3/6 Perfectoid Spaces and the Weight-Monodromy Conjecture

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3/6 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.
Point (geometry) Area Group action Perfect group Multiplication sign Characteristic polynomial Element (mathematics) Ring (mathematics) Many-sorted logic Logic Oval Ideal (ethics) Körper <Algebra> Series (mathematics) Set theory Descriptive statistics Spectrum (functional analysis) Condition number
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Point (geometry) Classical physics Complex (psychology) Functional (mathematics) Differential (mechanical device) Multiplication sign Direction (geometry) Translation (relic) Similarity (geometry) Mereology Power (physics) Element (mathematics) Subset Pi Many-sorted logic Forest Series (mathematics) Associative property Set theory Descriptive statistics Condition number Injektivität Process (computing) Moment (mathematics) Physical law Model theory Content (media) Limit (category theory) Equivalence relation Uniform boundedness principle Field extension Category of being Arithmetic mean Right angle Musical ensemble Family
Complex (psychology) Extrapolation Multiplication sign Image resolution Model theory Variable (mathematics) Sequence Equivalence relation 10 (number) Product (business) Arithmetic mean Many-sorted logic Reduction of order Object (grammar) Series (mathematics) Resultant
Point (geometry) Complex (psychology) Group action Differential (mechanical device) Multiplication sign Real number Image resolution Insertion loss Mereology Element (mathematics) Group representation Mathematics Many-sorted logic Term (mathematics) Position operator Area Uniqueness quantification Physical law Model theory Content (media) Mortality rate 10 (number) Category of being Ring (mathematics) Summierbarkeit Moving average Object (grammar) Family Identical particles
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and it. the last time i started performing serum that there's the. detailed and function or film of so flat and uses equivalence. between the five in tall extensions of k. and off cape let. so that in particular the have to cover groups have to feel so he can honestly i some or think and implicit years the statement that any fired extension of cable again be perfect avoid. and plenty of proof for this was a to use some almost most medics to recall that you find a cadre of almost cannot one year olds. this came at the more you will switch was the cutting of cannot models were you the watchdog by the six a pedigree of them told in objects and this was possible because we're an enormous creations henson square have to make small ideal sequel to itself. cancer redefines motion so full cannot a r r g ross and so on and so forth and the refined version of the serum send that. we have to falling chain of with. the province of categories that some of these fine to talk sentence extent almost was in trouble level. and. some of them there's. lifting serum stating that one can define to talk the talk extensions of was such a war new potent than once and then we can go to the other side. because of his eyes more for the most truncated. rings. and and. so part of the seas long equivalent of contiguous who generalize to own more general set up which is a kid said above have perfected algebra us and it's this season will introduce now so. on. first of what case of a perfect avoid. kayla jabre say. on a. knowledge bar. and then the first conditions of a typical nature saying that. the ring of paul bonnet elements in our this is always open but the condition is at its own it. and then come so it's a perfect condition was says said. reducing multiply. the subject. i have been used more poise subjective and so it may seem odd sort of in the classical set up of which to judge you also considers been a hard job ross and and this condition that sees the suffering of public sentiments of melbourne it is precisely true. the four. reduced audit process but this condition were basically never be the case. the second isn't over. the almost into a level of their it's a flat and politically complete. them. you know they are drawn. a such that. again why you apart from businesses should subject of but no additional require that. the current was generated by the piece would have pie so it was possible to someone chooses uniform miles to be equipped with a canonical are some a fee to the ends hoops and and. and. the last point to know also reduced version perfect on it. kaine want a month pilot euro. is a. that. in. not a month pilot row. a bar and demi again put this condition of books for venus such that. for the news. such a nice. i. i can. and so that we have the following series of occurrence of getting there is so that we know by caper.
as a country of perfect to appeal to both and then we have a purpose cool into. not a perfect. then it's of the ones who came on a piper and once we if this and this will be just the same as on the other side again. and then we can lift to the other side. i. i. and. so. the proof splits into two parts so first we have to prove physical ones and then this the. and. a first step let me explain what some of the conditions imposed on such a perfect it cannot algebra means that we put the condition is flat and piety complete and. so let me explain what this means and in classical commuted to fogey brown so. that m.b.a. cannot more you almost when you. and recall that then we had to find. is what you and your star. which was the whole more physicians in this almost twenty three from cable a two m. and says we cannot continue. in fact it's if you take this was almost what do we get back. so the first condition the first part describes the condition one m. us let's the. imus let it only if the last us let it only have lost a poison. and. for me because it's the same years a. called. because you have almost elements of them and and. for the foreign wins the good name so. c n n c almost more you associate a twin. and and slept for an awkward. then. as the season. almost elements of em all this crap in the following why are the elements accident and high such the fall. on an m. to abseil i'm off for the last summer to not really elements of and itself but. as soon as you multiply that by something makes an idea they become elements of the in this sense almost in the months and. if so this basically experience the effect this condition though we want to the explain what needs to be part of a complete and the various following them still assume that this let. then fall x. and cannot them. one can compute what xm time lost or isn't this just makes them more still so that's ok and secondly one can wander want them want to extend your stars and their voter know what that does contain.
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so first. remark about point three. i. so the. we call that for some have even to compute cannot more time i lost all that was a general formula saying cannot pay for your stalinist general from your saying that this is sort of from more films from the maximum ideal into can open point. and so more generally if you have any modular take its associated almost more you and it's almost elements to this computer by this formula takes a more firms from the makes money into their and so this contains. has already mentioned last some elements of the four. some of the of the pioneers of one of minus one repeatedly on its i were the x i are any elements and can occupy because as soon as you multiply by some small epsom on this will just be a finite some which was then make sense and so the problem. that these elements. i do not the line. you know what point which is so good. office and. the. cannot lift some tournaments cannot and became so but also we see that some of these sentiments were not lift any further because some oh there's this calculation point some of the exponents year and. there must not have any further and that some of the reason that here and three we get this law state and can and so. let me give approval says. so for one first we check that the and most of us let. than in the sport. well that's easy because we're told that i'm just that. means but to finish in the fall. thanks cannot manuals. and i've begun zero that's a tall i have. cakes and lost as almost a year. but of course is the most us let's a single busier serve that sort of. this implication is clear that in the second implication sent him a sled than and lost our small potatoes. how do we do this so first note said. again using this formula for the almost elements and settle for more prisons from m. and two and lost and the subscription imply said singh has snore almost zero months. i. because and square to the country and going on. he. and. the and. so then we use it. as a particular cases condition to use its toll one of them know a star was. cannot more point that this thing is almost zero. and this trust translates into the conditions that the colonel off multiplication by pie.
on and lost as almost zero. i. but again. colonel can have an almost irrelevant because of this intensive follow said the thing is you. and so which means that and most of us will put. and then there's the last step saying that he and also having a pie toys and. it was almost a third but of course is just the standard in his state in stone of commuters all draw and. i was stunned affect the game though no for the second statement. so again we use his formula that and lowest the can be computed this said a formal more firms from the make small ideal into in. and because and sleds and bets into the more films from k. into and in that part. and and heart of these crises more more films that line the image so the more firms are given by the element of phone which them and in the pie. and the condition is said as soon as you multiply by some have said on an m. you have to get an element of. but that's just one complaint. or. a. a certain point. if was that exactly is also be right next to last though. you can wanted us. said of elements and why in. most would say there's no standard pie. such said fall it's not an end. so its own text and more. most of its times more stone and this can be written this set of four x. times the center for water and the most of the universe. such said. forbes been an excellent times why doesn't the most are.
but using the talk he proved in part to for the special case of any put to an all-star was just six times style. and because them up soon. northstar us have left expect because that's why the joint to something. and we know get that. i'm also more you fix them the worst are lies in their model a somewhat star. and. so ok the last of surgeons left as an exercise. and if the and for the last point but for the node first said. both tears. and that's too and most aren't and nothing to an almost the middle of two joints. the giving the second case by the funkier that i've not introduce which is and maps to and or star which is. and or shriek which is in tens or in the austin. and. and hence commuters in recent months. i. and so in one direction is and get that if the. and if any aspiring completes an employer star is the most limited. prior to the un north star this commutes was immersed in it so it's the most intimate of them or point two million or star and now we want to see that this is also the same us. the two most limit of and also apply to the n.. and. so a pro you would think that this might be only an intrusion because he only has his inclusion but were quiet summer we have in the months and this must limit and so every saying that the criticism of the wrist limit test lift to something larger and then we use a statement here saying that then is actually ice as something so. i. and for the other direction it's even simpler some. for. the. can and so now we can through some propositions. the first is said. and if you have a. perfect on it. he also draw. and then. the fire and uses in fact in a small fishing between are not what i would use one title one of a p on of one peiser remember that the only required this fight to be subjective and but in all other rings we sort of required that we have such an eye small business. sort of year it's automatic and also. the if one considers the almost are dry associated to are not a and this is the perfect order. not the only. . and for the proof. just no incentive exists an element of on oct such where it's in our own on subsidies peace power. this and are not going to close it would have to fix this thing as paul bonus. then. obviously explode. divided by parties want to repeal is also a pub on it. and so i switched gives the first statement.
and for the other notes said. when knows it. are not is a certain players to compete. but the conditions that it's open and bonus and tense by the previous limit he sees it the almost a job or is also fed and pointing to complete. and so its last condition also. translates into the almost set up and. ok and no one to so we can opossum perfect ok i'll just affected cannot almost embarrassed and no we want to go the other way that the falling like mosul it. a be a perfect avoid. cane or older brother and me. a. that army soldier you get the unwitting pie. and you want to see that this gives you prefer to affect taught k i do wrong so first to give its apologists little was to forty. making the last are open and on it. and. i. then in fact a pub on of elements of a single just be a lost art and so in particular we see that. are you can reconstruct a by taking the almost larger associate to two are not which means that the two functions will be immersed in and there's also son was already on the plate court imply said. we get an eye some office in between. i lost some apart is one of the peons and also empire which might be surprising if first sight because a probe the only the were quite good to be an almost a small doesn't want to turn sold to the it's a nice more has more to magically and the power. it's perfect. i can and so spruces and. and. here's the first the show that forests injected. so. first we know that high. this medicine almost arsenal. my assumption. meaning that the colonel and coke own almost zero and so the first step is to show the fires in fact not only almost inject that injected. so what does mean so we have some. it's an enormous star such that acts as a p some point they know a stone and we want to prove said. i'm and infect x. less already imposed as one of a p an all-star but it almost inject to eighteen. says that there's at least you're often multiplying by some it's a long slow. i do. the. the so fall it's none of them we see that as long times access to be an element in parts of the one over peak times all-star. but this means that sex is almost an element of pie the. one of a p times a clause in this set of almost eight months but then the last woman tells us that this is just the same as. what we have so he gets a desire to conclude. and the next we need to forming small emma. that if exit some element of our. but said that spews power lies in a lower star then already explosive the north star and was approved. we. excess some intrinsic a such said. i took a repeat times. x. lies in the last are by definition of our. and if k. is at least one than in particular we see that why to the peace. some party a lower star. it was almost and to buy the interactivity that we have already proved to follow said that in fact why lies in what price the want of a p times and or star. which. it just says that or sold parts that came on as one of a p times x. still lies in a low star and then we do induction to get to claim. in a and e..
if. for. the song. next to prove said. the problem and elements are just a lost art and for this note said it's clear said. a last are contained in the pub on elements because all the parcel from the moment of a star will lie in a star and a stars and deficient bonus subset of our and for the other direction. this year that the x b. paul moment and then fall. that's on the make similar idea its longtime six is top notch new potent. and in particular the sexist some big and such that have sometimes sex appeal to the end will be in a last off or and not enough and this implies that. its longtime six some day in lost over the previous lemon and tense. x. again defines and almost element of a law star and then fix itself as a description of almost elements is the north star or. and sort of know we have to prove that. we have some falling equivalents. the. so in other words we have to see that. the images from the ball which is a tender. the point and so we have to see that. a bar its midst. unique deformations. and for this we use a series of sokoto engine complex. though. some nations which are laid out in the sea was awfully busy and then recall of the bit about school to engine complex. so too. the first in the classical music of arbroath set up some and so to any remorse isn't. r two s. we associate. the so-called quarter engine complex which. is an element of you can consider this an element in the drive to the real of as models whose call knowledge is concentrated in degrees and nothing positive to greece. the and some properties of this are said. of first of the age i guess it's more stunner to use from a logical attention. it's not also sings just. the differentials off as over our. and the for any. i r s t. one gets a triangle. in. dr to have two more use of connecting nor have to get this right. well as a war are. pull back to from his team goes to the cool tension complex from she'll tumor argos to contention complex of tea over as a. i. the.
. and some of this extended. extending the products like sequence of one know some of from for just let me go on. the. and. let me say one word about the construction of this saying so. construction once as follows. we've also someone new to this. and. the first to use of those khan equivalence. saying that. mr roffe could give us more deals with this is a bonus restriction is equivalent to the criteria of some political. s. more deals one of the problems. it. and so instead of to sing in the draft cutting rooms to construct some some fish us more fuel and some way we do this is to choose some. some pleasure. resolution. of s. by free are dross. the meaning just a porno me all draw and possibly infinitely many variables and there is an economical way of doing this if you like and then once sold for. for such important only knowledge was easy to define what see i'm so sick or tension complex should be as should be just one thing i wanted someone extrapolates from this case someone sets sail us over are to be. the simpler this is only the object in. associated two. own. only a one off simply show eligible which then gives us some political a stock more you and your. the timing is right and. a. yes and in the tens or was this was a bit. and. and so why are interested in this it's because of falling serum so. and deflation series so that obvious a module row wearing and i are an ideal such that i swear to zero. and and so said. r. zero to be almost i. and. that s b a flat are not old abroad. and we want to. lift as to have fled. all to grow. and we are asking what the obstruction to doing this is and what are some possible ways of doing this and that's on sept was falling serum. which says that if there is an obstruction class. in. thanks to have the coach engine complex low cost model were on old was where you sit in. the ideal tenants turns up to us not. and. the possible lifts are the tours are under extreme one of the worst same objects and see all to more physicians have one lift our thanks to row of same sex.
i. and in particular for example of c o two engine complex vanishes then there is a unique gift and the sort of there's no ambiguity whatsoever. a. it. and segments the formation results that we will need to said if we had already have s. and as prime two. lead ology bross and we are given a home or film. from the reduction so. on the reductions. two are not and were even have a home office in from that's not just not prime and. was the most are not as ross. and we want to lift this too. i. home office and promised to his prime and then there's a following. that there's an obstruction klaus. in extra worn off. the potential conflicts of this more. over on old was way isn't just not prime and there are not i.
and the possible lifts. foreman toys are on the group and extolled the same. thing again we see the difficult times and complex managed to send also home office seems if too neatly and so. and it would be interesting post temperature in for medical attention complex plan isn't as the following are not propitious. it also appears so i learnt his firm's book of carbon from year old and i don't know whether this was observed before so. there is falling varies quite serious of arson. perfect if the older brother. in the courts engine complex of are over fifty which is.
so long. and you'll need a slight enhanced version of says. so it. but all too as him off isn't. i have a theology ross. that our law foyer the. the already brought. are we as a whole month isn't. because the for the news. and the. as in the same would be defined as five and we assume that. the reds and from us. them. five as of all our is an awesome oath is in between. and even in the drive category between. this and the song it's i saw from the dr sense just mean said. it's was also it's not the most of them and all higher toll terms manage it. and. and then again the court engine complex wishes. k.. and. oh. so for example the first statement. and it. the. can be used to say that. ring a fifth victim so far just a unique. the formation of our to have threatened putin to complete. the p.r. the road just so much closer to home or think about some directors an unusual definition. and. and also some the. it's clear that the. so my got one of our over fifty is zero. why is this so. i think some are and ex-wife to live here. then the aches the food was the peons that's few times something and zero zero and then some of the crucial identity behind this position here. and. and so we just have to like the. go for slightly more enhanced argument and so it goes as follows. so. an. soled shoes. the simplest a resolution. and i and one. i'm so we also have sort of some relative from us. we're all which is a man from off by and there are s. daughter to a start five and the assumption just says that this is a weak opponents are. there's more to do whatever we call it. the. the. and and. so we have. to whether missions. of testify by a free simply show. as for high office. i was wrong. so both sides of this give you such a resolution and so it follows that they both compute the same hotels and complex which means said. our fire tender are. all know if you like. of the cold change and complex of as start over our course as lofty to call attention complex as it is also taught us by. over off five. but on the other hand. we can expect the computer was the das this thing to do on the differentials so the sum of one part of the simply chaudhry broads just some. the internet porn on the audubon general and. then fee s.k. over our. on differentials. that sticks i too excited to the p. which is again zero so paulo census murcia memphis which is causing a small film is also the aromas. the. and particular both sides of his last most of them have to manage it. and this is exactly what you want to. some of the. but some of the same saying just concern from a different point of us to call times in context. it's all wrong. ok now we have to use this wishing of sokoto engine conflicts but first we have to sort of debt to syria to the almost context of this was also done in the book of carbon time you'll soon. i. they show the following that in so if you aren't s. are cannot alter ross. then if you consider. the potential conflicts will fizzle more are but considered the almost way then begins as an object of dr kedgeree know if it's almost one year olds and show that this depends only. on the mall physician. from our way to essay of can almost always ross and soul and you get. i. you get l.a.. they'll be in the driest country real fame on the walls. for any. in. loss isn't. it should be of can almost always rose in this way. and. and in fact the gundam you want to find a sledge we find version of this contention conflicts which actually is an actual model some old an actual ring with the rolling and it's almost worsen and. then the previous serum stage for. syrians bought the formations. stay true. is almost world. and the. we all men need some of the following coal area of this mission of sokoto engine conflicts which is set of a bar is the perfect on it. kaine aunt they all want pilot abroad. then. the coach engine complex. of a bar over the saying to a zero. a. and while going to reduce it to the case that was already handled this sort of. the scene off too. represents. a bar by a flat. i'm not want pile draw are such that of for being is an awesome or prison. from our point to want to repeal was our and. again because the strange elements was simply earned the the someone puts rise this but this doesn't work. lot to this book there is defined as a different way of for presenting a ring boy and actual almost anyone actually so what they call a double or shriek which is to take a loss rate which was. i recall like small ideal tens or an all-star. this moon general be flat or eight is one that's better so the same might not even the flat and but it's not an audit by any more mccain not because does not contain the identity so we ought to officially sort of have to at the identity and of modeled and. and so is this works. them. don't want to get into details here. and i guess one could also just to do the same prove some low or is it possible for. oh no. so. so it.