 Thank you. Thank you for the introduction and thank you very much for the invitation to speak here. It's a really great honor for me to be speaking for this conference in honor of Arthur. So one of my favorite theorems from when I was a graduate student is the theorem in chapter 8 of Berthela August. And what I want to explain today is some joint work with Matthew Morrow and Peter Schultz, which sort of really uses the construction in chapter 8 of Berthela August in a crucial way. So I'll mention what the theorem was when we get there. Okay, so I guess it's a joint with Matthew Morrow and Peter Schultz. And the setup, so I want to talk about a certain integral columnology theories and relations between them. So the setup I'll fix once and for all as the following. So C is going to be some complete an algebraic closed extension of Qp. And so for example C equals Cp. Oc is the ring of integers and little k is the residue field of Oc. So this is a valuation ring and it's a perfect field. And then you also have the ring of vectors of k. And then finally the sort of main object is going to be a proper smooth scheme over Oc. So actually everything I'm going to say already works in the context of formal schemes. So if you like you can work in that context, but it's interesting enough in the case of smooth project. Schemes over Oc, so things with good reduction. Okay, so in this setup, there are certain co-mology theories that we know about eventually. But nothing I'm going to say. I think, yeah, I want to... The paper on the archive, you assume I think projective smooth rather than proper smooth. No, no, we assume proper smooth. Right? I thought it was a proper smooth formal scheme. Okay, so since everyone already knows the paper, maybe I shouldn't talk about it. Classical piatic co-mology theories associated to this picture. So there's three of them that I want to focus on. So the first one is etalc homology. And so by this I mean the etalc homology of the generic fiber. We'll denote by X of C and the coefficients are in Zp. And so this lives in the world of finite Zp modules. And so in the algebraic case, the fact that it's finite is classical. In the formal scheme case, it's more recent. And then it does a Galois action. All the constructions are going to be Galois covariant. So if there is a Galois action, then we'll get a Galois covariant statement. So this is a theory coming from the generic fiber. There's a theory coming from the special fiber, which is crystalline co-mology, which we heard about in the previous talk. So this is the crystalline co-mology of the special fiber. Relative to W. And then the Frobenius and X on the special fiber induces an endomorphism of this. So this lives in the world of, let's say, finite P modules over W. So it's a finite module over the ring of width vectors, and it has this endomorphism given by Frobenius, which will be called P. And then the third one is sort of... So I have the generic fiber, I have the special fiber, and then I'm going to use the integral model. So here's the Dirac homology. So this is the Dirac homology of this morphism. And so this is, again, a finite dimensional... Well, it's a finite OC module, and it has a filtration coming from the Hatch filtration. So it's a finite filtered OC module. So these are the three theories I want to focus on, and then we know that there are sort of various comparison isomorphisms relating different subsets of them. And I'm not going to spell them out right now. Like, for example, Fontaine's Dirac conjecture tells you that A and C are essentially the same information after you extend scalars to B Dirac. And A and B is the crystalline conjecture, and then B and C is the birth of August theorem relating crystalline with Dirac. Okay, so the goal for what I want to do today is... I want to sort of try and explain a single picture, or a single homology theory of which all three of these are specializations. So realize all three are specializations of the same, of a single theory. And already it's clear if you want to do this, you need something... So in each of the previous cases, the ring that I was working over was either Zp or W or OC, so it was a dimension one ring, it was a valuation ring. And if you want something that's going to specialize over two different valuation rings, it needs to be at least two-dimensional. And this ring is sort of the main sort of playground in this talk. So it's Fontaine's ring A and... And sort of going to be the object over which everything lives. And let me sort of... Let me tell you what it is. So the definition, it proceeds in two steps. So first we define what I guess nowadays we call OC flat. It's by definition, you take the ring of integers of C, reduce it mod P, and since now you have a ring of characteristic P, it makes sense to talk about its perfection in the sense of inverse limits. So this is the inverse limit over Frobenius of OC mod P. So it turns out, by non-trivial theorem, that this is actually itself a really nice object, so it's a valuation ring now in characteristic P with an algebraic equals fraction field. So C flat is going to be its fraction field. And so roughly what's happening is that when you take your valuation ring and reduce mod P, you sort of lose the dimension, so this is a zero-dimensional ring, but it has tons and tons of nil-potens. So when you pass through the inverse limit along Frobenius, those nil-potens sort of build up to give you an extra dimension. And that's what OC flat is. And then the ring A-inth is the deformation of this to characteristic zero. So the ring of it after is OC flat. And this is a perfect ring of characteristic P. So taking the width vectors is a really nice thing to do. And I'll call phi the Frobenius you get by thinking of it as the width vectors of a perfect ring. And so this is going to be the basic object we're going to be working with. And I want to explain how it's related to the three rings appearing on the first board up there. So in order to do that, I'm going to choose coordinates. And this is not strictly necessary, but it just makes certain constructions easier. So I'm going to choose P underline in OC flat, which is a compatible sequence of P power roots of P. So you just choose a compatible sequence of P power roots of P in OC. And this gives you an element of this inverse limit, which I'll call P flat. And then by taking Teichmuller lifts, you also get the element, bracket P flat, living in A nth. And then the basic slogan, which I'll try to make precise, is that A nth is sort of, it's kind of like a two dimensional regular local ring and the two coordinates are P and P underline. And I warn you right now that what I'm about to write is completely nonsense, but that's why it's a slogan. So it's a two dimensional regular local ring. And the parameters are P and P flat, sorry, P, bracket P. And there are theorems that say that it behaves like such a ring, but for example, it's not two dimensional. It's not regular because it's not an aetherian. And those two are certainly not parameters in any reasonable sense. But anyways, that's the philosophy. And the reason this is related to the previous slide is that because of the choice of coordinates, you can define these three explicit sort of rank one quotients. So you get three divisors. So the first one is where you kill P. So if I take with vectors of a perfect ring and then I kill P, I just go back to the ring I had earlier. So I have OC flat. The second one is where you kill P flat, so the other parameter. Sorry, I keep saying keep P flat, but I mean bracket P, bracket P underline. So this is not literally where you kill to get here. Here you use the fact that the width vector construction is factorial. And there's a map from OC flat to its residue field, which is k. And so passing to the width vector is you get such a quotient. It is the case that this element is in the kernel, but for example, so are all of its P power roots. So this element does not generate the kernel. So this is even worse things in there. And this is independent of any choices this one. Yeah, so the map is completely independent of the choice. I'm just to write it down. Yeah, the map is what I said. It's the functoriality of the width vectors. And then there's a third map, which is Fontaine's map theta, which goes to OC. And here you kill the difference of the two coordinates. So I'm killing the horizontal direction, the vertical direction, and then the diagonal direction. So you kill C, which is P minus P underline. And this literally is true in that this generates the kernel. Again, it's independent of choices, if you were right. Yes. So the map is totally independent of choices. This is just one convenient choice of representatives for elements in the kernel. Is there a functorial definition of this map? You can define it, for example, using the cotangent complex formalism. In this formula, it was E p and mod p. There is such a map, and therefore it lifts to characteristic zero. There are more explicit definitions. OK. So I want to work with this ring. And so I want to actually draw a picture of what it looks like. And so this is a cartoon of Speck-A-Inf. And so if you've seen the cartoon Sholza has been drawing at Berkeley a year ago, I warn you that this is different from the one he's drawing. I prefer this because it illustrates one feature that's relevant to what I want to talk about here. So this is really a picture of Speck-A-Inf. So I'm going to draw the generic point. So here's the generic point. Here's the close point. And here are these three quote unquote height one points that I'm interested in. And I will label all the points with the corresponding residue fields. So extensions of residue fields. So the residue field at the close point is K. This point corresponds to the fraction field of W. So W join one over P. This point, so it's the generic point of this quotient over here. The point in the middle is the generic point of OC. So C. And then the last one is C-Flight. And then there are specializations relating them. So I'm going to draw an arrow whenever there is a specialization. And oh, yeah, I guess I didn't label the generic point. So it turns out that the generic point sits inside a beater arm. So I'll just label it by beater arm. So that's a rough picture. I also want to go further. So I want to not only label the points, I also want to label the specializations. So for example, this specialization corresponds to the ring W. Meaning this quotient is to W of K. This specialization corresponds to the ring OC-Flight, which is the first one over here. And the other one is the other one. So this is OC. And so in terms of this picture, what's going on? So let's see. Let's write down. Maybe I want to label one more or two more. So this specialization, you can think of it as being W of C-Flight. Meaning those are functorial map from W of OC-Flight to W of C-Flight. And the quotient of the special point, the field of the special point is indeed C-Flight. So this is where W of C-Flight comes in the picture. And this specialization here, after completion, is what's called B-derarm plus. So this is a valuation ring. And its fraction field is B-derarm. And so the only thing missing is this guy. And this guy is kind of weird, so I don't want to name it. It's sort of closely related to the fact that the kernel of this map is not generated by just P-Flight. There's lots more junk in there. OK. So in terms of this picture, what is going on? So crystalline homology is living over here on W. So crystalline homology. D-derarm homology is living over OC. And a telecomology, well, it turns out that there's a pretty nice way of thinking about it is living over here, which I'll explain when I write down the theorem. And maybe, OK. So these are the existing three theories of living. And what I would like to explain is a picture of what sort of fills at this end. So we want to construct something that lives over all of A-ins. And it's going to have the correct specializations. Is there any interest in that? It just might be a theory of OC-Flight. Yes, it's very interesting. So I'll comment about that theory once I write down what the result is. That's somehow like where the new information is coming in. What about mysterious errors? So the point is if you label the specialization by the complete local ring at that point, then this guy just turns out to be W of 1 over P. Because the kernel of this map satisfies that M equals M squared. So that's why it's kind of weird. Are there any other questions about the picture? OK, so I'll have to figure out how to erase. OK, so here's the main result. I think this was section two. Well, I guess it was section three, probably. I think it should have been three. So there should be four. I'm following my notes. So results. And so the main theorem is that what I said we are going to do, we can do. But let me sort of say it in a rather precise and maybe slightly long-winded way. So again, I remind you that our setup is x over OC is proper smooth. And to this data, we can associate or can functorily attach a complex, which I'll call r gamma sub A of x. So this is a perfect complex of modules over the string A and f. And it has an extra structure coming from Frobenius. So plus a map phi sub x from the complex to itself, which is Frobenius linear, semi-linear. A is A and f. Yeah, sorry. Yeah, thanks. So it's phi semi-linear with respect to the phi on A and f. And it's an isomorphism after you invert this element that I had called psi earlier. And maybe this is not so important for now, but we'll come back later when I make a remark about this. And an isomorphism after inverting this element psi, which I remind you was a generator of the kernel of theta, with the following realizations. So satisfying. OK, so in my notes, I have the etal one first. So the etal realization is the fact that this construction recovers etalc homology. And the way it does is that you have to go to W of C flat and then extract a ZP module out of it. And here's how it works. So there's a canonical isomorphism putting this complex scalar extended to W of C flat. So restricted to that height one specialization over there. And etalc homology, ZP coefficients, also scalar extended to the same ring, which is compatible with Frobenius. So there's a Frobenius on the left-hand side coming from the fact that there's a Frobenius on each factor. And there's a Frobenius on the right-hand side from the stupid Frobenius on the right-hand side on W of C flat. And what this means is that you can recover the etalc homology itself as the phi in marines. So this is by a theorem of cats, I guess. So if you're interested in the etalc homology, not just a scalar extension, then you can recover it by the following formula. So you take this complex over A n, you extend scalars to W of C flat, and then you sort of do phi equals 1. And if you sort of invert P, then you really do phi equals 1 on homology. Otherwise, it's the derived version of that construction. Okay. I think that phi minus 1 is a surjective on... So when you go from ZP to W of C flat, it's a flat extension and the Frobenius minus 1 is... I think... So you're saying the derived construction is not necessary? Well, for homologies, it should get this... Yeah, that's... Yes. I just wanted to be safe. Yeah. So cats' theorem is that if you have any perfect ring of kerosic P, ZP local systems on spec of that ring are the same as modules over the width vectors of the ring which are isomorphic to themselves after pullback along Frobenius. And so, I guess, yeah, that's a good point. So I want to point out that once you make the scalar extension, this element C becomes invertible. Because remember, in this picture, the zero locus of C is over here. So when you base it over here, there is no zero locus. So the Frobenius is actually an isomorphism on the nose, not just after inverting something. And so you get a unit root crystal on W of C flat and thus it corresponds to the local system. Okay. So the others are more direct. So the crystalline realization is exactly what you might expect. So you take our gamma A of X and you scalar extend to W of K and you get the crystalline complex. Again, as females. And so, for example, what does this tell you? So if you know that on the left-hand side, Frobenius is an isomorphism after inverting C. Now if you stare at the picture, the zero locus of C is over here. So when I restrict over here, that just means it's an isomorphism after inverting away from the close point, which is the statement that Frobenius is an isogeny on crystalline comology. And then the drom one. So again, our gamma A of X answered over A of N. And now there's a choice. So I think the way we wrote the announcement, there's no Frobenius twist, but I'm going to twist it. So I base the angel on phi first and then specialize to O of C and then I get our gamma drom of X over O of C as filtered modules. And I will explain or maybe not how the filtration comes about later. And I guess I want to emphasize, maybe this is clear from the title, but that everything is integral and on the nose. So you get some statements about torsion from this. Okay, so corollary. So we got to write two of them. So the first one is if you can ask what the relation is between the dimension of etalcomology with Mach-P coefficients and the dimension of crystalline comology with Mach-P coefficients. So the crystalline comology of the special fiber relative to K, which is also just the dimension of the dromcomology because drom is crystalline. And then what you get out of this is that there's an inequality this way without any assumptions on ramification. Sorry, my shoe laces are coming and done. And then there's an integral lift of the statement. So this is a Mach-P statement and you can ask what happens integrally. And integrally, what you can say is the following. So if the crystalline comology is torsion free in degree I, so is etalcomology in the same degree. And x is x. Ah, yeah, sorry, thanks. Otherwise it would not be so interesting. Okay, so maybe I want to make one comment. So first of all, the inequality in one is strict. So there are examples where there's genuinely more crystalline comology than there is etalcomology. So you can't expect to do better. And this is more or less the only direction that is reasonable if you sort of have some defamation theoretic picture in mind. And indeed that is the case. And likewise, there are examples where the crystalline comology has torsion, but the etalcomology does not. In fact, we wrote one down in our paper. It's a three-fold incursive two. And so this is the only thing that could be reasonable. And maybe I'll sort of just explain the proof of one. Before that, can I ask you a question? Yes. The difference in torsion and crystalline comology and in the Dirom-Cohomology, there are Ecke-Dahl's example, which shows that the ramification is bigger than p minus 1. They need not be the same. Do you have any way of incorporating that? I guess the short answer is no. I don't know what the relation, like a precise bound for the discrepancy between the two. I've been asking people if it's reasonable to expect that the Dirom-Cohomology always has less than or equal to torsion than crystalline comology. But I don't know the answer to any of those questions. But if the Dirom-Cohomology is torsion free, it's not enough to know there is no implication, other implication. So all the implication between such and such in torsion free or something else is torsion free is in what you wrote that you have examples showing that other implication of torsion free is a lot more. The only implication I have a counter-example to is the other one, the converse to this statement. So the implication that if Dirom is torsion free then... I don't say anything about Dirom. Sorry, I mean I guess I don't say anything about Dirom. Because all of them are torsion free, it's just an American thing, which is the same. It is true, I can't think on the fly, I think it is true that if all the Dirom-Cohomology groups are torsion free, then in fact, so are all the HL-Cohomology groups. That follows from the proof. I don't know about a single degree how that argument would work. And the reason I want to mention the proof of one is exactly this point that Arthur raised. So you consider what happens over the new specialization that you gain, namely the one over OC-flat and use semi-continuity. So semi-continuity for this base change. So the point is this is a perfect complex over OC-flat and whenever you have a perfect complex over some ring, the dimension of the special fiber, the dimension of HI of the special fiber is always an upper bound for the dimension of HI of the generic fiber. And that's exactly what one is saying. But this ring is a notherian, right? OC-flat is not a notherian, but it's not so bad. It's a valuation ring of rank one valuation. It's still okay to use this notion. Yeah. I mean, for example, you can do some approximation argument to reduce to the notherian case. Okay. Yeah, she used these words. I'm going to make some remarks about the theorem. So maybe the first remark I want to make is that in fact, when the crystalline chromology is torsion-free in the setting of two, you can actually say something finer. So if HI-chris is torsion-free, then this HI of this complex over AN is itself a finite free module. So what you have is a finite free module over AN together with the self-map fee, which is an isomorphism after inverting C. And these gadgets have a name. We call them Broi-Kissen modules. And he proved a very interesting theorem about them. So the theorem far proved is that if you have a Broi-Kissen module, then the crystalline specialization is determined by the other information. So in particular, what that implies in our setup is that if you're in the torsion-free case, then the crystalline chromology of the special fiber is a functor of the generic fiber, which is not easy to see from the definition. Okay. And maybe for the second remark, I need the most space. So the second remark is about how the proof goes and the way the proof goes is that you construct the homology theory by first constructing an appropriate sheaf and taking its homology. So more precisely, we construct a complex which we're calling A omega x, which is a complex of sheaves on x of AN modules together with the Frobenius such that this homology, our gamma sub A of x, is just hyper-chromology of this complex. I mean, what kind of sheaves? A tall or pro-atall, I guess, since it's some limit thing. Yeah. So a constant sheaves with coefficients in A and f. And this complex of sheaves actually has very interesting properties. So there are two specializations that are, at least I find interesting. So the first specialization is related to what happens in the Diron specialization in the theorem. So if your base change along the map from A and f to Oc, which is Frobenius twist of theta, so the same one that showed up in the theorem, then the complex you get is really the Diron complex. So this is the Diron complex of x over Oc. In particular, it's a complex whose terms are given by differential forms and the differential is not Ox linear. It's just linear over the constants, which is Oc. But you can do something else, which is you can specialize when you don't twist by phi. So if you do that, at least if I got my twist right, then you actually get a complex of quasi-coherent sheaves. So in fact, you get a perfect complex. So it's a quasi-coherent complex on x, whose ith-comology group, or sheaf, is given by omega i. So here you are working with a formal scheme? Formal scheme. Formal scheme. Yes. So the x is a special phi? Yes. Yes. So this is the Diron complex of the formal scheme over Oc and differential forms for the formal scheme over Oc. And so this is a quasi-coherent complex. The differentials are all linear. And so this is kind of this mixed characteristic version of the Cartier isomorphism. So in particular, if you specialize both to the residue field, then you get the Cartier isomorphism. So you don't. It's not the wrong. It's not the wrong dip. OK. So I want to point out that this is the complex that somehow is responsible for the hot state filtration, menu invert p. So there's a somehow close connection between the two, which goes through this gadget over here. Again, then finally, the third remark I want to make about this is that there's a specialization in court. The complex totally decomposed in the sense of? So I don't know if there's an analog of the DeLine or Luzi theorem in general, but what is true is that if you just look at the obstruction to splitting off h0 from h1 in the one truncation of the complex, that measures the obstruction to lifting the formal scheme to a2, a-inf mark c squared. So the square zero extension of Oc relative to Zp given by this length two quotient of a-inf. And the obstruction to lifting across that extension is the extension class that's showing up here. So how can you be, you can have, when you tend to be small k, then the first line is there is an extra four venues, but how can you then have a quasi-current complex and when you tend to follow by some four venues, you get a drum complex which is not all linear, so I don't know, there seem to be something strange here. Well, it should realize the statement that the Frobenius push forward at the drum complex is that the total fiber has homology groups given by omega i of x over... And this is a generic fiber, so you did not... Well, this is integrally. Maybe we can discuss this after the talk. I get confused about Frobenius with, like, all the time when I think about this, so it's quite possible I have something off by a sign. Right, and then the third specialization which I wanted to mention is the a-chris specialization. So in this picture, there is yet another period ring that's floating around which is sort of the dotted arrow which maybe is not so visible, which is sort of a speck of a-chris, which is where you get when you join divided powers with the kernel of c to a infinitiatically complete. And there is an analogous statement over there. So r gamma a of x tends to over a... So again, there's a Frobenius twist, I believe, with a-chris is the crystalline homology of x relative to a-chris. So this is the homology theory that we already knew interpolated between the crystalline homology, the special fiber, and the Dirac homology, the generic type of the integral model. And the theorem is saying that there's somebody to extend it across all the things. So all these are isomorphisms or almost isomorphisms? Quasi-isomorphisms? No, almost. Okay, so those are the results, and now I'd like to explain what goes into constructing this object. So this is an algebra object? Which one? A omega x? It's an e-infinity algebra. Yeah, it's not a dg-algebra. So one of the main tools going into the construction is this gauge transformation business from Bertha-Logos chapter 8, so L-Ada. That's what it's called. It's a killing torsion in the Dirac category. And so the basic goal is the following. So this section is going to be kind of abstract, unrelated to the attic things. So let's say A is a commutative ring, and F is an element of A, which is a non-zero divisor. A question you could ask, sort of a naive question, you could ask is if there is a way to take a complex of A modules and systematically kill all the f-torsion in its homology. And this recipe should be somehow independent of the quasi-isomorphism class. So it should work at the derived level. So the goal is to kill f-torsion in some complex hi of k, in hi of some complex k, functorially in k. And this is not really possible if you try to do it using just nice exact functors, but I will nevertheless explain why it is possible. So the key definition is the following. So I'm going to define a functor which does this job, and to define the functor, I'll first specify what it does to nice objects, and then you resolve everything by a nice thing. So the nice objects, in this case, are complexes of flat modules, or really you just need f-torsion free. And then you do the following construction, which is denoted a to sub f. So this is an example of the filtration of the Kale for the f-addict filtration. And so you construct a new complex whose term and degree i is the set of all elements, alpha in the original complex and degree i that are divisible by f to the i, such that the differential is divisible by f to the i plus 1. You need the second condition in order to get a complex. And so this defines for your new complex, a to sub f. And then the theorem is that this recipe passes through the right category. So theorem, applying it to flat replacements, say, gives a functor from d of a to d of a, which I'll call l-ada with respect to f. It's not a phi with ki, it's a p and plus i or something like that. Well, you can define and you can think of the input as a filtered complex given by the f-addict filtration, and the output is a new filtered complex. And what I've written down is a 0th step of the new filtered complex. Do you need some boundedness in the direct category for the value of i? Not for this construction, no. But I mean, for all what you're going to do, it's going to be bounded below. It's going to be bounded. And it's going to be really nice, but... If I can do, like, f to the i, k, i. Of course, i is... you can allow formally i to be... You can allow it to be negative, which is why you need at least f torsion free. Yeah. Yes, actually, let me do an example where i is negative. And so the remark I would like to make is that this function is not exact. So it's a function between triangulated categories, but it doesn't take short exact triangles to exact triangles. And here's sort of y. So let's do an easy example. Let's take a equals zp and the element f to be p. And consider the complex k, which is just z mod p, placed in degrees zero. So this is... We have to choose a flat replacement, so this is quasi-isomorphic to this two-term complex. It's a multiplication by p on zp, where the right-hand side is in degrees zero. And so when you apply L8a to it, L8s of p of k, you're supposed to apply 8s of p to this replacement. But if you apply 8s of p to the replacement and you think about what the indices mean, it means that in the first court, in the first entry, you're allowing denominators of p. And in the second entry, you're not. So this looks like the complex, which is one over p, zp, mapping by the same differential to zp. And this is zero, quasi-isomorphic to zero, because it's an isomorphism. But if you do the exact same combination for z mod p squared, it's not zero. So you end up getting z mod p, just because everything looks the same except this differential is a p squared. And so there's a z mod p in degrees zero and so this is why it's not exact. If it was an exact function and it killed z mod p, it would have to kill z mod p squared, but it doesn't. And this is really crucial for what we are doing. I guess to go back to an earlier question. So you can prove that L8a is lax monoidal. And so in particular, it takes algebras to algebras. And that's why the answer to Martin's question from earlier is yes. Okay. So how do we use this? Lax monoidal. So a function is lax monoidal with a tensor f of b as a canonical map to f of a tensor b. It's enough to make algebras go to algebras. So the construction is going to depend on a choice. But I mean, the actual construction does, but the output doesn't. So I'm going to choose, sorry, I remind you again that x over oc is this proper smooth formal scheme and the construction is local. So let me just say it's smooth. Formal scheme. In order to do the construction, we are going to take a nearby cycle and a function applied to a certain sheaf and then modify it using this L8a construction. So I need to choose an element f to do the L8a and for that I choose the following. So epsilon underline is a compatible sequence of p power roots of one. So trivialization of zp of one, if you like. And like before, we can think of it as an element in oc flat. So earlier I had p underline and now I have epsilon underline. And the key element here we're going to use is epsilon minus one, which is an element of a. And so maybe I just make one remark about this. So one thing you could do, so I guess I should pick my normalization so I think the way I'm doing it, I want epsilon to start at a primitive p through to one rather than starting at one. So what this means, I think, is that if you do phi of mu, well, it's a compatible sequence of elements here. Yeah, but you just have to be sure where you're starting. Okay, so what I want is that when you apply this map theta to it, you get one minus a primitive p through to one minus one for mu. So it's not in the kernel of theta. And it's the closest thing that can be there, but it's not. So if you do phi of mu divided by mu, this makes sense, and this is actually a generator of the kernel of theta. And this is the relation to the previous, to that picture. So this could play, this is equally good as the element c I had earlier, which is p minus p under a line. Okay, so, I mean, if you're familiar with computations of nearby vanishing cycles in the Piatic-Hoch theory approach sort of due to faultings and then Peter, this element shows up quite prominently because you're essentially trying to do group invariance under some actions and the way you can do group homology is by gamma minus one or gamma as a generator. So that's roughly where this object is coming from and what we're going to do with it is the following. So step one. And the construction is in two steps and the first step is you get an almost correct thing and then I'll correct it in the second step. So this is almost correct version. Yes, of A and oh, A omega, X. Okay, so we're going to use the nearby cycles map, so let me give it a name. So nu is the map that goes from, or either the per-atial side of the generic fiber to the etal side of the formal scheme just given by the fact that you can pull back an etal sorry, this is a map of the corresponding tau-point. The pullback function just says that an etal cover of a formal scheme gives you an etal cover of the generic fiber. And what we're going to do is we're going to set A omega X to be the following. So this is A omega X prime because it's almost correct. So one thing you can do is you can do our new lower star. So you can push forward along this A nth, but really the sheaf-ified version of A nth on the site. So the construction of A nth makes sense for any ring. It doesn't have to POC. You can always reduce any ring mod P. You can pass through the inverse limit perfection along for bannies and then you can do the width vectors. So you can perform that construction as a sheaf on the per-atial side and I'll call that sheaf A nth comma X and then you push forward. And okay, so that's complete nonsense if this is the first time you're seeing it. I'm sorry, but I wanted to give an actual definition. And then this guy is sort of not good because for example, for bannies is an isomorphism on nth. So for bannies is an isomorphism on this complex and it certainly is not going to be an isomorphism on our A omega. So you're going to modify using L eta. So do L eta sub mu off this. And the for bannies on this guy which was an isomorphism now induces a map which is not an isomorphism. It just induces an endomorphism. So this is an object of the dry category of the formal scheme of A nth modules and I can So here the et al et sat al is the special fiber of the formal scheme? Yes. And the left is the rigid part? Yes. Yeah. So we also get a map phi which goes from phi upper star of this complex to itself. And if you sort of understand how L eta works, essentially when you do apply phi upper star with L eta it basically commutes except the commutations off by an element here. Instead of doing L eta mu you do L eta phi of mu. And so therefore the cononical map is going to have a kernel and which are killed by phi of mu divided by mu which showed up over there. So this is this map. So it's an isomorphism outside the kernel of theta. And that's just a completely formal calculation based on the fact that phi of mu divided by mu generates the kernel of theta. The point is that L eta doesn't quite do the phi star can you say that? Is it something really fast? I said if you do L eta sub little mu of phi upper star that's the same as No. Let's see. Hello. I think L eta sub if you do it in this direction then that's the same as doing L eta phi of mu phi upper star. Just because phi is an isomorphism so I'm just transport of structure. And so yeah. So in fact by the similar formal argument you can actually prove the following. So we have this we produce this map phi upper star A omega x prime to A omega x prime. And on the other hand you also have a cononical map here from L eta with respect to this kernel XE which was phi of mu divided by mu to A omega x prime just because this was a sub complex of things satisfying certain divisibility properties and there's a cononical isomorphism like this. And so this tells you in a precise sense how far this map is from being an isomorphism namely is the highest power XE you need to make this map an isomorphism and that's determined by the number of non-zero homology groups. So this is the morphism is on the one in the capital this isomorphism is just phi I think. Yeah it's the way phi works with L eta so that should be if Fabian is here. But it's completely formal based on how L eta works. Yeah so it's in the essentially. Oh sorry you're saying it's in the book. Oh okay. Excellent. The main thing is that we will get it actually. The L eta and the phi. Right. But I guess in their case well phi of p is p so yes thank you. Okay I lost track of which board I was. Okay. Right so this is the almost correct construction and this is almost correct in the sense that this is almost isomorphic to what we want and to get something that's honestly isomorphic we have to fix this almost non-sense. So I haven't actually done any almost mathematics but what is really going on is that this is living in the almost well you can think of it as living in the almost world and you modify it by something supported on the special fiber to make it correct. Is it clear why you needed to kill the torsion? What's the purpose of the L eta? Why did you put it in? So if you try to compute nearby cycles in this framework of perfective spaces say there's just a lot of bad torsion that shows up when you compute these nearby cycles you use. So for example if you did it instead of a nth just for the structure sheaf over here the torsion would be sort of really like infinite but the observation is simply that whatever that infinite torsion is is actually killed by epsilon p-1 and so when you do this L eta it just goes away. Okay and then the second step is you fix the almost non-sense so step two so I mean the way you do sort of you have something in the almost world and you want to get something in the real world the way you do it is that you modify it by something that lives in the quotient of your ring by the ideal of almost mathematics so in order to do that you're going to do the following so you're going to construct a net well I'm just saying you can do this so you construct a natural map which goes from the Diron-Witt complex of the special fiber thought of as a complex on the formal scheme to this complex A omega x prime scale are extended to W so the ideal of almost mathematics is the ideal cutting out spec W inside spec A and so all I'm going to do is I'm going to restrict to this slice and modify it by this map so let's call this map alpha and I'm not going to say anything about the construction of this map but once you've constructed this map you can sort of define A omega x as the homotopy limit of the fiber product of the following so here you have A omega x here you prime here's A omega x prime tends to W this is sort of the quotient in the special fiber and then here's the Diron-Witt complex and I'm out of time so I guess step three is that you check that this works ah what? it's 5 minutes ah 5 minutes is it an almost iso this map which map? the natural map some map is almost iso this one I'm not sure it's related to what we were discussing earlier so if I guess, no I don't know I would like it to be and I would simplify some of the constructions but I don't know if it's an almost isomorphism is that right? wait sorry which one? this one? ah no no alpha it doesn't make sense to talk about almost isomorphisms because it's taking place over W yeah sorry there's a version of this over W of O C where it doesn't make sense I got confused yeah no so maybe I should so we're really doing almost mathematics with respect to this closed immersion which is whose ideal as I said earlier it's cut out by p underline and all of its p power roots and then you also have to worry about piatic completions and then what I'm saying is that once you want to specify something over here you first specify something in the almost world which is a omega x prime and then specify what happens when you restrict to the special fiber and the way you specify what happens when you restrict to the special fiber is that you're allowed to arbitrarily modify it in this way okay and I guess I can write step 3 over here which is kind of stupid but it's where all the hard work happens so check all the specializations and so maybe I'll just say which ones use what so to use that the etal specialization is correct to use Scholes's theorem which is some version of Falking's theorem comparing X plus comology with etal comology and then the other two you check by hand so crystalline is more or less by construction so the crystalline specialization is what happens when you restrict to W and the way I'm modifying it I'm forcing it to look like the derambe complex so this is by construction and then the derambe one you'd really have to check sorry? so it follows from the fact that the specialization along the kernel of C gives you the derambe complex which is a perfect complex and so I guess I should have said this but you also had to theoretically complete everywhere so everything is complete along kernel of theta and then when you specialize along theta you get a perfect complex and so it's a perfect complex okay so that's all I wanted to say thank you there are some comparisons between the etal and etal comology from this picture sorry can you derive the comparison theorems from the crystalline and etal comology from your picture in this picture but in the same way that it's sort of I don't think the proof you get would be any different from the one that you get by faulting this machinery I have two questions first one, classical story in Bertrand August the generation on the special of Hartree-Gerrand E1 and the special fiber that's an important quote together with torsion's finesse of etal comology to get a nice representation and it doesn't appear in this sort of in Bertrand you discussed torsion's finesse in Gerrand but they didn't discuss the generation so is it important or are you talking about not the personal so there is a new general meta-august object you see which is really major still but then you get that the for example the Hartree-Gerrand with the special fiber abstract the Hartree-Gerrand the filtration under these conditions not the generation that you want plus torsion's finesse so I wonder if there is some analog I don't know the story so I can't really comment on it but what I want to say is that in chapter 8 you mentioned I'm getting old the only thing I want to mention is that if the crystalline comology is torsion free then each comology group is in fact finite free so you really have a lot of things you can do with it so anyway my second question after this question maybe related to the previous one so is there any relations between this a omega dot x and the complexes constructed by Sasha a natural a natural derangue a natural craze you give the I don't think so I think the complexes constructed in the H-localization picture are what you'd get if you did not do this modification at the end so like if you do our gamma of a script A derangue then you would have this Sasha's then here there's no such a thing I mean some of the comparison theorem is hidden formula similar so there should be some relation between two objects so I guess maybe one thing I can say which is related but not an answer to your question is that you can ask if there's a way to construct this comology theory using the H-localization picture and I currently do not know the answer to this although I suspect the answer is yes if you use topological H-cell homology but this construction won't work because it's a local construction okay so yeah why do you get an almost isomorphism between a omega x and a omega dish x you set it oh is this because the way I'm modifying it I'm modifying it by something that's almost zero so this side is almost zero so when you pass back to the almost world you just get a nice like any module over w is declared to be almost zero okay yeah probably probably I don't know but I mean I would strongly suspect yes okay so I have two questions maybe I will ask only one and then the other one right now so yes can you recall also the integral that we had you see when so look at the h i with i is less than minus one it should be so what you do get is that so this map this Frobenius map specializes to the comparison isomorphism and it's an it's invertible up to this element c so if your numbers if the dimension is small enough then you get some mileage out of it like so for example faultings has these theorems that say the integrally the comparison maps you constructs are always invertible up to beta to the d and that's this fact that this map is invertible up to c to the d up to c to the d where d is the dimension of x and in particular for more of the groups can you get c to the i or something like this that should be a formal question about this gauge construction and I can't think right now but I would guess yes