 Thank you for the introduction. I would like to thank, I think it works. It works. It's already done. Yes. Thank the organizers for this invitation to speak in this conference in honor of Fontaine and the memory of Fontaine and Jean-Pierre Vintemperger. It's quite emotional for me, you can imagine. I will revisit joint work I had with Deline, which is now over 30 years old. In fact, this work, to some extent, was complementing results obtained by Fontaine Messing and Cato just a couple of years earlier. So let me just recall what it was about. So we start with the field K, which is perfect of characteristic key, as usually denoted by W between K and WN. Now if you have a scheme X over spec K, though it is not really necessary, but it's convenient to look at its pullback by the Frobenius automorphism sigma of spec K, which I denote by X prime, so that the absolute Frobenius of X factors with relative Frobenius F from X to X prime. And if you look at the complex of X over K, then this direct image by F becomes, in fact, complex of O X prime modules. And I recall that if X over K is smooth, then its converging sheaves are described by F as a morphism, omega i X prime to H i of X. Now one of the main results, X is a smooth scheme over K and dimension is smaller than P. Then with any lifting of X or equivalently X prime to W2 is associated with decomposition in the derived category of X prime, sum for omega i in Z of omega i X prime minus i to F to R star omega X. I will drop the over K, so in D of X prime over X prime, inducing the Cartesian morphism on the child. So this complex turns out to be the direct sum of its converging sheaves, sheaves suitably shifted. In fact, this theorem one was the consequence of the following prime, one prime, where you start with X over K, smooth, but you don't put any assumption on the dimension. And the statement is that to any lifting of X to W2 is associated a decomposition, but a truncated decomposition, so O X prime plus omega 1 X prime minus 1 to the truncation here, the truncation, the canonical truncation where you cut in degree greater than 1. I will just write D of X prime for D of X prime, inducing Carti of course. Of course, this is a simple matter to deduce one from one prime by multiplicative properties of both sides in here. And my goal today is to show some new thing about this, which is theorem 2. So let X over K smooth, so no assumption on dimension, no lifting, then there exists a canonical isomorphism. In D of X prime, between you take the truncation in degree at least minus 1 of the cotangent complex of X prime over W2, you shift it by minus 1 and again you take tau at most 1 of the Durham complex. This is strange because there is no lifting and not so obvious how cotangent complex and Durham complex can be related. But let me show you why theorem 2 implies theorem 1 prime. So first you look at K, we have W2 and W. Then looking at the transitivity triangle for cotangent complexes, you find that LK over W2 is K over 2 plus K over 1. Now suppose you have a lifting of X over W2, say X tilde, so this is smooth, the square is Cartesian. Then L, the cotangent complex LX over X tilde is the pullback of the cotangent complex of L over W2. So it will be OX of 2 plus OX of 1. And now if you look, because this is smooth, the square is a torriny pendant, and the cotangent complex of X over W2 is the sum of the cotangent complex of X over X tilde and cotangent complex of X over K, which is that omega 1 X over K since X is smooth. So you take X prime here, you apply tau at most minus 1, so this part disappears until you get OX prime over 1 plus omega 1 X prime over K, and then you shift by minus 1. So you get OX prime plus omega 1 X prime over K of minus 1, and you have by 0 and 2, you have tau at most 1 over star. So you get 0 and 1 prime. So this is quite the simple observation. And in fact, of course, as you know, the decomposition theorem I recall here has many applications to when X is proper to the generation of a hard-to-derum spectral sequence, both in characteristic P and characteristic 0, and also to codera type vanishing theorems, both in characteristic P and characteristic 0. But concerning this improvement somehow in theorem 2, there are several remarks I should make. First of all, there was another result in my paper with Deli, which I will recall. This is a special case of this result. So you can look at the local liftings of X to W2. So they exist because they exist smooth over K, and any two local liftings are isomorphic locally. So they form a gerb, so lift of X or X prime of W2. And similarly, this truncated-derum complex here, which has H0, which is O, X prime, and each one, which is omega 1 X prime, admits decomposition in derived category, at least locally, and any two static compositions are also locally isomorphic. So then you get also a gerb of the splitting of 2 at most 1 of star omega X. And in fact, there was a theorem 1 double prime in Di that there is an equivalence like this. But this is also a simple consequence of theorem 2 because this gerb of the local liftings of X prime is in fact naturally identified to the gerb of splitting of 2 minus 1 of L X prime W2 by associating to X prime, the decomposition X prime W2 is a truncation here omega 1 X prime plus X prime of 1. And so you have an isomorphism here, so an isomorphism between the gerb of splitting. The other thing I would like to mention is that there have been several variants of this in generalizations. So first of all, there is a variant of this result of some base S of characteristic P, where S is over spec Fp and flatly lifted to spec W2. So in this case, if you start with an X which is smooth with an X, then you have to be careful to distinguish between X and X prime. And the gerb of liftings of X prime to W2 is 2 S tilde is X prime to S tilde is in fact tau at most over F star omega X over S. So this is smooth of course. In fact, that can be proved along the same line as theorem 2. Though it's a bit different but not so difficult and in fact, more or less the lifting here corresponds to a truncation of the cotangent complex and this isomorphism in theorem 2 can be generalized here tau X prime over S tilde is isomorphic shifted by minus 1 isomorphic to this. And two statements are somehow equivalent. So generalizing to a more general base S is not much. And now the other which is more interesting is that bat gave a prismatic variant. So I don't know much about the prismatic homology but roughly it goes like this. You have X over S and S is in T and then you T is a prism. So it's a spec R and inside you have the ideal I in R which is periodically so you assume that R is periodically complete and I complete in the derived sense and then you assume that some condition about the Frobenius. So also you have some delta structure so you get a lifting phi of the Frobenius and then S is just a variety of I and you have something formally smooth. Then prismatic homology looks like a prismatic homology so you have a map U from X prism over T. These two X Zariski and I think the statement is that you take L X over T which will turn out to be in this previous case the same as the this truncation of L X prime over W2 if we explain that in a moment then to then this affected by minus 1. This is more fixed to our new low star O X over T to take the truncation that was wrong. All these and here you tensor over R with I mod I square. So now if you take T to be T to be a spec W and I to be P then by austere comparison theorem or crystalline comparison theorem you will get again theorem 2. So in fact I got this in a mail from Batten and now I see that it proof has appeared in the paper of untruth and lobra. So this is one of the connection with the current topics. As a generalization you have log variant. You find again some results of improvements and results of Cato and some more recent results by I think Shung and Shung 2. Anyway the most interesting for me is the LCI generalization. But if I have time at the end I will say some words about it. But you replace smooth by locally complete your section. So let me explain the proof of theorem 2 because this is so simple somehow. So a preliminary remarks. I looked at the cotangent complex of X over W2 but you see you have X over spec W. Here you have spec W2. And here you have spec W. So actually X is smoother. So this is a locally complete intersection. So L, L, X over W is a perfect amplitude in minus 1, 0. And in fact it goes on to at least minus 1 of L, X over W2. And if you look at the corresponding triangle here for H minus 1 is 0. Then you will find that this is an isomorphism. So it's much nicer to work with L, X over W which is in fact L, X over T which appeared here. So this is the first thing. And second how do you calculate concretely the cotangent complex of X over W? Then suppose you have a closed embedding of X into a smooth scheme over W. So this is a minor restriction of X. Then there is a very simple definition, a simple calculation of this cotangent complex. So suppose you have the X or also spec K. I think the other chop was better. Yeah, you have spec W. And here you have, so this is an embedding. So ideal J, embedding, not a lifting. Then the cotangent complex of L over W of X over W is just a complex concentrated in degree minus 1 and 0 which is J mod J square goes by D to O X tensor omega 1 Y over W. So this is in degree 0, in degree minus 1. So in fact you have this. And actually this is this way that the cotangent complex was discovered by Groten Dijk I think in probably 1964 or 1965. You know he was in the GA4. He introduced the so-called module of imperfection. And this is important in separable extensions and conditions and excellencies of the local algebra, the deep local algebra in the GA4. And you observe that this module of imperfection was somehow paired with omega 1 and it was much better to look at complex with homology omega 1 and imperfection modules than with modules of homologies themselves and that led to construction of this truncated cotangent complex. So anyway, now I will give the proof of the theorem 2. So first special case, a special case, I will assume that not only I have an embedding of X into Y is smooth W but also I will assume that I have some endomorphism F over Y which lifts the Frobenius of Y tensor K. So now how can you compare this cotangent complex and the truncation here? Well, curiously it will come through the envid complex of X over K. So this is clear because there is no, this is a strange object and there is no connection, obvious connection with both sides. But still. So first of all, you have this Y with endomorphism F lifting the Frobenius. So then by, you can assume if you like that Y is only smooth over spec W so periodically complete. So then you have the Cartier section of the Y to Y. Note that this is not the characteristic P but still the vectors exist and there is a local section, there is a unique section which is compatible with the lifting F here and the Frobenius F here. So you take this section, morphism of ring and then from it you deduce of course a map from omega Y, Durham complex to the Durham complex of WX. Can you go to WOX or WOY? Oh sorry, to WOY. Thank you, thank you. Yes, of course. Thank you. Oh X, oh X, yes. Oh Y, yes, yes, yes, yes. Thank you, thank you, yes. So you have the SF like this, so you compose with WOY and so you have this and so you look at this composition, so omega Y goes to here and then this of course goes to there is canonical map into the Durham complex of X. So you have this way you find canonical map C. So now if you look again at this map from OY to WOX, so Y to WOX, so it goes to WOY and like this and here you can look at the projection onto OX and here you have the Y and here this is identity, here is the projection, so ideally here is J and ideally here is WOX. So then this map C, C0 if you like, sends in fact J into VWX. Okay, so now you want to analyze more closely what happens to the Durham complex. So then you will consider the following sub complexes of omega Y. So this will be jR goes to jR minus 1 omega 1 Y goes to etc goes to omega R Y goes to omega R plus 1 Y etc. So you have a filtration here with the omega Y is G0 contains G1 etc. Now, what is the first associated, the associated graded in degree 1? So, well, the G1 for G of the omega Y, what is it? So you take degree 0j mod j square and then what do you find? You find omega Y modulo j omega Y that is OX, that's omega 1 Y, so degree 0 and 1 and after that you find 0. So this is the potential complex of LX or degree shifted by minus 1. Now we have seen that C sends, you have here OY goes to WOX, so J goes to VWX. Now, if you look at X, VY, you will find the P, VXY. So more generally, jR will go to P to the R minus 1 VWO and so it's natural to introduce the following filtration of the Durham bit complex which was first considered by Nygard in 1981. So N0 W omega contains N1, W omega etc. and R W omega where N0 is just W omega and 4R is just 1. Then an R W omega is, well you start with the P or minus 1 VWOX, then you go to P or minus 2 or minus 2 VW omega 1 etc. and you have the VW omega or minus 1, W omega R and W omega R plus 1 etc. So W omega is a filtration by, so this is V, does V apply to W omega? Yes, yes. It's not the V plus D? No, no, no. Okay, so then we see that C maps jR, omega, Y to NR, W omega. Now we have seen the gr1j, omega, calculated. So now we want to understand the gr1 here or more generally grN, W omega. So this is the result of Nygard which calculates this. So let me recall it. You consider the map from NR, W omega to tau at most R of F1 star omega X which is defined as follows. U for I, so actually it will be a map from, I should keep track of the sigma linearity, I should put an X prime here. So for I is smaller than R, so you send P R minus 1 minus I VX to X. For I equals R, you send X to FX. And for I greater than R, I mean the class in the height inside, you send X to 0. So actually it factors through a map from grN of W omega to tau at most R star omega X. And the theorem of Nygard is that star is a quasi-asomorphism. So now we are in good shape to prove our theorem. First of all, I should say that there is simplification to this proof now in BLM by Lurie Matthew. But still this is basically the same idea. So now what do we have? Well, we have this L X prime over W. So it goes by some isomorphism derived category to gr1j of omega y. So now this goes to gr1 Nygard of W omega X. And then this goes to tau at most 1 by quasi-asomorphism here, star omega X. Now the construction here involves the choices, the y and the f. But the composition here does not depend on choices of y. Because if you have two choices, y1, f1 and y2, f2, then you can embed into the product y1 cross y2 with lifting f1 cross yf2. And then you have the compatibility between the composition. They will be the same. So now we want to show that the composition is an isomorphism. Then this becomes a local question. So then you can assume local questions. So you can assume that y being not only lifting, not just an embedding, but in fact lifting. So y lifts x. And then you have to unravel what is this map here. And in fact the map which appears here, the composition, will turn out to be, then you have oy, ox, sorry. The infiltration becomes, you have p, so you get ox, and you have zero here, and then x prime. And then here you get omega 1 x prime. So this is this part here. And the map goes to, well, the truncation that is ox to the cycles here. And what is the map? Well, in degree zero, this is the Frobenius. And in degree one, this is the divided Frobenius induced by dividing the operations on omega 1y defined by f. f star is divisible by p, divided by p. You get the lifting of Frobenius. You get the lifting of Cartier. And so this Cartier induces Cartier. So that disposes of the case where you have an embedding. And if you do not have an embedding, then still you can find an embedding system. And you work upstairs on the suitable co-skeleton. And then standard techniques of comgecal descent will give the result. I would like, because I have still some time, to explain some new viewpoints about this. Stemming from Dirac-deram complexes. So we are going a little more in the Dirac geometry business. So inputs from an omega. So I have to recall what is the Dirac-deram complex. So suppose r is an a-algebra. Then you can consider it's a standard free resolution, a simple resolution over a. So p of r, standard free resolution. So p0 of r, so it's an enormous resolution. I mean the components are enormous. p0 of r is a of the underlying set of r. So p1 of r is a of a of r, et cetera. And the cotangent complex of r over a is defined by applying omega1 to this and tensoring with r. So the Dirac-deram complex, l omega r over l omega r over a, is defined as, so I denote by s, what is sometimes denoted by tot, the simple associated complex, to l to omega p0 r over r. So this is a simplicial in this degree. And then there is a Dirac differential in the vertical direction. So you get here, you get essentially a complex which is concentrated in the second quadrant. And you take a simple associated complex, defined say by sum, because a priori there might be infinity number. So the product gives a different definition. Different definition, I take the sum here. Because in fact I will truncate so then. So anyway, this comes in fact equipped with the filtration. Here you take a sum associated complex to omega at least i over p of r over r. So this is called a field i, hodge. We sometimes drop the hodge over l omega r over a. And this is completely funtorial. This globalizes some schemes and then you get, for any x over s, you get the Dirac Dirac complex. So globalizes on schemes. And so you get l omega x over s with filtration fill i, fill i, which is essentially l omega at least i. And the associated grader is i for hodge filtration, omega. Well of course it's omega i, but placed in positive degree i. So it is omega i of this simplicial resolution. So essentially the cotangent complex of which you take the exterior power, derive the exterior power. So this is l omega i x over k minus i, where l omega i x over k is lambda i of l x over k. So you have that. So now the COM3 is the following. Suppose x over k is smooth as before. Then there exists. So I hid this expression, there exists a canonical but they don't have any. So in fact we will define, we will construct, etc. A canonical isomorphism. Well, in what? Well, I will write in the Df of x and w's, the filtered Dirac category of w modules on x. But in fact there is a refinement of this Dirac category when you take complexes of differential operators of order at most one. Where the filtration is a grader, the associated grader is linear. So this category is introduced by the link and used by Dubois. And this is sort of a category we have here. The grader will be all linear. So the isomorphism is like this. You take an omega x defined by field p for the hot filtration. And here you have w omega x divided by np. So the nagar filtration. So let me, so in Df, so it means that the filtration is here. It's compared to the filtration. Here you have field i mod field p, which corresponds by an isomorphism to n i mod np. So a few corollaries of this. I will say maybe later a few words on the proof. But a few corollaries. By the way, if you take the product complex and mod it by field. Sorry, I'm sorry. No, no, no, no. W mod field p. Yes. Sorry. This is very important here to take w. Yeah. Sorry. So when you mod by field p, it doesn't matter if you take the sum of the product in the total complex. Yeah. Yeah. Okay. So there's only a finite number there. Yeah. Yeah. Because then you get this. In any case, I chose the sum, right? But I think this is in this case. Yes. Because then the omega i plus 1 is there. I remember there was some question. Okay. So here it's, of course, i omega x of w mod field p. Of course, when x is, when r is smooth as there is, i omega is just a Durham complex. And if I had put k instead of w here, I would just have found the Durham complex, which is not what I want. The corner one is that for i smaller than p, then the omega i x, the l omega i x of w shifted by minus i, which is isomorphically to the grow i for the Nagar filtration. That is by Nagar theorem to at most i of f star. And here I should put x prime here. So this is the statement on the grow i. Now, if you take, in particular, if you take i equal to 1, you get l omega 1. This is l x prime over w minus 1 to at most 1, f star omega. So you recover the theorem 2. Now, corollary 2 is that, suppose x liftable to w2. Then you get a decomposition, not of the truncated at most 1, but the full truncated up to p. Then get isomorphism in d of x prime or x prime. The sum for i smaller than p of omega i x prime minus i to smaller than p minus 1, that is, to smaller than p of f star omega. So this is the result in deline. So the reason is that then the liftability, if you take x tilde, then you get that l x tilde of the lx of the w will be a sum of o x 1 plus omega 1 x. You have a decomposition like that. And then when you take the exterior powers, which produces the minus i here, minus 1. So then you have somehow a refinement of theorem 2. Again, the proof is a bit similar to the proof of theorem 2. I don't think I have time to give it here. But the main point is that you first work on the affine case. And then you look at this hot filtration, fill i. And it is somehow similar, and in fact turns out to be quasi-asomorphic to a filtration like the JR filtration on the drum complex of an embedding. So the embedding will be here, played by you have your r, let's say your k. And you have here the w, and then you take the p or the w of r here. And so this is somehow the embedding. So I will skip the details here. So this is details for what? What? What you are sketching here is about. So the proof of theorem 3. So you take r smooth as your k, and then you take the free polynomial algebra over w, and then you look at the, so you consider the drum complex and then several filtration. OK. So I won't give the details here. There is also some prismatic variant of that, which is due to bat and for which, I'm afraid there is no reference of the moment, so prismatic variant. So you have this same situation, spec, let's say s, we have t. And then the statement is that you take a omega x or there are t. So think of this as w and this as k. So this is my prism. I think I forgot in the previous statement, I forgot some condition on s. The prism has to be bounded, bounded, which means that here I have an arm of i. So arm of i infinity is arm of i. p to d n force n log. So you have to. So then this field p was to what? So you had our new lower star of o x or the t, the prismatic c in the upper star of that. And here, so phi is this, and the morphism produced by the delta structure on t. And here you have some Nygert filtration, Np, which is a little bit technical to define. But in the case of t is spec w, you have the comparison between this and crystalline comology. In fact, this turns out to be crystalline comology, the lower star x over w. Here, this phi upper star is important. So prismatic comology descends crystalline comology. So then you will find here mod Np, and this is w omega. So you have this sort of thing. But in the remaining minutes, I would like to mention some things about LCI variance, because it has some connection with some paper by B, B, L, S, Z, Bat-Blickler, Lube-Blake-Snick, Sing and Tongue, with vanishing theorems, Alacodera, in singular cases, those are complex numbers, in characteristic zero. So LCI variance, in fact, we have the w omega, but, of course, we like to derive this. We have the w omega. So then you have the C1, C4, that you have x over k. A new assumption, x is in a scheme. Then the omega x over w, modulo phi p, we turn out to be Lw omega x divided by omega Np. And this will give some degeneration and partial degeneration plus vanishing, hopefully, not yet... I'm not deriving the moment, the degeneration, obtained by Bat, in characteristic zero by the usual spreading out argument. So the... Yes? In theorem four, you have the same conclusion as theorem four. Sorry, theorem... But this is derived, yes? X is LCI or smooth? No, nothing. So this is very weak here. So no assumption. No, no. In fact, it works. It's a trivial thing, actually. But at the end, the result for the LCI case are like this. So of course, if x is lifted, then you get some decomposition. If here on the x lifted, plus LCI, you get decomposition of the conjugate filtration in the degree P-1, conjugate P-1 of L omega x and the sum of L omega x prime over W i to the minus i. And then this, in terms of degeneration, so one result is that x, now k, now a characteristic of k is zero. So you consider the complete theorem complex. So the r-limb of L omega divided by fill, fill r. Then for x over k LCI, then n proper, dimension d. And you assume that the dimension d, and you assume that the singular locus of x is s. Then for n, I will write it here. Or maybe I'm over time, yes. Over time, yes, I should. I started at 35, yes. So then the theory is that the dimension for n, smaller than d minus s minus 1, then dimension of hn of x and this complication thing. So this is hn of x and c if xn and c, if k is c. So this is the sum from 0 to d of hn. So this is sort of a hodge to the random dimension. Partial one, x L omega j, x of minus j. And the rest is 0 for j greater than d. This turned out to be 0. So here's the sum of the dimension. And we expect some codera vanishing when you put some line bundle here. Anyway, so thank you very much. What's the status in the logarithmic setup if you have some simple normal crossing divisor at infinity? Can you represent? Yes, so of course you have the... So then it is 2 minus 1 of L. So you take the Gabber-Rolson codation complex, x prime over k, this should be minus 1. There is a morphism to almost 1 of f plus omega x over k. But everything taken in the log sense, and also here log sense, for x, x over k, log smooth, and kate type. So in particular, x prime will lift over w2, the Taichmeler thing, the Taichmeler lifting, if and only if the ham complex is decomposed in degree at most 1. So this was one of the result of some motion in shunt 2. Can you write the name? Yes. You mentioned the people who did... I didn't catch the names of the... So yes. So there's one firstly made, so there is sheng, sheng ma, and sheng 2. Yes. Sheng ma, sheng 2, yes. Maybe my accent is not good. Yes, yes, yes, yes, yes.