 Thank you very much for inviting me to speak here. It's a great pleasure. Well, we have collaborated with Arthur on one loan project, I think, for about three years. And this was, I think, the most fruitful time for me, mathematics. And in addition to that, I learned from Arthur what a log scheme is and what Betty had a homology of a log scheme. And in fact, the work I'm going to talk about is, well, grew up also from a question that we discussed a while ago with Arthur. And so the question is a full in. So I have four log schemes. And then I have various homology theories. Well, for example, I can look at a TAL homology, or Betty, or say, the RAM. OK. And so if my log scheme happens to be defined over a finite field, so then this TAL eteladic homology of x with QL coefficients, well, it's a vector space which is equipped with an action of the Frobenius. And one can check that for a reasonable scheme x, the eigenvalues are all real numbers. Of course, they are of different weight. But in particular, because they are real numbers, this gives me the weight filtration. And so the question is, so let's see whether I would use it. So can one define this filtration geometrically? So I want a definition which would make sense for all these three theories. And well, more precise, well, so you can ask for rational Betty homology, or you can ask even whether it's well defined integrally, as for usual singular homology of algebraic varieties. So one can ask whether one can associate a motive, or maybe even a finer question, material homotopy type. Well, I want to tell you a little bit about this question, so what kind of homotopy type you expect to have. So the answer lies in the construction due to Cart and Nakayama. So OK, so I will recall it here at Nakayama construction. So they defined Betty homology. So what's OK? So that's Betty homology of a log scheme x. So this is a coefficients, any coefficients. Well, to be by definition homology of a certain topological space. So this is just usual, well, so this is x log. So that's what Arthur explained to me. So this is just a topological space. Well, in many cases, it's a, yeah, yeah, sorry. My x is now over complex numbers, of course. Thank you. Thank you. So x is a log scheme over the complex numbers. And so this x, well, it's just in general, well, it's a topological space, but well, in many cases it's a manifold with boundary. It has some additional structure. So what is it? So this is a topological space, which is equipped with a map. It's a proper map to x. And so, well, so the construction is the following. So, well, I have my log scheme. So I will denote by x underline, then the line scheme. And m is the sheaf of manoids. So, well, so I have the sheaf of manoids. I have a star that sits inside it. And I have a map to o, OK? So, well, the only thing I care about, in fact, this, well, it's true for the whole, for the entire talk, is the group completion, m group. So this is a sheaf of a billion groups. It has a star sitting in it. And it has quotient, which, well, I will assume I will look at fine log scheme. So it's going to be a constructible sheaf for, well, for a tiled apology, or if you wish, you can consider a version of it for the risky apology. In fact, in my talk, it will be more convenient to consider all these sheaves for the risky apology because, at the end, the object that, well, yeah. So let's stick to this. So this is, you can think of this, well, a good example of such lambda is just the constant sheaf supported on the device, for example. So you have such an exact sequence. And, OK, so to a scheme and to such an extension, I want to associate a topological space. So how I want to do this, well, if I have a point, well, I can sort of restrict my, so that is here, evaluation map, point x. And so from this extension, I can derive an extension of the stock of this sheaf lambda at point x. And this will be by c star, OK? So this is just a finitely generated abelian group. And this is c star. So it's extension. OK, so now. And so x doesn't mean the stock. Yeah, it doesn't mean the stock. It means pull, push out. So and here I have map from c star to s1, which is called the argument. So, well, let's give a name to this map. Let's call it gamma. So I look at all possible sections like this. We'll call it sigma x. So satisfying the property that if I compose gamma with sigma, I get argument. And so x, look. Which is the same section, so this is splitting of this extension. Well, push out of this extension, right? And then this space, as I said, is just set of all pairs, x and c. And your lambda is torsion free because you assume fs, right? It need not be torsion free. I'm going to assume, in the second, I will assume that it's fs. But for the moment, I don't even need this. So, OK, so I consider all such splitings. Of course, there is a map to x. And I want to, so, and the fibers, fiber over point x is a torsion over from lambda x to s1. So in the case of nice log scheme, fs log scheme, this is just a real torsion, OK? And so then there is a topology on, so any section of m group defines a function on x log is values in s1. Namely, it takes point x sigma to sigma of x applied to m. And I want these functions to be continuous. So I consider the weakest apologies such that the pullback of any continuous functions on x is continuous, as well as these functions are continuous. OK, so that's a topological space. And I want to see whether, well, I want to, the ultimate goal could be to realize this homotopy type, well, to leave this homotopy type of this log space into a martyric homotopy category. So, well, the possibility of such construction was suggested to me by a long time ago by Maxim and also by Nori, somehow independent at the same time. So they explained to me one single example, which is kind of really the key example. So I want to tell you about this example. OK, so what is, so this log structure, the log space that comes from a special fiber of a semi-stable degeneration. So this will be x0, this will be xt. And so semi-stable degeneration, so it's sticked. And moreover, so I will assume that there are no triple intersections, very, very simple. So in my picture, there will be only two divisors. It can be generalized in the case of many. So I have these divisors. This is d1, 2. And so then I have two line bundles on x0. Namely, I have l, well, i, which is o of di, restricted at 0. So this is, so I have i is 1, 2. And so this li comes with trivialization. If I restrict it, here is another piece of notation. ui is the complement, x0 minus di. So and the y is, when I restrict it to ui, it's canonically trivial, right? So these two structures, these two line bundles together with such trivialization can be organized. This is given such two line bundles. This is the same thing as given extension of z, d1 plus z, d2 by o star. This is everything over x0. So I have on x0, I consider x0 as a log scheme. Well, the log structure is just the pullback of the log structure and everything given by this divisor. But while the same curve has this very simple description, so these are the constant shifts on d0, on di, extended trivially to everything. And I want to, in this case, I want to look at x look and realize it as a element of the a1 homotopic category. So well, what you do is the following. So well, I need another piece of notation. So let's call li-circ. This will be gm-torsor associated to li. So li is a line bundle. I take the total space, I remove the 0 section. And then so what I want to do, I want to consider the following complex of. So I want to construct, well, Wojewódzkii motif. And well, in fact, it will be even an element of stable homotopic category, and even unstable after a certain modification. So OK, so what is this? So let's first define it as a motif. So well, I want to do something very, very simple. I will take l-circ. So again, this is regarded as a scheme, leaving over x0. And I want to restrict it to u1. So sorry, I think I want to restrict it to u2, of course, because it's trivial on u1. And then the second thing that I want to do, I want to take l-circ2 and restrict it to u1. So when I write square brackets, it means that I consider this as a motif. Just that it's a plain scheme, so it makes it well. And what I will have here is l-circ1 times l-circ2 over x0 restricted to d1, 2. OK, so and this d1, 2 is intersection, OK? So well, so my motif is going to be sort of the cone of this map, this complex. Now, I need to tell you what this is, this arrow. So and here is a reminder. So well, this construction is possible because, well, in the category of Wojewódzky motifs, you have identification between punctured tubular near-bordhot of a smooth manifold with punctured normal bundle. And namely, if you have so smooth y and z, this is closet, smooth closet, then there is a canonical map in from the motif of the punctured normal bundle to y minus z. This exists in differential geometry. This can be lived in the motif of the category. And well, the map that I have here is exactly what comes from, is precisely that map. So here is the picture. So if I have, let's see how we get it. So OK, so here is my d1, for example. And here I have this line bundle, which is l. So and here I have this is d1, 2. So I apply this construction. So I have this line bundle, this GM torsor. It leaves over the whole d1, 2. And inside this, the total space of l1, l1 minus, I have this fiber over d1, 2. And I apply this construction to y being this GM torsor over d1 and z being its fiber over d1, 2. Then you get precisely this first map and the second map is constructed similarly. So you do this. And do you put a sign or it is not important? I don't think that there is any sign here. It must be symmetric at this moment. But if you have several things that intersect each other and then say, go in a loop. No, no, in here. Well, so I don't think that there is any sign in this picture. So maybe it seems to me that at least here you don't need sign. But maybe it's possible. So OK, so in fact, this map that I used here exists even in just in a one-homotopic category, but only after suspension. So you can kind of do the same gluing except that what you'll get, you cannot get homotopic type of X log itself, but you can get homotopic type of the suspension of X log. So at least you'll get something in the stable homotopic category. So the fact that that map exists after suspension is proven by Morell and Weyholz. So OK, so and well, one more remark about this whole picture. So in fact, this ELI have additional structure if they come from the semi-stable degeneration. Namely, EL1, TENSORL2 is trivial. Well, strictly speaking, this trivialization depends on the choice of a parameter here. You choose it T, coordinate, because this is OD1 plus OD2. So it's of special fiber, it's lifted from the base. And therefore, well, if I have a map from everything in this picture to this is 0 to complex numbers. So there is always there is a map from these guys to complex numbers, from invertible complex numbers, right? And therefore, so let's give a name to this. Let's call it motif of X0M. So it's motif of this log scheme, just definition. And in fact, in this case, this motif is not is a motivic shift, motivic shift over C star. And it's fibers, for example. Well, you can take fibers over. So every scheme here is a scheme over C star. So you can take fibers, you have still this map, to get what's called the limit motif, right? And the picture here is, I cannot really draw a picture of this gluing of this X log. But this X log here, it actually maps to log space associated to the log point. So it maps to the circle. And I can draw a picture of a fiber of this over a point of the circle. So imagine that, for example, you have the generation of elliptic curve. So OK, so here is my, so it generates an interrational curve with two double points, right? So this is non-singular elliptic curve. And this, now, it degenerates. So what you do, you sort of remove these two points, and then you glue along punctured tubular neurons. And then you get something back. Well, I'm going to pick it all into the original space. OK, so I want to kind of extend this construction to all of the schemes. And that, even if you have, in this semi-stable case, if you have many components and multiple intersections, it becomes really, well, direct the generalization. It's really unpleasant, because, well, you can kind of write down a similar complex. But you have to do it, lift it in somehow on digital level. And then the square of the differential will not be 0. It will be the homotopy equivalent to 0. And these homotopes are given by some double normal cone constructions. And so it's going to be completely useless, though possible. You can't really do anything with that. So instead, one should look at the dual thing. One should look at the motivic homology of this motif. And this has a very simple geometric description. I will show it to you at the end, in the case of this semi-stable degeneration. But first, I want to formulate the main theorem in a kind of in-abstruct form. So the main theorem is the following. So I need a bit, first introduce a bit of a certain category. I will call it the category of log motives. But, well, then you will be able to forget about this. I mean, so first I introduce a category that I will call the category of log motives. And in order to do this, so let me just write down the definition of usual category of Vodka motives. And then I will list two more kind of relations and get the category of log motives. So I will work with log motives. So I will assume for simplicity that the characteristic of my base field is 0. Well, if it's p, you can do the same thing, but you have to invert p in the coefficients. And then, so for Vodka motives, you do the following. Motives. So you consider a category of schemes, say a finite type over a field. And then you form an additive category, just objects, schemes, all schemes of finite type. And morphisms are a linear combination of maps, no correspondences here. And then I want to take complexes, finite complexes over this. And then I mod it out by certain relations, by subcategory. So this is a, well, if you can do it on triangulated level, pass to homotopic category, then take Verdi-Aqvoshan, or you can regard it as a differential graded category. So and what are objects of teeth? So objects. Well, of teeth. So, well, I will list them. Let me start here. So first object is, well, it's going to be class of objects, right? So it's X. So objects are complexes of schemes, right? So this is first. This is called A1 homotopic. And then the second class of objects is, well, perhaps you have to let me do it here. So this is the second class of objects. I will refer to it as CDH topology. So these are generating kind of sequences, covers. And so there will be two parts. Well, so first, suppose I have the risky open subset and I have in a tile map, this is a tile. And so what are the conditions? Condition is that the map P inverse of from X minus U to U to X minus U, this map P, the restriction, is an isomorphism of schemes. So that means that this is a tile map. And on the complement to U, it has a section. It's actually a bijection, right? So this is the risky open. This is a tile. So in particular, U and W cover X. But it's much, much, much more than that. And in this situation, I want relation. And this relation has the form X, U, W. And here I have the fiber product, U, W. So all these maps. And so this is part A and part B. So I want the following. I consider maps P from X prime to X. These are proper. And I assume that there exists a closet sub-scheme here, closet sub-scheme, such that over the complement, this map is an isomorphism. So P induces an isomorphism between P inverse of X minus Z and X minus Z. For example, Blob. So OK. And in this situation, I want to have the following relation that motif of X prime plus motif of Z. And this maps to motif of X. And here I have motif of prime of Z. And here you must put a sign inside one of those things. Yeah, I have to put sign. So either it doesn't really matter where. So yeah, you're right. So I have to take the difference, otherwise the square will not be zero. So and this calls it right. So and also A, B, C, D. OK. C is that the map from X is used to X. It must be an isomorphism. That's it. It follows from B because you can take Z to be there. OK. So but you're right. So many things in fact follow from others here. So and well, you can so this is not that's not. So you can have. So this is not what's called, usually called, the category of Wojewódzki motifs. It's also defined and studied by Wojewódzki. And it has a very complicated notation, this category, this quotient. So let me give it here. So it's H, A1, CDH. And then here I take Z of schemes. I need Karubian completion. OK. So what is the relation of this category with the category of Wojewódzki motifs? So there is a, well, at least there is a functor. So this Z, the Wojewódzki category, which is obtained exactly in the same way except that you add transfers. So you start with, consider the same relations, but this category is different. You consider schemes and where maps are finite correspondences. And also only smooth scales. Don't think that if you do CDH, the poetry, it makes any difference. You're speaking about now the, OK. So you can do it with this. Yeah. So if you consider smooth schemes, then CDH, the poetry, then B can be derived from A and other schemes. OK. So now I want log motifs. So what I do with log motifs? How I, so let's just consider the same, similar category, but let's start with log. So by log schemes, so I consider, I mean, F has finite saturated log schemes. OK. And then I want to take, so I want to consider quotient of the log scheme by T. Well, and what is, so what is, what is this T log? Well, so T log consists of, it's a full subcategory. Objects, well, I did not mean objects of type 1 and 2, 2 and the following. So I need to do, to impose a bit more, a few more relations. So, OK, so what is, what are the relations? So, well, maybe just before we explain the relation, so what I have here, this, well, there is homotopy equivalence relation. So in this homotopy equivalence relation, X is allowed to be any log scheme, as well as here. So I want everything in this picture, X, well, this must be log schemes. A1 here is just usually A1, but I will add axiom for log A1. And if you speak about closing merchant, perhaps you want the exact one. Yes, yes. So, OK, so what I want is the following axiom. So I have A1 log. So what is it? It's a, the underlying scheme is A1, and the log structure is given by one point. So that my M group is just corresponds to the line bundle of this point. And here I have, so log point, it's kind of the origin. And here I have GM, it has trivial log structure. And I want these maps to be isomorphism in my quotient category, but not only this. I want to multiply this by, and again, X is any work scheme. So, and also this GM times X is valid. Well, there is one additional axiom. It's strange, it just has to do, we already observed that all the construction we have made so far depend only on M group, not on M. M is irrelevant. And I could consider the category of our skin just by the category of extensions. What do you say about them? They are in my category T. So you require them to be here? That's all. That's all. Yes. Do you want the A1 log, is that one isomorphic? Or you, too, like A1 log? Yes, so these are object in T, like, for example, is this those. So, and finally, 4 is very, very strange. Well, so it's very, very, is the following axiom. So suppose I have any map P from X prime to X. These are log schemes. And the underlying map between usual schemes is an isomorphism. And suppose that it induces, the map induces it on M group P up a star to M prime group is an isomorphism. Then I want the following relation, that X prime goes to X. It must be in my category. So this kind of way to say that everything depends on the group. But again, so I could consider just the category of pairs X plus this extension of star, this star. So now, very good. So here is the theorem. When you do block blow-ups, what seems when you do the operation of block blow-up, I think that the space, I don't remember now, the space X log doesn't change? No, it changes. OK, space X log, yeah, it changes, but it's the same problem. So here is the theorem. So, well, I have a category A1, CDH, this is my category. And here is the name for this. So this is Z of schemes. So there is, of course, a function to this larger category. We are going to consider A1, CDH, and here I have Z log. OK, so this function is an occurrence of categories. So what does it mean? So what are, for example, functors from this category to, say, the category of complices? These are homologous theories, right, that satisfy these basic properties. The claim is that any homologous theory can be extended uniquely to log schemes provided that it has all these properties. OK, so let's see. So in particular, well, of course, here you have map to functor to usual Weyvotsky category, and then you can compose the inverse and get Weyvotsky motif. Now, so the construction, the proof is very little, uses very little geometry, in fact. It's more or less linear algebra. And so, well, first of all, no, you have to prove two things, that this functor that I have here, the obvious functor, let's give a name to this functor. Let's call it here. Yes, fully faithful, homotopically fully faithful, if you think of this as G-functor. So fully faithful. And B, so essentially, subjective. Well, and in fact, so easy step is that A implies B, and therefore, so all you have to do is to prove A. So and that's because you can do it by induction on dimension. So you need to show that any motif of any log scheme is in the image of this functor. So we have enough relations to express a motif of any log scheme in terms of motifs of usual scheme. So that's easy. So you do it again by dimension of your log scheme. So if it's zero-dimensional, then, well, it's fine saturated. So it means that my Manoid M group is just z to the n. And so the corresponding, so if I have point and take its, well, its motif, for example, is motif of GM. And if I consider Cartesian product of any scheme of log point with itself, then of course it will be GM times GM. This follows from this axiom. And so using this property, any zero-dimensional log scheme has the same motif as product of this. So now what you're doing, one-dimensional case. So you have some curve. Well, now, generically over an open set, the log structure is trivial. It just corresponds to trivial extension. M group is just trivial extension of z to the n by your star. And therefore, OK, so you know what to do over an open set. And then you can use this CDH axiom to see that this log motif of this curve is also in the image. So what do you think? You have missed my phone, sir, have you? No, I used because, so let's see. Well, if you look at this CDH axiom, how it looks like. So you have some key inverse of z. And then you have z plus x prime and this maps to x. So I want to show that this object, so knowing that this object and this object are in the image and also this, I want to conclude that this object is in the image. But then I need a map from x from this square bracket x to this guy. And I need to show that this map exists in the usual category of Levi-Wadzky category. It exists by definition. So because it's an exact triangle, there is a map from square bracket x to this guy which is shifted by 1. And I need to show that this map actually exists. So what do you do for? Do you assume that the log factor is a risky local trivial? I use it as a risky local trivial, right? So I could, yeah. And so you don't have motives for a tile? Then I need to work with a tile typology rather than be a risky and, well, at the end, I will get a Natal Levi-Wadzky motive. Well, some applications that I had in mind have to do with this integral weight filtration which exists only for realization of usual Levi-Wadzky motives, not a tile. So, okay. So what you do for A? So what do you need? It's enough just by formal nonsense. It's enough to prove the theorem. It's enough to do the following. So given any object of this category, usual category, A1, CDH, maybe I will just say the following. So let me, because my time is short, so I claim that it's enough, it suffices. I will sort of suppress the completely formal part, so it only uses the fact that the category of Levi-Wadzky motives is rigid, duality. It suffices to show that the functor, home, Z of n, from this usual category of motives CDH schemes extends larger category, this category A1 CDH. And what you do here, so you just sort of, you want to define motivic homology. It's not quite motivic. It's home in this category with no transfer, so I don't want to call it motivic homology. It would be motivic homology if I can consider this category with transfer. So you want to do it for any log scheme. And the idea is very simple. So you have the same group. It can be evaluated, it can be considered as a motivic shift over this motivic shift, shift over the underlying scheme. That means simply that you can evaluate it over any scheme which is smooth over this guy. You simply pull the log structure. If I have a math like this, then you can pull the log structure, this log structure here, and take the same there. So and then what you do, you just compute, you take a homology of x, these coefficients in symmetric powers. These are operations in the category of motivic shift symmetric powers. Well, they're defined by Weyvotsky. It's characterized by the properties that if you take x as that motif of a scheme x, is the symmetric power of motif of a scheme x is the motif of the symmetric power of x. Is it n bar group or n group? Just n group. So for example, what happens if you take n as 1? Well, when you pull it back, you don't make it a log structure on Y, you just take, you don't enlarge the units. I mean, usually when you pull back n. There are n large units? You are a large unit. Yes. So for example, what is first, well, motivic homology is n equal 1. This is just homology of n group. It's kind of analog of the p-carve. So, okay. Now, I want to finish by- Which category? This is the homology of what? So, well, you- From the base of this homology works. Yeah. This is nisnerich homology of this material shift. So, well, I want to finish by a very explicit formula that I promised at the beginning, just very geometric, for the motivic homology of the limit motif in the case of semi-stable degeneration, and also for the, for the tripler-nubrum. It will be just in the case of semi-stable degeneration. And it comes from this, I just, well, it's- This formula is here. I just want to- I will make it explicit, which is trivial from this definition. So, the formula is the following. So, here is my setting. Well, now I will have, again, I have the log structure coming from semi-stable degeneration, but now I can have multiple intersections. Well, but it's still semi-stable. Maybe that's not a good picture. So, this should be planes rather than lines. So, okay. Probably I should not put it here. Yeah. So, it's confusing. Okay, semi-stable degeneration, and it's strict, right? So, I want really, that's, I want log structure for the risky topology. So, strict, strict, strictly semi-stable degeneration. So, I have components, D1, D2, well, so, so, DI. I, I, belongs to sunset. I have this GM torsors. So, first of all, I have Li, which is of DI. I just want to, I want to, I just want to, I just want to, I just want to, I just want to, I just want to, I just want to, I just want to, I just want to, I restrict it to X0. And then I have the corresponding GM torsors. And I have UI, which is the complement of X0 to DI. So, okay. So, now I will produce, construct the material homology. So, well, first I will produce a complex of smooth varieties over X0. The fibers will be algebraic tori. So, what are these varieties? So, so, step one, complex, very simple, complex of smooth varieties. So, this will be product of all this Li. So, this is product over X0. It's a, it's a, it's a torsor over, over an algebraic tori of dimension for cardinality. Now, what is the next term? So, you take a sum. Let's consider it as a, what's it kind of, I want to, it's not really, it's, in fact it's a, I want to consider kind of an additive category. I want to make sense of sum. So, it means that it's, you can think of this dejoint union. So, of products of i in i, i is not equal to j. And here I will have L-circ i times Ui over X0. So, what is this map? Remember that Li is trivial over Ui. Therefore, I have such a map. It's just given by, well, it's a section. Now, it's a long complex, and here is its last term. It's sum i over times intersection j not equal to i Uj. It's again over X0. And so, all I use here, all the differentials come from the sections. So, for that Li is trivial over Ui. So, this gives me such a, again, this should be dejoint union. So, this is step one. It's very simple complex. Now, form, define a complex of pre-sheets on X0. Well, what you do? You call this z dot, and then you take the following. So, if you have a scheme over X0, then, well, the value of my complex of pre-sheets on X0 on y is the following. You call this correspondences or X0 from y to this complex. So, well, informally, you just take sections of this complex of varieties over y, and then you make it homotopy invariant by applying the system construction here. So, let's call this f, and my material homology of X0m, z is just CDH homology of this X0 with coefficients in f. That's it. That's all I have to say. So, there is one, well, there is, as you see, you can ask whether the log geometry here, and in fact, there is no log geometry. You see it's just linear algebra, and the log geometry appears if you want to prove, for example, that if you have a smooth log scheme, then it's motive. It's isomorphic to the motive of the open part where the log structure is trivial. This I don't know how to do. I know how to do it in the case of, like, normal crossing. Normal crossing situation, but in general, it really requires some geometry. Thank you very much. Was there anything in notation of the logical homology did you say you have several zn for different n? Yeah. So, here I defined only this one index, right? This index is absolute value of this set of indices, i. So, if you want to define it for larger, for larger n, right? You have to just add, in the whole construction, empty divisor formula. And to define it for smaller, you don't... Yeah, yeah. You don't... You don't need to define it for small by consolation. It suffices to define this for sufficiently large numbers. This follows from the books. Yes. No, no, no, just material homology. So, if you want it for... You have consolation. So, you can express material homology of something with coefficients in z of n, in terms of homology of the twist, these coefficients in z of n plus 1. Which is the product of this GM? Yeah, yeah, product of this GM, for example. Yes? So, I just was curious. Where is n? Should we have... Yeah, yeah. This twist here, right? No, no. This n means the lord of monodome. Yeah, it's... That's a very good question. It's this whole complex of schemes. It's this over GM. If you have... It's exactly as in the picture I started with. This gives you the tubular neighborhood. If you want the vanishing cycles, you have to take the fiber over. So, it's a... Yeah, it's a unipotent material shift over GM. But how can you define the map to GM if you're even just... You're in the fiber. No, there is a... The map to GM comes from the trivialization of the tensor product of the swine bundles. So, each of these smooth schemes over X0 admits a map to GM, for example. Because the other one... The product is... Yeah. This is implicit when I said that you were going to... You started talking about how to define the weight filtration. Yeah. Does this come out easily from this? Yes, so there is a... Well, yes, it comes thanks to the work of Bandarka. So, if... Betty homology, or the Ram homology, or the Tal homology of any wave-wise motif is equipped with integral weight filtration. So, that is equivalent to... Rational. Yeah. To what? Yes. He has defined the weight filtration. Yes. But after, you have to show that it's equivalent to the others. Well, but here you don't have... Well, a priori, you don't have the others. What you have to show is that the basic homology of this log-motive that I... of this wave-wise motif I defined coincides with the homology of this... As defined by Kat and Nakayama. But this is obvious because Kat and Nakayama homology theory satisfies the list of all my axioms. And therefore, this factor from log motifs is uniquely determined by its restriction to usual motifs. And for usual schemes, there is nothing to prove. Yes. And... Sorry. Last question. In characteristic P, this was everything characteristic... Yeah, you have to invert P. And always with zariski? Well, yeah, for zariski again. So, the only reason why I work with zariski is that I want wave-wise motifs really for Nisnevich topology, as opposed to wave-wise motifs for Taltopology. So, it is enough to help Nisnevich locally... Nisnevich charts for the work start. Yes. Yes, yes, yes. Though I don't really know... No, maybe that way. There are... Yes. You need to figure it out. Any more questions?