The p-adic Simpson correspondence and Higgs isocrystals
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Tsuji, Takashi
The p-adic Simpson correspondence of Faltings between small Higgs bundles and small generalized representations, and another approach for it by A. Abbes and M. Gros both depend on the choice of a certain smooth lifting of the relevant variety. I will give a way to eliminate this dependence by working with a kind of (iso)crystals, which I call Higgs (iso)crystals, instead of bundles. I will also discuss cohomology of Higgs isocrystals and comparison of cohomologies.
 So today I will talk about the PIEA correspondence introduced by Falkings and I gave one a slight variant of his theory. Okay, so first I'd like to explain what Falkings 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 as LP and assume that the residue field is algebraically closed and 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 yeah 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 project limit is taken with respect to Frobenium and this is a ring used in to construct the periodic period rings and and as it is well known we have a natural subjective ring homomorphism and its kernel is generated by one element usually denoted by Guzai and Guzai is in fact nonzero divisor in this ring and and finally I define a sub N V bar to be the reduction of this module of N its power of Guzai so N is positive integer okay so so so this is a short subjective homomorphism so for if N is equal to 1 so then this is isomorphic to this completion yes so this is a notation and so I first explained the periodic Simpson correspondence by our things and so we consider a proper smooth scheme over a few K and and and we want to compare the two objects one is a finite dimensional C representation of the fundamental group so C representation of a fundamental group and and we want to compare this with X a bundle on X K bar and so so and we want to compare these two objects and of identity recall the theory over C but over C we on the left hand side we consider the the finite dimension of C represent C is a complex number in that case of a pi 1 and here we consider a Higgs bundle satisfying certain conditions or semi stability and so 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 on the so-called faulting site and it's a bit it's something like this and so this is a faulting site and and so what is so I don't want to give a precise definition of site this site but the shift is it looks like this can be given by the following data so she from this site is the following data so so for each etad scheme we are given some shift on the finite etad site of the geometric generic fiber and and also for so for morphism between the two etad scheme over X so say G then we have a map from the pullback of f u to the f u prime and and plus some a growing condition with respect to ital covering okay so I actually I I'm thinking of a projective system of the module over as a reduction mode p power p yeah so maybe I don't know how to write yeah and the rationality because we work with C here and yeah this is a category and so first he proves that the so the theorem of faultings is that we can easily construct a functor natural functor from from here to here and the faultings so that this is fully faithful yeah and then so and then he compared this cut some compared some vector bundles on here with Higgs bundle so next I will explain this part ah so I have to I'm sorry I I forgot to define the what is the structure shift the structure shift is okay so what is o x to define now o x using that that description it is enough to define o x u for each etad scheme u and to define this sheaf on finite etad sites it is enough to define its section over finite etad scheme over the geometric generic fiber and so this is just given by the section of this scheme and the B norm is integral closure of u in v okay how to you I don't know how to write okay why why I buy is used to yeah thank you okay so and for that part actually faultings established local theory local theory for this part okay so so yeah so so maybe yeah so we consider the affine etad scheme over here and assumes that coordinate consisting of invertible functions so this is sometimes called small affine and then we consider a it's field of fraction and it's 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 parting site we consider the stalk of our bar is the stalk of o x at the this generic point geometric generic point yeah so this is yeah this is in fact say integral closure of R in the and maximal unrhyme phylic extension of of the generic fiber in k bar okay this is not finite the charge not finite so it's a union of finite it's not no no no no so fire so the union of finite yeah on the generic fiber so if you invert p then this is a union of finite the 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 hodge theory okay and so delta u is a Galois group of R and then so excuse me x is over is only over k the fraction field or it's more than oh what I'm sorry I made a mistake actually so this is I have to be but I'm sorry so that's a mistake so I have to start well from x over v yeah I'm sorry yeah I made a yeah x over v yeah so yeah smooth over v yeah so good reduction yes and that was so we pick we pick some model yes okay so then so far things prove that finite free module with continuous semi-linear delta u action and such that's so the some smallness condition the trivial module p to the alpha for some yeah so 2 over p minus 1 so this condition is usually called he calls the condition small and then he proved that there is only equivalence with also finite free R V bar hat so maybe this V by hat module M with theta theta is something like this let me see R R V bar omega 1 R V bar over V bar and because I inverse I will explain the meaning of x I inverse later so this is just so for his field we consider just a linear map and and if you take the which product so then this become such that which product is 0 so this is a definition of Higgs bundle at the integral Higgs bundle in this case we consider like this and the xy is the same the xy appear there and okay 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 after so the 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 by 1x so it's become the bigger category and then he proved that yeah under some smallness is it an exact equivalent but I think I saw in some talk that you lose a little bit in the art that is but you claim it's an exact equivalent you don't have to go to alpha we don't I don't fix alpha so okay so and then so he also established a global theory and also also Qp theory yeah Qp theory and also some comparison of kohomouji 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 x to the bar v a smooth lifting of x x 1 x 1 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 aves and grove give an another approach to this theory but still they need to choose a smooth lifting and so in this talk so today I just here 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 that's 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 so 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 fix crystals or fix ice crystals and then explain how these theories interpreted in terms of fix crystals okay okay so to do that first I I will explain an analog so crystals is of course defined using a certain PD thickening and 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 so no no so first we consider the following category I just write by C and so the object is that sequence of periodic formal schemes over a n V bar so sequence of excuse me immersions excuse me it's like this and the content satisfying the conditions that so for n bigger than 2 are closed immersion and and its reduction mode P is just new potent yes and so then we consider the full subcategory which plays a role of PD thickening in our theory but we have a category in on various levels so r is a positive integer or infinity and so this is a full subcategory consisting of y n such that's the following satisfying the following four conditions the first is such as this is close immersion okay and also flatness over the P so this is P torsion free and the third condition is in some sense I'm not sure whether it's it's really the same but in some sense I did want to also define the flatness over a n of V bar yes and so but I define it I'm not sure but it's the same as flatness but anyway so this condition is necessary so exist factorization like this and that is multiplication by g's eye gives a map from why n to y n and it factors through the that's why n minus 1 and also so factor through the injective for more morphism okay and the last condition not exactly the same as flatness so 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 so and in the last condition r appears so okay I have no place to write so so we define the filtration is just by kernel of y n to y small n if n is smaller than n and 0 if n is bigger than so so this is a decreasing filtration defined by the six sequence and then we have that this filtration is generated by all because I n is power of psi n if so this is if r is infinity and in the if r is finite so then the condition is like this so p times o y n is containing xi n o y n for n is for n lies between 0 and r so if i is finite so this is a condition so that's okay the reason of why we consider c okay I will give an example later okay so so then by condition you can easily check that the c infinity is the smallest one and and it becomes it becomes smaller and smaller if r big becomes bigger and bigger and so or c and all are containing c okay so then first proposition is the analog of divided power envelope and so the claim is inclusion functor from cr to c has a right as joint right right right yeah 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 I give an example which we use in the theory of Higgs-Rissault 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 a 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 the another object of c so so we consider object of c which is y1 y prime and y1 prime is the same as y1 so and and I consider the yn is just convergent power series say t1 to td so then you can take a joint so for the for the infinite level and then what we obtain is is something like this yeah so this is just just in some sense just taking a blowing up yeah blowing up of y1 prime inside yn prime for any speaker than two yeah and partial blowing up yes and make it the ideal defining y1 prime in yn prime invertible yeah oh 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 okay and and so this Guzai in the denominator corresponds to this one in fact later okay okay now we are ready to define the site to define Higgs crystals so so so yeah and r is the same as before and so the object is just the just the object of cr plus a morphism of t1 to xr so x1 so x1 x1 is I defined somewhere I forgot so I recall so this is a this scheme yeah no format scheme and so this is just a pair and the morphism is this is obvious morphism in C compatible with z and topology is we can choose either the risky or at all but here I choose etal yes so x1 is a scheme you see it as a form of scheme no no I have the same as taking a formal compression along the special fiber yes so this is I'm sorry this is defined over spec v I'm sorry yes and so then okay and then we define a structure shift a structure shift is just a let me see x1 over a v bar plus tz is just a global section of t1 so I just look at only look at the structure shift of the first component of t and we ignore the other component but but it's still this has good property I don't know the module over this gives interpretation of px is crystals I will explain now okay so anyway so this is just a shift of course you can define a shift of using tn but I didn't discuss here only work with t1 excuse me what so more morphism over over yeah morphism over v bar yeah yes so yeah it's something like we have okay so just just a morphism over okay so then in the similarly as crystal insight the module can be defined by a compatible system of shift on each this each t that is so the module over this structure shift is equivalent to give for each t give o t1 module on of ft on on on the tail side of of the t1 and and also for 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 a n v bar so we have many morphisms so the so this data gives some structure corresponding to Higgs field okay so and so Higgs isochrisis so I only discussed with Qp theory in this talk so so Higgs isochrisis so this is defined so this is defined Higgs ice crystals so Higgs ice crystal on this site is but you can if you know the definition of crystals so you can easily guess so this is so this is this is an O x1 over a v bar tensors Q modules F on this site now we consider this such that first condition is that I assume that this is finally regenerated projective so but this I don't explain precisely but et al locally 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 the I didn't assume that these netherian 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 for all g okay so this is a definition of our Higgs ice crystals so this is very simple okay so this is okay so now I will explain how this Higgs ice crystals are related to Higgs bundles I explained here okay so no this is not such this is not a big theorem so maybe it's better to write proposition yeah I gave one yeah so so to have a comparison with Higgs bundles we have to assume the existence of lifting assume that we are given a compatible system of smooth lifting of x1 then it gives an equivalence of category so here 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 I will write the so this category so HC QP of r of x1 over AV bar I just write this category like this and then so this category is equivalent to the to the category of Higgs bundle with some condition okay so I will explain so what is this okay so this is so the object of this is a so locally finitely generated projective OX1 tensor module E plus this Higgs field OX1 tensor Q linear and I have to add two conditions the first condition is just a definition of Higgs field so and the second is one is a convergence and for for for local so the condition is local for local coordinate if you choose a local coordinate on some on some affine so this is a tau over x and define no no no put then so we we can describe theta by d endomorphism so I go 1 to d and because I inverse d log t I then so this theta I has some conversions yes satisfying some conversions so the conversions is is something like so m bar is in the element of this set and so multi index and over r times let me see theta m bar x so this is a product of theta Imi so this way goes to 0 for OX so this is a condition and so and so and so this this symbol means the smallest integer not smaller than less than or no no bigger than no bigger than or equal to a yeah so this conversions yes so this is our first proposition and and to prove this I have a little time so okay so I only have 15 minutes I look right yeah I have to I'm sorry so so yeah so so this is just so proof is so I have to go to my yeah it's difficult to finish everything I know 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 a big symbol of this one so so the first component x1 and the component and 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 you using this description you get this equivalence yeah through this stratification yeah okay so so now so so I will explain the homology and so so here this 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 so if you are our values we can this consists gives a project system of site and so you can take the project limit of this site and you I just write by dagger and so then so if you take the project of limits so then this give a good homology so in the sense that this gives this gives a homology of Higgs spectra bundles if we choose a lifting of x1 over an of p bar so so the first so that's here where means that so if you as before if you have a so smooth lifting there and F is a Higgs crystal of Qp of x1 and if you take a lift we have a corresponding so this is proposition by proposition to we have a corresponding Higgs bundles no x1 over V 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 homology one is the homology of this site F dagger and the other is just a x let me see x1 of this the complex defined by this Higgs field yes so so the weight of the theta is zero so theta which theta is zero so you you can make using that you can define a complex whose differential maps are linear and so you can take a hypercognitive and then also on this we can consider this homology on this project limit 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 Chris saying case so we have to we take a resolution on of F on our or some are prime bigger than our and so then so on each on each are the resolution as a no no linearization we we consider in analog of linearization and linearization is does not give a resolution on each step on each level but if you take a limit so then you get the resolution and the resolution is in yeah in some sense complex of some overcompensant power series and so and so that's why I write dagger in this site for this site yes so this is the reason why we consider 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 so first we consider a fine object as before and consider our bar and also we can consider our inf bar yeah so this is a bit ring of the projection limit of bird and and Frobenius and in our bar is its reduction most n and the and so so this rings actually give an object of this site so how so that is so this becomes a compatible system of sequence this gives a compatible system of scheme over a and 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 something's you one yeah and so no no no x1 so we have a natural map and so we are just obtain an object in this site yeah and also there is a natural action of the fundamental group of delta so the fundamental group delta u which I introduced in the beginning and so so so having this just by evaluating the crystals you get the representation of delta u okay so yes so so if you are given x crystal yeah so then one can evaluate this on on this object yeah so evaluate on this object I just write like this so this is a actually r bar one 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 crystal so I have this one so then so this naturally gives so about module plus semi linear delta u action and this is the same type of representation appear in the fountains theorem okay so now by very very so if you're considering all sufficiently small u we get so varying u we obtain a functor or a course he he's correspondence in one direction so so this one to to where so so I write the r higgs actually so there is a so we want to go to the fountains topos and so here as I explained before so this is a compatible system of module over o x module some power of p a projective system of o x module some power of p so so to construct a functor we have to choose a lattice of of f but it is not this is not very easy because actually if you define a isochrisis for yeah so anyway so yeah so the final lattice I have to use the the level infinity isochrisis but 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 from this full subcategory of from here so the integral version inverting p yeah so this is a functor and so now I staged the two serums and and actually so I didn't write but I don't write but but so if you vary r so then this category is the same as the category considered by fountains actually yeah and and also the functor becomes the same as the the functor considered by fountains and so then the and the first serum and is that this functor is fully faithful yeah fountains also if this studies the essential image but I don't discuss here and the second serum is is a comparison of performance and so ah so now so this is the last serum so if you if if if you consider a crystal of finite level so and if dagger as before that that is a pullback to to the dagger site so then ah is this canonical isomorphism between the cohomology so that is f dagger is ah so to have a good cohomology we have to go to dagger so take a pullback to this dagger site and then take a homoji so then the this coincides with the so I don't know the r-hex of of of f so this is so this is a fountains site so the homoji for the supporting site so then we can compare and we can we have this isomorphism and these serums are actually proved in the fountains paper so basically the results are the same but I formulated I just we formulated using this this uh hex crystals and so then yeah that's all the questions so that's your construction of the factor dr refers a choice of lifting of x to a n no no no replace are you mean ah just to construct the front we need to choose a compact flow system no no no just evaluate so yeah if you start with a vector bundles so then you it depends on the lifting but I start with the hex crystals so then it doesn't depend so if you so so the lift so you need a lifting to to to compare the hex size crystals and the hex bundles other questions ah so so yeah actually so yeah so I didn't explain here but to construct this factor if you choose a one lifting then you can describe this factor in terms of certain period rings and and and you can define period rings using this site and also abyss and grow also consider similar period rings and it looks that they are canonical isomorphism but actually we didn't check it yet I have some idea but I didn't I haven't done yet so maybe just one small question so how can you describe the smallness in this picture so decided because compared to findings ah yes the smallness uh is uh uh corresponds to the convergence uh where uh yeah this convergence yeah so so if you vary so if you so the issue if you are given a hex uh vector bundle uh satisfying this then the the condition that the convergence holds for some r is equivalent to the smallness of our things yeah so this is exactly the same category yeah okay so there's not another question let's thank you speak again
 So today I will talk about the PIEA correspondence introduced by Falkings and I gave one a  slight variant of his theory. Okay, so first I'd like to explain what Falkings 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 as LP and assume that the residue field is  algebraically closed and 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 yeah 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 project limit is taken with respect to Frobenium and this is a ring used in to construct  the periodic period rings and and as it is well known we have a natural subjective ring homomorphism  and its kernel is generated by one element usually denoted by Guzai and Guzai is in fact  nonzero divisor in this ring and and finally I define a sub N V bar to be the reduction of this  module of N its power of Guzai so N is positive integer okay so so so this is a short subjective  homomorphism so for if N is equal to 1 so then this is isomorphic to this completion yes so this  is a notation and so I first explained the periodic Simpson correspondence by our things and so we  consider a proper smooth scheme over a few K and and and we want to compare the two objects one is a  finite dimensional C representation of the fundamental group so C representation of a  fundamental group and and we want to compare this with X a bundle on X K bar and so so and we want  to compare these two objects and of identity recall the theory over C but over C we on the left  hand side we consider the the finite dimension of C represent C is a complex number in that case of  a pi 1 and here we consider a Higgs bundle satisfying certain conditions or semi stability and so  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 on the so-called faulting site  and it's a bit it's something like this and so this is a faulting site and and so what is so I  don't want to give a precise definition of site this site but the shift is it looks like this can  be given by the following data so she from this site is the following data so so for each  etad scheme we are given some shift on the finite etad site of the geometric generic fiber and and  also for so for morphism between the two etad scheme over X so say G then we have a map from the  pullback of f u to the f u prime and and plus some a growing condition with respect to ital  covering okay so I actually I I'm thinking of a projective system of the module over as a reduction  mode p power p yeah so maybe I don't know how to write yeah and the rationality because we work with  C here and yeah this is a category and so first he proves that the so the theorem of faultings  is that we can easily construct a functor natural functor from from here to here and the faultings so  that this is fully faithful yeah and then so and then he compared this cut some compared some vector  bundles on here with Higgs bundle so next I will explain this part ah so I have to I'm sorry I I  forgot to define the what is the structure shift the structure shift is okay so what is o x to define  now o x using that that description it is enough to define o x u for each etad scheme u and to  define this sheaf on finite etad sites it is enough to define its section over finite etad  scheme over the geometric generic fiber and so this is just given by the section of this scheme and  the B norm is integral closure of u in v okay how to you I don't know how to write okay  why why I buy is used to yeah thank you okay so and for that part actually faultings  established local theory local theory for this part okay so so  yeah so so maybe yeah so we consider the affine etad scheme over here and assumes that  coordinate consisting of invertible functions so this is sometimes called small affine and then  we consider a it's field of fraction and it's 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 parting site we consider the stalk of our bar is the stalk of o x at the  this generic point geometric generic point yeah so this is yeah this is in fact say integral  closure of R in the and maximal unrhyme phylic extension of of the generic fiber in k bar  okay this is not finite the charge not finite so it's a union of finite it's not no no no no so  fire so the union of finite yeah on the generic fiber so if you invert p then this is a union of finite  the 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 hodge theory okay and so delta u is a Galois group of R and then  so excuse me x is over is only over k the fraction field or it's more than oh what I'm sorry I made a  mistake actually so this is I have to be but I'm sorry so that's a mistake so I have to start  well from x over v yeah I'm sorry yeah I made a yeah x over v yeah so yeah smooth over v yeah so  good reduction yes and that was so we pick we pick some model yes okay so then so far things  prove that finite free module with continuous semi-linear delta u action and such that's  so the some smallness condition the trivial module p to the alpha for some yeah so 2 over p minus 1 so  this condition is usually called he calls the condition small and then he proved that there is  only equivalence with also finite free R V bar hat so maybe this V by hat module M  with theta theta is something like this let me see R  R V bar omega 1 R V bar over V bar and because I inverse I will explain the meaning of  x I inverse later so this is just so for his field we consider just a linear map and and if you  take the which product so then this become such that which product is 0 so this is a definition of  Higgs bundle at the integral Higgs bundle in this case we consider like this and the xy is the same the  xy appear there and okay 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 after so the 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 by 1x so it's become the bigger category and then he proved that yeah  under some smallness is it an exact equivalent but I think I saw in some talk that you lose a  little bit in the art that is but you claim it's an exact equivalent you don't have to go to alpha  we don't I don't fix alpha so okay so and then so he also established a global theory and also  also Qp theory yeah Qp theory and also some comparison of kohomouji 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 x to the bar v a smooth lifting of x x 1 x 1 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 aves and grove give an another  approach to this theory but still they need to choose a smooth lifting and so in this talk so  today I just here 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 that's 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 so 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 fix crystals or fix ice crystals and then explain how these theories interpreted in  terms of fix crystals okay okay so to do that first I I will explain an analog so crystals is of  course defined using a certain PD thickening and 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 so no no so first we consider the following category I just  write by C and so the object is that sequence of periodic formal schemes over a n V bar so sequence  of excuse me immersions excuse me it's like this and the content satisfying the conditions that  so for n bigger than 2 are closed immersion and and its reduction mode P is just new potent  yes and so then we consider the full subcategory which plays a role of PD thickening in our theory  but we have a category in on various levels so r is a positive integer or infinity and so this is a  full subcategory consisting of y n such that's the following satisfying the following four conditions  the first is such as this is close immersion okay and also flatness over the P so this is  P torsion free and the third condition is in some sense I'm not sure whether it's it's really the  same but in some sense I did want to also define the flatness over a n of V bar yes and so but I  define it I'm not sure but it's the same as flatness but anyway so this condition is necessary so  exist factorization like this and that is multiplication by g's eye gives a map from  why n to y n and it factors through the that's why n minus 1 and also so factor through the  injective for more morphism okay and the last condition not exactly the same as flatness  so 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 so and in the last condition r appears so okay  I have no place to write so so we define the filtration is just by kernel of y n to y small n  if n is smaller than n and 0 if n is bigger than so so this is a decreasing filtration  defined by the six sequence and then we have that this filtration is generated by all because I  n is power of psi n if so this is if r is infinity and in the if r is finite so then the  condition is like this so p times o y n is containing xi n o y n for n is for n lies between  0 and r so if i is finite so this is a condition so that's okay the reason of why we consider c  okay I will give an example later okay so so then by condition you can easily check that the c  infinity is the smallest one and and it becomes it becomes smaller and smaller if r big becomes  bigger and bigger and so or c and all are containing c okay so then first proposition  is the analog of divided power envelope and so the claim is inclusion functor from cr to c  has a right as joint right right right yeah 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 I give an example which  we use in the theory of Higgs-Rissault 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 a 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 the another object of c so so we consider object of c which is y1 y prime and y1  prime is the same as y1 so and and I consider the yn is just convergent power series say t1 to td  so then you can take a joint so for the for the infinite level and then what we obtain is is  something like this yeah so this is just just in some sense just taking a blowing up yeah  blowing up of y1 prime inside yn prime for any speaker than two yeah and partial blowing up yes  and make it the ideal defining y1 prime in yn prime invertible yeah oh 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 okay and and so this Guzai  in the denominator corresponds to this one in fact later okay okay now we are ready to define the  site to define Higgs crystals so  so so yeah and r is the same as before and so the object is just  the just the object of cr plus a morphism of t1 to xr so x1 so x1 x1 is I defined somewhere  I forgot so I recall so this is a this scheme yeah no  format scheme and so this is just a pair and the morphism is this is obvious morphism in C  compatible with z and topology is we can choose either the risky or at all but here I choose  etal yes so x1 is a scheme you see it as a form of scheme no no I  have the same as taking a formal compression along the special fiber yes so this is I'm sorry  this is defined over spec v I'm sorry yes and so then okay and then we define a structure shift a  structure shift is just a let me see x1 over a v bar plus tz is just a global section of t1 so I  just look at only look at the structure shift of the first component of t and we ignore the other  component but but it's still this has good property I don't know the module over this gives  interpretation of px is crystals I will explain now okay so anyway so this is just a shift of  course you can define a shift of using tn but I didn't discuss here only work with t1 excuse me  what so more morphism over over yeah morphism over v bar yeah yes so yeah it's something like  we have okay so just just a morphism over okay so then in the similarly as crystal  insight the module can be defined by a compatible system of shift on each this each t that is  so the module over this structure shift is equivalent to give for each t give o t1 module  on of ft on on on the tail side of of the t1 and and also for 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 a n v bar so we have many morphisms so the so this data  gives some structure corresponding to Higgs field okay so and so Higgs isochrisis so I only  discussed with Qp theory in this talk so so Higgs isochrisis so this is defined so this is  defined Higgs ice crystals so Higgs ice crystal on this site is but you can if you know the  definition of crystals so you can easily guess so this is so this is this is an O x1 over a  v bar tensors Q modules F on this site now we consider this  such that first condition is that I assume that this is finally regenerated projective  so but this I don't explain precisely but et al locally 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 the I  didn't assume that these netherian 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 for all g okay so this  is a definition of our Higgs ice crystals so this is very simple okay so this is okay so now I will  explain how this Higgs ice crystals are related to Higgs bundles I explained here okay so no this is  not such this is not a big theorem so maybe it's better to write proposition yeah I gave  one yeah so so to have a comparison with Higgs bundles we have to assume the existence of  lifting assume that we are given a compatible system of smooth lifting  of x1 then it gives an equivalence of category so here 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 I will write the so this category so HC QP of r of x1 over AV bar I just write this  category like this and then so this category is equivalent to the to the category of Higgs bundle  with some condition okay so I will explain so what is this okay so this is so the object of this is  a so locally finitely generated projective OX1 tensor module E plus this Higgs field  OX1 tensor Q linear and I have to add two conditions the first condition is just a definition of Higgs  field so and the second is one is a convergence and for for for local so the condition is local  for local coordinate if you choose a local coordinate on some on some affine so this  is a tau over x and define no no no put then so we we can describe theta by d endomorphism so I  go 1 to d and because I inverse d log t I then so this theta I has some conversions yes satisfying  some conversions so the conversions is is something like so m bar is in the element of this set  and so multi index and over r times let me see theta m bar x so this is a product of theta Imi  so this way goes to 0 for OX so this is a condition and so and so and so this this symbol means the  smallest integer not smaller than less than or no no bigger than no bigger than or equal to a  yeah so this conversions yes so this is our first proposition and and to prove this I have a little  time so okay so I only have 15 minutes I look right yeah I have to I'm sorry so so yeah so so this  is just so proof is so I have to go to my yeah it's difficult to finish everything I know 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 a big symbol of this one so so the first component x1 and the component and 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 you using this description you get this  equivalence yeah through this stratification yeah okay so so now so so I will explain the  homology and so so here this 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 so if you are our values we can this consists gives a project system of site and so you can  take the project limit of this site and you I just write by dagger and so then so if you  take the project of limits so then this give a good homology so in the sense that this gives  this gives a homology of Higgs spectra bundles if we choose a lifting of x1 over an of p bar  so so the first so that's here where means that so if you as before if you have a so smooth lifting  there and F is a Higgs crystal of Qp of x1 and if you take a lift we have a corresponding so this  is proposition by proposition to we have a corresponding Higgs bundles no x1 over V 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  homology one is the homology of this site F dagger and the other is just a x let me see x1 of this  the complex defined by this Higgs field yes so so the weight of the theta is zero so theta  which theta is zero so you you can make using that you can define a complex whose differential  maps are linear and so you can take a hypercognitive and then also on this we can consider this  homology on this project limit 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 Chris saying case so we have to we take a resolution on of F on our or some are prime bigger  than our and so then so on each on each are the resolution as a no no linearization we  we consider in analog of linearization and linearization is does not give a resolution on  each step on each level but if you take a limit so then you get the resolution and the resolution  is in yeah in some sense complex of some overcompensant power series and so and so that's why I write  dagger in this site for this site yes so this is the reason why we consider 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 so first we consider a fine  object as before and consider our bar and also we can consider our inf bar yeah so this is a  bit ring of the projection limit of bird and and Frobenius and in our bar is its reduction most n and  the and so so this rings actually give an object of this site so how so that is so this becomes  a compatible system of sequence this gives a compatible system of scheme over a and 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 something's you one yeah and so no no no x1 so we have a natural map  and so we are just obtain an object in this site yeah and also there is a natural action of the  fundamental group of delta so the fundamental group delta u which I introduced in the beginning  and so so so having this just by evaluating the crystals you get the representation of delta u  okay so yes so so if you are given x crystal yeah so then one can evaluate this on on this  object yeah so evaluate on this object I just write like this so this is a actually r bar  one 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 crystal so I have this one so then  so this naturally gives so about module plus semi linear delta u action and this is the same type  of representation appear in the fountains theorem okay so now by very very so if you're considering  all sufficiently small u we get so varying u we obtain a functor or a course he he's  correspondence in one direction so so this one to to where so so I write the r higgs  actually so there is a so we want to go to the fountains topos and so here as I explained  before so this is a compatible system of module over o x module some power of p a projective system  of o x module some power of p so so to construct a functor we have to choose a lattice of of f but  it is not this is not very easy because actually if you define a isochrisis for yeah so anyway so  yeah so the final lattice I have to use the the level infinity isochrisis but 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 from this full subcategory  of from here so the integral version inverting p yeah so this is a functor and so now I staged  the two serums and  and actually so I didn't write but I don't write but  but so if you vary r so then this category is the same as the category considered by fountains  actually yeah and and also the functor becomes the same as the the functor considered by fountains  and so then the and the first serum and is that this functor is fully faithful  yeah fountains also if this studies the essential image but I don't discuss here  and the second serum is is a comparison of performance and so  ah so now so this is the last serum so if you  if if if you consider a crystal of finite level so  and if dagger as before that that is a pullback to to the dagger site so then  ah is this canonical isomorphism between the cohomology so  that is f dagger is ah so to have a good cohomology we have to go to dagger so take a pullback to  this dagger site and then take a homoji so then the this coincides with the  so I don't know the r-hex of of of f so this is so this is a fountains site so the homoji  for the supporting site so then we can compare and we can we have this isomorphism and these  serums are actually proved in the fountains paper so basically the results are the same but  I formulated I just we formulated using this this uh hex crystals and so then yeah that's all  the questions so that's your construction of the factor dr refers a choice of lifting of x to  a n no no no replace are you mean ah just to construct the front we need to choose a  compact flow system no no no just evaluate so yeah if you start with a vector bundles  so then you it depends on the lifting but I start with the hex crystals so then it doesn't depend so  if you so so the lift so you need a lifting to to to compare the hex size crystals and the hex  bundles other questions  ah  so  so yeah actually so yeah so I didn't explain here but to construct this factor if you choose  a one lifting then you can describe this factor in terms of certain period rings and and and you  can define period rings using this site and also abyss and grow also consider similar period rings  and it looks that they are canonical isomorphism but actually we didn't check it yet I have some idea  but I didn't I haven't done yet so maybe just one small question so how can you describe the  smallness in this picture so decided because compared to findings  ah yes the smallness uh is uh uh corresponds to the convergence uh where uh yeah this convergence  yeah so so if you vary so if you so the issue if you are given a hex uh vector bundle uh satisfying  this then the the condition that the convergence holds for some r is equivalent to the smallness  of our things yeah so this is exactly the same category yeah okay so there's not another question  let's thank you speak again
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