 So today I will talk about the p-edge correspondence introduced by Foutings and I gave one a slight variant of his theory. So first I'd like to explain what Foutings did. And to explain it I first fixed several notations and so V denotes not a representation but this is a complete DVR of mixed characteristic 0p and I assume that the residual field is algebraically closed. And K denotes its field of fractions and as usual K bar is its algebraic closure and V bar denotes its ring of integers. And also C is the completion of K bar so it's this field and I also use ring A in V which already appeared in several lectures. And so this is just defined by the V-ring of the projective limit of the reduction mode P of V bar and the projective limit is taken with respect to Frobenius. And this is the ring used to construct the periodic period rings and as it is well known we have a natural projective ring homomorphism and its kernel is generated by one element usually denoted by Guzai. And Guzai is in fact a non-zero divisor in this ring and finally I define A sub N V bar to be the reduction of this module of N's power of Guzai. So N is a positive integer. So this is a short projective homomorphism so if N is equal to 1 then this is isomorphic to this completion. So this is a notation and so I first explained the periodic Simpson correspondence by outings and so we consider a proper smooth scheme over a field K and we want to compare the two objects. One is a finite dimensional C representation of the fundamental group. So C representation of fundamental group and we want to compare this with X a bundle on X K bar. And so we want to compare these two objects. I didn't recall the theory over C but over C on the left hand side we consider the finite dimension of C, C is a complex number in that case of pi 1. Here we consider a Higgs bundle satisfying certain conditions or semi-stability and so on. And so it is natural to ask the periodic analog of the correspondence. And so to study this correspondence faultings use the category of modules on the so-called faulting site and it's something like this. And so this is a faulting site. And so I don't want to give a precise definition of this site but the sheaf looks like this. It can be given by the following data. Sheaf on this site is the following data. So for each etal scheme we are given some sheaf on the finite etal site of the geometric generic fiber. And also for morphism between the two etal schemes over X, so say G, then we have a map from the pullback of FU to the FU prime. And plus some growing condition with respect to etal covering. What did you write in the substitute? Which? Mod chi faulting site and then for X mod P. Okay, so actually I'm thinking of a projective system of the module of reduction mod P, power of P. So maybe, I don't know how to write. And the rationality, because we work with C here. And this is a category. And so first he proves that the theorem of faultings is that we can easily construct a natural functor from here to here and the faultings shows that this is fully faithful. And then he compared some vector bundles on here with Higgs bundle. So next I will explain this part. So I forgot to define what is the structure sheaf. Okay, so what is OX? To define OX using that description it is enough to define OXU for each etal scheme U. And to define this sheaf on finite etal sites it is enough to define its section over finite etal scheme over the geometric generic fiber. And so this is just given by the section of this scheme. And B norm is the integral closure of U in V. So this V is the initial concept. Ah, okay. How to U? I don't know how to write. Okay, why? Why is it used? Thank you. Okay. And for that part actually faultings established local theory for this part. Okay, so yeah, so maybe let me see. So we consider the affine etal scheme over here and assumes that coordinate consisting of invertible functions. So this is sometimes called small affine. And then we consider its field of fraction. Ah, so I have to also assume that R is, so this is connected here. And okay, so yes. And we consider the algebraic closure. And then, so in order to study the sheaf on partings site, we consider the stock of R bar is the stock of OX at this geometric generic point. Yeah. So this is, in fact, the integral closure of R and the maximal unrhymified extension of the generic fiber in K bar. Okay. But this is not finite at the top. Not finite. So it's union of finite. Ah, no, no, no, no. So the union of finite, yeah, on the generic fiber, invert P, then this is a union of finite scheme over the generic fiber. Yeah. And then we take the integral closure of R. So this is a usual ring we used in the periodic approach theory. Okay. And so delta U is the Galois group of R bar of R. And then, so. X is only over K the fraction of V, or it's more than that. Oh, what? I made a mistake actually. So this is, I have to V bar. It must not be doing that. Just the generic. What is the generic? Ah, I'm sorry. So that's a mistake. So I have to start with from X over V. Yeah. X over V, yeah. Smooth over V? Yeah, smooth over V, yeah. So good reduction. And also we fix a model. Yes. Okay. So then, so Farthings proves that finite free module with continuous semi-linear delta action such that some smallness condition in the trivial module P to the alpha for some 2 over P minus 1. So this condition is usually called, he calls this condition small. And then he proves that there is an equivalence with also finite free V bar hat, so maybe this V bar hat module M with theta. Theta is something like this. Let me see. R V bar, omega 1 R V bar over V bar. And xi inverse. I will explain the meaning of xi inverse later. So this is just, so for Higgs field, we consider just a linear map. And if you take the which product, so then this becomes such that which product is 0. So this is a definition of Higgs bundle, the integral Higgs bundle in this case we consider like this. And xi is the same, xi appear there. And also we assume the divisibility, theta is divisible by P to the alpha for alpha for some alpha bigger than 1 over P minus 1. Yeah, this is a local. No, no, so just say, yeah. So, yeah, so actually the middle category, in the middle we can consider a kind of vector about those, but the category is much bigger than the representation of the fundamental group of pi 1x. So it's become a bigger category. And then he proved it under some smallness. Is it an exact equivalent, because I think I saw in some talk that you lose a little bit in the alpha, that is. If you go, but you claim it's an exact equivalent. So you don't have to go to alpha. For some alpha? Yeah, for some alpha. We don't, I don't fix alpha. Okay, so, and then, so he also established a global theory and also QP theory, yeah, also QP theory. And also some comparison of cohomology for QP theory, QP and global theory. So, yeah, so for comparison we also have a local version. So, yeah, so, yeah. However, so the, so in this construction, so we need to choose a lifting, global lifting, a smooth lifting of x2 over x2, a smooth lifting of x1, x1 is the base change of this to the formal scheme of V bar, the very computational V bar. And so this is the same as A1 V bar. And we need to choose a smooth lifting in his theory. And actually, so, so based on growth, give another approach to this theory, but still they need to choose a smooth lifting. And so, so in this talk, so today I just give a formulation without x2, so without this lifting, yeah. And, but so to do that I work with a kind of crystals instead of x modules, yeah, okay. And so, yeah, I forgot to give a definition of Guzai inverse. So this is just a tens ring, this inverse to something, yeah. And so this is a free V bar hat module of rank 1. So just normalization. So this is, yeah, this is free. But there is no canonical choice of basis, so we have to write like this. Okay, so now, so I will explain first the definition of Higgs crystals or Higgs ice crystals. And then explain how this theory is interpreted in terms of Higgs crystals. Okay, okay, so to do that first I will explain an analog. So crystals is of course defined using a certain PD thickening and speedy envelopes plays an important role. And so I first explain what is an analog of PD envelopes for Higgs field. And so instead of PD thickening we consider the following, no, no, no, I'm sorry, no, no. So first we consider the following category. I just write by C. And so the object is that a sequence of periodic form of schemes over A and V bar. So a sequence of, excuse me, immersions is like this. And satisfying the conditions that, so for N bigger than 2 are closed immersion and its reduction mode B is just new potent. And so then we consider the full subcategory, which plays a role of PD thickening in our theory. But we have a category on various levels. So R is a positive integer or infinity. And so this is a full subcategory consisting of YN. So that's the following, satisfying the following four conditions. The first is that this is closed immersion. And also flatness over ZB. So this is P torsion free. And the third condition is in some sense, I'm not sure whether it's really the same, but in some sense I want to also define flatness over AN of V bar. And so what I define it, I'm not sure, but it's the same as flatness. But anyway, so this condition is necessary. So if this factorization like this, that is multiplication by XI gives a map from YN to YN, and it factors through the YN minus 1 and also factors through the injective homomorphism. Okay. And the last condition, not exactly the same as flatness. Flatness after inverting P or something, I guess. I'm not sure. Anyway, so I assume this condition. And the last one, so here I don't use R. And in the last condition R appears. So, okay. I have no place to write. So we define the filtration is just by kernel of YN to Y small n if n is smaller than n and 0 if n is bigger than n. So this is a decreasing filtration defined by this sequence. And then we have that this filtration is generated by all n to the power of XIN. So this is if R is infinity. And if R is finite, so then the condition is like this. So P times OYN is containing XIN OYN. For n lies between 0 and R. So if R is finite, so this is the condition. So that's all. Okay. The reason why we consider C? Okay. I will give an example later. Okay, so then by condition you can easily check that C infinity is the smallest one. And it becomes smaller and smaller if R becomes bigger and bigger. And so all are C. And all are containing C. Okay. So then the first proposition is the analog of divided power envelope. And so the claim is inclusion from CR to C has a right adjoint. Right, right, right. Yeah. So I wrote by Higgs. Okay. So this is the first thing. First proposition. So this is not very difficult to check. And so I just give an example of which we use in the theory of Higgs crystals. So suppose that we are given an affine object in this category. C for simplicity I only explained in C infinity. Okay. So this is just the compatible system of algebra over A and V bar and satisfying that condition. Yes. Yeah. So the affine object. And then we want to consider another object of C. So we consider the object of C, which is Y1 prime. And Y1 prime is the same as Y1. And I consider the Yn is just a convergent power series, say T1 to TD. So then you can take a joint. So for the infinite level. And then what we obtain is something like this. TD. So this is just in some sense just taking a blowing up. Blowing up of Y1 prime inside Yn prime for N is bigger than 2. And partial blowing up, yes. And make it the ideal defining Y1 prime in Yn prime invertible. Yeah. No, no, no. The ideal defining Y1 prime in Yn prime. We want to make the ideal is generated by Guzai. And so to do that we have to blow up and then you get this kind of things. Yes. OK. And so this Guzai in the denominator corresponds to this one in fact later. OK. OK, now we are ready to define the site to define Higgs crystals. So R is the same as before. And so the object is just the object of CR plus a morphism of T1 to XR. R, so X1. So X1, X1 is I defined somewhere. I forgot. So I recall. So this is the skin. Yeah. Formal skin. And so this is just a pair. And morphism is, this is obvious, morphism in C compatible with Z. And topology is we can choose either the risky or etal. But here I choose etal. Yes. So X1, it's a steel speckle. So X1 is a risky. Do you see it as a formal skin? No, no. It's a shape of a skin. Are you good? No, I'm good. Yeah, they're the same as taking a formal compression along the special fiber. Yes. So this is defined over speck V. I'm sorry. Yes. And so then, OK. And then we define a structure shift. A structure shift is just, let me see, X1 over AV bar plus TZ is just a global section of OT1. So I just only look at the structure shift of the first component of T. And we ignore the other component. But still, this has a good property. I don't know. The module of this gives interpretation of Px isocrystals. I will explain now. OK. So anyway, so this is a shift. Of course, you can define a shift of using TN, but I didn't discuss here. I only work with T1. Excuse me. So Z is any morphism? So morphism over V bar. Yeah. Yes. So it's something like we have. OK. So just morphism over this one. OK. So similarly, as a crystalline site, the module can be defined by a compatible system of shift on each T. That is, so the module of this structure shift is equivalent to give, for each T, an OT1 module of FT on the third site of the T1. And also for each morphism between this object, just give a map from FT to FT prime on here and plus some cycle condition. So this is the same as the usual crystals. And the point is that here we only look at T1, but here we have the morphism is defined over V bar. So we have many morphisms. So this data gives some structure corresponding to Higgs field. OK. And so Higgs isochrisis. So I only discussed with QP theory in this talk. So Higgs isochrisis. So this is defined Higgs ice crystals. So Higgs ice crystals on this site. If you know the definition of crystals, you can easily guess. So this is an OX1 over A V bar tensor Q module. Q modules F on this site. Yeah, we consider this such that first condition is that I assume that this is finally regenerated projective. So what's this? I don't explain precisely, but et al locale on X. So actually, so I just I give this strange condition because I don't know well about the theory of modules because we consider I didn't assume that T is necessary and so I don't know well about that module over on such very formal scheme. So I just give us a little bit strange conditions. This may not be necessary. And so the second condition is that, so this is an isomorphism for all G. So this is the definition of our Higgs ice crystals. So this is very simple. So this is, okay. So now I will explain how this Higgs ice crystals are related to Higgs bundles. Higgs bundles I explained here. Okay. So this is not a big theorem. So maybe it's better to write proposition. Yeah. So to have a comparison with Higgs bundles we have to assume the existence of a lifting. Assume that we are given a compatible system of smooth lifting of X1. Then it gives an equivalence of category. I have to choose the lifting over all n. So it's a little bit, the assumption is a bit strong. But anyway, so we have an equivalence of categories like this. So I will write the, so this category is Hcqp of r of X1 over Av bar. I just write this category like this. So this category is equivalent to the category of Higgs bundle with some condition. So I will explain what is this. Okay. So this is, so the object of this is so locally finitely generated projective OX1 tensor module e plus this Higgs field OX1 tensor Q linear. And I have two conditions. The first condition is just a definition of Higgs field. And the second is, one is a convergence. And for local, so the condition is local. So for local coordinate, if you choose a local coordinate on some affine, so this is tau over X. And define, no, no, no, no, put. Then so we can describe theta by d endomorphism. So i equal 1 to d and xi inverse d log ti. Then, so this theta i has some convergence. So the convergence is something like, so m bar is in the element of this set. And so multi-index over R times, let me see, theta m bar X. So this is a product of theta i mi. So this guy goes to 0 for OXA. So this is a condition. And so this symbol means the smallest integer not smaller than less than or, no, no, no, bigger than or equal to a. So this convergence, yes. So this is the first proposition. And to prove this little time. So, okay, I only have 15 minutes. No, right? Yeah. Actually, I have to, I'm sorry. So yeah, so this is just, so proof is, so I have to go to my, yeah, it's difficult to finish everything. No? Okay, okay, I will try to explain a little bit about the proof. So this is just use the stratification using d1. So d1 is the envelope of this one. So the first component x1 and the component bigger than 2 is just a product. And then if you take this envelope, then you have a description like this and some modification of this. And so then by using this description you get this equivalence. Yeah, through this stratification, yeah. Okay, so now, so I will explain the homology. And so here, so I didn't explain the reason why we use, why we consider various levels. But here I need to take the projective limit of this site. Yeah, the projective limit. So if our values, this gives the project system of site and so you can take the project limit of this site and I just write by dagger. And so then, so if you take the project limit, so then this gives good homology. So in the sense that this gives homology of Higgs spectra bundles if we choose a lifting of x1 over an of p bar. So the first, so that's here where it means that, so if you, as before if you have a, so smooth lifting, and f is x crystal of qp over x1. And if you take a lift, we have a corresponding, so this is proposition, by proposition 2 we have a corresponding Higgs bundles of x1 over p bar. And so then, so yeah, this is corresponding. And also I write f dagger is the pullback of f on, on this dagger site, so x1 over something. So then, so we have a canonical isomorphism between the two homologies. One is the homology of this site, f dagger, and the other is just x, let me see, x1 of the complex defined by this Higgs field. Yes. So the weight of the theta is zero, so theta weight theta is zero, so you can make, using that you can define a complex whose differential maps are linear, and so you can take a hyper-homology. And then, also we can consider this homology on this project-limited site, and then there exists a canonical isomorphism, so this gives, so if you have a lifting, so then this gives, yeah. The proof is that we, that's basically the same as a crystalline case, so we have to, we take a resolution of f on r, or some r prime bigger than r, and so then, so on each r, the resolution, no, no, linearization, we consider analog linearization, and linearization does not give resolution on each step, on each level, but if you take a limit, so then you get resolution, and the resolution is, yeah, in some sense complex of some over-convergent power series, and so that's why I write dagger in this site, for this site, yes. So this is the reason why we consider a site with various levels. Okay, so now I have to explain the periodic Simpson correspondence in terms of this, yeah. Okay, so this is actually not very difficult to define this function, so first we consider a fine object as before, and consider r bar, and also we can consider r in for r bar, yeah. So this is a bit ring of the projection limit of r bar and flow venous, and a n, r bar is its reduction, and so this rings actually give an object of this site, so that is, so this becomes a compatible system, this gives a compatible system of scheme over an v bar, and so this is an object of c, actually infinity, and also we have a map from, so this is a periodic completion to spf, so this is some sense u1, yeah, and so no, no, no, x1. So we have a natural map, and so we just obtain an object in this site, yeah, and also there is a natural action of the fundamental group of delta, so yeah, fundamental group delta u, which I introduced in the beginning, and so having this just by evaluating the crystals, you get representation of delta u. Okay, so yes, so if you are given x crystal, yeah, so then one can evaluate this on, on this object, yeah, so we evaluate on this object, I just write like this, so this is actually r bar 1 module plus semi-linear delta u action, and so then, so this is, so if you take a sufficiently small u, then this gives actually, no, so this is, I work with ice crystals, so I have to answer this one, so then, so this naturally gives so r bar hat module plus semi-linear delta u action, and this is the same type of representation appearing in the Fowl-Tink's theorem. Okay, so now by, so if you're considering all sufficiently small u, we get, so varying u, we obtain a functor, or the Higgs correspondence in one direction, so this one, two, two, where? So I write dr Higgs, actually, so there is a, we want to go to the Fowl-Tink's topos, and so here as I explained before, so this is a compatible system of module of OX module some power of P, a projective system of OX module some power of P. So to construct a functor, we have to choose lattice of F, but it is not, this is not very easy because actually if you define an isochrisis, yeah, so anyway, so yeah, so to define a lattice I have to use the level infinity isochrisis, but anyway we have certain integral version of this one, and we have to tensor q, and so the category may not coincide, so I didn't check it, so we have, actually we have a functor of this full subcategory from here, so the integral version inverting P, yeah. So this is a functor, and so now I state the two theorems, and actually so I didn't write, but I don't write, but so if you vary r, so then this category is the same as the category considered by Fowl-Tink's actually, yeah, and also the functor becomes the same as the functor considered by Fowl-Tink's, and so then the first theorem is that this functor is fully faithful. Now Fowl-Tink's also studies the essential image, but I don't discuss here, and the second theorem is the comparison of performance, and so this is the last theorem, so if you consider a crystal of finite level, and F dagger as before, that is a pullback to the dagger site, so then is this canonical isomorphism between the cohomology, so that is F dagger is r, so to have a good cohomology we have to go to dagger, take a pullback to this dagger site and then take a cohomology, so then this coincides with the r-higgs of F, so this is the Fowl-Tink site, so the cohomology for the Fowl-Tink site, so then we can compare and we have this isomorphism, and these theorems are actually proved in the Fowl-Tink's paper, so basically the results are the same, but I just re-formulated using this Higgs crystal, and so then, yeah, that's all. Any questions? The construction of the functor dr refers to a choice of lifting of x to a n? No, no, no, a replace, are you mean? Just saying to construct a functor, do we need to choose a compatible system by lifting it? No, no, no. Just evaluate, start with the vector bundles, so then it depends on the lifting, but I start with the Higgs crystal, so then it doesn't depend. So you need a lifting to compare the Higgs-sized crystals and the Higgs bundles. Any questions? Any questions? Any questions? Any questions? Any questions? Any questions? Any questions? Any questions? Ah, so, yeah, actually, yeah, so I didn't explain here, but to construct this functor, if you choose one lifting, then you can describe this functor in terms of certain period rings. And you can define period rings using this site, and also Abes and Gros also consider similar period rings, and it looks that they are canonical isomorphisms, but actually we didn't check it yet. I have some idea, but I haven't done yet. So maybe just one small question. How can you describe the smallness in this picture? Besides, because compared to funtions? Ah, yes, the smallness corresponds to the convergence, this convergence, yeah. So if you are given a Higgs vector bundle, satisfying this, then the condition that the convergence holds for some R is equivalent to the smallness of outings. Yeah, so this is exactly the same category. So there's no other question. Let's find this picture again.