2/6 Perfectoid Spaces and the WeightMonodromy Conjecture
Formal Metadata
Title 
2/6 Perfectoid Spaces and the WeightMonodromy Conjecture

Title of Series  
Part Number 
2

Number of Parts 
6

Author 

Contributors 

License 
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. 
Identifiers 

Publisher 

Release Date 
2018

Language 
English

Content Metadata
Subject Area  
Abstract 
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 FontaineWintenberger, 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 weightmonodromy conjecture.

00:00
Manysorted logic
Network topology
Characteristic polynomial
Körper <Algebra>
Complete metric space
Position operator
01:43
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
Manysorted logic
Root
Einheitswurzel
Körper <Algebra>
Series (mathematics)
Condition number
03:16
Proof theory
Group action
Root
Manysorted logic
Körper <Algebra>
Distance
Extension (kinesiology)
Absolute value
Condition number
Element (mathematics)
Maß <Mathematik>
Power (physics)
05:11
Elementary arithmetic
Perfect group
Manysorted logic
Term (mathematics)
Characteristic polynomial
Körper <Algebra>
Series (mathematics)
Inequality (mathematics)
07:01
Pi
Arithmetic mean
Perfect group
Ring (mathematics)
Characteristic polynomial
Valuation (algebra)
Limit (category theory)
Element (mathematics)
Condition number
08:58
Ring (mathematics)
Manysorted logic
Uniformer Raum
Correspondence (mathematics)
Game theory
Limit (category theory)
Mereology
Absolute value
Element (mathematics)
10:33
Uniform boundedness principle
Goodness of fit
Similarity (geometry)
Limit (category theory)
Mereology
Element (mathematics)
11:52
Uniform boundedness principle
Mathematics
Fluid
Process (computing)
Characteristic polynomial
Autoregressive conditional heteroskedasticity
Ideal (ethics)
Maxima and minima
Körper <Algebra>
Mereology
Local ring
Element (mathematics)
13:26
Proof theory
Goodness of fit
Insertion loss
Limit (category theory)
Determinant
Element (mathematics)
14:52
Field extension
Manysorted logic
Term (mathematics)
Projective plane
Moment (mathematics)
Summierbarkeit
Limit (category theory)
Inductive reasoning
17:17
Point (geometry)
Proof theory
Pi
Sequel
Manysorted logic
Term (mathematics)
Modulform
Game theory
Mereology
Limit (category theory)
Approximation
Directed graph
19:23
Curvature
Pi
Perfect group
Root
Manysorted logic
Weight
Characteristic polynomial
Normal (geometry)
Modulform
Körper <Algebra>
Mereology
Physical system
00:00
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00:53
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.
01:50
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.
02:24
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..
03:23
this summer's this condition years should. the exclude.
03:33
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.
04:07
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.
05:12
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.
06:36
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.
07:02
he came so its concerts. and.
07:08
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.
07:35
as a ring. in just one tenth construction takes interest women were for being used as a problem and elements menu pie.
07:45
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.
08:18
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.
09:02
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.
09:16
in this sentence out to be in this movie. the. so in particular.
09:29
it. to get a map.
09:36
from this investment. of cannot what pied two.
09:46
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.
10:45
i. mccain and then we let you know this to build the it takes in muslim it.
11:01
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.
11:53
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.
12:37
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.
13:08
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.
13:28
he soon. a. a. or maybe a good show proof of this and so on.
13:40
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.
14:22
and. we lived in each of the deck sidebar just arbitrarily to some.
14:34
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.
14:54
and so we have to see that this makes sense and so we have to check said.
15:04
the courts enough to check that them fix and. the school were into exam prime what you do park and.
15:18
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.
15:52
it's ok and.
15:57
right so that we have this map and then we get.
16:08
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.
17:27
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.
18:40
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.
19:24
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.
19:45
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.
20:15
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.