equivalent at the end. So it's a theorem saying that they are equivalent. So even if you don't choose. Okay. So there's no proof of this embedding that doesn't use? No. Probably you can find one. I don't know. The question was, is it possible to prove this embedding without passing by Vojvowsky's category? Like proving this embedding directly. From the non-effective, now it's not. Now the non-effective chart, okay. Effective or non-effective. So the question makes sense in both setting. If we can prove without using finite correspondences. Okay, so let's see. In fact, to give you some motivation, I'm not sure. It's a good idea because I'm always not good. Never good in giving motivation. So let's see. So why would one be interested in these kind of notions? So we have this embedding now. Effective or non-effective. Okay. So if you go back to the statement of the conservativity conjecture. So it's a statement about maps in this category that we would like to show that they are invertible in this category. So in principle, we can say in principle, we can transfer. So let's see. Given gamma as in conjecture. So we deduce a morphism. Let's just maybe do it in the non-effective setting. So we'll get a morphism of t-spectra. And let's just put gamma tilde and tilde and tilde. So this is just the image of gamma by this embedding. And now, so saying that gamma is an isomorphism here is just saying that is exactly the same as saying that gamma tilde is a stable A1 et al. local equivalence. So now the task is knowing that gamma induces an isomorphism in convergy. The task is to show that gamma tilde is a stable A1 et al. local equivalence. So, but then of course one would say why would this be easier than before. And one could argue that maybe there are more techniques for constructing stable A1 et al. equivalences than just constructing a cycle which would be an inverse. To gamma. When you say as in conjecture, do you mean that it gives the impression of a isomorphism in a morphism? Yes, yes. I'm just assuming. Let's just say that I have a finite correspondence and I want to show that this is an isomorphism. Okay. So in principle, I could just assume that it is a homologically correct. Yeah. It's okay. So then we can pass to this setting here and then we have a morphism of t-spectrum. We could try to see if we can show that this is an A1 et al. local equivalence. And so there are indeed some examples. I mean, I tried to think yesterday about some example of new equivalences that they don't have an analog in the context of charmottes. And I can give you two examples, maybe just to illustrate that this is potentially a good idea to do this. So maybe two examples on et al. equivalences. New in the sense that we don't have them, we don't have an analog for them in charmottes. So one is not really, so this has to do with the localization theorem of Moritz Wojcicki. So this is a slightly different setting. We are not working anymore over a base field, but rather over some base scheme. So we have s, a base scheme. And I take u inside s and open and z its complement. And then the theorem which looks very obvious, but in fact it's not so obvious, is that if you take x over s, a smooth s scheme, then you can look at the map x modulo x u. So the inverse image in x of u. This is a quotient of a pre-sheaf offset, and then u tends to lose q. And there is a map to the direct image i lower star of x d times q. And this is an a1 et al local equivalence. And it is not an et al local equivalence. So if you remove the a1 here, it's not true anymore, it's wrong. This is one example, another example which is maybe more interesting maybe. And this has to do with the cancellation theorem that I just mentioned a few minutes ago. So another example of such phenomena of a new equivalence is the following. So you take any f complex of pre-sheaves on smooth varieties. Then you have a map from f to r et al home. I can take t and t tensor f. So this is a map, and it's very far from being a quasi-isomorphism or even an et al equivalence. But it is, thanks to Wolkowski, we know that this is an a1 et al local equivalence. So these are examples of really non, I mean all these theorem are quite tricky to prove. They are not maybe extremely difficult, but they are still, one still has to do some work to get. So they are not complete trivialities, very far from being complete trivialities. And they give maybe some kind of a new example of, so the right, oh yeah, this is, so r et al, this is the right, the right counter for the et al local model structure. So you replace this guy by an et al fibrant object and then you apply. So these are two maybe, two reasons somehow to, maybe to expect that one can go a bit further in the conformity conjecture. But in fact, more importantly, so more, and this is really what I will be trying to use later. We can do some new construction in the setting, in the et al, that we cannot do charmotives. For a simple reason, because this construction will give you some, some objects which are not charmotives, that will be really mixed motives. And also there will not be necessarily compact, there will be of infinite dimension and so on. And we will see an example of this construction later. So try to give maybe some motivation for working in this setting. Okay, so I want to, yeah, before the break, there should be a break, right? So, and, oh, it depends on me. Okay, so I'll give now some compliment about this language that I've just introduced. So compliment. So, yeah, so what you think about, so, so this vibrant object. So being fine is something like being some kind of a complex of injectives or injective resolution. Of course, this is not correct, but it's a good approximation. And so, and so we have this, and so in particular, so the consequence of this weak equivalence is between vibrant objects, or just quasi-isomorphism. So this is why it's nice to have a vibrant object because... Which models, which... I'm just speaking in general. I mean, it's not a theorem. It's just... Because usually, the equivalence was defined for complex of precepts, like for the etalcine, for the defined as being quasi-isomorphisms. Yeah, I mean... For the homology shapes. Yes, yes, so... So quasi-isomorphism of precepts, of complexes of precepts. So here are some examples, like, so if I have f to g and a1 etal local equivalence, let me give a name to this, with f and g or a1 etal fibrant, then this implies that f is quasi-isomorphism. Again, similarly, if I have a morphism of spectra, so spectra I usually write them in bold letters, let's say now it's a stable a1 etal local equivalence, and where e and f are stably a1 etal fibrant, so it implies that f is level-wise a quasi-isomorphism. So this is... Yeah, quasi-isomorphism of complexes of precepts. It's equivalent, yeah. I mean, the other direction is clear. Yeah, so this is maybe the first compliment to say, and then the second one is about... The notion of etal fibrant, sorry, a1 etal fibrant, is in practice maybe is too strong. In practice, you would have something slightly weaker than this. But yeah, so a good weaker notion is the following. So we say that f is complex of precepts, is a1 local, there are too many local in this setting, I'm sorry, but this is not my terminology. Okay, so f is a1 local if an etal fibrant replacement if f is already a1 etal fibrant. So it's a complex that if you just make it etal fibrant, like if you just take an injective resolution of etal sheaves, it becomes a1 etal fibrant. You don't understand the sentence, you have these, these... So f... So it's a definition, f is a1 local if... Oh, yeah, thank you. If an etal fibrant is okay. And this is actually equivalent to say that, that if you look at the etal hypercomology of x with value in f, it's the same as the etal hypercomology of a1 x with value in f for every x. There is a similar notion also for stably a1 etal fibrant. So maybe three. So again, a definition. So we say that, so a t spectrum is called, is an omega spectrum. So this is a definition. If, or maybe omega t, if a level-wise a1 etal fibrant replacement of E is already, is stably fibrant, stably a1 etal fibrant. So it has some of the properties of an a1 etal, of a stably a1 etal fibrant object. And so again, this has, this is equivalent to say the following for every n, the map from en to r etal internal home t en plus 1 is an a1 etal local equivalence. So it's, if you are familiar with the notion of an omega spectrum in topology, this is very similar to it. So if you want, it's a good time to make a break. All right, so next paragraph is about the DRAM spectrum and the main conjecture. So as I told you at the beginning, the approach I have to the conservative conjecture somehow relies on a stronger conjecture, which I will now state. So this is what I call the main conjecture. Okay, so in order to state this conjecture, I need to introduce an object, which is the DRAM spectrum. So let's, so I denote by omega, so the complex, so the usual DRAM complex they're considered on smooth varieties. So if I, if I choose for simplicity, let me just take t to be a1 minus 0 pointed by 1 q. So this is, this will be my model for the tape model. So we can construct a t spectrum that I denote by omega underline, which is given by, by the DRAM complex but shifted by n, this and the, the map gamma n is actually gamma 0 shifted by n, where gamma 0 is the map that goes from t tensor omega to omega shifted by 1 tensor omega. So this will be the usual D log, the identity and the multiplication. Okay, so very simple construction and it gives a spectrum which is at each level essentially given by the DRAM complex. So this is called the, this is what is, what I call the DRAM spectrum. Okay, so let me give you some properties of this. So we have the following, easy La Maouche. So the, the complex of appreciate omega is a1 local in the previous sense. In fact, we have, we have more. In fact, it is essentially a1 et al fibrant and more precisely, in fact, if I, so the restriction omega to smooth varieties which are affine, affine smooth varieties is a1 et al fibrant. So it's like a complex of injectives in this setting. And we also have something nice, similar, also similarly for the spectrum. So omega is, is an omega spectrum. So I'm sorry for this. These two omega have nothing to do with each other. But more precisely, so in fact, the restriction of the spectrum to, again to the affine smooth varieties is stably a1 et al fibrant. So this is a very simple, La Maouche, just, just refraining some well-known properties of DRAM comology. So I'm not really good at the truth, but this, this follows from amotopy invariance in DRAM comology and maybe something like projective model formula. Or if you want, or so differently, you can also use the current formula and the comology of, of GM. Okay, so it's, so we have this end. So as a consequence, so whenever I want to, to compute the set of morphism from, from an object to, to here or to here, I can just do this up to quasi-isomorphism. So, so this has the following consequence. Okay, so, yeah. All right, so maybe I, okay, do it as a remark in general. So if you have, if you have an object F, which is fibrant, so fibrant, so it could be either A1 et al fibrant or it could be stably A1 et al fibrant and so on. So morphism from A to F in the amotopy categories is just morphism up to quasi-isomorphism. Okay, just, just, okay, just this, the complex of morphism. A is an, any object, so just take a co-fibrant, maybe. A co-fibrant, all right, so. Okay. An object, usually an object of the site is a co-fibrant. Yes, yes. So you see, you see that if I apply this, what I said here to, to the motif of a variety, which is, which is, which is co-fibrant by definition. X, X, X sensor Q is, is co-fibrant. So as a, as a consequence, I get that the, the RAMc homology, so for any X, X plus K, the, the RAMc homology of X, then identifies with the set of morphism in the A effective, between the effective motif of X and omega, maybe shifted by star. Yes. This is, this is the homotopic category. And similarly, this is home in this table. So we have this formula, which just follows directly from, from this lemma. So said differently, these objects, they represent the RAMc homology. So these objects, omega and omega underline represent, of course this is completely tautological. I'm not really doing anything here, represent the RAMc homology. But at least, so it give you, give you some nice characterization of these objects. Okay. So, so up to now, there is no reason to prefer omega or omega on, on, on, on, on the line. But the next, the next proposition is really special to the, to the, to the, the RAMt spectrum. It's not, it's, it won't be true for the RAM complex. So here's a proposition, which is due to Sezynski-Diglis and say the following. So the t spectrum, omega underline is a, is a field, okay. And field in the sense of competitive algebra. Namely, every module. So this is of course, I didn't say it, but, but omega is a ringed object. I'm saying that every module over omega is free. So omega is like a ringed object or an infinity, something. I'm not considering this as an infinity. I'm just, I'm, I'm working in the homotopic category, and I'm just saying that this is a, this is a monoidal, this is a monoidal category. Okay. And I'm just saying as an, as an object here, it is a field. So it has a product structure? Yeah. No, I mean it is also an infinity ring spectrum. Yeah. But I'm, I'm, I'm, I'm saying that it is a field even in the homotopic category. So you, but you have to consider the ring, some, some ring structure in the naive sense in the homotopic category. Yeah. So it has, it has a, it has in the strong sense and therefore also in the naive sense. Okay. And in the naive sense it is a, it is a field. So any module in the naive sense is free. Yeah. Yeah. And maybe I should say, because there, there might be two, two, two, two, two possibility for what, what is the free and the setting. So let's say more precisely, if, if M as a module over omega, then there exists a canonical isomorphism between M and the global section of M that is tensored with omega. Sorry. I explained this in a moment. Canonical isomorphism. Or, or maybe I should say that there is a natural morphism here and it's an isomorphism. So gamma M is a, is a complex of take vector space. Yes. Either directly take the vector space. Yes. Yes. Okay. So it is free with, with grating. Yes. Yeah. Yeah. Yes. It's a graded field maybe. I don't know. Yeah. So I sketched the proof of this proposition. Yeah. And so I should say that this is, this proposition is completely, completely wrong for, for the Durant complex, for, for, for that, for that guy here. In the sense, every module is free of the whole vector of omega. Yes. In the sense, every module is free of the whole vector of omega. Yes. Yes. Yes. It's actually true for any vector of omega. Yes. Yes. Yes. And, and duality. It's really very simple. Right. So I will not, maybe let me first tell you what is gamma m, for, for any motive m. This is, I think, think of this as a global section of m. And it's, it can be defined as the complex of morphism r home from, from the unit to zero. Okay. In, in the m et al. Okay. More concretely, what, what is this? So if m, so remember m is a spectrum. So what you can do, you can first make, make it fibrant. So stably one et al. Fibrant. Replacement of this. Take level zero, take global section. This is what it is here. Okay. So this is a complex of vector spaces. So I, I, I prove this only for somehow semi free, maybe module. So I, I, I assume. So in fact, the, the general case reduce, I will not do the reduction. So the general case reduces formally and easily to the case where, where m is of the form, omega tensor, some motive M zero, some M zero. Okay. I just do this case. And it's really an easy reduction to, to pass to the general case. And in fact, I can also assume M zero to be compact. And this is because every m is some kind of a collimate of compact objects. Um, right. So, and in this case, we can use, you know, the formality. So, uh, so we have, remember, recall that, uh, because M zero is compact, M zero is strongly normalizable. And as a consequence, this means that, uh, tensoring with M zero is like taking also internal home from the dual of M zero. So we have this formula, omega, tensor M zero is the same as internal home from the dual M zero to omega. Okay. So this is, this is exactly the place where I cannot, uh, do this argument for, uh, in the, in the effective category because then, uh, we don't have, uh, we don't have strong, strong dual for, for compact objects. All right. Now, so now this is, uh, if you think about it, um, how, how do I compute, uh, this object here. So, uh, if I, if I want to evaluate this if I evaluate, if I, okay. And if I use the, this, uh, this, uh, this remark I just made here that, that this, this guy is representing the Rammkommology. And then if I also use the, the Kuhnett formula, all right. So, plus, uh, previous observation plus the Kuhnett formula. Okay. I see that this is just the same as, um, the home of M0 to omega, R-home, uh, tensor. So, maybe I, I want a bit too fast here, but, uh, you see, so if, if I evaluate at X, what, what do I get here? I get the Kuhnett of X tensor with M0 dual. Okay. So, by the definition of the internal home, uh, I do it here. So, again, if I look at, evaluate it at X, uh, this is, uh, the Rammkommology, star here, this is the Rammkommology of X tensor M0. No, no, everywhere is derived. So, this is R-home. R-home, yeah. Okay. So, so if I map X to, to this object, I get the Kuhnett of X tensor with M0. But this is, by the Kuhnett formula, this is the Kuhnett of X tensor of the Kuhnett of M0. Okay. Which I can also write as just omega evaluated at X, tensor of that. Okay. So, this is why we have this formula. Okay. So, this, this, this show that, so, and this is a vector space. So, vector spaces, spaces, tensor omega, and this is, and also, should, and this, yeah. So, there is still one, maybe one small step is to remark that this complex is indeed gamma of, of this. So, gamma of M or gamma of omega tensor M. Remember, this is the, the, the group of morphism from the unit to this tensor product. But, but M0 is dualizable. This is also morphism from M0 dual. Okay. So, this, this guy here is exactly gamma of, of that. And this is what I wanted to show. Okay. Okay. So, I use, I use this to make a definition also. So, we can define the, the Dharam realization as follow. So, if I have a motive so I, I set R Dharam of M to be by definition this global section of M tensor omega. So, this, this defines a functor R Dharam which is which is a monoidal functor from dA et al k to dk of k vector spaces. And so, this is called the Dharam realization. Just maybe, let me just say that so if you apply this, it's a homology, yeah, homology. So if you apply this to, to the motive of X, you get really the dual of the homology. All right. So, okay. So now I want to to, to state this conjecture that I mentioned before and I need a small construction before doing, before doing this. So, so, maybe just a remark. Let me construction. So if you have, if you have an object A in some monoidal category, and let's say it's a, it's a ring object in some monoidal category, you can form the so-called church complex. So this will be a costimplitial gadget, which is, which looks like this. So it's somehow the, the dual of the church complex that you get from the cover. So the map here are given by, by the unit. So this is unit tensor identity. This is identity tensor unit. This is multiplication and so on. So there is such an object. And I want to apply this construction to, to the DRAM spectrum. Okay. So, and, but one has to be a bit careful here when it has to, to make a co-fibrant replacement. So it's not really important for what I want to do, but just in case okay. So choose a co-fibrant placement of omega, which is also, which is still, a commutative or, yeah, which is again, which is, which is also which is still a commutative. No, no, no, no. I mean, okay, omega is a commutative ring object in spectra. I didn't say it maybe clearly, but it is without, without passing tomography. It's, it has a multiplication. It's which still means that the compatibility. Yeah. So there will be a morphism from here to here. Yes, yes, yes. In this spectra. Without, without passing. Yeah. On the nose. Okay. So once, once we've done this, it's, we can, you can apply this construction and we have the following. So I can now stay the following conjecture. So this is what I call the main conjecture for the map from I have it like this from K zero to whole limit omega co-fibrant is an isomorphism. So I, I, I choose to say this connection in, in motives with coefficient in K, which is, I think the natural thing to do here. So, so K zero is just the unit object. So it's K, it's Q zero times X, X and it to K here. Also the tensor product is taken over K. So the definition. And yeah. So this is, this is the main conjecture and so here's a remark which is kind of important. So the, the homotopy limit is computed with respect to A one, the stable A one model structure. And this is exactly what, what makes this conjecture very difficult. So if you, if you would allow to, you're set to compute this homotopy limit in some weaker or some, some different structure. For example, if you, if you remove stable from here, if you just compute the homotopy limit in just in the, in the level wise A one at a local structure, this will be a much, much easier thing to do. In fact, I think one can prove it quite is really coming from this, from the stability here. But, but on the other hand, if you, if you just, I mean, if you replace whole limit by, by a whole limit compute computed in a different model structure, like if you remove the state stability here, then the second that you get is kind of not interesting at all. You don't, you cannot prove and you cannot do anything with it. So, for example, if you, so the next time the conjecture implies the conservativity conjecture and we will be using really that, that this whole limit is computed in this model structure here. Otherwise, it doesn't work. All right, so let me state this lemma. So, if you take the homotopy limit and somehow object by object, I don't know if it makes sense. What do you mean by object? Maybe I'm not sure what, because usually your, your things are three complexes, three shifts on something. Now I'm lost. I don't know exactly what you are. Everything is a t-spectrum. Yeah, yeah. So it's, everything is a t-spectrum here. So, so it's, it's components are a complex sense of. Yes. Pre shifts. Yes. On smooth varieties. On smooth varieties. So you can evaluate each of them on. Yes. Okay. Now, so suppose you consider session evaluation. Yes. And so you can take the Yes. And is the open limit that you take by taking it? No. No. No. You said something about it to this. Huh? I said something. So, so I, I, if I have time today, I'll say more concretely what this mean, this motor pyramid. Okay. But I, I, I, I, I had to convince that I didn't try to, to write down the detail, but I'm, I'm, I'm, I'm, I think it's, uh, one can, one can prove it, uh, without much, much difficulties if you remove stable. And then it corresponds to what they said. No. So what, what, what you said would be re, remove everything. Yeah. Yeah. And it's fine. Yes. It's, it's, it's, it's very easy. Yeah. No, I mean, it's not, not very easy. It's, I mean, it's not, no. So, so the statement for the effective is very easy. And, um, you, you can, uh, yeah. So if, if we don't work with spectra, if we just work with effective, uh, it's, it's fine. You, or you can, or you can remove this. Uh, all, all of these seven are true and are easy to, easy to prove. Okay. Right. So I just, just wanted to say that somehow the, the main problem is coming from, from this, uh, objective here stable. All right. So I, I, I, um, I think I, I have now like 15 minutes, maybe. I should maybe hurry up a little bit. So, um, yeah. Okay. So, so here's, here's a lemma. So, uh, so this conjecture implies, implies conservativity. Okay. So I want to explain this. In fact, it implies more. It implies that, um, um, so here's proof. Uh, even more, it implies that, uh, the realization from dA et al. to dK is conservative on compact object. Okay. So this is strong, strictly, I mean, stronger than, uh, conservativity for Chao motive because we have this embedding. Um, and, and in this, in this triangulated setting, it's enough to show that, uh, uh, uh, so enough to show that, that if you, if you motive realises to zero, then m is zero. Okay. Because then you, you just, if you have a morphism, you take its cone and showing that, that the cone is zero is the same as showing that the morphism is invertible. Okay. So let, let, let me explain this implication here. So, so if we, if we know that the, uh, that are the, are the Ram of m is zero, uh, this implies that, in fact, that m tensor omega is zero. And so why, before, uh, this is just order Ram. If you, if you, if you look at the, at the, at my notation, this is just isomorphic to this. Okay. Because this is a field and, okay. So, so we can upgrade this, this, uh, vanishing here to a vanishing of a motive, m tensor omega. Um, and then you see immediately that this implies that, uh, m tensor chech of omega is zero as a costimplicial, uh, motive. Um, this of course implies that, you can pass to the, to the homotopy limit and get, and still get zero. Okay. So here of course, this requires the fact that you are completing homotopy limit in, in the right model structure. Otherwise, you cannot do this implication. Okay. So here, here it's one point where you, you need to be computing the homotopy limit in the, in this model structure. Um, and then let me continue here. So because m is compact, uh, m is strongly dualizable and, and therefore, tensioning with m commute with homotopy limit. So we get m tensor homotopy limit equal to zero. Okay. And now, now we just, uh, so now it's clear. So if, if we have this conjecture, then this, uh, so this is m equal to zero. Okay. So this is, um, how one, how one get, uh, the conservativity conjecture from such a statement. Okay. Now, there's an observation that this, so if you just, if you are just interested in the conservativity conjecture, uh, in fact, this conjecture here is, is too strong. You, you don't, you can, you can, uh, you can adapt this argument with, to explain this now. So, uh, so we do not need the full strength of, of this main conjecture for conservativity. So there is a weaker version of this, which is sufficient for, for conservativity. Um, and to explain this, I need to introduce, uh, one more object. So, um, so maybe just, here's a remark. So, uh, we know that, uh, this category has a T structure, so called the homotopy T structure. And, um, it does the following. So if you, so the truncation function are, so if, if E, uh, is established, in fact, I don't need established. So if E is level-wise, A1 et al. Fibrant, then the truncation, uh, of E is given as, I'm sorry. Um, so you just truncate at each level and, and it gives you the right object. Um, but it's important in fact to, to have this objective here and, uh, and again this is not completely, it's not, not an obvious fact, but it's, it's true that you have this, this T structure. Homological notation for quantization. Yes, yes, yes, yes. So this is the complex that have, that have only homology in positive degrees. Okay, so this is, okay. And so I, I apply, I will apply this to, to omega. So, uh, so in, so for example, if I apply this to omega, I get, you get the spectrum which has, um, the following shape. So it's given by, uh, you truncate at minus n omega, the, the round complex and then you shift by n. And it's easy to check that, that, that, the map gamma n that define the spectrum somehow, uh, induces maps on, on these, on the truncations. Um, yeah. So, so this, so at level n, this is the spectrum which somehow sees, sees, um, the round comology up to, up to degree n. Okay. So, okay. So with this, I can now state a, a weak version of this conjecture. Um, let's say the following that exists, amorphous ring object in dA, dA, um, between the homotopy limit. So of course this is, this is weaker than, than that because, uh, certainly, uh, there is a map from K0 to, to this. So, so this is a weaker statement. And so maybe here's a remark. So, so you could ask why, why this object? In fact, you can, you can weaken the conjecture, uh, you, you can choose any, any spectrum and, and you, you could ask yourself if there is a map from this guy to, to your ring object. Um, so the answer for this is, this is, uh, this is the, uh, how to say, the, yeah, this gives, so this gives the weaker, weakest form of main con, of the main conjecture, which is, which is still strong enough for conservative. Okay. So, this is one reason. And the other reason is that it's, it's, it's kind of a nice statement because it involves only one object somehow. Um, and one could hope that, that indeed one can relate this, uh, homotopy limit of omega, of, of Church of Omega with, with the truncation of omega. Okay. So I, I don't know. I try, I can try to explain very quickly why it is sufficient, uh, why, why this is sufficient to, to obtain conservativity. Um, so, lemma, the last conjecture, also conservativity. All right. So again, we, yes. So, uh, so the point here is, um, yeah, so the point is, is as for, so if you have, so, uh, if, if M compact, uh, such that, the, the sterilization is zero, then, uh, as, as we have seen before, this implies that, uh, M tensor is a homotopy limit. Okay. Uh, so if, if you have the strong form of the conjecture, this is, this, this enables us to conclude. Now, we have some things slightly weaker. Um, so this, this will give us, this would imply that, uh, M, uh, mapping to, um, so you see the, so the map, there is a map from M to M tensor, um, tensor, this truncation, just, uh, tensor with the unit. And, thanks to this, uh, weak form of the conjecture, uh, this map factors through this object, which, which is zero because, I assumed that, the, the, realization of, of M is zero. So we conclude that this is a zero map, um, from M to M tensor, tau bigger than zero, but since this is an, this is, um, this is an algebra, this actually implies that M tensor, this ring is also zero. Okay, so it's, clear. Okay. And, um, tensor, tensor, truncation bigger than zero, omega is zero, because you are tensor, right? Yeah. So if the unit map is zero, then the, the whole, the model is zero. So the, so you, we see that to conclude, it's enough to show, therefore, it's enough to show that the functor, tensor language, omega, that goes from D, D A et al, K to D A et al, K, but, uh, with a structure of a module over, over this. So this is a category of, there won't be a category of modules over this algebra. It's conservative. So if your module's in the right sense of... No, no, no, no, no, no, no. In the right sense. No, no. In the right sense. Yes, yes. No, I mean, this is an, this guy is a, so this is, this T spectrum is a, is an algebra in, in, in a category of T spectra. And I can speak about module over it, okay? All right. So, so I, I'll tell you why, why this is conservative. Uh, but only for Chaomotives. Uh, it's also actually, it's actually true in general, but the argument somehow requires, uh, uh, more time. So, for Chaomotives, it's, it's followed from the following diagram. So, um, so we have this embedding. And then, turns out that, that there is, uh, this is, this is also an embedding here, but, but with Chaomotive, uh, for the algebraic equivalence. Okay, so maybe I should transfer with K everywhere, but, uh, so cycles up to algebraic equivalence. Okay, so transfer with K also, so here is, coefficient in K. Um, right, so there is, there is such a, such a diagram, uh, commutative, and this is a fully faithful embedding. Um, and then, uh, and then it's clear that, so this implies that, that this is conservative on Chaomotive, because this is conservative on Chaom, this is conservative, uh, by, by, by Wojcicki theorem. So this is conservative, Nile-Poten theorem. Okay, so the only point is really to, to understand why, why do we have such an embedding, but this is also something which is kind of classical. This embedding follows from the block August formula. Unfortunately, I don't have much time left to, to, to explain this in details, but, uh, maybe I can, I can take, like, 10 minutes just to, to say a few words about this homotopy limit. So how do we compute this homotopy limit? Maybe this is, um, okay, so, um, okay, so, um, yeah, so I mean, I should say that, so this conjecture, like, like even the main conjecture seems, seems pretty approachable. I mean, when you see it, the, the first reaction would be the homotopy limit. And, and somehow one can do, one can do this in some sense, uh, at least one, one can understand concretely what it means to, to take a homotopy limit in this setting. So, um, yeah, okay, we have the following lemma which is pretty standard. So, um, so let, let, I do it in general because the notation are maybe slightly easier than, so, let, let, let E, okay? So, for example, the cheddar con, the cheddar conjecture object associated to, to omega underline. Um, so, let's assume that, well, this, my index already is spectrum and another index for the cosimplation coordinate. So this is the cosimplation coordinate. And each for each cosimplation衛 is spectrum. It is spectrum, yeah. And everything commutes through, yes. Yeah. The cost-efficient of the category of this big cloud. Yes, yes, yes, cost-efficient t-spectrum, yes. So we assume that at each cost-efficient degree, this is a stable A1 et al. Fibrant. Then the whole thing is easy to write down. So then the t-spectrum is what you expect. So let's see, in level end is given by, you take the total complex with a product, not with sums of the normalized complex, or you don't even need to take the normalized complex. So at each level you have a cost-efficient complex of pre-shifts. So at level end you take the normalized object, so this will give you a bi-complex of complexes of pre-shifts, and you take the total complex, but you take it for the product, not for the sum. Why did you assume that EED? EED, so this is a cost-efficient object, so that is, this is a degree, yes. So the normalization is relative to the cost-efficient structure? Yes, yes. And of course usually there are two ways to view the normalizes like... Caution or sub? Sub. It doesn't matter, no, they are the same, no? I don't need to take end, I mean, I just... I don't want to take end, I just want... Okay, I didn't want to put a dot again here, but... It's the same, I mean, it doesn't matter. Yes, yes, yes. Just you pass through the bi-complex, and then you take the total complex. You can apply this if you restrict omega to... You can. You can do this. These are some things that satisfy if you restrict to... omega to... Yes, this you can do, you can restrict to different varieties. For omega, yes. But then not for the second one, for the tensor product. Yeah, so... Right, so this is how you compute the... the motor pyramid, but this is in the case where you have this assumption here. And in general what you need to do is to... to make a five-minute replacement, or a stably five-minute replacement at each level... at each... degree before applying this lemma. Okay, so... So if the ED are not stably fibrant, then you need to take a replacement first. You need to make a five-minute replacement before applying this. And again this is why it's a complicated thing to compute because it's complicated to make this replacement and... Yeah. Now let's go back to this... to the problem that we... to the case that you are interested in. Maybe before doing this let me save a few more words about how to... how you make this replacement. So let me still be working in this abstract setting. So, yeah. So... So if... Fortunately, we have nice ways to make a five-minute replacement in this setting. So... So here's... Maybe I should start with the preposition. So there are two parts. So if you start with f, a complex of pre-shifts, and you want to make it A1 et al fibrant. So there is a way to do this, which is as follows. So then... then an A1 et al fibrant replacement of f is given by the so-called singular construction on... So you take your f, you make it... you first make it et al fibrant. There is no way here to... there is no formula for making something et al fibrant. But then if you admit this bit here, then to pass to the A1 fibrant, you just apply some concrete construction. So here this is... This is so-called Sussin-Weibowski construction, which is like the singular construction in topology. So I just write... I write it now. So this is total complex. Now you take direct sum instead of product of internal homes from the algebraic simplices or these guys, so spectrum. Okay, so this is... And the second part is about how you make something and establish fibrant. So if... if E is a level-wise et al fibrant spectrum, d spectrum, then establish fibrant replacement of E is given by something that is called lambda infinity of E. And this is a co-limit over the integers R of something like this. Internal home from T tensor R to S minus components R of E. So I just say it orally. So this is... Maybe I can write it like this. So one can shift the level by R in the spectrum. So at level zero it becomes E R. At level one it becomes E R plus one and so on. And there is a map. So then you take the loop space in some sense and take a co-limit. This will be lambda infinity and this is et al fibrant replacement of E. So just as we have concrete ways of understanding what is... how do we get a fibrant replacement? So it involves first applying this level-wise and then applying this to make the spectrum and then the spectrum. Okay, so this is said, then the question becomes very kind of concrete. So if you put all these together, maybe I just write a formula. It's not really very useful. So if you sum all this up, so at the end, so at level N, this homotopy limit that I want to compute. So the co-limit is just... the naive co-limit. There's nothing... No, here it's a filtered co-limit. So it doesn't matter. Does a filtered co-limit present this kind of... Yes, yes. It's because of finite commercial dimension. Ah, so the commercial component is filtered. Right, so this guy that I would like to understand at level N is given by this kind of formula. So it's a total complex. But for the product, then there is a co-limit or in N, there is an internal home, T and so R, total complex for the plus. Okay, something like that. And this is a tar-fibrant. And so somehow the difficulty in trying to compute this object, or I mean one way to compute this object is to show that it's okay to prove that you can commute. So there are these two kind of co-limit that are involved, that appears here. So a co-limit... So this is a co-limit for computing the A1 et al. fiber replacement. This is a co-limit to make the spectrum an omega spectrum. So if you could commute these two co-limit with this limit here, then you would be able to compute this complex. It would be easier to compute the complex here. So if you want, this is a co-limit over... It's a co-simplice... It's a simplicial space, and you are taking the co-limit of this simplicial space. So this is a co-sim... This is a co-simplicial object. This is a simplicial thing. And I pass to... Yes. So we will see that there is a way to deal with this co-limit here. But what is really the trouble is this co-limit here. This is the thing that is responsible for the omega spectrum. All right. So I think I'll stop here.