2/6 Perfectoid Spaces and the Weight-Monodromy Conjecture

Video thumbnail (Frame 0) Video thumbnail (Frame 2565) Video thumbnail (Frame 4895) Video thumbnail (Frame 7765) Video thumbnail (Frame 10525) Video thumbnail (Frame 13459) Video thumbnail (Frame 15823) Video thumbnail (Frame 17801) Video thumbnail (Frame 20153) Video thumbnail (Frame 22293) Video thumbnail (Frame 25927) Video thumbnail (Frame 27984) Video thumbnail (Frame 30191)
Video in TIB AV-Portal: 2/6 Perfectoid Spaces and the Weight-Monodromy Conjecture

Formal Metadata

2/6 Perfectoid Spaces and the Weight-Monodromy Conjecture
Title of Series
Part Number
Number of Parts
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Release Date

Content Metadata

Subject Area
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.
Many-sorted logic Network topology Characteristic polynomial Körper <Algebra> Complete metric space Position operator
Point (geometry) Correspondence (mathematics) Characteristic polynomial Valuation (algebra) Similarity (geometry) Routing Lattice (order) Complete metric space Numerical analysis Element (mathematics) Uniform boundedness principle Goodness of fit Many-sorted logic Root Einheitswurzel Körper <Algebra> Series (mathematics) Condition number
Proof theory Group action Root Many-sorted logic Körper <Algebra> Distance Extension (kinesiology) Absolute value Condition number Element (mathematics) Maß <Mathematik> Power (physics)
Elementary arithmetic Perfect group Many-sorted logic Term (mathematics) Characteristic polynomial Körper <Algebra> Series (mathematics) Inequality (mathematics)
Pi Arithmetic mean Perfect group Ring (mathematics) Characteristic polynomial Valuation (algebra) Limit (category theory) Element (mathematics) Condition number
Ring (mathematics) Many-sorted logic Uniformer Raum Correspondence (mathematics) Game theory Limit (category theory) Mereology Absolute value Element (mathematics)
Uniform boundedness principle Goodness of fit Similarity (geometry) Limit (category theory) Mereology Element (mathematics)
Uniform boundedness principle Mathematics Fluid Process (computing) Characteristic polynomial Autoregressive conditional heteroskedasticity Ideal (ethics) Maxima and minima Körper <Algebra> Mereology Local ring Element (mathematics)
Proof theory Goodness of fit Insertion loss Limit (category theory) Determinant Element (mathematics)
Field extension Many-sorted logic Term (mathematics) Projective plane Moment (mathematics) Summierbarkeit Limit (category theory) Inductive reasoning
Point (geometry) Proof theory Pi Sequel Many-sorted logic Term (mathematics) Modulform Game theory Mereology Limit (category theory) Approximation Directed graph
Curvature Pi Perfect group Root Many-sorted logic Weight Characteristic polynomial Normal (geometry) Modulform Körper <Algebra> Mereology Physical system
when ould but was stopped by the. defining what a perfect it feels so. for. so rocking the set up so full of and not human enjoyment tree so we're working this document a few throat so we call that in on a communion field. it is a couple logical. the field. i must apologise and use play a non-trivial.
a new spin on trivial. the way she offering phone. so. so in particular you have sort of an all met from k. to positive for us which is essentially unique and. i don't require that is complete it. but it's not for perfect it feels i will make this week why mental a perfect outfield. it is a complete. not a communion. i feel ok. he was falling crime months it's whether the characteristics of the positive.
so a six point number of people going to than zero. one s. to require that the when one valuation that this sort of. corresponds to it is not a screwed up. the. the end to the condition is said for being years. on the pub on the elements model new pew so expect six appeal.
is subjective. i. the cane and so they're sort of examples are. i. in that case the completion of course the following types the field. the tainted take your p.c. but. then the for being is what the subject if you because it's just a few good. this sort of you don't want this school. we have to join lots of people routes for example or. one could also joined the people roots of unity and. or you could just go all the way up to see peace. or you could also him do similar construction in characteristic peaceful take a long series field and enjoy all the roots of g..
this summer's this condition years should. the exclude.
and when we fought extensions. of q p i don't want them to. so one of the one thing it implies for example the following. or. and so the way your group of the cake cross in such a poll to fields this is a pity was a group that so and distance and now you've sent said.
any of them into the piece rude and it's very easy to proof was forced. so let's come a business where your group. then i got most not just. the powers of the introduce just because sort of he excludes the enemy for extensions of keep your use units on the screen here. and. so it follows that got most generated by. the absolute value of x. four. it's elements which satisfied that the absolute value is tricky lodges in peace between the wind. i. in so we only have to find peace roots of such elements and than that. there is some why such that the a few biopics one is why to the p.. this in the uk today of peace.
by this with crime and it's really is a subjective more you people and so than if you buy your flight to the p. is exactly the opposite way of exposing. strong trying and trying or inequality them. the. in the summer the perfect it feels separate into into two cases so they're the that's the case a characteristic of case here in this case the subprime and simply month saying that case a complete perfect company perfect on a communion field. i. so it's something i'm pretty easy to avoid them. and some characteristics year it's sort of. in the most subtle notion. that's now we want to define its tilting funkier. which takes any. perfected fuel to the perfect if you have characteristic the and then we will also have in this whole series of talks see that the whole syria but though was not the most perfect if you have characteristic zero has sort of the spell of to the serial the characteristic the field for everything can be expressed in modest elementary terms so we're supporting.
to from the all perfect it feels. to you. those of characteristic zero. of interesting people. so it will be noted came its to pay for that.
he came so its concerts. and.
so let us to some. the sort of uniform either but as a valuation is known as creepy see that it doesn't quite make sense to talk about your for my is what gives you some element of. that is not too large to its that most and not too small and this and one and consider it.
as a ring. in just one tenth construction takes interest women were for being used as a problem and elements menu pie.
so this condition precisely interested. as a swing of characteristic peak. meaning that piece year on the swing to make sense to take some risk limit year and so this is a perfect ring of characteristic peak. you and it also supported the so to see and with limited apology.
golding said the sings naturally have so discreet apology. the u.. ok. and. so. now we could up to tilt out of it so would say it's on them as about those behaves. i. the moon. there's a will to pluck it to have a. me i'm off isn't. from.
you can take serious limit of support wanted elements now and it's a piece poem that which no in general is no during almost as many more. and you can project is to this ring.
in this sentence out to be in this movie. the. so in particular.
it. to get a map.
from this investment. of cannot what pied two.
it. into them cannot which i do know take steps to. the extra are. and some all by taking the winners of this as more physicians in projecting to the first cotton and and. the second part is that the game they're saying some of you want a corresponding sort of uniform was on the other side's and so we to some. there's an element which have been called by fled in this it must limit. such said. corresponding element into not stop him and it's not necessarily equal to piebalgs said as at the same absolute value will be enough.
i. mccain and then we let you know this to build the it takes in muslim it.
over fifteen years. and then sending goods element life that. the and can do. and then third part is that. if it is indeed a perfect if you have characteristic peak. the. i. and and. i still have says a small business case that can be. get it by taking his interest limit over the piece palm up on k. and so in there we have again the pub on the elements. they are similarities in this limit.
ok for cirque and which by this size more things in also a swing year and if you want to be can also sort of consider the maxima ideals offseason local rings and fluid which is he said of elements of. no messin one and is also sending with movement. one or correspond some of the new. and what else can we say them so we this provides us with a map. if that's the case.
expects to extort some extending the steps that we have here in the pub on the elements and. i. if one takes his arch brown takes the still to dodge a bar and i would use again would use his uniform either then this comes can only the ice more fake two. the small part some o four characteristics zero local of four characteristics your perfect right field.
by then finally some a creaky of the characteristic of cables already p.. i. then the whole process didn't change anything so. they said it's just a.
he soon. a. a. or maybe a good show proof of this and so on.
i. so for wanting to first we construct. i. the mets from these in this limit. or for being years. to not lose ecliptic stop it. and how do we do it so we have. i. some element of this in was limited.
and. we lived in each of the deck sidebar just arbitrarily to some.
next i called on determines. and. the we define. explore. as the limit is it then goes to infinity six and two p.c. in the.
and so we have to see that this makes sense and so we have to check said.
the courts enough to check that them fix and. the school were into exam prime what you do park and.
the next into the park to see an escape from one tricks and prime to find the end. when you apply to the end of one for example and. the sum of each term of the sort of approximates a sex shop to larger extent and so this can be sort of proven induction and. i.
it's ok and.
right so that we have this map and then we get.
i know i met also to the inverse limit of what was a piece parliament. i have to sing and are just sending. at some moment x. year too. pixar it's one of the peach top and so on. so. and it's clear some of that this met sharpest the multiplicative in continuous some say this. i. but then so does the metaverse the projection. and when usually do you do support one.
and. the. and. the. and so for proud to be in the. the. four. part two we. have to. find a sentiment and still. if we first take some pie one such that a normal point one k. not as such it's a normal fly want to the p.. sequel to the normal by this exists because or where you get this proposal and. then we take any. i fled to sort of form zero point one and then something in this interest limit over for being years in the sink. or and.
those in the proof of pod one show said. and if you want to calculate the spy fletch sharp then. as a first approximation we can take some peace corps of the spy one and saratov term will be sort of pie squared and still is already implies that this is desired. the game. and. the bone. it.
they want to see that. or as these things are satisfied but because. he said as multiplicative it's easy to see that it extends to them this map on a flat.
given this was the most isn't and. they can also see that the norm on. they fled was given by. some of the norm to the notebook thought said this given by.
the. the weightings enormous problem and then it's easy to see part the other things and no. the. muslims for. i. the and so i mean to see there's a perfect and field we only have to see that by the me it's perfect it's clear and if she is complete but it's also pretty clear and then no no that's perfect if you've characteristic p. and the last towards it. if it's already have characteristic using it's the same as kazin the. yes in some of this is the investment. where the piece paul map which know this case perfect in this case just came. can you. be. the. the and. i'm so you'll be no redefined. all uniform as opposed to be off this form type that sharp so it has the same way you asian food doesn't quite change anything. but this is the advantage that. the pie sort of has a canonical system a few to the un's roots so. given as first taking the p. to the institute of pipe that which exists because has lived in the perfect field and then taking.