5/6 Perfectoid Spaces and the Weight-Monodromy Conjecture

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5/6 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.
Axiom of choice Perfect group Process (computing) Projective plane Basis <Mathematik> Open set Mereology Equivalence relation Element (mathematics) Pi Root Uniformer Raum Network topology Right angle Körper <Algebra> Series (mathematics) Object (grammar) Associative property Thomas Bayes Spacetime Condition number
Point (geometry) Group action Rational number Algebraic structure Basis <Mathematik> Total S.A. Sequence Element (mathematics) Subset Product (business) Proof theory Many-sorted logic Ring (mathematics) Causality Natural number Network topology Energy level Algebra Spacetime Social class
Covering space Group action Network topology Sequence Resultant Element (mathematics) Subset
Complex (psychology) Group action Pi Energy level Insertion loss Netzplan Resultant Power (physics) Thomas Bayes
Covering space Finitismus Group action Perfect group Process (computing) Correspondence (mathematics) Direction (geometry) Characteristic polynomial Expression Physical law 1 (number) Parameter (computer programming) Limit (category theory) Sequence Subset Many-sorted logic Modulform Family Identical particles Associative property Resultant Spacetime
Probability distribution Point (geometry) Finitismus Direction (geometry) Multiplication sign Characteristic polynomial Insertion loss Open set Student's t-test Mereology Subset Power (physics) Mathematics Causality Many-sorted logic Term (mathematics) Profil (magazine) Energy level Antiderivative Algebra Posterior probability Physical system Engineering physics Covering space Rational number Process (computing) Forcing (mathematics) Limit (category theory) Complete metric space Sequence Category of being Uniform boundedness principle Order (biology) Universe (mathematics) Quotient Spectrum (functional analysis) Spacetime
Compact space Group action Netzplan Limit (category theory) Analytic continuation Spacetime
Covering space Positional notation State of matter Price index Limit (category theory) Sequence Spacetime Product (business)
Axiom of choice Perfect group Characteristic polynomial Sheaf (mathematics) Algebraic structure Basis <Mathematik> Parameter (computer programming) Mereology Sequence Power (physics) Explosion Category of being Goodness of fit Many-sorted logic Extension (kinesiology) Algebra Spacetime
Category of being Perfect group Many-sorted logic Natural number Modulform Spacetime
Existence Scaling (geometry) Process (computing) Perfect group Forcing (mathematics) Metrik <Mathematik> Insertion loss Open set Product (business) Category of being Arithmetic mean Order (biology) Modulform Arrow of time Series (mathematics) Local ring Spacetime Condition number
when ould so the start so continue. a perfect choice basis that they let me give some show for minor so again fix some perfect old field. so something the should be going all the core roots of trees to it and complete and of course and we have a few the court to flit just given with clinton construction taken seriously but there was a piece for mobile bay in this case. is this would be. some of the field of a characteristic p.. so he takes a long series field to join all people was also a uniform wiser and to a complete and so in general we have some a new firm was a year which would be something like she and some uniform was on the other side which would in this case be cheap and. then we had the perfect avoid killed across. i. on the field brought our. such that. the problem the elements open bonded. and the crucial condition as it's ever been used so effectively. the new high. the and. and it proved to sorting equivalents. that could resort trouble and so. perfect to kill job ross. were equivalent to perfect avoid if that old ross. so also escaped that's perfect a few of them the same definition applies and this just sense any such perfect all be brought to its so-called chilled which is off that into its just again the phone ten construction of tickets in wrestling with his part of what our and in particular as this. gives us the met from our to offer to our which is much quicker to have but not at two of which sense. the just to something called the shop so it's just a projection to the first four hundred. if. oh dear. it is. that's the automatic because of its opponent elements so if the piece possible by pies and singh said itself which will apply to one of the people because you're concerned the prominent on. um. right. so we have this and then we went on the one to define space is associated with such across and the basic objects that one associates space to and this was serious was. and so called a few dollars ross and then we defined a perfect or different audra. is a pair. are all costs were always perfect toilet. and of course some opened into really close sobering.
of the problem of the elements and in most cases is in many cases it's really just a set of called on the elements and then also. such are applause the cutting of such as also pulled into could agree on offer that offer plus one way to get his offer classes is again to just takes his resume at the world peace point. problems. and. and so we looked at the space associated to them so the so-called expect from which consisted of continuous what the asians on our which are at most one on this the suffering of sort of internal elements and we sort of have the same thing on the other side. you and then there is a map from each year sending x. two x. lead which is defined by work. barring that if you have a year to speculation at this point. so this exploit this way should not in this way so it's should send any element of off let some. you and. you just takes you up to jail just drop represent bulls shop a month which will be an arts and you can apply this way asian year and what we proved last time this. the that is in fact this home office and. which identifies rational subsets. so the original subsets for certain natural or a basis for the topology. and there are certain freesheets to find on this or to political space and three showed that fall of the original subset you. which correspond to. then under this home office into a subset of the tilted space in its true that's all acts of you. or explosive use again perfect toward a few it. and. it's still just just given by evaluating. and she's on the other side appreciates. some of the situation said starting from any factoid algebra with this additional data and we get a space which is the same on both sides in effect we get preachy switch also related on this procedure of tooting.
and. the first thing i want to prove today. if is that it. and. in fact the likes and hence also explores is a chief. and that's a higher called maggio. explain issues but even better or that. as a higher quality of service into roast subsurface m. tall and four so it's almost you're basically for on bigger than zero. so as i did not insist that is all plus piquet not eligible it does not. fall he does not make sense to say almost year at this point because it's not ok more than one you that it's a very minor point the. ok. and slow at the first do some through mirror from some first reductions. you. so the statement is equivalent to the following. that fall covers. the u.. i. but finally many. rational self sets. lawyer next. as a sequence. oryx fix some perfect audubon's and you can look at it it's the almost into a little. so this is basically the same as for explosive takes a. so this is a plus rings but almost the same this is wearing a fall one of them and six and two in general and. then we can do sing sing you and and to the product. these are again rational subsets and dance and so on. so this long six sequences long complex that this is a extinct. sequins. if you know they want you. so that's a more general nonsense. and. but because. he's always so if you. these are process. see almost into the level they are let empire to compute. answer is also equivalent. i'm the same sequences expect after reducing more pie. the uk but now because we have proof that he was tilting summer we get the structure chief on the other side. here's just really just. the same sing for the other side. when you buy flat and. this and we just a product of sea oil explain it. a few i fled your way when you apply fled and so on. and so it's enough to produce its the sequence which was uncharacteristic he is expect and sewed follow said it's enough to do with uncharacteristic zero and could disappear. the. ok. and. so we have to prove it. i. yes she's and their the species in this piece was waging call majid and so reduce production summit to the statements that are known and just original much we sold every call this the to. i. i. and it is his last year a reduced. if you know calibre. topple to find a child. so we're cause that's this means said. there exists some any such that. this is a cold front. also a total of two concerns and said this as plus we just the suffering of paul one of them.
so the single never be perfect audra except in very trivial cases and. but then we as a foreign results so we have. serum which is basically tate's six it is your room and in full details and six falling so first is that infect because he assumed that the saudis rosewood used them. the sub said the results have to fall down and elements really is just the this defines the topology so it's open and bone. the. and secondly. if. you an ex which is spices plus an additional subset. the. and then evaluating. the. the thing.
and then we get a again a reduced a few children have to put too fine a type. but. and thirdly and there's no finally the tate essex is to see room. or call knowledge groups. of just four weeks. sequence are so it takes. it's a cover behind him any rational substance. i. then. so that's not quite the standard way of saying it but that's it for one and a.
or cornwall your closest complex are in a native by been the power of pie. it. and so usually won their says said. the complex results as iraq's years to really on the rational level of six sec but additionally says its he you know transition maps a strict and strictness the equivalent by been also connecting three room to the fact that the comedy groups of service.
the complex are the nearly at bay the circumpolar fi. a. i. let's just say that this is basically to it as a city. loss. year.
the. ok and now we want to sort of start from this case and deduce what you want and so they're sort of an intermediate kind of perfect or drinks which i called the finance a form to finish. soon the characteristic of case p say that. a perfect or if you have killed her brother. are all plus. and i don't know whether it's a really good name a quarter p. finite. and i. if there exists some as espoused the is above. such that and. are pluses just take the perfection of curing so it takes a direct competitor for venus this process is an computing klee.
and of course artist to be a plus in reply.
i. i can. ok and then we have fallen proposition. the. i assume that this is the fine art. i am coming from. but yes it's also because it wasn't. the. that's the fun and and so coming from some of the is a plus. i. then the first statement is said. the space associated to are all process is just the same as the space associate to two. and some it goes in this direction. i. this is in the oil law firms in which again identifies virtual subsets. i. the. a. and simply or for any rational self said. human expression which thing corresponds to a million void. some will suggest this again the same that if you consider likes of you express a view. in this is really just a complete to perfection in the sense of its ones left. off. why have we want plus a fee. the. and finally. fall covers. as above some or a fire at the menu arsenal subsides. the sequence was we want to prove his expect this expect to work six. i. and. i. a. he said. i. but the. it is. the. the uk and. now this is easy to do. so. i. i mean just the same way as you prove said for being isn't just an eye some off isn't on spectra and uses. and i some off isn't. so it's just the identity more prison. i. and. some of it's also clear that if you take. you can defines us. expect more so for long can take things and then you. some are gentle fake said. pa seeing taking a few the argument corresponds to. taking the fruit interest limit on the space is so you get. this. the. the result of this but because all transition of snow as you get identity that's just. despite all of us us plus and it's a journal fact you two were. what. it. i. it. the. the. i. well ok as less so if you wouldn't consider to pull to drink but usually rinks it's sort of general to the statement and sending there's some small small step showing that some of the continuously ages except to correspond so is this maybe you have to do by hand but it's not difficult. the it. i am no no i am sorry i don't claim that there is this the general statement i don't claim that there is so this general statement in this gentleman to but in this case it's ok.
but. an end. and but in general it's true that if you possibly completions and it won't change the space and also not a rational subsets a. it. the. ok and then. therefore to this gives one for two. it's enough to use in your universe of property. you. sos a complete professional swing will be affected if you have to pay algebra and it's clear that it's the universe have perfected the futility of most of the diet properties that some of the small business sectors will you. and for three notes that now. the it takes a document or for being is also sequence was i have why. and so we know that if he said. so he and this complex senate by a power of party and if you takes them it over for been yes this power of private again shrink by piece powers so who is almost extinct in the end. so again the statement that if something is true up to a bone of people aren't use any to write for venus long enough then it will be almost true and and then we can compete. i. and that's what he wanted to prove. the. the. a few. for a fee. came and in all the general case some have followed by presenting and the perfect a caldron characteristic p as a direct limit completed totemic of such be fine at other bross and assume that students so i. again assume that the characteristic of case people. and and. and in perfect order. if you don't pay algebra. our applause was a slight technical sumption that are classic a lot of the bar. to which i was sort of ignore this. they completed. direct limit. through to direct. the first such said all. the. all right last occupant. a. i. the second part says said that in this case. the the expect from of our of last year's again just as a direct limit as interest limit the expect trough. the swings. and again this the basically identifies rational subsets in the sense that the. any rational self said. you in this cause fix this. the cyclone. when you're next is a pullback. have a rational subset u.i. next eye for some i. the. the. the the it. he is. and. in this case. like so if you were expense of you is simply to derek limited wall j. which are bigger than i. of. all i have exchanged a few j. j. plus of u.d.a.. he said. and. i. it's one thing i want to say as you jays a primitive. you all i can exchange. the. and sings no comes the final part that is yes. the. he needs the following technical statement said if. the annex i slunk was a compact open. such said. the image of xsun x. i is contained in view. then the excess someday. such that also the image. of system that will be contained and j.. so this will lead to some most shows that if you have a cover up next to each individual term as cover can be.
seen as a public profile that level of the also need to see that some of the cover to find a level and this will be to this part due to this part of a lemon and then finally we can shows it. schieffer pretty satisfied. the. the. but. which. a. the. ok. so its previous. so in some sense the heart of sports part one which was in a very difficult so maybe do just what we expect to do so. as for example one can present entering is a future don't limit of address a fine a type so for any subset. i nod loss we consider. if somebody broke. it's all in our given us the image. of. and the question that. from that point only algebra if it's a finite. and the. the netting to our own. and. that aside thousand s i. the the pub on the elements. but conversion posterior something so to go fishing go to zero. and samoa so. so this gets induced apology. the forum. no them. this was not strict so as i does not have to subsidise apology of our. it's a question to apology to quotient is a. so in this way between become say. this was appalled wanted an immense really becomes a total find a type and its reduced because it's the suffering in some them. we'd used a few nikkei algebra. of top notch a different time. and. and now. but the. if the. the. the. i. then we let our i are a plus the. i. it was completion and. one can show that. in this case then are all costs. his way just to compete directly middle seas. i. and. of course some of because our offices perfect you really get some eps from our eye into our and you can check said some of this process will remain to this pos when year and. and the case give some of his presentation. and. as for to uganda nodes it. the spa justice dark the middle c.r.i.. consider a chemical c.r.i. plus and shia sort of have to do the same sort of verification that the continuous variations expect to correspond with cocaine and. so yes this would asian and then again. completion doesn't change anything so it's the same as expect from of are up to us. i mean i mean what. it. but i get it. it's worth it. yes yes. so we suspect the space is a spectrum apps that will become important in the second. so. part three years said. again use universe of property. and the and pop force slightly interesting so. and. it's a general statement about spectral the spaces and spectral maps that i want to show of sketch so let's call ai this compliment excite minus the so this is a spectral space. because it's this view is a constructive subsets and tense as the compliments construct will subside and it's the general statements it's constructive subsets of spector space again spectacle and that a j.v. the image.
woods. because all transition missile specials is again spectrum.
and the assumption. it is said see interest limit.
of course the ai a j s and. i. but. if you consider the age able to construct able to apologise.
and this is a really a compact apologise space so was a conflict and tolstoy of.
guinness general statement about the spectral spaces and because the transition maps a spectral and photo said.
if you give the a j just apologize on the transition ipsa continues. for the construct apology.
so now we have an interest limit of conflict a political space which is empty intensive has to be empty to find a. the. the.
the. some. it was limited. the space empty. ns and there's a general statement which is probably to one of their kids into to find that. the. and so you get it. the muslim and. and i think it. it. i. so to stay in some financing j.. i. j m t and then you take some cable just because injury.
and at this index which exists because we're taking food in their equipment. the single the empty and finally we want to check so she for two and for this note said. but we can all take. but conflict notation so it's uk be a fine a cover. i. i. white uk. his pre-match. of uk all i know where i was large and if you choose a very large and you can assume that it's same i fall k. and. then the union of the uk i. a.j. covers exchange a foursome. j. which is again dodges an eye and. so you may assume finally that. the u k i p image. of a cover. the u.. k i have some exciting by renaming. the stay as i possibly and. then fall. a bigger you call to our we have. he said that we know that the direct limit or the j. big open to all i have to sequence the euro maps to all x j a fixed a. just a product over a k.f.c. public state. have you k. ga going abroad and so on this is the exact. because it's said she was fine and to each of these conflicts as they can expect intensify equipment is expect and then complete can get the that statement.
ok. so maybe i should say two things one might also try to prove so this was all boy directly presenting any perfect avoid are the broad as a direct limit of the. arduous a finite type and then you would also get to some extent sequence like this but was also the almost because we're not getting the perfect situation and such said at each step its komaci the sold in india to buy them some power of party but in this argument is part of court course explode and two. i have no control of what happens and hence you sort of have to first go to the perfection as he challenged brought to see that it's almost zero as in so says almost often not explode anymore and you have it under control. can. and samoa again it's approved said. the truth and characteristic peons and understood the procedure. gets the resulting characteristics euro but i don't have any direct way of proving that this structure fishy for some preach this actually achieving characteristics you so. for. the. of it. but it. but if you are. about it here. at it. it is it worth it. they're so goes the thick of it. so you to view the sections to know that i. all the dust. i think they're to the island. to be sufficient to be using what good is no caned the and. i was trying to do. how can i. so far we have all the basic properties about half a few not perfect and spaces and finally we can define what a perfect to its basis so what did we do so for any. in fact ok algebra. our up first. we have an infinite expats. x. spa for us to work a. and. would be fine. and that a perfect choice space ok.
it is a team expects all ok.
so it's just a full subcategory of secret you have any space as was to a great man said it's locally sort of us have hinted factoid nature such that. in . and. but it is now a form ltd said. them. the. we have. thinking again a typical and so the category of perfect art spaces.
ok. succulence we could you have perfected spaces. work a foot or so.
a. it may very well. but first let's just have to find what i mean by an attic space of a case so it's basically one which lives of us bought kg and the plus cane old and she will be closed.
if you see it this way and so i only requires that this its local force bar a plus walk where are all class. if you're challenged on the censored it was a scale some of the kale jabbar was an open bonnet suffering and to stop loss is as usual. the. so we have this compare to take organs that is now we have waited kind of john metric spaces on both sides and the characters are going to mix characteristic is a series of just equivalent and so have some expects to make fled and.
then you have to form properties that the underlying technology has spaces are the same. and. the it.
it's as if you're perfect toilet. meaning that it's also forms ball are all plus were all process who prefer to the new killed in broad if and only if it's still this is so.
and. in this case.
the oink so thanks to express a fix. is saying the too old to its two. the was actually us. a. and it's really just a form a consequence of what we have proved it. it. the. but. if it. the. i. i. and. and we all need to follow proposition. namely that fire products exist in cutting of expect perfect its bases. the. the. and this is remarkable in that sense because for usual attic space is what always has to impose a fight this condition on one of the two arrows in order to get them. a year the existence of our products because some are usually takes place as a. to me you always see some find his condition to ensure that say this species actually achieve but not all situation the us where general also resulted corporate she was a sheath and the just locally years following so. if you consider the five a product of all plus.