 As usual, I will start by recording a few facts from the previous lectures. Recall the following categories, smooth, okay, so this is the object or diff-smooth variation. And on this category, so we have many natural topologies and that would be today considering mainly two of them. So the first one is called the C et al topology. And then there is a finer one which is the C foliated topology. So this is approximated by C et al and then C ft. And so these topologies somehow were used in a not very explicit way before. So I tried to somehow make the role a bit more transparent today. So record the following facts. Fact one is that so we have the the drum spectrum that naturally lives on this category. So this guy turns out to be equivalent, so that is equivalent to something which is something like this. So it's a p1 delta localization with respect to this first topology of the infinite suspension spectrum on the drum complex, not the drum spectrum. And this is in fact, it's a very strong isomorphism. This is a level-wise quasi-isomorphism. This is the best you can hope for an equivalence. And I could also write this as follows. This is also localization, again the same kind of localization of the infinite suspension spectrum on some much smaller sheaf which is called or delta which is in fact the h0 of the drum complex. But here I have to make an ft-fibrant replacement. I have to make this object level-wise, fibrant for this topology, and then apply this thing. So again it's also a level-wise quasi-isomorphism. And so this was behind this model p2 that I introduced, model p2 for omega underline. So more precisely p2 is the out or the computation is obtained by computing the restriction of this guy here to smooth varieties. In fact, yeah, from this one can also, as I explained before, one can get, one can map the the church complex on omega to something which is the following. And then again computing the restriction of this guy to smooth variety. Yeah, this thing that I was calling p2 dot. And this somehow, I tried to use with some inputs from Hodge theory to say something about the limit of this course in sugar. And then there was the second model which was, which relied on a similar fact, but which was the following. So there is also an equivalence from omega underline to something very similar but slightly different. And it is the following. So it is again a P1 delta localization for the same topology of infinite suspension of O delta, except that the only difference here is that I take a fibrant replacement for the PcA topology at the end. So we also have such an equivalence, but this is somehow a much weaker kind of equivalence. This is really a stable local. So nevertheless, one can again also compute, so computation of this guy restricted to smooth varieties. Yeah, the third model P3 of omega. And then this third model was also extended to a co-simplicial object P3 dot, which I also tried to use using this input from Hodge theory to say something about the motor P limit of this guy. And so I said that here one can go a bit further except that because of this problem, because of this is only a stable, not as strong equivalence as before, because of this point here, one get to a problem. And more precisely, one need to somehow stabilize the spectrum before applying the motor P limit. So one can go a bit further, P3 dot, but has to face the fact that we don't know that these guys are omega spectrum. And therefore, a priority we need to stabilize them. And this creates some serious difficulties. Okay, so at this point, there are two possibilities. So either one can, one is clever enough to somehow extend this Hodge theoretic argument, so that it also work after stabilization, or the other possibility is to show that there is no need to stabilize these guys, that these guys are essentially on the gas spectrum. And somehow, this somehow leads to the following maybe conjecture that I will state at the beginning a bit in a vague form and then try to make it precise, which is, we say the following is that a version of these T spectra or n are actually or close to being omega spectrum. So I try to make this a bit more precise later. So if you have something of this kind, then somehow this will enable us to finish the argument, as I explained in the last lecture. Okay, so this is what I would like to discuss today. See if there is, this is really a reasonable thing to hope for. So I will first try to make this conjecture a bit more precise. So I will explain what I mean by a version of this. I will also explain what I mean by close to being omega spectrum. And then I actually then give a more stronger statement, which might look more approachable, or at least more reasonable, which would imply this conjecture. Then I explain that this statement is actually a bit too optimistic. It cannot be true as stated. And then maybe try to propose a statement which is weaker and which maybe still be sufficient to imply such a statement here. So there will be a lot of speculation today and nothing I will state as a theorem will be actually really proved. So all this, consideration are somehow somewhat a bit new. I only started to think in this direction only a few weeks ago. So I have to say that I didn't really get very far. And so even if I state a theorem today, this is not really to be considered as a theorem, but maybe something that I think I have, that I know how to prove. Okay, so this is somehow the plan for today. So I'll start maybe by saying really what this is the simplest thing that I can do is to make this a bit more precise, what I mean by close to be an omega spectrum. So I start maybe with this, make a definition. So let's so let E be a spectrum on smooth varieties. So this can be actually also easily extended to any category, for example, to smooth variation. I would just maybe say it in this case. So okay, so we say that that E is a weak omega spectrum in the following course. So if after a level wise, A1, A2, are a fibrant replacement. So the spectrum satisfies the current condition. It satisfies. So I maybe state the condition on the other blackboard. All right, so it's it's what would define something like something which is a variant of an omega spectrum and it's natural somehow to, of course, not change the level up to up to homotopy. So this is why I allow myself to do a level wise A1, A2, fibrant replacement. And then the condition will be about what happened after doing this. So we want the following condition to be to be satisfied. So for for x, smooth variety, for P, an integer, there should exist an integer n0, which depends on the choice of x and the level. And such that the map r gamma from x times, so this is gm pointed by one, wedge n, the spectrum E P plus n. So there's a map from this to the same thing, but it was n plus one. So we want this to be a quasi isomorphism. This is for n bigger than zero. So the idea is that these are the map that you see when you when you apply the stabilization procedure to a spectrum. Remember, there is this lambda infinity of a spectrum. This is this will be the omega spectrum associated to E. And if you evaluate this co-limit, so this is a co-limit, if you evaluate this co-limit at level P at x, you end up with these maps here. And so what I'm just saying that this co-limit evaluated at x at level P somehow stabilizes. It does not really is not really a true co-limit. It becomes quasi isomorphism for n big enough. I hope this is clear, the meaning of this definition. So this is if you want a slightly weaker version than the notion of an omega spectrum. And it's it obviously somehow suffices to solve the problem here of commuting homotopy limit with the stabilization function. So it's a reasonable definition in this context. Okay, so all right, so what I will so the conjunction that I will I want to press to propose today is that again a version of this guy will be actually a weak omega spectrum. Okay, so this is what I want to do today. So I start with. So yeah, so the next thing I want to do now is to explain this version. So I don't think that these guys are will be weak omega spectra is but I want to propose a version of this guy, which I hope it will have this property. Okay, so I want to introduce a new spectrum. So this is again it will be it will be a spectrum which will again give a model for omega. I will not do the computation. So I will not tell you what is P, what is the what is the restriction of this new model to smooth varieties and how it looks like. I will just claim that you can do the same thing as as we did before was with P3 with this new model. Okay, so I want to start with a with a with a general construction, a very simple general construction. So we fix a foliation, so maybe let's call it SE, so it is smooth foliation. So to this we have we have the category of smooth foliation over this guy. So I denote like this or this is smooth or diff smooth maybe see the reactions. So the object are just morphism of foliation which are smooth in the usual sense. Inside there is a smaller category which is the category of diff et al once. So this is I denote this like this diff et al foliation. Yeah, so this this inclusion defines a morphism of sight. This is the inclusion of diff et al into the smooth defines that goes in the in the opposite direction. Okay, so of course we think about this as a small site and this is a big site. So I can put here any topology. So it could be for diff et al or diff et al both work equally. And this morphism of sight I denote by Yota SE. Okay, and so if if I have a K, a complex pre-sheaves from FDES, I can form so I can consider its inverse image. The thing I get a complex of pre-sheaves on smooth foliation. And in fact this guy has a very easy description. So I give it just in case you might be interested. So if I have X over F an object of smooth foliation over S, let's assume for simplicity that X and S are affine, simplifying that assumption. Then there is associated to this guy there is an algebra A X over F, which is actually an algebra in algebra in the category of module with integrable connections over the foliation of S mod E. And it can be defined as follows. So at least I'll tell you what is the global section of this thing. This is simply given by the algebra of constant over X times F fiber product with a discrete foliation on S, okay. So and in fact so this so therefore the so because this has a structure of a module with integrable connection then it's the spectrum of this guy is actually naturally an object in FD. So it's naturally a diff etal foliation over S. And then with this so then the claim is that is that if I want to evaluate my high SE upper star of K at XF this is given by K at spec. And this is actually not completely true. So this is really up to or at least locally let's say the etal topology. It's a good enough approximation for us, okay. So all right. So I want to apply this construction in the following situation. So recall that we have this so in the previous lecture I was considering pairs of Lexus where r is itself a couple of integers up to up to n for some some n. All right. So these are all positive integers. So if I fix such a pair then there is associated a foliation X bar R underline times E bar m. All right. So it's a I don't need to recall the definition of this but it's really a very simple kind of foliation. But right. So we so I want to apply this construction with SE being this guy. So this will be my SE. And then I want to then take for K. I want to take the Duram complex over a diff etal foliation over this guy. So that's noted also by omega or maybe Chetch n of omega. Okay. So I want to I want to apply this. So okay. So then let me introduce some notation. So we denote by by CRM. Maybe it was a dot to parenthesis. This is my definition I up your top or star X bar times and maybe yeah I forgot the delta here. I want this to be discrete of the the Chetch Duram complex. Okay. So I simply take the Duram complex but on the small foliated side not on the big one and I extended it somehow by this functor instead of just taking the natural extension of the Duram complex. Right. So this is one thing. Then I now define the spectrum. So we define the T spectrum on smooth polarization by the following formula. So I denote by pi prime or M the projection to the point. Okay. So I define the spectrum. So I call B be maybe B bold. This is and it's actually a co-simplicial object. So it is a dot here. Or M. So this will be the the push forward along this map of simply this infinite suspension spectrum on these complexes. And this is made five. Okay. So for the moment this might seem a bit arbitrary. I try to motivate this construction in a moment. But yeah once we have this then the next step is to as we did as we see as we have seen before passing to the motor peak limit in the indices R and M. So then next thing we define again it's a co-simplicial spectrum but without indices this is a motor peak column. Maybe just before continuing let me just say one thing. So what I did here is essentially a variation of something I did before. You see if this may be a remark. So if I replace the complexes let me just do it in degree zero. That's so poor. So if this is replaced by the pre-sheaf or delta or by the Durand complex then we get simply and we get respectively two things that we have seen before namely the P1 delta localization again for the Pcft topology Pcft topology of sigma t infinity of or delta respectively the same thing but of the Durand complex. Okay. So I did this I just said today or I just reminded you about today these are things that we have been considering before and somehow they are responsible of the models P2 or P3. So what I did here I somehow introduced a family of complexes which we'll see in a moment somehow sit in between or delta and omega and I use them somehow in the same manner as I've been using or delta and omega before to get a model for the Durand spectrum. Okay. So I hope this at least gives some motivation for this construction. So let me say a bit more about this spectrum. Maybe also just also a remark. So the spectrum again B0 maybe will be the subject of the conjecture I want to address today. Yeah so again the reason why I have introduced this this object maybe even the dot here it's simply because the conjecture I want to address today is seems to be false for that one here. So I had to somehow to do something slightly more complicated and this is the thing that I came up with. All right so this yeah so but before getting to the conjecture let me let me derive some easy properties of this spectrum. Okay so yeah so the small lemma so this is an easy fact. So so we have a natural morphism from or delta to to the complex CRM maybe the zero part of it. So without the church construction which is CFT local equivalence. So this is actually very simple that I just want to do it so that somehow we get a bit get usable to this construction. So remember so the way I constructed this guy was was to start with the RAM complex but only on on small on the small side of this particular foliation x bar. And of course here here we have also a natural map from or delta to omega this is just the h0 of this complex. So all this is living on on this small side but this is in fact as we know this is a local or CFT local equivalence very this is before this is just a version of the Poincare lemma which is actually even true on on the big sides but but I just want to use it now on the on the small side. And now the next thing is to remark that this function I upper star and delta preserve equivalences. And this is a totally general fact whenever I have a morphism of sight one has such such a property. So this is this is a proof. Yeah so in particular one get in fact a chain of weak equivalences locally. So maybe I put I can put another church complex and this is again okay so this is a sequence CFT equivalences on the big side of x bar. Okay so as I said before somehow these two things I used before and now I'm somehow looking at something in in between. Okay so maybe I should also say but not but not Seattle local equivalences. Okay so this is somehow the thing that this is why somehow I I will use this thing in the middle because it will give me something a bit different locally for the Seattle topology. Okay and so again let me somehow go back again to this remark. So by passing to the co-limit of r and m so what what we what we get is the following sequence of spectra on smooth voliation. And then there is this me let me put let me do it on the cosimplitially. All right and then this is as I explained before this is just the model for for the spectra omega. So it's not the case that this is a model for omega this is a model for omega after you you further take a level-wise CFT five-inch replacement. All right so we have we have we have this thing so in particular there is a map from this guy to omega. Okay so and let let me just say it again so a corollary of the of this consideration is that the morphism is spectra b dot CFT five-inch replacement level-wise to to the church and here is a church also. So this guy is again the stable a1 C at local equivalence. Okay so I have been using the same fact but with b replaced with this guy before and I'm just saying that it's also true for this guy and therefore we can do as the thing that I explained in the previous lecture but with b instead of this guy and get compute the restriction to smooth varieties get get a new model and then it turns out that this new model is as good as the model p3 that I that I discussed last time. Okay so so so this consimplisher guy give rise to let me call it p4 dot which is which is as good as p3. Okay so we can apply really the same method for to this. Okay so I and and so therefore I somehow I now try to understand if this guy is is an omega spectrum so that I can somehow complete the proof. Okay so let me so I can maybe now state the conjecture so here's the conjecture that I want to discuss today in precise form. The restriction be at each consimplial degree that's simply the restriction of this to smooth varieties is a weak for every for every. Okay so this is what I would like to discuss and this as I said will be sufficient to somehow complete the proof. So I would like to discuss I would like somehow to introduce another conjecture and explain why this this other conjecture uh relates somehow to something more classical and also then explain why this other conjecture is actually too optimistic. So so here's maybe remark so so how one can one prove such a conjecture the one that I just stated there. Um so the the first thing one could try to do is try to to guess a formula for for the spectrum bn. So we could we could try first there's a formula for bn the spectrum even just level wise. So here here we will be working so working psietal locally okay I mean the the conjecture already give us a guess locally for the topology of cft but but this is maybe hard so let's let's try to first uh I mean this is what we want to prove so but to prove this we try maybe we can try first to guess if you have a formula for bn uh somehow locally for the topology. And uh so so the next conjecture is somehow a guess or a plausible guess of what what the spectra are and as I said as I said this this guess will turn out to be too optimistic but I want still somehow to uh to give it for you and try to discuss this a bit more and and see um so hopefully somehow it it makes this conjecture a bit kind of maybe more reasonable or more natural okay so to to and to explain the guess I introduced one notation um so from now on when I write c uh church omega I really mean uh the fibrant replacement of this guy so uh this means in fact the stably A1 et al fibrant replacement the church construction on omega cofibrant okay so I hope this will not create any misunderstanding this is notation and and and therefore and and uh and also I want to introduce a truncation of this so when I write this this is the usual truncation of this of this of the t-spectrum okay so uh truncated version I shouldn't put here psi eta so this is uh this is on the level of foliation I'm working now okay so these are notations and with this I can state this uh this guess or I don't want to write it as a conjecture but because I know that it's wrong so maybe I just hold it to guess instead of conjecture um so that that is level wise a1 c et al vocal equivalence between uh the spectrum b n okay so of course in degree n uh and uh this truncation of uh okay so I'm saying that so this this spectrum that I have introduced uh which looks a bit weird is actually computing something much more natural and simple uh simply the the truncation of this uh spectrum so for example in degree zero this is b zero is just the truncation of the Durant complex that we have been that I also introduced before um which is also somehow the subject of of of the main conjecture that I'm trying to to prove so this is this is a first guess so I try to explain why this is guess is not so uh I mean why why one would guess something like this and even give you some evidence and and some partial result uh so that this actually is is uh at least expected uh in or maybe a weaker or a version of this is actually expected um yeah so I try to explain this but maybe maybe first let me uh let me state the following lemma just to be um maybe first proposition which is actually not not so obvious and in fact it it will lead to some to introduce some interesting object that I will need later so let me maybe do this uh some care so uh so the natural map to omega is a level wise okay is a level wise uh CFT local to give some details about the proof of this so uh and I start with n equals zero first this is the simplest case and this is actually completely clear uh so in case n zero what you are what I am claiming here is that if I if I take the Durant complex I truncate it goes to the Durant complex and this is level wise CFT local equivalence so to to prove such a thing we we look what what are the level of this of this morphism but so in level level p this is up to a shift simply the truncation at minus p of the Durant complex I'm not the spectrum now to the Durant complex okay but so since p is uh is bigger than zero this is uh this is actually contains uh Odelta right so Odelta is still the H zero of this complex yeah and then then then we are done because we know that these two guys are indeed CFT local equivalences by the Poincare Lemma okay so the the case n zero is really somehow clear but I want to discuss somehow the the the case of larger degrees which is more interesting okay so uh in higher degree we have to use uh a bit more so so now for for n zero so I will use the fact as a black boss uh which is the following so so in fact we we have some information about this this Durant complex uh we know that it is uh it's isomorphic to uh to the Durant complex itself and tensored with over Odelta with some uh sheath h maybe put an n here where h n uh is a minus one connected sheath maybe or complexes of sheaths or pre sheaths of OX mod u Odelta mod u okay so this fact in fact will imply the result by using the same same reasoning all right so if he knows this then so fact implies result again by by the Poincare Lemma so the point here is that uh h uh also go to the truncation omega because of because because h is minus one connected okay for for for p for p so this is because h connected and therefore uh so this is this is this is what we have in level level p and these guys are equivalences because of the Poincare Lemma okay so let me maybe just say one word about about this fact so what where it comes from so I will not really say much about it but just tell you that that this for example if n equal to one so this h1 is something which which has been considered before or more precisely so if I evaluate h1 at some let's say uh yeah in fact take okay let me just do it simply I evaluated at the base field this is actually the the help algebra or the motivic help algebra of the field uh k for the deram fiber functor if if if I take an extension and and think about it as as a as a discrete variation so if I look at k but but k uh so k is an extension base field with the discrete variation so I think about about the element of k as constant then this is also the same thing uh this is actually yeah this is a motivic of uh in fact algebra maybe uh again of of k over k and so so this is actually naturally a k times k algebra okay so the the if you know about uh help algebra it so it's like like a groupoid and and and the the space of of object is actually the spectrum of big k so this is why uh so this is like a groupoid uh where the object is spec k so it lives naturally on k times that's okay so and and the fact that this that this is minus one connected is somehow of course related that we know that the motivic help algebras are minus one connected okay so this is said it's I can now prove the following lemma so so this guess imply the conjecture that is that is still there uh yeah simply because so if you if you know that this is a level wise anyone see it at equivalence uh then we deduce that if I take this make it vibrant then again this is also a level wise otherwise okay quasi isomorphism uh to the this guy here again made cft five but uh because of the proposition that I just erased uh this is just in fact omega right because uh this is already a cft vibrant and we know that the map from here to here is a cft local equivalence so we get this uh and so we get that this is the same as this but this is of course this guy is an omega spec so it it proves the connection uh so this this looks like it's giving much more because we are somehow telling us what what is what is this guy but in fact we know that if if we know if this if this gadget here is an omega spectrum then we know what it is it is it is this and here of course I I forgot to put the chech everywhere okay so this is an omega spectrum by by by convention by the notation that I just maybe remark we know that you know what is it is a stable cft vibrant it's it is this so so this is not really uh giving more so so in fact so the guess is not giving than than expected more than than than given by the conjecture except that uh really the conjecture is uh saying that the spectrum is not an omega spectrum but but only a weak one um right so so this lemma is actually saying that this is uh an omega spectrum in this strong sense and somehow this is related to the fact that that the guess is not really true as as as I will explain later and so in fact we we it's maybe too much to expect that this guy is indeed an omega spectrum all right so um I don't know I it's maybe a good time to take a break now and uh I continue next thing I want to discuss is uh some analogy who is so-called radiansome left and bound okay so let me maybe uh first write again this this guess there is so the guess is that uh so I have introduced uh uh some spectra uh I call them b uh it's actually a cross-implicial spectrum so there's index n um and I so my guess was that there is uh an equivalence between this guy and the truncation of the the nth church object on omega so this is is uh level wise Seattle local this was a quite strong equivalence between spectra so this is this is uh this was my guess um we also so I also said two things that that uh so two things worth mentioning is that we know that uh if I if I take a fibrant uh replacement for for the psetopology on both sides then this is essentially true so okay so this we know that it's just the same thing as that guy um it's not and yeah and you know that the the composition is at least uh a stable local equivalence okay so we know this I guess if you want uh locally uh for the a1 cft model structure but somehow we we we we we would like to have something much stronger which is which is which is this one other thing is that uh yeah so so so we know that this is uh this is the same thing so in fact I can write this uh guess in a slightly different way that's following there's a different formulation just saying that if I if I take this spectrum be at a degree the uh conceptual degree n then in fact it is equivalent to truncation of the same guy but made fibrant this is actually stable not necessarily just level wise okay so this is uh again this is the same guess but written differently and when I when I write it like this we see easily uh a relation with the Bayeson-Lichterman conjecture in fact this guess somehow was indeed motivated by by this conjecture so maybe maybe you don't know what is the Lichterman the Bayeson-Lichterman conjecture so let me remind you it is in fact now it's it's a theorem due to Wojcicki and Rust so recall which is now the theorem that's it so it says the following so it's it says a slightly different setting but so we we are working here uh with integral coefficients so so let let me write one for the unit object m k with integral coefficients so this is Wojcicki's category of motives so this is one this is the object representing motivic homology then there exists it's a level wise in fact a one Snevich equivalence spectra between one and one made etan fibrin and truncated so this is maybe not not the initial formulation of Bayeson-Lichterman but it's it's the equivalent okay so and then you see of course that it's somehow kind of a similar statement so so the of course the analogy maybe it's worse doing table of analogies so this is the the BL conjecture and this is the guess so here we here we are working on smooth varieties here we are working on smooth variations here there are also two topologies there is the Nisnevich topology and then there is etal topology and somehow it's a statement is about the interplay between these two topologies here the what plays a role of the Nisnevich topology is this etal topology and then this is c-foliated topology here of course we have the unit object which is replaced by the spectrum b maybe let's say b0 at least yeah and then then it's clear that this is somehow an analogous statement and in fact it turns out that there is also another conjecture classical conjecture which somehow is related to this guess and also actually people do consider this conjecture as an analogous as an analogous to one this is it's it's a conjecture which is due to Susslin concerning Lawson homology so I also want to discuss this so this is actually kind of more direct directly related to this guess and we will see that in fact this guess is a generalization of this of Susslin conjecture but somehow it's how to say so so this guess is wrong but but as I said it part of it is true and and this part or part of it is expected and this part is really just the thing the Susslin conjecture so I want to explain this also yeah first explain what is Lawson homology so here's a definition so yeah let's let's start with an effective motive so maybe with rational coefficient and let's let's let's fix an integer positive integer Q this would be the weight of of the Lawson homology so I define a complex so it's called called by L, D, R of M and Q and this is by definition it is a DRAM realization so this is R the R this is DRAM realization of the following motive the internal home of M with Q all right so and so so somehow yeah Lawson Lawson's homology at least DRAM version of it so the I think the original version is done with Betty homology but of M in weight Q is homology of this complex so this is Lawson homology so maybe I just the word what why what is the idea behind this so let's assume that M is is an effective motive of some variety X then this object this object here the internal home of M into QQ is is really somehow the motive of the of the space of co-dimension co-dimension Q cycles in in X okay so the idea is that this is so this is home maybe up to shift but let's say Mx N QQ is the motive maybe the monoyed or the group completion of the monoyed I don't know the group completion the monoyed of co-dimension Q cycles and therefore we are just somehow taking the co-homology or sorry or the homology of the space and this is somehow the original definition of Lawson all right so so there is this classical invariant and something conjecture somehow is about this these these groups so I can try to state it now and maybe first a notation so so yeah sending a variety X to to this complex around the effective motive of X comma Q defines with some care defines the complex of pre-sheaths LQ sorry LDRAM Q on smoother ideas and we can also make a spectrum of this so find a spectrum LDRAM to be the collection of these LDRAM Q maybe shifted by Q yes one has to define a bounding map it's going to be done and so we have we have a T spectrum and the R and this is the spectrum of course represent the Lawson Converge yes so by construction LDRAM represent Lawson Converge or the the RAM version of it okay now I can state the Sussling conjecture again it's a the RAM version of of it just in conjecture saying simply that there exists a level wise or just again reformulation that exists level wise a1 et al or a1 et al local equivalence in fact it's even just an et al local equivalence but okay between between Lawson the Lawson spectrum and truncation of the DRAM complex so this is contrary to the to the Baylinson-Lichtamon conjecture this is still open and probably will still open for some time in fact it may be worth mentioning that Baylinson Baylinson has showed this conjecture and the Sussling conjecture in fact implies the standard conjecture in characteristics so it's really a deep conjecture so yeah so the thing I want to say now is yeah the following theorem so this is as I said it's I didn't really prove it it's just I think it's true and I think I know how to prove it but didn't write anything yet so it's the following so say that the restriction of this weird spectrum of let's say in in cosimstial degree zero so the decision of this guy to smooth varieties is actually nothing but the the Lawson spectrum so it's actually is equivalent to and as a consequence of this theorem we see that the our guess is it's true if we assume it's true under under Sussling's conjecture and after restricting to smooth varieties okay so in some senses this guess is could be considered as an as an extension of Sussling's conjecture to foliation of course a wrong extension but nevertheless an extension okay so maybe I try to say a few words about about this the proof of the theorem I don't really have time to do this but yeah let me maybe very quickly so just to maybe to to motivate this this construction of the spectrum B just maybe with weird sketch okay so so we fix the smooth varieties and we would like to understand the the global section of X of this of the spectrum B okay so we would like to show that this guy computes Lawson homology this is our goal in weight in weight okay so we we go back to the construction of this spectrum so remember this guy is given as a homotopy column of some pieces like this and each one of this is itself a direct image along some some map okay so right so this guy here and I can write as no I don't want to write this guy so so I want to know so therefore I look at the r gamma of x each of these pieces so okay so this is given by just go to the definition r gamma of x with r psi et al pi prime of t tensored with q tensor this this complex appreciative CRM now okay now the point is that this CRM is actually remember it was pulled back from from the small foliated side to the to the large one and so because of this and and with some computation and one one can see that this is given by one can somehow commute maybe let me write it and then I explain so so there is a trivial commutation so by adjunction you can put somehow x here putting but then yeah because right we are tensoring this the state motive was something pulled back from a small it's small foliated side it turns out that that that the internal home from x to this guy somehow commute with this tensor product and at the end we get something like this okay so just again again once once let me say it once so because of the of the nature of this of this complex because it's somehow so simple it's one can commute the internal home of x with value in here and and somehow taking outside the the parenthesis so you get such such a thing and once we have done this this also can be written like this like I could replace this CRM by omega because on the on the small side of of my scheme and and because I'm just computing now homology over this side these two guys somehow they coincide so we get this and then from this it's it's kind of easy to conclude so one know that that taking the co-limit over this object is just essentially computing the drum homology or the drum homology of this of this is passing to the co-limit homotopy co-limit or m get the homology of this object and this is yeah look at the definition of a gift of host homology you see immediately the link all right okay so let me how much time is left okay so I still have some time so I want now to explain why why this guess is wrong yeah up to now I didn't I just pulled it was wrong but I did not give you any reason to believe that it's wrong on the contrary I somehow I explained to you that it's maybe linked to some deep classical conjectures but somehow the problem becomes apparent when you somehow go beyond smooth varieties and start to look at more complicated variations you see that there are this is somehow a too optimistic guess to be true okay so this is why the guess is wrong and how to fix it so the guess is actually um wrong in two directions in some sense so it is already wrong in if I take the the conceptual degree to be zero so and I start by explaining this so it is already in course simply shall degree zero okay so I will just somehow give you a heuristic argument I don't really have more to offer but I'm really convinced that it is too optimistic and let me tell you why so let me fix some notation so um so take an integer d maybe positive let let k be be so let me write delta for a set of derivations so it's a set of commuting derivation for example acting on in the usual way on on the field of fraction k t1 up to td and let us uh fix k an extension differentially delta x so so delta is also acting on this field and it is differentially closing in some sense okay so to this I can so so the point that the spectrum of k is naturally a affiliation k affiliation which behaves like which behaves like a point for example uh say uh with respect to the CFG topology it's like an algebraic closed field for for the for the topology now let's look at uh let's see be a smooth curve and let us look at uh yeah I want to look at the r gamma of c uh I think for the c at topology of c with values in b d0 but in level one okay so this this guy actually maps to the ramp homology of c uh maybe shifted by one but it doesn't really matter so maybe let me I want to write it like this so the the ramp complex okay so that I mean this guy in level one it maps to the the ramp complex in level shifted by one and and this induces such a morphism now so one can somehow use similar argument in the the sketch of the proof I gave to to show the following fact so maybe it's heuristic but so how to say so so the the only classes in h1 the ramp c that one can reach okay reaches uh by this map are those that are that are pullbacks h1 the ramp c0 uh where where we have a map from c to c0 and c0 is a just a curve smooth curve over over k small k okay so what I'm saying here that even if we take uh so if we if we take a big differential extension and take curve there this somehow commercial commercial theory is built out of uh the ramp converging classes but of smooth curves over the base feet so so so something like this could be somehow uh I don't know uh proved in some sense and but but then of course you see that as a problem because there are indeed classes in the in the h1 uh the ramp of c which are not of this form uh so maybe I should say what what what is the h1 the ramp of c so the way you can compute it is as follow so you can the the h1 the ramp of c is naturally um a k a big k vector space so because so you look at the the ramp so you maybe I should say it so so what what what do I mean by h1 the ramp of the foliation c so this is uh I can show it it's given as follow so you look at the at the h1 the ramp in the usual sense but uh of the curve over k okay so this is a k vector space with a connection okay so this is k vector space plus a connection and then you restrict this connection to uh to k uh considered as a foliation and then you compute the you can you compute the commodity but but here because k is is differentially closed um so maybe I should practice this so delta zero um yeah so because k is differentially closed in fact these two have um so the the rank of of this guy is actually the rank of of that thing okay so I don't know if this is clear I mean it's very simple I'm just I'm doing a bad job in explaining this all right so so you have a multiple integral connection over big k which is uh differentially closed and so in this situation we know that the the space of constant has the same dimension as as a rank of the module okay so this is a and so there are a lot of sections here and and uh not not all of them really are coming from in some sense from from geometry from so this is of course it's it's it's very uh it's somehow telling that that this yes is really very very wrong very far from from being true but let me just give again one more uh positive result and about about this again about this guess in degree zero so however I believe that however I believe that the following is true so this is conjecture I was not able to really to prove it but I think it's it should not be very difficult I think uh is the following that so if you don't take curves over k but but just look at our gamma uh k with value in in b zero any level I believe that this is uh just as as expected namely it's just the constant in k uh shifted by I do I do believe that this is uh this is true which is a good indication for uh so I think this this this is should imply that if you look at b zero and take a fiber replacement but this is this is the spectrum omega because the spectrum omega has also this same property so it's uh uh on on field like this you only have h zero and so this formula is true for for omega uh and so the idea is that uh if you work locally for this topology it's enough to check that two objects have the same values at fields like that so this is not of course precise but I think this conjecture has this as an application um and then of course if if we could so then the then uh the hope would be that that this conjecture is also true for uh all cosimplisher degrees but uh which would then imply what what you want in some sense but but actually this is uh also wrong but unfortunately this seems to be wrong so what what seems to be true is a slightly weaker version so I'll try to explain this okay so what I believe to be true is the following conjecture so again let's let's take k as before so an extension of qt1 up to t up to td which is uh differentially closed um then I expect that uh again the r gamma of k value in bn by q uh is what we expect so what do we expect here actually we expect this to be this h and k so if you remember there was this uh sheaf uh of of odd delta modules hk which which appeared uh in one of the proof in the previous lecture in the previous hour which uh so if I if I replace small k by big k here sorry if I if I take small k this is just mutilicope algebra so this is and so this is what what what what we would expect stably so uh if we uh yeah but so the conjecture is saying that this is true uh level wise but but for some for q larger than the dimension d is somehow the differential dimension of the field k so this is the conjecture which I think it's true and which I also I think it's not so difficult to prove and let me maybe explain why I believe that this is true but only for large for large q's okay so here's an explanation oh yeah I want to explain why why in degree zero I expect this uh conjecture to be independent of the weight but that uh in in higher conceptual degree I need the weight to be at least the dimension so it's actually it's actually very simple as the explanation is as follows so so again one look at at the natural map uh in any way let's see and and this we know because so since k is is differentially closed this is simply h and k shifted by q by the way uh I should have right so we look at this maps and so uh one can show that uh this map or the image of this map spanned pullbacks morphism of differential fields with dimension of l at most q or or you can take it to be equal to q okay so so this is a this is a very explicit vector space but now I can show that the image of this is generated by uh as follows so you take you take you look at all morphism from spec k to spec l where l is also an extension or or also a differential field but also maybe differential closed but but here it has differential dimension less than q so the number of derivation is at most q uh and then you look at the constant here and then you you pull back the constant there no no no no not the constant but but rather these these chiefs okay so you look at the value of this chief h n at l and you take its restriction to h and k and and look at the sub vector space which is generated like this and there's no no reason to expect that if you take q smaller than d then you can somehow span the whole thing but of course if q is larger than d then then obviously you can take k here and then you you you can reach everything somehow this is a the idea and you see that if if the if the consumption degree is zero then this is just k delta and for k delta it's clear it's easy to to span the whole thing with with any with any dimension there is no no no difficulty in doing this so this is somehow this this is why somehow this should explain why in conceptual degree we have a different slightly different conjecture so unfortunately this is this conjecture which i think is not too difficult uh that one would would not suffice to uh to have to to to get a similar implication like this right so so i don't know i mean i don't yeah this is not sufficient uh because of this uh assumption here uh this this does not imply that b b n uh made fibrant level wise is an omega spectrum or is level wise okay so this would be sufficient here but but yeah we will get the problem for highly and so but but then there is a natural uh a natural conjecture which is stronger than this which would be which suffices unfortunately i don't really see how one can prove it yet but let me state it so the slightly stronger version of this conjecture is that follow so so take k as before uh and then so i expect that the r gamma now i have to put a topology here i i don't need to put a topology if i had only k because k is a differentially closed but here i'm looking at k times e one one wedge p and again this value in b and q let's see this this should be again h uh each n of k maybe shifted by uh q minus p and this should be true for all p positive and for all q bigger than the dimension okay so uh yeah the difference between that and this is that we allow now uh we we put this uh the state motive here um all right and in fact we don't really need to have an isomorphism it's kind of enough to show that you can get the map in this direction so it's probably very hard to compute the homology of such a thing with such a complicated object but it's maybe much simpler to find uh a good map from from here to to that this is this seems maybe more approachable but anyway so if if we assume this conjecture uh then this would imply that uh that this guy is a weak a weak omega spectrum okay so i think maybe i could stop here i think i have something else to say but maybe just say what one one last thing so here uh you you might wonder what what is the end so if you fix x and smooth varieties you might uh and the level p you might wonder what is the n zero in the definition of a weak spectrum which we we need so but it's uh any n zero which is larger than the dimension of x minus p plus one will actually work so it's it's an easy exercise somehow to to figure out uh how this how such a conjecture would would imply so if you have questions think we still have some time i'm sorry i i i was i was hoping to to to do something nicer today but uh yeah it turns out to be also quite technical even to speak about this but