 Thank you very much for the invitation to speak here. It's a great honor and a great pleasure. So everything I'm talking about today is maybe I'll start here. So it's joined work with Andrew Snowden. And the goal is the construction of integral structures on Diromco homology of varieties over number fields. So I would like to start off with an apology. This is work in progress. We haven't yet completely finished writing everything up. I ordinarily do not like talking about things like this, but there are many people in the audience that have seen me give talks recently. So I didn't want to repeat myself. OK, so as I said, the goal is to construct integral structures in Diromco homology over number fields. But for motivation, let's sort of, and this is how we came to this problem. Let me try and recall the situation over the complex numbers first. So delta will be an open unit disk inside C. It contains 0. Delta star is the punctured disk. And I give myself the information of a family, which is smooth and projective over the punctured disk. I sort of don't specify its behavior on the boundary. So here's a picture. And the central fiber could be horrible. I don't want to think about it. But away from the central fiber, it's nice. OK, so in this situation, one has the following fundamental invariant. So it's the Diromco homology of the fibers. To each point of the punctured disk, you associate the Diromco homology of the fiber. As the point varies, you get a vector bundle over the punctured disk. And this is, in fact, a vector bundle of the flat connection coming from Gosmonin. So the question that motivates this does this bundle have a natural extension to delta? So question. All right, OK. I'll try to follow Laurent's strategy, where I only use two words. So let me give this a name. I'll call it Romani. H i Dirom of x over delta star have a canonical extension to delta. I mean, of course, it extends as a bundle. But we want the extension to be sufficiently functorial. And the answer, I mean, this is not a new question. And the answer is also not new. So the answer is yes. And in fact, there are sort of many constructions. I'm going to review a couple of them first, because the pediatric ones will be direct analogs. So the first one, it's more of an abstract construction. It doesn't use the geometry of the situation. So you start off with this local system, or bundle with connection on delta star. And you use the geometry to conclude that this associated local system is a quasi-unipotent monotromy, meaning when you sort of look at the representation of z that you get, the matrix is unipotent. And then it's a fundamental term of delaying, I guess proven here, that there is a unique extension and one can characterize it in a nice way. So it's a bundle on delta. It extends e. And of course, there are many such choices, so one has to say what one wants. So it satisfies two properties. I'll try to write them here. So the first property is that the Gauss-Mannien connection extends to a logarithmic connection. So log 0, in this case, I'm working over a one-dimensional way, so it just means I look at the divisor of 0 and I allow sort of poles of order 1 there. And then there's a second condition. So I guess I, all right, this is the answer. As soon as the family extends to not quite unipotent monotromy, or do you assume that the family extends to something to a purple thing over delta to know that there is no. Yes, I'm putting myself in the algebraic situation. That's the one I have in mind. OK, so there's one condition and the second one sort of, there's still a z-worth of choices at this point, roughly corresponding to the fundamental group. So one needs to make a distinguished choice. And the distinguished choice is that's the monotromy operator, so n, which is the residue at 0 of delta. So it simply means you look at this map and look at its fiber. I mean, you look at the matrix coefficient of dt over t, if you like. So this is a matrix. And you assume it has eigenvalues and half open interval 0, 1. Oh, yeah, sorry. No, the matrix is a matrix. And so there's a unique one with this properties, and the construction is actually functorial. So I'm going to make a few remarks about it. So this is what is called a lower canonical extension. This is quasi-unipotent, since it is quasi-unipotent monotromy, automatically you don't make the statement. The statement is still correct. But yes, so this is the lower canonical extension. And I say lower because there's the other choice, which is you still use the other interval and the other side is 0. That's called the upper extension. And these are not the same, but they're in fact dual to each other. The most interesting case geometrically n is actually no-potent, or this is the typical case. And this corresponds to this bundle E having unipotent monotromy. I mean, this is after all the logarithm of. And finally, I want to recall one last property, which is that this construction is sort of you can do it after a base change. And this is why I use the lower one as opposed to the upper one. So if you choose a finite cover G, so you just extract an nth root of the uniformizer, so I get a pullback bundle G upper star of E. Now it's a bundle on this, the copy of delta star. And I have the associated lower canonical extension there. And then I can intersect it with my bundle E downstairs. So just take the inverse image lattice. And this is E as subsets of E. So in other words, you can do a finite base change, compute the lower canonical extension there, and come back down. So this is sort of what we'll be using in the geometric setting. Sorry, in the arithmetic setting. OK, so I mean, this construction is purely abstract. It works in the land of differential equations. One sort of might ask for more geometric construction. And it does exist. And so let me sort of briefly just say what it is. So for this, you need to be involved here in ACA. So now I'm really placing myself in a situation where I can do that. So I choose some model x over delta where the reduced central fiber is simple normal crossings. So I no longer had that horrible picture I sketched. Sort of much nicer. And in this situation, you can sort of regard x to delta as a log morphism. I mean, in particular, one has an associated log-deram complex where you allow logarithmic poles along these components. And it's a theorem of steambrink in generality and cats, when the multiplicities are one, that the log-deram homology of such a model computes the lower canonical extension. So it's HI. I guess I should give. Let me go with this morphism and name F. So it's, well, I guess it doesn't matter. So you look at the relative log-deram homology with logarithmic poles along this central divisor. And you look at its log-deram homology. And there are two assertions here. One that it's a bundle. And secondly, it's sort of this extension. In particular, it has the sort of correct unipotence properties. OK. So these are the two sort of abstract constructions in our punctured unit disk, which give us natural extensions of a diram homology to the boundary. And I want to try and mimic them in the arithmetic setting. No, by virtue of the comparison, it doesn't. But it just gives us a way to compute it. So I mean, for example, you can see those eigenvalues geometrically. And this is going to be, of course, is going to be a big problem in the phiatic setting. We don't have models, and dependence is not clear, and so forth. OK. So in the arithmetic setting, let me sort of write down the result and make some remarks about it. And then we'll sort of see the actual construction. So let's say k over q is a finite extension. And x over k is a nice variety. So here, nice is actually a technical term. I learned it from Bjorn Poonen. It means proper smooth and geometrical, projected smooth and geometrically connected. I like it. It's quite short. And so the theorem is that in this situation, if you look at the, and so you fix an integer, sorry. If you look at the diram homology group of x over k, algebraic diram homology, then there is a canonical lattice in there. There exists a canonical. So I mean, of course, everything here is hidden in the word canonical. It doesn't depend on the model, actually. Well, there is no model in sight. But so it's, yes. So there's a canonical lattice. It's functorial in x. And there are some other properties, but maybe I'll sort of, I'll state them under remarks. And we'll see what they are. So the first remark is that this is sort of a lie in the spirit of Jared's talk. It's kind of a white lie. If you don't actually have a canonical lattice, we have at least two. I mean, it's easy to see that if you have one canonical lattice, you can scale it by five, and you'll get another one. But that's not what I mean. There's really two genuinely different constructions. So we actually get two, well, probably different. Certainly, I don't know how to prove that they're the same. Two different such lattices. Both somehow corresponding to the two constructions I had earlier. So you can copy Doolene's abstract construction and using integral Piata-Cosh theory. And I'll give us one lattice. And then we can also copy the Steinbrink construction in some way. And I'll give us a new lattice. So there's the Galois-Theretic one. Q, Q, Q. I'll make a comment on that. So yeah, I mean, no, no, no, these two are different. You can define duals, but you get a lot more. I mean, you can just, I mean, yes, I'll say what they are. So they do have some properties. So if you look at sort of all of them inside the dram-comology algebra, simultaneously, then this is actually a subalgebra. So it's closed under cup products. But it is not closed under Poincare duality. I mean, this is the same thing as what happens in the geometric setting. So one shouldn't be too surprised. OK, and then where does all of this come from? The construction is essentially local. I mean, one can state it like this. But you do it locally at all primes, and then you make sure that your construction for most primes is compatible with some standard one. So what do I mean? I mean that if P is un-ramified. So P is a prime of the field K, and it's un-ramified over Q. Let's see what else. I need x to have good reduction at P. So meaning there is some model with good reduction at P. And so to assume that dim-x is less than P minus 2. And I think actually, so we need different constants for each lattices. So let me be safe and put a 3 over here, because that's the maximum one we need. So the dimension is sufficiently small compared to P. Most primes are going to satisfy these properties. Then this lattice is the lattice coming from the smooth model. So script x is the smooth model. So in other words, if you're trying to do such a construction and you are able to create such local lattices and they satisfy some property like this, then they are going to patch together. Oh yeah, sorry. Oh, oh, oh. So it really is just a local thing. But there are global questions one can ask. For example, I don't know if the determinant of this lattice is an invertible is the zero ideal in the class group. So I expect not, but it's somehow impossible to compute. For each i, each comodal g group. Yes, and all of them. I mean, yeah. It's not a good one guy, I guess. It's not a good one guy. No, directly each group. Yeah, just each group, each group, h i. Well, so by lattice, I mean a projected module. So yeah, so it makes sense. Well, I guess the determinant makes sense for any module, because the ring is smooth. No, no, that's not what I mean. It's projected. Is it clear that it's independent of the 20x? Well, it follows from the theorem that it is. But I'm saying you can define it like so. You just, a priori, you pick some good model over all of the, over like most of spec z's, eh? Bad at a few points. When you said that lattice is canonical, what do you mean? I just mean functorial. All I'm inserting is that there is a functorial lattice. And it took us a lot of work to actually see that. So in the geometric case, you have the sort of the upper one and the lower one, and they're dual. But you also know the upper one contains the lower one, and the difference is not very large. Right, so do you have a similar statement here that the upper, sort of the upper Galois theoretic one contains that it's dual, and the difference is bounded by p? Well, it's a power of p that I think depends on the hostage state weights. Oh, OK. But yeah, you have that kind of. In one of the constructions, yeah. And for, OK. Yeah. And so the wonder mark I wanted to make is that, I mean, one shouldn't really view this as a final result along this problem. I mean, there should be a better lattice. We just don't know a good way to sort of ask what we want from this lattice. We want it functoriality, and that's what we're able to get so far. But for example, you can ask if there is sort of natural torsion. We're completely ignoring torsion in this discussion, and that I don't know. So some functoriality in time. Yes, in the sort of obvious sense. Well, the question was, is there functoriality in the field? OK. But to make it clear, it's not cool. If you ramify delta star, I mean, if you ramify delta, then the pullback of the canonical extension is not so common. So I mean functorial in the stupid sense. There's a map. I'm not saying it's the whole thing. Yeah. So you can try to do a construction instead of models, take the projective limit of all models, and take something like omega log log. If you wait for 10 minutes, it's more or less what I'll do in 10 minutes. So I want to first talk about the Gallo-Atheriotic construction. So this is why we'll have to wait for 10 minutes for the other one. So this is the cleaner one. It comes from sort of good integral piatic harsh theory, which is my excuse for talking about this here. So this is why I left the two boards. So I'm going to have a board of notation now. It's the standard notation, but one has to say it. So p is a prime k over qp is a finite extension. I fix an algebraic closure k bar. Let's see, what else do I need? I'll need many things in a moment. This k is different. So the entire discussion is going to be local from now. And so I start off with x over k nice. Yes? And we want to construct this canonical lattice in there. And so the idea is sort of naive. And it's rather simple. You want to create a lattice in Dirac homology. That's functorial. We don't have that yet. We do have a lattice in etal homology that is functorial. And you use that. And you transport it across the piatic comparison isomorphism. Zp, what torsion, is a lattice in qp. And so we take this lattice. And so everything is gk-equivariant. Gk is the Galois group. And so you just transport this lattice along the period isomorphism. So I'm going to think of the period isomorphism as saying that d-derom of this object is equal to a Dirac homology. OK, and then, of course, I mean one has to actually go through the machinery and see that this can be done. If the field is unramified, the dimension is small, and you have a good reduction, then this is due to Fontaine messing. If you assume that all of the above, except the field is no longer unramified, then one can use Brouille's theory of S modules, and in complete generality, we use Kissen's theory of sigma modules. So I'll very briefly remind you of the main assertions from that theory. So pi, I need to fix a uniformizer. The functions depend, a priori, e of u inside, oh, sorry. One needs another ring first, which is w. So k0 is the maximum ramified subfield. w is the ring of integers. So OK, or w is totally ramified. Pi is the uniformizer, and e of u is its minimal Eisenstein polynomial. And this is still not enough, one has to make more choices. So I'm going to use the notation pi underline to denote a compatible system of p power roots of pi, sort of my chosen extension. And k infinity is going to be k adjoin all these roots, and g infinity is the Galois group of the resulting field. OK, I apologize, but so maybe one thing to remark is that this is not the same k infinity as in the previous talk. This is not the Galois extension. OK, so in this setting, Kissen defines the sort of following story. One has this ring that I'm going to call sigma, because I don't quite know how to say that. Maybe the audience can tell me what the right pronunciation for that symbol is. There's going to be another s later, so let's stick with sigma. Yeah, it is s in tech. So it's going to be the power series ring in one variable over w in this parameter u, and phi is going to be sort of u goes to u to the p, and it's semi-linear over the phi of w. OK, and so one is interested in modules over the string, and one can specialize them in sort of two ways that are going to be interesting to us. So there's w, and there's OK. And the specialization to w is u goes to 0. Specialization to OK is u goes to pi. So this is just a quotient by the ideal e of u. So with respect to this ring, Kissen defines the following categories. So mod phi of sigma is going to be pairs m and phi, where sort of m is a vector bundle over sigma. So I guess it's free, because it's local. And phi is a map from the Frobenius pullback of m to m, and it has some properties. So the property it has is that its co-kernel is killed by a power of this polynomial e of u. And similarly, one has a rational category, which I won't write down. And sort of the miracle of this theory is that this category is rich enough to detect integral semi-stable representations. And as yet, somehow, we're a much smaller ring than one might naively expect. Oh yeah, sorry, power, power for a sufficiently large n. OK, so theorem, this is Kissen's theorem. After you make all these choices and you define this ring, sorry, there's a functor. And I'm not going to state the theorem in the most generality. I'll just state it in the context in which I'm going to use it. So it's a semi-stable QP representation of GK to mod phi sigma join 1 over p. So I'm going to denote the functor as v goes to m of phi. And it satisfies the following two compatibilities. First, there is a canonical isomorphism between a specialization of this module and d dirom. So fix your v in this representation category. So there's a canonical isomorphism. It's not between just this u equals phi specialization, but you have to sort of compose it with the Frobenius. So when you go all the way down to the fraction field k, get d dirom of v. So it's some integral object that lifts d dirom of v for the purposes of my discussion. And it's not just any integral object. There's actually a nice correspondence between integral objects on both sides. So there exists a natural bijection, which goes lattices in this representation v that are stable under not the full Galois group, but G infinity. So this is a rational Cissin module, m, and you can study integral forms of it. And that's what I have here. So sigma forms of m. So integral Cissin modules are equipped with an isomorphism between the thing with p inverted and your given one. So bijection. OK, I'm sorry. I'm trying to make one point with all this notation. So the point is we can just use this to get a canonical lattice. So in our application, is there a question? So if you start off with x over k, which is nice, and you set v to be the ethylchromology of xk bar, coefficients in qp. And inside here you have this lattice t, which is the zp ethylchromology, modular torsion. Then you're in the setting where Kissen's theorem applies. So well, first need to assume v is semi-stable. So v is semi-stable. I have a lattice in there, so I get a natural sigma form of the associated Kissen module, m. So we get some m in this category, and an isomorphism with d dirom of v. Up to you, specialize to k in this manner. And by the periodic comparison, isomorphism, one knows that this is equal to hi dirom of x over k. So it's sort of obvious what to do. You just look at the image. So you define this lattice hix that we seek as the image of composite. So this is the semi-stable picture? If you change the iris and start, I mean, the uniformizer, or you change k, this is compatible or not? Yeah, I'll say that. I mean, yes, but it's a theorem. So for general x, so here I assume that the representation was semi-stable. Our priority, all we know is that it's potentially semi-stable, so there's some extension over which it's semi-stable. But then, this is why I went through this geometric construction earlier in the complex setting, where you can always sort of, the construction of the lower canonical extension is compatible with intersecting from field extension. So you just do that. Let me choose L over k finite such that we restrict it to L with semi-stable. And you define hi of x as hi of sort of XL intersected with this Dirac homology. So OK, that's the definition. The point I want to make with the definition is that it's sort of a very easy consequence of integral pi out of Cauchy theory that one can define such an object. What is not clear is that it's independent of all the choices that were made along the way. And this is, in fact, true. So I chose a uniformizer pi. I chose this polynomial e. I chose a combative system of p power roots pi underline. I chose this field extension L. And it's independent of all those choices. So we had actually proven this about two years ago in sort of restrictive conditions, meaning we needed to assume something about the hostage rate of v or the sort of difference between p and the ramification index of the field. But recently, Tonglu just proved this in general. So it's completely canonical. And moreover, under sort of good conditions, it is compatible with the sort of easy lattice, if you like. So x has good reduction. k equals k0. And the dimension of x is less than p minus 1. So you can use Fontaine messing arguments to end some sort of compatibilities between Kissen's theory, Breuys' theory, and Fontaine-Lefaye's theory to get that this is the usual lattice. OK, so that's the construction of the lattice in this sort of abstract Galois theoretic framework. It relies, I mean, obviously, sort of the main work is being done by this theorem over here. Yes? Yes, yes, there's no x needed, it's just v. Yeah, no, thanks. Yeah, so that's why I somehow wanted to think of this as an analog of DeWain's construction, where it somehow works in the world of differential equations over the disk, and it doesn't sort of think about the geometry. I mean, if you're sufficiently familiar with these aspects of integral piatic harsh theory, it's not very hard to come across this construction. At some point, I took it to my advisor at the time, DeWain, and he sort of immediately recoiled. And he insisted that there should be some sort of a geometric way of seeing such a functorial lattice that doesn't go through all this machinery. And so this is what I want to describe in the rest of the talk. So this is the geometric construction. They're both local precisely because of assertions of this sort. But if you're in a sufficiently generic situation, then it agrees with the lattice coming from a model and is independent of the choice of that model. So you can always sort of construct it as the finitely many bad places. So once again, x over k is nice. I have my integer i. And the idea, which was already pointed out maybe more than 10 minutes ago at this point, is you just want to look at semi-stable models. So if you want to copy Stienbring's construction and say, OK, pick a semi-stable model, look at the Dirac homology of that and try to prove it's independent of the choice. Of course, there's many things wrong with this. You don't have a semi-stable model. It's not going to be independent. But you can do the next best thing, which is you quantify over all models. So just look at those differential forms which become integral on every possible semi-stable scheme over your variety. And hope that this is actually large enough. So once again, I'm trying to define a lattice in this k vector space. And I want to look at those omegas which are integral on all semi-stable schemes of varieties over x. So at least this is how I understood your suggestion of doing a bigger field, bigger field. You need the bigger field. No, I thought to do it, but I'm not sure it can be done. It's a fine study of this risk-remin space. We saw, of course, there you have a lot of resolution of singularity. So I'm not sure about it. The idea was, from experience, that if you assume enough resolution, you should mimic it. But consider the log structure relative to the special 5-deck omega log relative to the log structure bound. So this should mimic the log construction for semi-stable. The problem is to prove things about it. But it's not what you do in any case. Yeah, I mean, in the arithmetic setting, another problem is the multiplicities. I mean, you can have sort of peas in the multiplicities of the central fibers in, if you assume, resolutions. But OK, so I mean, this is the idea. And I would like to formalize it on state of proposition. So let me make the following, define the following category. So ss of x, it's just going to be, I mean, this, semi-stable varieties over x. Except I want to sort of state it formally. So it's going to be diagrams like so, where x is your fixed guy, this is some morphism, and y is nice over some field extension, which is finite over k. And it has a semi-stable model, namely x. For a lot of what I'm going to say, you can actually replace semi-stable with the log smooth of Cartier type. You mean OK with it? OK with it. Oh, yeah, sorry, OK with it. Thank you. But let me just sort of stick with the more intuitive terminology. OK, so these are all the collection of pairs I want to work with. And the one sort of naive observation is that whenever you have such a datum, y, y, and x, you get a log scheme, which is this guy, y over OK. In particular, you get its log-deromcomology. And this gives you an integral structure on the algebraic-deromcomology of y. F can be any morphism. F can be any morphism, absolutely. So you can sort of, there's two games you can play. You can either put conditions on F to make it easier to show that you get a lattice, but then you have to work with to get functoriality. I'm going to just work with all morphisms of functoriality we built in. And then the hard work is to show it's actually a lattice. So I'll use different notation, Mi, because it's not equal to the lattice from the previous discussion. So this is going to be all classes here. So F upper star of omega is integral on each of these triples. I mean, this is obviously functorial. If you have a map from x to y and an object over x, you get an object over y. So this subgroup or sub-OK module of my deromcomology is clearly functorial. It's also clear that it's compact, because there is some lattice on which it's integral, so you can intersect back down and you get something nice and small. The only non-trivial content is that every, so this is large enough. In other words, whenever you have a deromcomology class, say omega, there's some multiple of that class, which is integral simultaneously on every possible semi-stable scheme that lives over your x. And so that's the proposition. Maybe I say this is one and two. It's the analog of what I had in the previous construction. So if x over k is good reduction, k is equal to k0. And here I need three dimension of x to be small. Comes out of the proof. Then this is equal to the lattice coming from a model, a modulation. So in other words, if you do this globally, then you actually do get a lattice. The age of gluing of the integrations of helicopter base extensions? In a very sort of stupid sense. I'm not doing any homotopy theory here. I just fix my rational vector space and I just search for things that are integral in there. I don't sort of care about if they're homotopic over the double overlaps and so forth. So I mean, I don't know a nice shifter at a grade of saying this. OK, so in the remaining part of the talk, I would like to explain what goes into this. And at least from my point of view, there's a lot of technology that goes into this. So some of it is known and we had to sort of discuss, develop extensions of some known things. So I'll try to explain what they were. So the idea of the proof is very simple. I mean, you want to show that every, as I was saying, you want to show every class as a multiple that becomes integral on a model. Now, we have no idea how to do this a priori. Like, you have some model and there'll be some multiple that works and then there'll be a larger model and there's some other multiple that works and there's no compatibility in this process. But there is compatibility on the side of piatic etalcomology. And so the idea is to have an integral piatic comparison isomorphism between Dirac and etal and then try to use that. So does what depend on the vanishing theorem? No, no, it's not going to depend on my vanishing theorem. It's going to depend on a slight extension of De Jong's alterations work. So you use integral piatic comparison theorems. And I said theorems in quotes because one doesn't have such isomorphisms. They're only maps. But the maps have sort of bounded kernel and co-kernel and we can try to use that information. OK, so here are the main tools. And I will sort of work in a slightly different setting. So I'm going to assume now that this x is semi-stable. I mean, after all, all the work I'm going to do is not going to be on my given x, but it's going to be on this category, ss of x. So I'm going to assume x over k is nice and has semi-stable reduction. So semi-stable model, say. Script x over OK. And d is going to be the dimension, the relative dimension. So I want to tell you first what these integral comparison theorems are. In order to do that, I can't just work with Dirac homology. One has to work with a slightly richer object. So first, I work with log crystalline homology in sort of the ramified sense. And let me sort of tell you. So I have this log scheme, x over w. x is given the log structure coming from the fact that it's a semi-stable scheme over OK. w is given the trivial log structure. And that gives me a site, the log crystalline site. And this site has a morphism down to the log crystalline site of just the base. We'll give this a name, f. And so associated to this datum, you can do homology. So in particular, you have the push forward. So rf lower star of ox over w lcris is an object. And so it's a crystal. It lives on the site. And it's sort of very instructive to think of it as a crystal, but just to understand the functoriality. But to work with it, I'm going to use a specific specialization, which is Brouh's ring. So I'm not going to need Kissen's ring anymore so I can delete this. And so I'll do the analogous thing. So once again, let me say in words what I'm trying to do. I have this crystal, and I want to access its values. To access its values, I have to tell you what it looks like on a really large PD thickening of OK over w. And that's what comes from this choice of. So s is you take w adjoin u, the polynomial ring. You join divided powers of the kernel of the map from w u to OK, which is given by setting u to pi and you periodically complete it. And so this is a Versailles PD thickening of OK over w. So in particular, it makes sense to evaluate this crystal on that thickening. And the result is what I'm going to call e chris of x. It should really be script x. So this was a sheaf on the crystalline sites. So when I evaluate it, I get an object on each ring. So I get an object of the derived category of s modules. And this object has really nice properties. So one needs some of those properties. So the first property is that it's perfect. I mean, it is very tempting to just guess that each of the individual comology groups are nice. I don't know if that is true. I certainly don't know how to prove it. One does know, just by sort of easy derived arguments, is that if you look at the full complex, then it's a perfect complex. But since I don't even know if this ring s is coherent, I can't say anything about the individual groups apriori. Secondly, by functoriality, it has some natural structures. So it has a filtration, has a phi action compatible with the induced phi action on s. And it has a connection, a logarithmic connection with respect to parameter u. Yes, yes, yes, sorry. So yeah, this object certainly depends. Yes, I mean, the structure you've upstairs is an object of the filtered derived categories. I take the push forward in the filtered sense. OK, so I mean, these are some abstract properties. And what does it actually look like? Well, to actually look like, I can only tell you what it looks like when you specialize. If you specialize from s all the way down to OK, so this is u equals pi, then in fact, you get Dirac homology. So E Chris of x, 10 set over s with OK, is the log Dirac homology of this model. So in that sense, it's reasonable. Once again, if I had to say something at the low of individual homology groups, I would be lost. So let me stick with the complexes. And similarly, you can do the other specialization, which is u goes to 0. And there you get the log crystalline homology of the central fiber. So it's the usual situation one has when one studies a Bray modules over s. You have two specializations. One of them is related to crystalline homology, and that's where the Frobenius is coming from. The other one is related to Dirac homology, and that sort of detects the filtration. OK, so what do we use about these objects? And what do we prove? So the fundamental fact that makes the complex amenable at all to calculations is the Hiotokata isomorphism. So one needs to know something about the homology groups. And so this is what I'm going to call Dwork's trick. And the assertion is that, OK, this complex would be horrible, but after you invert p, one has some control on it. So theorem. So I mean in the non-log case, Sperthelogus Hiotokato in the setting I'm in. And Baylinson gave a really nice explanation of this without somehow using explicit complexes. And it asserts the following. So as you take this complex and you invert p, then, in fact, this specialization, the 1 du equals 0 detects everything. So OK, this can be literally true. This is living over w. This is living over S1 over p, so you just extend scalars. So in other words, first of all, you can find a section back from the u equals 0 specialization. And then you can do Dwork's trick to conclude that it actually extends to an isomorphism like so. But it has the following consequences for us. Now one can actually say something about the groups. So at least this one is nice. Here, I mean nice in the non-mathematical sense. I just mean it's a free guy. And the second consequence is that if you look at the map at the integral levels from integral H i of E chris to H i d'Rom, so the u equals pi specialization, then the image is a lattice. So if I'm trying to prove something about d'Rom homology, I can always live classes to here up to a fixed error depending on this model and try to work with crystalline homology instead. I'm sort of running a little short in time, so I apologize. OK, so the main ingredient, which is sort of absolutely where everything all the control comes from as you go across models, is the integral piatic comparison theorem. So I will simply say in the interest of time that there is a natural map like so, S to A chris. A chris is a Fontaine's A chris. And the map is you send u to pi underline, coming from my choice of a compatible system. This is not Galois equivalent. It's only G infinity equivalent, but that's fine. So the theorem, so I mean I learned this formulation of the theorem from faultings. This is work, but I mean I'm not an expert. I think it follows from many of the other approaches as well. So it says that there exist maps back and forth, which relate E chris of x to piatic etal homology. So Cx goes from E chris of x, tends it over S with A chris. And on the other side, you have etal homology called that dx. So the first map is the one that comes from the comparison theorems. It's the one that's compatible with the algebra structure. The other one, the backwards one, is induced by duality. And the composition is just multiplication by t to the d, where t is the usual period of 2 pi i. So it depends on the choice of a compatible system of p power roots of 1. So Cx depends on the model? Yes, yes, yes. Sorry, I keep doing this. So swing it up. In this entire discussion with E chris, it depends on the model. I mean I wouldn't know how to define E chris without the model. No, well not because of the theorem, but other than that. I mean I don't know what the question would be if you didn't take a model. Yeah, so my question is about left answer depends on the model or not. Well not because, but no, but because only because of the theorem. Well, left answer depends on the model, right? Oh, sorry. It doesn't depend on the model. I don't quite see a direct way to say this, you use the rational comparison isomorphisms. You have Piata-Gittal-Comology, it doesn't depend on the model. You get filtered phi n modules and then that's what that one is. Yeah, sorry, thank you. Okay, so I'm out of time and I was just about to get to the things we did. So I'll sort of simply state them in a few words. So once again, to prove this boundedness, we first live from Duram-Comology to Crystalline-Comology, which is possible by Dwork-Stragg. You then move from Crystalline-Comology to Piata-Gittal-Comology, which has a bounded error thanks to Fulton's integral theorem. And then you sort of try to argue on the side of a telecomology. To actually do this argument, we need sort of two more ingredients. And these are both of geometric nature. So the first, in Fulton's theorem, the error depends on the dimension of the variety. If you're working with HI, you want the error to depend on I, not on D. You could be working with H5 of a thousand dimensional variety and you want the error to depend on the fact that it's H5. So one needs some sort of a left-shits theorem for these groups. So there is a very weak left-shits for Ecriss. When I say very, I mean, I don't assert anything for a specific hypersurface. I just assert that there is some hypersurface where the usual injectivity and surjectivity properties hold. And this really uses the perfectness and a recent result of Jensen and Seito on Bertini theorems for lock schemes. And then the second ingredient in the Nalstow is sort of Deong's alterations plus something. And so the result we need is that in the proof, we want to make some alterations in a way that they're etal at a given point. Now, one can do that, but one has to actually prove it. So some control on the etal locus, meaning if you have a variety and you're trying to prove semi-stable reduction, Deong will tell you there is some alteration which has semi-stable reduction. But what we want is actually, when you fix a point downstairs, you can choose that alteration to be etal at that point. Find that etal even, which has semi-stable reduction. And you can do this for sort of all points. And that's what we have to prove. So, okay, I guess I'll stop here because I'm out of time. Sorry. So we wanted to talk about a question. So the semi-stable category, you make trajectory because you set proper and later on you do directions. Yeah, I should have said trajectory. Thanks. Two, two. Is there any distinction with a question about the A and the E? Not this construction, no. But what I, what one can say is that, so if you have this improvement of Deong's theorem, in Balanson's paper, he uses the H topology on the generic fiber. And what this allows you to do is work with the etal topology on the generic fiber. So you only ever do blow-ups and alterations along the boundary. And this brings it a lot closer to the original. You have always been using it as a model and she's been trying to do it in topology. So what do you do? Isn't there any relation? I'm sure. You could recover by taking the organa, A and sense. I mean, this is what the other show was asking about. This is what I wanted, I'm interested about waiting on what you showed. I couldn't get it to work because of the homotopies. I mean, you're not just saying that you have your class which is integral simultaneously on all models. If you do our gamma, you're somehow saying it doesn't have to be integral on all models with some homotopy on the fiber part and a double homotopy. At least, I mean, I didn't try very hard, but I couldn't see it. It's not enough to take all alterations with the misstable model or the original model. It is enough to get a lattice. But it's not, obviously, functorial unless one has something like this. Because. It's not flowing, but no, it's the same. No, I don't know if I got the same thing. We can control the error depending on the dimension and the ramification index of the field. In fact, that's how we do it. We first work with that category, you show that that gives you a lattice and then show that this other one is sort of off by a bounded amount. Painfully. You go through all of De Jong's proof and observe that at each step you can do this. I mean, Gabor told me that he knows how to do this and maybe he knows a better way. I don't know. So it's over, no, but actually, if you do it over one dimensional bed and characteristic pick as you have to introduce the separable thing. So this is a tile means after extending to. Yes. But here it's a number of field case. But it's, I think if one looks at the case over a field it's not more or less easy to see the choices in the young original paper over a perfect field case. And then the one is just choosing generally not. Yeah, okay. So I mean, this is. It's really. Yeah, I mean, that's what we did. We just go through each step of the argument and observe that you can make those choices correctly. Yes. So if I understand, so your first course of time you must be used in comparison between the tile comology and the arm comology. And the second one is rather between the arm comology and the arm comology. The second one, it seems natural that maybe it's out of reach to get the derives. So maybe that is a better complex. Yeah. I mean, one can try to define something locally by saying you look at Piata-Gatalcomology, you tenser up with a Chris and then look at the right Kalwa invariant. Oh, but you can rather use a complex like this, I mean, that's what it starts to believe in. The torsion is completely different and the pistons are the same as the arm comology. All right, no, I don't know. I mean, yeah. So you get two constructions. So one line is bigger than the other or do you have some relation between them? Do I have a relation between them? Yeah. I have totally everything I know which is that if P is sufficiently large in the, but other than that, no, I don't know. Well, okay, I mean, there's containment for a stupid reason because this one is simultaneously semi-stable on every model. And for the other one, I pick some semi-stable model and then intersect. So, it's containment. And they're genuinely small. So I think if you compute the case of an elliptic curve with hospital reduction, you get a difference of P. Do you need more Chris? Okay, this one's got a speaker here.