4/6 Perfectoid Spaces and the Weight-Monodromy Conjecture

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4/6 Perfectoid Spaces and the Weight-Monodromy Conjecture
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We will introduce the notion of perfectoid spaces. The theory can be seen as a kind of rigid geometry of infinite type, and the most important feature is that the theories over (deeply ramified extensions of) Q p and over F p((t)) are equivalent, generalizing to the relative situation a theorem of Fontaine-Wintenberger, and also implying a strong form of Faltings's almost purity theorem. This method of changing the characteristic is then applied to deduce many cases of the weight-monodromy conjecture.
Geometry Perfect group Military base Multiplication sign Open set Limit (category theory) Complete metric space Subset Element (mathematics) Proof theory Category of being Root Ring (mathematics) Game theory Körper <Algebra> Object (grammar) Absolute value Family Condition number Physical system Spacetime
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when ould so today i want to speak about it. only about the fact its bases. it and saw that murray call a little bit about what we did so. first off we defined the notion of perfect oil field.
is a cook it's a political field. k. which is complete. the. the. was first put paid to a nondescript. way shun offering one. such said. whether you feel to dispose of characteristic p.. i. an end to for being us. subject when you thought you. and over such fields and saw him as an example you can take say something like the completion of. q p two which you enjoy all the people roots of p. for example. and. over such fuel to consider the following type of to put eulogy ross. these were complete knowledge across they were one hundred ross auditor are with a bloody good condition. that. are not which is a sort of public elements. i. this is a bonus subset so it's always open for to the bone it so compared with just two geometries a will smith this would mean that something is reduced and also in this case it implies that something will be reduced and crucially condition this said. again. it. soon you apply so immune from my system. the star. we wanted a new from what. and if we chose samoan just business of when i discussed perfect it feels that this message active. the game and. what proof last time basically wall said if you introduce the fuel which i call k. fled which can be it. yes some of years you need to. the. the boy to say this its absolute value is bigger than the opposite view. i. if you take the earth limited. of the piece parma of cases this can be given the natural to restructure its construction different to find tenants and of them this is a perfect a field. of characteristic you were just means that it's the perfect incomplete. and to any ring are associated with which any perfect to kill draw the associated another perfect tickets that o.j. drop off let which can be describing the same way as the most limited overs a piece pamuk of our senses the perfect road. if that older brother. and again in characteristic the perfect toilet so. the condition that. the foyer. from an aunt. the subject of. it's equivalent to the condition that are is perfect. soon this case it's probably. also it so and so depressing characteristic be a perfect ok all. well it's a one off here said i was brought such that the public elements open bone at answering is perfect so it's something it's very easy to write don't want this and so. it's amazing sokol tooting serum to as use it. the two categories are proven so could you have predicted celtic boss which it will take a perfect is equivalent to the category perfect a case that knowledge bross by the procedure by which you can translate some of the film another to true uncharacteristic zero two phenomena and you could trust. a p which are much easier to understand that some of the way to think about it. it will. well yes was almost as it continues most isn't it. and so today we want to associate spaces to such objects and recall that some of them the first told that. when you find a notion of the tate calibre. it's a logical failed to grow. being called are such that there exists and open and bone that suffering. on oct and or and so in particular we see that perfect all it killed ross automate full it's not as and. then we will consider the following type of if you know it all to cross. reflect on it. if you know it. failed wrote. so again it's a pair.
are all close to where our specific toilet. and. all plus in on autism and the interval the coast so it's just a few no tail draw such that's ring or is perfect toward. and so two sets rings be associated stop a logical space. takes which was called to expect from of our applause which is a set of all. continues way shuns. own such it fall. functions in this class ring the wages at most one and pick them up to a pool and on this would be fined appreciates. boy accent wakes plus which samoa on global sanctions would give you bake aren't all plus because here we are a complete situation. ok. and who first lemon we need today is that. but we also have an influence looked at us between perfect avoid. if you know it failed ross. which is all checks that perfect its base lucky a build although there's a miss it. the same thing over the other side. so moving are all for us to you. off that off the us and soul so what we need to see is that we can identify these open it into a close up brings and for this known to that. the. first it. necessarily these elements are not thought to cost a total she knew potent. the. and because all process open some power of some will be in there and got something really close they will themselves lie in there and. then we have set up a lot of applause and or not. his budget she moves into a closeup rings. of are not what he was make smaller deal and. because of some on this procedure of tilting you. someone had an awesome off isn't between this seeing. and seeing where some on here pie and pipes that. some a corresponding uniform misers in. these two feuds and so in particular you also have said are not not make small ideal. it's the same thing as on the other side. and tense. this thing it really is the same as it usually still suffering sold off to note what. but. which again deacon than if i was possible a plus offset does in it. so this way you get the desired identification between such scarce and. no i can formulate the serum that the what approved today and says falling so let our applause. be a perfect choice. if your children grow. was. chilled. off let off the post. and and let's consider the associates bases expenses expect from of or a plus ins also an explicit. the. mccain and of course we also have all these structures she's on when this basis structure produce a prayer or. and the first part says that we have. canonical home office and. the. which identifies. rational subsets. so we called its apology on these spaces was defined by choosing a so-called traditional subsets as a basis for the apology and a rope recalls a description the second and it's very easy to write down the met in one direction namely.
effects maps to fix that under this correspondence than i should say that some of it. in general there's a map from offset to our by projecting to law school or this year and dismayed by the notebook the left to a sharp because sort of course and the other direction. and. then the formulas given by. the signing. the aviation sending of function f.. to the way shun. offices a function of shock. i. the. it. now. the. the. he said. but there are. yes so you mean if you day identify the political space is in the original subsets year ago except as a russian subsets you. in most elections some of their four years and the image will be one here and the other way. they are. but it is. one and. the very first version of my true for in sort of the only proof is in one direction but it's true in both and it can now get approval. we have done. ok so the second part. you have someone its formulation the first part and its s. it. which. for any rational. you an ex corresponding than under his correspondence with some you fled an expat rational. um. you can form. what's so if you were explosive you. and by definition of the structure she's received this was some oh and if you know it all to bronze a claim is that it's perfect toit a few of cuba. it. i. the. was to build. given by sort of what we get by evaluating the prescience on the other side. i. some of this relation she fights between are a plus and it's too. and then the surcharges that infect these things are you choose. and this implies it also has other things that she's. the. and. and lost the war working hard to heaven and a local firm takes a cyclist the serum saying that the higher commodity groups mentions is true so a choice. thanks go x.. fierro for i began zero. but in this perfected situation is even better lamy the statement which a statement about the generic for some more extensive as he slept a generic fiber and the sense that. them if you take c. plus chief than this will be m. towards an. for positive thoughts. which is something that is not true and physical which a german tree so this could just assume stating that than on previous year in stating that this any lighted by some bonded power of so you know from my the but it might be some all knowledge. so i can. and the proof will occupy a house that these for today. and so let me give an alternate to prove first. so i. first one years was one and two and proof them completely and then afterwards some troops run for completing. so. so the first step. will be to show a week form. of part two. the. i. but with the addition though the. prisoners precision that we are no explicit. i. i have to expect a description of well almost. both will expose a few. which will be important and taking steps somoza crucial step so. the problem is some know that the map ref groups to have sharp from ought to lead to our. is far from subject. so some hour. in the definition of some of it as a slipway shun year on the other side somebody only see the way she's on the strabane woman's and so we want them to determine everything some of the way she but this medicine very far from being subjected in general and soul but still we need some kind of approximation them as it. you can approximate any function by function of the form of shock and so that who before the station them. any of them are. can be approximated but some sharp. what gene offset such it's a falling holes. the. the. the. the firm. the. the fall xsun x.. the absolute value of it is at least the same as the absolute value of the sharp. except possibly have both sides were small. the. so you cannot expect the approximate a function of the sort of its absolute way you can be approximated a nice way. and then. after you have so if it's easy to conclude with pot. wanted to. and as this will take some time i will not to its gets the author of the other two parts instead let's start with approval one and two so. what you see is that this meant from x. to explain it. well defined. because all you need to check that this actually find a way she on on the swing off that but that's not so difficult. so we need to see that it's continuous. and for this first recall that. the original sub said. is something else a form you. given by some s. and the gene it. subset of.
so it's a set of all xsun x. such at the absolute layoffs yet eyes. but mostly have to pay off. do you. where's the f.b.i.. for certain elements of our that generate are snowed you. and g. is also some element of our. and. this point he will need to falling remark that if we can. we can always assume that all the ups and g.'s or an a plus for example or not just by multiplying them by a large power of the uniform either and we can also assume that. if and it's just some power of paul and some will number. and. the. because. they exist in such that apply to the anderson. applause. if on. plus if in a subset of our because if you're pies and the rule you generate are so sick system such end and then we can be just as a spy to the end. to set of s. and since last condition of the sort of automatic dissatisfied some particularly sort of see that on rational subsets. if today of geophysics is at least. so it's some was nine zero the bone from. the. and the kinno. or does this imply that the mets continuous. he says. the. if he take a rational subset of the tilt. them with these conditions. then the pre-match. his you stop giving in the form that it's a rational subset associated to all these top of the month. the side effects. and so that some of this is correct as a set of skiers someone from the definition are and what its we have to take us it's really rational subset. and for this we have to see that the fog still generate. are about some oh if any was some power of the uniform either. then if and shop would just be. this pipe to the ends in already just one element generates. it generates on to its. but. i'm not completely each year i'm not sure i would expect yes but some way of unchecked. well it. it is true. that if they do not generate are then there's the when probably use it. so. even if they're not next year in. so ok and it is that this. i am. and it's. you. then one could sort of do this. ok i must admit to just seven shake that sort of this way. you get your race with the east and and. so now we come sort of the first part of this prove so weak form of two so here we sought. with some rational self said. the expression that rational. and. yes you these conditions so that all of them. the victim enough that their the pub on that and said if and it's just some power. you from wiser. and so let you shop in xp the print. and i want to get this description and so. it was euro. but we want to define a perfect or drinking for this summer you have to join all of these people roots but then i think about these elements that we have to join anyway if it's become a crypt this can only kill people rude summer because it come this problem once and. i. the so that's a speedy. political position. i. of. the ring generated by all of these elements. inside of. are what you invert shot. and then. maybe i should give this monster name so let's be. and a. a. then the almost all your boss will see it today is a. perfect ought to. in almost all genre. but. the the the. now. the. the second part is said. wakes of you shop is again a perfect toilet. the here and abroad. and because that perfect yards ross work google into such perfect it cannot also draws and so. was associated. i. you know almost origin road. which was to find in this way just really being this saying that we have some own not only described structure she even know what we have sort of almost described it owns into global in this way so. i mean a pre-order some oh some the possums in sugar crawl your stuff that we have to do what sort of its enough to join just people roots of these elements to get almost into closer of sourcing. and the last part is that. what makes a few sharp. two to two. fix that. i. the the. and. with the day. that view. but. but. the second part implies that is independent of the presentation almost.
because this thing we know is independent. many choices. so when i recall the definition of this truck to appreciate it. but. it. it. well it. about it but that. but what the hell. but you will. and that. so the. maybe it will be continuing to do. i brought the. the. yes but after pausing to also drogba lot you can because of for two and that's what i have proof. let me just procedure food ok so so. the first we do pod won in case it's a characteristic of case peace to recall that this implies said. if that's just k. and so on so i can get all the sets and sharks and so on. and so on. it's clear that a is the lead and poetic you compete. i. the show. this that summit. this map as an almost as more fun. i. ok. and for this we can look at sea. i am. what to do putting on you. to this theory is a perfect toward sold here this product is true and the subjects on to. judge whether to looking at. the. and the others. so as a subject of so she i was all were peta the and sent to a boy would you be. one over key to the end. to subjective. it's a colonel. contents the ideal i which is generated by oil months also for the eye to eye of a p t m times g.. minus a foreigner i. the. and to claim is that the induced met. the. the two. this thing. is almost and it was almost as often. the. i am totally proof this so. the first person i was more puzzling. after inventing pie. the cause. in this case. we also invert end and. and street because energy divides if and and once you have divided.
converted g. and t. i swear he just become the question that gets you in a small office. some. and so that has been the cone. the index is some cases that putt to the quay of. it's an eye. and so in particular. he to the way off. peter j. is not. because it's just a multiple of this one but this implies that. the thing of service and i because i was perfect. but to finish. and so the receivers almost inject. but because so many animals in the koran or already almost inside your eye but by which you have to it. i. the.
you. the. ok. and so. s.. it's clear that if you. consider now this all the ring which is almost the same. then the statement that he wanted to tour the truth. that if you would use more pop a one of a p what party that gets it. the. so this is to get clean. . next to prove said one plus two. and for general case. the. the. so let me call one definition of the structure she's so. i. i'm so. first we take something some opened on the suffering and in our case we can just take. those of the paul wanted elements because no case they are open and bone it and then. to take. the single just apply to completion of. i. i. i. this up audra. and. then lakes of you will just be.
this thing into a new report. the game and we already have to see a summer that we get the shot. i. if you can also expressed in terms of or all draw a so. so what we have it. his said. i. if we only any reason a month and of course we get something of this one where we. join all the people roots. but i claim that this is also contained in by two months and times and times. i. this thing. and this is because. any element. if year. the richness. some of terms. in one over. the show up to the and towns or not. why don't emerge from much. because some are so they may be some denominators in was sort of people are in the human nature of sharp but so if you can take the closest integer to this and then the additional parts will just have sort of for every f.. i just a little bit less than a one time to shop in the nature. with. the.
but one over the shop to the n.. there is equal to. prior to the minors and times and times. if and sharp. would you show up.