 Thank you very much. So let me start by thanking the local organisers and the organisers on the other side of the world for the opportunity to speak here and in China and in Japan all at the same time. It's a really unique experience. So as I've said, I'm going to be talking about some very mechanical homology of formal schemes and characteristic theory. So, I don't... The table doesn't move. Ah! There's a lever. Ah yes, because it's very hard to adjust. Can you move it to the other side? You can use it, exchange it, but this thing there, it doesn't go there. So you can move it to the other side? Pardon me. Fucking stop. So the government will precisely be to discuss some definitions of continuous mechanical homology and this continuous is going to be important because we're going to imagine very non-continuous variants as a theory and as how they compare. But that won't be the focus today. For this material, we're closely related to some continuous k-theory of these formal schemes but the purpose is not just so much to offer these equivalent definitions but rather to derive some nice applications from the equivalences involved in the definition. So let me start by setting up then a little bit of the notation we'll use. So my rings... Everything's going to be in characteristic P today. Everything's characteristic P. Everything will be in Ethereum. And even more, everything is going to be so-called f-finite. So we say that a ring is f-finite precisely when it's finitely generated over a subring of P powers. So as it has a finite P basis essentially. But it doesn't mean to be a basis. It just means to be finitely generated. So for example, most of the time you can just assume that we've got some variety over a field with a finite P basis. But what's important is that you can also take the completion of such a variety along some subscume. And in fact that's really the reason that it's convenient to work with these three general hypotheses. One needs to reduce certain calculations for the case of complete local rings which are of course not treated as classical results concerning finite type problems with server fields. So I can move this up here. But now they can't see that, right? Yeah, but we have peak energy. Okay, I think I've got the hang of this. So in such a situation we're going to be considering the Whitman. This is the usual P typical Whit vectors. And the random achieves to be in the etymology, but we can move them where we like. We'll continue here then to make a video review I suppose. So let me now explain then, now with this basic setup, how we can define the typical homology of a usual smooth variety in about five equivalent ways. Do you assume that this is... No, in fact in the moment X is going to be some thickened subscume of definition of a full-mosecume. But let's now suppose that X really is a smooth, to a view of what's true classically, let's suppose that X is a smooth variety over some perfect field of characteristic view. Then we can define a tau. So I'm going to be primarily today in a tau motivical homology, but most of what I'm going to say would be verbatim if you want to define usuals as a risky motivical homology that's something we're using in this nabbage topology in between. But for the applications it's more useful to consider it from the point of view of the etymology. So let me say that we can define the etymotivic homology. So this is sometimes called the Lytzenbaum homology, each star or I don't really have a standard subscript that's sort of the motivic etymology, something like that. So I'm going to be considering it periodically because that's the really interesting case. So I'm writing capital P, so I'll consider the finite coefficients, weight n like this. So let's say this can be defined, the correctly shifted etymology on X of any of the following, and I'll present you with a bunch of isomorphic fields that are now known to be isomorphic and hence provide a equivalent definition of the motivical homology. We can firstly, so I can move that up a little bit, like that looks okay. The most explicit definition is that we take the etymology of the logarithmic hardwitch sheaves sitting inside the deramite sheaves. So these are these WR, omega X, log n, as I said by definition the etym... Ah, so alternative notation, this is sometimes written as a new maybe sub R brackets and sub X, where one puts the R, the N and the X seems to have evolved in the literature, so I'll use this notation instead. So this would be a tau sub-sheaf of the deramite sheaves, and we are omega X and N generated locally by the log forms, that is to say the log of some type model lift F1 multiplied together to delog some type model lift Fn. Almost succinctly, which I write just as a convenient opportunity to introduce this delog map, I mean to say this is the image in the etymology of this delog map from G and X, then in the etymology to make things clear, N times to the deramite sheaves. And as I already mentioned, these usually go under the name of the logarithmic, and I wanted to attach some names to it, I can just squeeze them in at the bottom, so this was a classical definition of a termotonic homology that was considered before material homology existed. I think that Carter suggested this would be included there. Exactly, so the names I was going to include are precisely Carter, Milne and Lichtenbaum. I don't know how the independence works, perhaps Carter and Lichtenbaum independently suggested it might be the right definition, because Carter in general, Lichtenbaum was more interested in special values of zeta functions for smooth projective varieties of affinite fields, and he proposed that these values should be detected by this homology theory and Milne developed this theory in some extent. Others could be included, but at least people thought it was true. So as I say, that's somehow a classical definition of the Piatica termotonic homology in characteristic P that doesn't require you to know anything about the termotonic homology. So an alternative definition, then, is I can, instead of taking the convolution of these logarithms that's hard with sheaves, I can take the homology of some Milne-Malcay theory sheaves. So this is by definition. I take gmx. Again, m-copies. Let me not worry too much about the topology, probably being the Italian topology for most of today, and I kill the relation that some symbol f1 to fn is 0 whenever there exists an indices i not equal to j for which fi plus fj is 1. So it's a classical Milne-Malcay theory sheave on y and the log map from gmn to the hard-wit sheave factors then through Milne-Malcay theory. So I can do just a little bit of this on the board as these disappear factors like so. Thank you. Instead of Milne-Malcay theory, we can also consider Cullen-Cay theory. Let me just call that knx, Cullen-Cay theory sheave, which I won't define. Let me just remark that we have then the natural map from Milne-Malcay theory to Cullen-Cay theory. If I'm in the case, for example, n equals 1, which will interest us in particular, then everything degenerates to gm. That's true, but then the next statement that my gm won't reach you. I'm going to do that, I see. So as Arthur was saying, it's also the case that the Milne-Malcay theory coincide in Degree 2 as well. And maybe for experts, maybe the only expert is in fact, Arthur, this would really be improved Milne-Malcay theory in the case in which I had, in case in which I was working with a smooth idea of the finite field. That could be some small problem. And then some people studied who did the characteristic p-dollars quite well. Yeah, yeah, that's what it's going to come. That's exactly what leads to these equivalences of the different definitions. Yeah. Yeah, we'll turn it out then to flesh out his comment that since we're honestly divided by characteristic p, modular any power of p, one knows that the Milne-Malcay theory actually coincide. That's what we'll see in a moment. And that's what I started out by saying that logarithmic odds which heave is in some sense the most naive definition that you can give even without knowing anything about material cosmology. And then the other end of the spectrum you have some general definition that you can take the correct, weight and correct shift of the Vajrosky's material complex. I think it's more p to the r. And let me then add one more to the list. So if n equals 1, this is perhaps the most closely tied to the old part of the story involving Milne and Litsenbaum's study of the logarithmic odds which heave, they also considered the cosmology of the P-Pirates of unity sheave in flat topology. And it's known that this is then equivalent. So if n equals 1, I can add to this list of isomorphic sheaves. Now I'll get complex sheaves. The divided direct image of mu p to the r pushing forwards. So I'm going to add topology down to be a tau topology and I'll shift it to minus 1. I think if I want to coincide with my indexing, yeah, so that means I'm looking at h. With a degree matching, we'll be in the rhetoric homology. Yes. So can I use... No, because then I cover up my screen. I want a new board. Okay, but I shouldn't move to the other side. No. Okay. Right up. Okay, so we now have onto the boards, for example, we have four or five then possible definitions of the materialic homology of a smooth variety in characteristic people. Okay, and also you have to take the miller and the Krillin model of power of people. Oh, sorry, did I forget to... Ah, yeah, sorry. Yeah. I also want to say that it can take model of power of people just in the naive sense or in the direct sense, but in fact the kernel is zero. Yeah, in fact the kernel is zero. No, the first part of those important questions, the miller and the Krillin K30 sheet, should of course be taken modulo p to the r, if I want to define that. Homology with coefficients mod p to the r. So as I say, these are all isomorphic. Let's start with the case when n equals 1, stop there. I just want to say bye. So y, so one firstly uses a, let me call it a fundamental exact sequence due to inner z, stating that the sequence of sheaves given by zero goes to gm, goes to gm by raising to the p to the r, goes via the delog of a type model lift to wr omega x1 log, is exact. And one combines this with a result from Goetendic's study of the chronological Broward group, which tells us that this derived direct image of gm in the fppf topology is nothing other than gm in the total property that I want to add one more, and maybe we add to the list three, Vrzovsky's identification of z1, weight 1 being nothing other than gm. And so if you combine 1, 2 and 3, you can easily check that when n equals 1 the least possible definitions of the weight 1 paedic, town, the typical homology are the same. The middle and the equivalent k-period definition already coincides because everything collapses to gm, then Vrzovsky's general definition just collapses to some gm log p to the r, but we see from Ilozine's sequence that gm log p to the r is nothing other than the logarithmic hodge-witch sheaf, and then you easily extract from Goetendic's result on the direct image of gm the FPPF topology is a similar result for the p-period of unity sheaf in the FPPF topology, which again tells you that the FPPF definition of p-period homology will just collapse to the homology of the logarithmic hodge-witch sheaf. In the end, everything collapses to the same definition. The situation then for higher weight homology is rather more complicated. I'm not trying to get the history completely correct here. So let me say that again that she's all isomorphic in positive weight by results of let's try to get chronological to the right we have the Bloch-Kartogemper theorem is Bolden's result with a normal cation of fields in characteristic p, have no p-torsion then perhaps Goethe and Suhrer, who established Gersten resolutions for logarithmic hodge-witch sheafs that you deduced was also logarithmic hodge-witch sheafs from the values of fields combined that with Bloch-Kartogemper to get information about Milner-K-theory sheaves in terms of logarithmic hodge-witch sheaves through the field case Kurtz's resolution of Gersten conjecture from Milner-K-theory and Greyser-Levin finally with Milester so anyway, this is the name of the path to the experts if you add together everything they've done you can deduce in quite a straightforward way though it's actually hard to find these precise statements in the literature that firstly, one has a generalization of this fundamental exact sequence of it as z but now from Milner-K-theory in higher degrees stating if I did it in exactly the same way stating, as I said, that the exact analogous sequence is indeed exact so that is to say that the Milner-K-theory sheaves have no torsion and that the I know the kernel of the logarithm is exactly multiple and then what you can deduce mainly is the result of Gersten-Levin as also already mentioned is that mod p to the r the Cullen and Milner-K-theory sheaves coalset moreover we can identify Milner-K-theory mod p to the r where the Weibovsky is much more complex mod p to the r yes this is Greyser-Levin it's Greyser-Levin plus Gersten-Koncharcter you'll use it in the case of Field and Greyser-Levin in the case of Field so as I say one struggles to exactly find statement 4 and 5 in literature but rather it's a straightforward quarrel of assembling of all the hard works you would have done so you get from that that then even in weight and bigger than one all these equivalent, all these possible definitions of mechanical homology of smooth-dividing calculus of p would again be the same so the goal of the main theorem that I want to say something about today is to extend this identification of possible definitions of mechanical homology and calculus of p from the setting of smooth varieties to the setting of regular formal schemes before I should perhaps remark that these various isomorphisms of foam plus smooth variety would fail as soon as x is non-smooth of course it could be regular because we're comparison still it's a limit of smooth things it will be true for regular so in fact if I take x to be any regular fp scheme then the statement will make true yes, this is true but I would guess if I insert any even very mild singularity into x they fail to be true and in particular if I make x non-reduced then they will definitely fail to be true and what we're going to see in a moment though is if I take a formal scheme and I consider what happens for all sub-schemes of definition then although for any fixed sub-scheme of definition all these isomorphisms will fail that they'll somehow hold in the limit then is that your right sense? is it okay? you mean I take some hypercover by smooth variety I think this will then depend on the sense if you interpret it to mean that I start replacing my singular variety by some smooth hypercover then it's totally logically true I guess if you use a ring you make a simplistic resolution by some rules and then you apply the theory for those and then you get something you get something, it's not clear what you get you get something but then you can prove it's independent like for the quotations problem it's independent of the resolutions the key maybe the capabilities will change yes I think it's not going to be something to look at so let's say there's my theorem that I want to discuss in an analog of these identifications for formal schemes so let me take y to be regular formal fp scheme y1 some sub-scheme of definition I keep going too far on that I take y1 to be some sub-scheme of definition and then I'll write y sub s to mean the f's infinitesimal thickening of y1 so it's defined by the s power or the s plus first power this is a well I want the notation to be consist depending on what is the first infinitesimal yeah in the literature sometimes it takes the n plus first I want y1 to equal y1 okay so it's this convention that's a good convention no you're wrong I see the possibility the first infinitesimal thickening could mean I okay for me the first infinitesimal thickening means no thickening which is maybe not a good convention okay so I make it s minus 1 so what about this I want y1 to be y1 just to keep the notation sensible the same I stopped there so then we're going to assemble together the following possible definitions of the Piedmont-Britjewik homology for each infinitesimal thickening of this sub-scheme of definition to get some pro et al schief on the formal scheme or equivalently on any of these sub-schemes of definition ah sorry I didn't mean to act like a hyphen in the one place I mean it's a no that's why I'm trying to get this hyphen right because it's it's a pro open bracket et al schief yeah this is a problem let me now have yeah first you write it for it first you write it for it, okay it'll be clear on situation let me write it all down and then pass it for the whole bit okay so that means what's going on now that it's all on the board so for each of these infinitesimal thickenings I can consider the previous definitions of the Piedmont-Britjewik homology which work for a smooth variety I can consider for as I say for each s-first infinitesimal thickening I can consider the logarithmic of Wittschief on y sub s or I can consider the Milmore k-fairy-schief on y sub s or I can consider the Quillen k-fairy-schief on y sub s or if n equals 1 I can consider this the entire direct image from the fpbf topology on y sub s of the Piedmont-Britjewik of the Piedmont-Britjewik of the Piedmont-Britjewik of the Piedmont-Britjewik and here r is fixed so let me first emphasise the point that for any fixed infinitesimal thickening I can consider any fixed s I think none of these maps will be isomorphisms absolutely none except if the s equals 1 and y1 is smooth otherwise my ys it has some nil-potent stuff and all the maps will fail to be isomorphisms so the statement then is that however as a maps of pro-sheaves so that is to say I neglect to ask and release the zero-system something called mix-like-left as zero-systems and I assemble them into pro-sheaves over all of the infinitesimal thickenings I do get isomorphisms of pro-sheaves at least maybe under some mild hypothesis in practice is satisfied in the case n equals 1 there is no additional hypothesis in the case in which we can weight strictly bigger than 1 we need to assume that y1 the underlying reduced sub-scheme of this sub-scheme of definition is not too singular so what can this mean certainly if it is regular then there is no problem regular things are not too singular but it could also be it could be for example a simple normal costing divisor so I could have started with some smooth variety and taken its completion along a simple normal costing divisor and take this as my sub-scheme of definition this is also allowed in fact all that is required is that y1 is somehow generalized I don't think there is a piece of terminology for this I want to assume that I can cover I want to assume that the reduced components of y1 are regular and that any of the reduced intersections is also regular any intersection or the only intersection is the reduced structure the intersection with the reduced structure so you can have arbitrary so they can contact you can have contact between so you can have several regular things with some contacts as long as the intersections are themselves regular with their reduced sub-scheme structure this is just because the method uses certain computations which you cannot generalize to do it's not exactly for computations it's because one needs to be able to decompose the study of the case theory of this sub-scheme of definition into the paving on each of the irreducible components and in order to do that you need some excision techniques in case theory which only work when you have some sufficient regular conditions so it might not even be true when you have a very non-regular sub-scheme of definition as in the case of long-scheme does not be well understood so by the way for the comparison of Milner and Laveret with the R&B if you are just doing it for the pathology it is simpler than to work at the gravel so one can give more or less a much easier proof well I don't know in your case but it seems that it should be possible to work it's not written balanced anyway but it's easier so it's just doing it for a separate flow so even the generation was pulled by cartels but then one can work on one can build on this but I will not in any case so I think that at least without the Quillen part it should be possible to do it without the assumption we discussed that actually it's the Milner and Laveret part which causes the most trouble we know in good techniques Quillen K theory so in fact the way that the theorem is but I'm even lying a little bit in that this isomorphism I can't establish in quite the full generality that I have written down but that isomorphism I can one directly constructs an isomorphism between Quillen K theory and the Milner K theory sitting in between but there could be some very slight difference which is the same difference so any direct technique to actually study the non-locator in this messy situation we'll talk about that later so the conclusion of this then is that we can use any of these three or four definitions to define the linked and bound methodical homology of a formal schema characteristic p which is sufficiently regular formal schema characteristic p so the conclusion of this is that we may then just define h star notificatel of this formal schema with c mon p to the r coefficients in way n to be the correct shift of the continuous etal homology on this sub-scheme of definition of any of these pro-etals if I'm given some pro-sheaf there are now maybe two or three ways that I can make sense of its continuous etal homology and either you can take the answer with a little pro or you can just take some derived limit of the system of sheaves a more categorical approach there are several ways so in any case the theorem tells you that any of these three or four sensible definitions of the continuous etymotivical homology of a formal schema characteristic p under some mild regularity hypothesis coincide so let me say just a few words about the proof of this theorem but let me focus on the much more down-to-earth case that n equals 1 so I wonder if it's still there ok so it's disappeared so the key to proving this when n equals 1 is an analog for formal schemes of what was the fundamental exact sequence of it was e which has now disappeared in other words if I assemble together on my formal scheme all of these gms for the infinitesimal thickenings and I go by raising to the p to the r to the gms for the infinitesimal thickenings and I go by the delong map to the logarithmic hodge-witch sheaves in degree 1 of all of these infinitesimal thickenings then I get an exact sequence of pro-retile sheaves even though it emphasises again for any fixed value of s this will be non-exact so let me explain the non-trivial step I mean the delong map is surjected by definition there's nothing to do there and one then analyzes some filtration here very similarly to what ilz did in the classical case of smooth variety and using that you can I mean when you first follow ilz's proof as close as you can to reduce to the case of r equals 1 you make the but now observation that the log of something is 0 the log of some unit is 0 if and only d a bit is 0 and so then you see that exactly what you need to do is to prove a Cartier isomorphism for formal schemes in characteristic p stating that so I guess this under these same conditions I've got some regular formal fp scheme I've got any sub-scheme of definition because I'm in n equals 1 case so no regularity hypotheses are required and so you need to check then that the inverse Cartier maps from the I mean they said to use the wit things from the the ram sheaves or from the hob sheaves to the homology of the ram complex so we always have to find even if the ys is a non regular one does have inverse Cartier maps to find as so and these are not isomorphisms because ys is very not smooth and the result states however that the inverse Cartier maps induce an isomorphism of pro sheaves when as usual I assemble everything over and once you have that type of formal Cartier isomorphism you can really remember the original probability to check that you get an exact sequence and that precisely gives you the equivalences in the weight one case of the theorem you also get that w are omega one and ys you know the co-object is the same as w omega one ys and how I do this is the hardest part actually I'll mention this in a moment this is the most painful part in fact you don't need this though in the weight one case of the theorem you can avoid it there so it only comes up in establishing the higher weight case so when any bigger than one is to be honest I was against this because I'm very reasonably straightforward for the equivalences in the weight one case I think when n is strictly greater than one because then you need to understand these Quill and Catery sheaves on all these thickened sub-schemes of definitions of the formal scheme where of course one needs to use some methods from topological cyclical homology and then exploit the fact that topological cyclical homology is itself related to a certain derive which sheaves via so-called Hockschild constant version work theorems to first to Hassell-Holt and then the variant that we need in this setting for formal schemes was developed in joining with Bjorn Dundas and Bergen Hassell-Holtz to refer to topological sorry? when you mention Hassell-Holtz because you are topological yeah yeah yeah yeah so the classical Hockschild constant version work theorem relates cyclical homology to Durant homology in the case of smooth algebras usually overcapacity 0 field where there is somebody in the system and then from topological cyclical homology the analogous result was established by Hassell-Holt relating certain pieces of the topological cyclical homology spectrum to Durant which sheaves and then Bjorn and I established an analog of that for formal schemes so again if you try to complete the topological cyclical homology of all these types of thickenings of some sub-scheme of definition of a regular formal scheme we managed to link that to the Durant which sheaves of again all the infinitesimal thickenings of the sub-scheme of definition and so that's then what provides the fundamental link between K theory of all these sub-schemes of definition and the Durant which sheaves and that's some of the starting point for the argument and then once you've got it started you need to have to put it, let me see you pick up your well-thumbed copier in these treaties on the Durant with complex and you try to prove an analog of every single result for regular formal schemes which is not always so easy and I just want to mention one such result you want to extend basically every result you can find about Durant which sheaves and logarithmic hardware which sheaves for smooth varieties of normal schemes the most tricky bit of which is just what you asked the moment ago that if I try to increase the level of my logarithmic hardware which sheaves say from r to r prime over here takes up some space then I can consider the logarithmic hardware which sheaves of all these ys's of level r prime take it from on p to the r compare it to the logarithmic hardware which sheaves of level r on ys and then for any fixed level of s actually I don't know this I think for any fixed level of s this is not an isomorphism but I don't have a count example so it is you prove this in the case with my ys's smooth variety and what is at least good enough and what already requires some amount of work is that as prove it to our sheaves over all the infinitesimal thickenings this is an isomorphism that's actually really a really essential step because without that one gets some description of the pro systems of the k theories of the ys's modulo p to the s as a pro system index diagonally over all the infinitesimal thickenings and over all the modulo p to the s's much weaker results that way and you have to somehow cut down this study of the k theory then to mod p to the r and the key essential step in doing that is precisely that isomorphism yeah and if you're not careful you end up with two you end up with very messy diagonal both indices at the same time and you've got results but they're much less pleasing results and then to cut it down to the way I stated it that's the key isomorphism so you try again you try to imitate what you did but there it doesn't really work there you have to do some extra work so that's the end of the sketch of the proof of these equivalences and one can now ask what's the point of the conclusion why do I want to write down a definition of continuous programming and typical homology of formal schemes and characteristics p and the answer is perhaps that I don't really what I want to know though is that these previous definitions are equivalent because well I won't point because it won't be on camera the k theory controls certain problems controls certain deformation and cycle theoretic problems and the logarithmic hodge-witch which is they really do control some typical homology of this floating around so having a relation between them leads to some nice applications which is what I want to mention to finish I'll try to present two, they won't take long so the first of these is middle f sets for channel groups so let's suppose that x is a smooth variety of a puff field characteristic p as usual in fact it's even going to be a smooth projective variety, excuse me y into x is some smooth n called divisor let's say hyperplane section and I take n to be at most the dimension of y so then I can consider channel group could I mention n cycles on x channel group could I mention different cycles on y consider this restriction and this is conjecturally an isomorphism after tensoring by q that's exactly this weak left-hand conjecture for channel groups so this no one can prove at the moment so what we do is to make our life a little bit easier we use the block Cullen formula to identify channel group could I mention n cycles with the n for the risk in carbonology on y of let's take Milner-Cachery just for sake of comfort of the n k theory sheaf we do the same for x and that then permits us to factor this restriction map from cycles on x to cycles on hyperplane section through some type of channel group of the formal completion of x along this hyperplane section and just introduce this as some temporary notation it's not supposed to have too much meaning which is by definition the limit of what we would get from the block Cullen formula so it's a conmological channel group of this formal scheme so the limit over the conmology on all these the y s's are again the infinitesimal thickness of my hyperplane section inside x all of the conmological channel groups you can also take the continuous conmology I can also take continuous conmology and the result will be the same yeah, the result will be the same the result will be the same so now I can factor this restriction map that we want to prove is a nice morphism about which we can't prove is a nice morphism and application one rather states that this is a nice morphism up to banded P P so this is the sort of result which is now going under the name of infinitesimal part of the weak left set conjecture for chat groups so say the overall goal is to prove weak left sets namely that cycles on x are the same as cycles on y but in fact all we can prove is that cycles on the formal completion of x along y are the same as cycles on y and then to prove full weak left sets you have to prove the so called algebraization part of the assertion namely that cycles on x identify with cycles on the formal completion and as it seems to be out of me choosing contents to have two minutes or okay, so then what I wrap up let me just say a brief word about the proof the first step is to check that everything I said in the main theorem continues to hold in some suitable sense for the ziriski topology you can see my chat group see what was defined in terms of ziriski conmology so first we need to check some analog of the main theorem in the ziriski topology where the difficulty is knowing where the logarithmic hardware achieves a generated ziriski locally by delog forms and so one proves that in fact one proves a very general result that logarithmic hardware achieves on an arbitrary fp scheme are always generated ziriski locally by delog forms seems to be new, but it was not to offer ah, it was not to offer I'm raising it to everyone else then I did not have under not in your definition so I also had in the catalogue there is a p-basis so with before Gersten conjecture I think I had a proof for rings with p-basis in the strong sense and the local so I had to say with infinite resguping so a proof of the block Gabor-Kartow with Sauer-Gersten that is avoiding the so this was in the 80s so I had this so on the other hand in your definition I don't know the so you claim that when you do like this you define it as a tallsheeps and then take sections over the ziriski local ring of these tallsheeps and this is locally generated this is generated by symbols but I didn't think about this that's what seems to be true so it seems to be true and then as you extend the previous term into the ziriski topology and so then there's assertion about some isomorphism between the ziriski chronology of K-sheeps and the ziriski chronology excuse me, it's isomorphism between the ziriski chronology of K-sheeps reduces to some similar isomorphism about the chronology of certain logarithmic hardware-sheeps and you leave them again, then applies in the vanes and Liechtenbaum to pass between the ziriski and the tall topologies and so it then reduces to establish some isomorphism between the tall chronology of logarithmic hodg-witch-sheeps for which you can use the known weak-left-shed in crystalline chronology identifying, so it's supposed to be some algebraization trick to identify then this type of object with a chronology-sheep on X which will be true for K-theory but which will be true for the certain logarithmic hodg-witch-sheeps and then the two pieces you obtain are isomorphic to a certain fubinous eigenspaces inside the crystalline chronology of X and the crystalline chronology of Y and these will be isomorphic by usual B-clutches for crystalline chronology but you're not modding by power of P in this no, that's okay, that's okay I thought I made a mistake by having that's right, so then you also need all of these logarithmic hodg-witch-sheeps not only into some limit in the S-direction but also some limit in the R-direction which takes account more and more of the difference between these two K-theory constructions no, but your results are about the mean of K-theory modulo-power of P correct, but the difference between these two sheaves is entirely P-power torsion but then you also need to control the kernel of P in those things I think I've got to finish there and I'll explain the second application to anyone who cares in private, thank you very much so we have a few questions from Tokyo and then BG are there some more questions so so you so your former skill is over FP, but can you study mixed-tocaster case to compare mean of K-theory and K-theory yeah, the mixed-tocaster case is very interesting but I can't say anything concrete at the moment ideally some similar theories will be possible in mixed-tocaster there's some perhaps synthesis of what Block and O-kerks did in the mixed-tocaster case together with this recent work with Barton Schultzer putting these two things together should get some good control of the equivalent in the middle of K-theory in this characteristic setting but there's nothing concrete at the moment any more questions in the case of a spec former power series do you have a concrete description of the grand B2C so I don't think I understood the question in the case of a spec former scheme of former power series over a former power series ring or just like spur-force or a former power series ring do you have a concrete description of the grand B2C well in that case the logarithmic could be written down very concretely and you mentioned the former Cartier isomorphism yes and can one expect the existence of some non-Aberian hot-theory for a former scheme oh I don't know do you want some non-Aberian hot-theory what did you say August Robotkins I don't know okay because for this question for the second one that's indeed the correct answer isn't it you just stay online for another hour I don't know I like questions from thank you any more questions can you can you consider a weak leftist system for a child group with modules modules everyone likes modules child groups with modules I mean we already have some big leftist system for child groups yes I don't see why not it seems to me the technique should generalize to the case of I suppose you want to take some close-up scheme of why and examine some child group modules on why relative to some thickening of the divisor setting on why and it seems to me that an logarithmic result could be proved yes but again they be of this type of infinitesimal nature are there any more questions thank you so is there any analog of the Voivodesky complex in the formal case wow you can just define one using the theorem I mean you can in some formal sense just by some cone construction glue them I mean let's assume that the sub-scheme of definition is regular even let's assume that the sub-scheme of definition is smooth let's assume we've completed some smooth variety along a smooth sub-variety then I have the Voivodesky's motoristic complex sitting on this sub-scheme of definition which just by some formal cone construction I can glue with any of the isomorphic pro-sheeps and that then gives me some type of pro-motivic complex on the sub-scheme of definition for which the associated piatic cosmology is exactly what I've been writing down and for which the associated allatic-motivic cosmology would just collapse to the allatic-motivic cosmology of the original sub-scheme of definition so the answer is yes you can define some type of pro-motivic complex but it doesn't give you anything more than the piatic cosmology that I've been writing down and the original motoristic cosmology of the sub-scheme of definition okay thank you no more questions from Beijing any questions the second application yeah that's very kind of you so the second application was going to be some in fact the second application it was charcoals and modulus so I hope Tokyo is still online so the second application is to some conjecture Kato and Shuji Saito this goes back to 1986 so the motivation for this is that effects is a used variety let's take my usual Saito a perfect field of characteristic p that's not really necessary then you deduce either from Gerson's conjecture for K theory or from some homotopy invariance results then you can compute well the zewisky cosmology of K theory and the nisnevitic cosmology of K theory are the same so if I simplify K theory either the zewisky cosmology and then I compute the cosmology I get the same thing this is nisnevitic and so then Kato and Saito made a conjecture then if I take some let's say simple normal crossing divisor and then I can try to do something similar concerning some K groups of modulus or relative K groups in which I look at what they were maybe only interested in the case of the dimension of the variety so I can look at the relative K theory sheaf of x relative to some thickening of y and I can again compare that to the nisnevitic cosmology I guess my t should be and and you could naively ask so having now modified K theory by some infinitesimal thickening along the simple normal crossing divisor whether the same was always true maybe now that conjecture is in fact to be true as part of this development of reciprocity pre-sheaves by Kato and Saito and Yanazaki but what they conjectured in 86 was at least after taking the limit over the thickenings then this should be true the motivation for which because it requires some motivation what's going on here and so at the time they were studying high-dimensional class field theory they wanted to propose the left-hand side if I understand the history correctly as the definition of a high-dimensional class group controlling the abelian extensions of x in which some ramification along y is allowed but it turned out that the calculations the work they needed to use in this nisnevitic topology but they really wanted to use the zorsi topology so the hope was that the two would be the same so we would pull the left the zorsi class group right-side nisnevitic class group and the conjecture is that the zorsi and nisnevitic class group should come inside and was it with quillen or millen it was with millen they said that you shouldn't use quillen but I think I disagree I think in fact we can replace millen by quillen the use of the symbolic results on Kato and study of the filtration of these pieces so the result states that at least if I go back to quillen then this is an isomorphism after periodic completion that's the second application thank you for the question ok so maybe just one question to finish because we have another lecture we should stop just the generation by symbols locally for the zorsi topology for the old bit logarithmic sheets is there any difficulty with small rescue fields or you can because usually the ok so there are two steps first you need to check that it's true for the arbitrary smooth let's start with the case of smooth variety yes the question is whether the logarithmic hot rod sheets are generated on a smooth variety is it risky locally by the long faults yeah this is true actually in fact the main location of the ship is in the risky not everybody yes for this you need this result of alba's vansans and a lot of stuff in fact I can explain the details of the proof to you in private afterwards yeah it's a messy proof but I can explain to you the details ok so maybe let's let the speaker