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

Title of Series  
Part Number 
6

Number of Parts 
6

Author 

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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. 
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Release Date 
2018

Language 
English

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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
Point (geometry)
Addition
Functional (mathematics)
Group action
Perfect group
Multiplication sign
Projective plane
Characteristic polynomial
Line (geometry)
Graph coloring
Equivalence relation
Power (physics)
Field extension
Category of being
Root
Modulform
Körper <Algebra>
Algebra
Resultant
Spectrum (functional analysis)
Curve fitting
Spacetime
04:46
Classical physics
Point (geometry)
Multiplication sign
Propositional formula
Category of being
Ring (mathematics)
Logic
Order (biology)
Object (grammar)
Boiling point
Resultant
Condition number
Spacetime
06:59
Logic
Closed set
Limit (category theory)
Physical system
Spacetime
08:40
Residual (numerical analysis)
Pulse (signal processing)
Group action
Ring (mathematics)
Total S.A.
Körper <Algebra>
Limit (category theory)
Condition number
Physical system
10:17
Category of being
Multiplication sign
Physicalism
Maxima and minima
Limit (category theory)
Fiber (mathematics)
Spacetime
Product (business)
13:07
Mathematics
Perfect group
Ring (mathematics)
Link (knot theory)
Diffuser (automotive)
Netzplan
Limit (category theory)
Extension (kinesiology)
Resultant
14:59
Goodness of fit
Variety (linguistics)
Moment <Mathematik>
Right angle
Condition number
16:27
Torus
Point (geometry)
Projektiver Raum
Group action
Term (mathematics)
Modulform
Right angle
Open set
Flow separation
Subset
19:46
Causality
Sigmaalgebra
Moment (mathematics)
Content (media)
Square number
Mereology
Numerical analysis
Element (mathematics)
Stability theory
21:54
Torus
Point (geometry)
Functional (mathematics)
Group action
Cone penetration test
Sigmaalgebra
Variety (linguistics)
Uniqueness quantification
Open set
Quadratic form
Element (mathematics)
Natural number
Modulform
Right angle
Spectrum (functional analysis)
Spacetime
27:37
Cone penetration test
Algebraic closure
Variety (linguistics)
Multiplication sign
Musical ensemble
Open set
Line (geometry)
Associative property
Orbit
29:25
Multiplication sign
Moment (mathematics)
Line (geometry)
Physical system
31:00
Sigmaalgebra
Multiplication sign
Expression
Sheaf (mathematics)
Special functions
Product (business)
Spacetime
Hand fan
33:11
Point (geometry)
Covering space
Closed set
Proper map
Set theory
Spacetime
Condition number
34:58
Pairwise comparison
Group action
Root
Variety (linguistics)
Projective plane
Körper <Algebra>
Inverse element
Limit (category theory)
Spacetime
2 (number)
Hand fan
37:19
Point (geometry)
Axiom of choice
Group action
Perfect group
Variety (linguistics)
Characteristic polynomial
Propositional formula
Open set
Proper map
Pi
Diagram
Körper <Algebra>
Fiber (mathematics)
Covering space
Multiplication
Theory of relativity
Surface
Projective plane
Neighbourhood (graph theory)
Basis <Mathematik>
Total S.A.
Limit (category theory)
Cartesian coordinate system
Approximation
Equivalence relation
Proof theory
Tower
Logic
Order (biology)
Mathematical singularity
Right angle
Spacetime
00:00
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04:50
between the toss side and of expenses he tossed out of its too old and so some o x was just the points would just mean some of that the of sugar a good size some often know we have extended sizeable for them too. spaces and. ok and but some oh we have to relate sees it all to boil for affected spaces know it too tall to a boil for more classical spaces in order to make. and he uses this result and still we have some have to compare them with the tall to employ of classic original to cry cheese and soul. we have also some of the proposition that if. why is locally in the syrian. experienced work a. so this means these of the expenses said who were usually works was so there's some this year in this condition and post on the ring you and.
05:58
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07:01
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07:20
of this year an attic space is so mysterious local in the syrian close. while the company can call the separated. express this. ok.
07:45
i. then we ride. that takes a similar to the inverse limit of the x. i so no the first you have to give some apps.
08:03
that. makes mistakes i'm being a. you may have to resume it to the new system. the ride. that if the next summer to the dairy interest limit of six i. and if. first of all onto the logic of space and uses nice and what isn't.
08:42
the home office. well as your condition and secondly we need a condition some on the rings and it's enough to impose a rise of the condition which is that fall xsun x..
09:00
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09:10
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09:16
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09:24
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10:19
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10:44
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10:50
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11:02
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11:08
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11:19
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13:07
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13:19
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13:35
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14:10
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14:23
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15:01
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15:08
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15:16
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15:19
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15:43
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16:32
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16:41
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19:51
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20:09
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20:54
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21:58
to think my a. tory variety.
22:04
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22:33
so some are not contained in a country. putting its own. if the engine. the. and. i. the weekend. so here to signatory great excitement this goes as follows so. our what i want to infect. i. c n n. well let's see what happened so. first for huge stigma. and for each don't we have. i. the stool which lives in intense are. remnants of you will have and. and when he said. you signal to be the spectrum. so i guess you're so just one sigma one engine of. spectrum of. the. the saying so that some. a few more fun to type. these. these days. these. for. and then came the. the. also face of singer. one gets and you stop emerging. the from you sigma in the form utah into a signature somewhat to get stung to realty we really have to do is to it as asian procedure and the. the. some of this allows us to blue everything together and get. this right the excitement. and note said. if you take. the. the cold which is just the and zero. then it's the cygnet you will be everything intense this is really just a spectrum. of k. of em. so the suffering of being groups as we just say it toro rosso assist the tours. so this call this team and notes that the two naturally x. on. the next six months. and there's a natural base point insights years from one in t. gives a point. this year and then this we become some of the open dense always said he wanted to have. and then this year ms said. and he tore it right she. the form. signal and if we fix. the space point xsun x.. and then stigma is a unique. i'm. and. you need for ten. in. the coaker because of she turns around the. the. ok so this. ford has probably was a well known should say wants more saying some of that. i have an element in you and this gives rise to function now on this open subset she intends to rational functional to write to you and the lead kite to say you'll be the original function. the associated which will function and. the unexpected. the. and but you will need some the few statements about the virus.
27:42
on toward varieties and the nice thing about them that they as wall justified in a community or your men and so that's the following so efficient or proposition. full and there are some some devices which you can the discrepant come to toil he so. but she i said no easy one dimensional cones. i only want to first associate to reach such one dimensional cone somme veiled adviser on on the story varieties and.
28:27
namely the open subset corresponding to top i have to change of paulson's maybe it's a small pick two. the a fine line times g.m. to something. the u.. on humans when the music but intention and the. d.i.. it's exciting to be easy. the closure of the. the euro times you. some of the open substance were zero s. were the first corner this one zero six acres us open then stores orbit and then now we have something of course i mention one adviser. and for us. with us.
29:29
i guess for years.
29:35
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30:37
and. if you fix a generator. office. one moment while system line top i as an.
31:01
one yes the following expression for the global sections off. and and the use of some of the idea i. as an. h. lot of excitement was ways and off the. is he. direct some over all you an m. such that to scan a product of fewest we i was at least one is ai off. kuyt pay times a special function kind to you. and he said.
31:47
that. came. and because some all what we need about for groceries and now we want to define some of the attic space isn't affected spaces worsens of these righties and so let's fix a fan. and assume those of case a person on a comedian feet. then we define an experience. i'm a. which i was. surely x. sigma. add. because it's not in general the edification some of the genetic the if the education of says scheme x. signal which is good out of. you signal which us. he said. this thing and. i saw some all of them.
33:12
of stuart broad she was just defined space in this would not be all have to find space but just too close to the fore. the. some. it takes thing as a n.. than is clearly x.. his.
33:39
anyone could be and so it's easy. close to the fore. meaning that if you for example consider its cable your points. then instead of triple six one of two as a set of two point six one up to its nk city of today you have all its eyes that last one. the and bought. if. exiguous proper. which by the way is equivalent to the condition school recommit totally again.
34:19
as said a signal covers. and ten there are. then. the is clearly acts as it is really just the space associated to fix it. the. the. he's. the and if moreover case a perfect or field.
34:59
and be fine.
35:06
felix per.
35:08
as being blue dog of. the you cigna. her which is. the space when all we some will join all of the people or roots of see caught in that city have so we allow if you know when it isn't he. the thing. and then we have to fall in comparison results. i. the. five. by the way i have for the second this examples that i gave have sussed specific fan the associated towards right huge would just be. as to imagine a project of space. he said. do ok so it's a point zero. the. the. so we take effect on field. there was to build. they fled. and then we have. so was perfect or version of this toy variety. ok and distils two. the same saying work a flat and. secondly. if that is perfect or version. of the space. his basic is the inverse limit of sea.
37:22
the usual versions where's this transition map. fires and used by multiplication m p on him. then we get set in fact if one takes. the attic sing overcame fled and it's a someone allowing top a logical space and it's just the interest limit over fires to topple logical spaces socio to two. i sing over case on to send some of the singing characteristics start writing characteristic p can be seen as a pro cover of the same towards writing characteristic zero. in fact this is not only true into political space is but it's also true on the total point. takes the total purse schools he did then. it's the indus limit of the fiber top course. i was into just of. so. the three days and that is a pro career of this is true on to a logical basis and the total point. and you will need to following the. when most statement so for any open. you a downstairs some also over a characteristic and makes characteristic of. i was pretty image. the. on the other side you have a community's diagram of top lawyer. was it you can. rejection epsom oh from. a year too. i. inside u.f.c. open sub to post your tall and then there's the map here because again you can drive the toss the most an interest limit of two point you. a. for. for. for. so in particular so now. but if you take just a p n over. this characteristic field. then in some sense it's equal to. the inverse limit over five of the project of space over k. where this map on calling this is just given by these powers. the and. so this and uses a projection that some apology from the pin of ok fled to piano ok which someone doesn't really exist but somewhat least morally exists and for example it's exist on to watch because basis and intel to pour and some of this is on calling that's given by sending such as. to pull to shop representatives. the. some of its relation between the project to space as it were easy to fields unfortunately there's not completely from toy also. his identification somewhat depends on the choice of coordinates on p. and. slim someone characteristic p there's this canonical way of passing from this right into a perfect road space but in order to prosecute or to a six year from for the from a variety to a perfect art space you have to make some choice. and. i. ok so. this basically for us directly from what we have proofed and. we also need to falling proposition that. the. we can compare the komala jesus have to work towards right is now also assume. k.. this separately when you break the coast. and. then become ology of. the store friday and as unworkable m p. one in k.. the. the only known if that. so if you take. the commodity prices she said what else was young. then this so of years as a projection the pie. of those are some off if you have to become orgy of. back. the. and. for the proof just note that. so because of this equivalent of toys that the one is seen as the middle of the others you only have to see that's a tradition medicine this tower and uses more films. which is also something to eating characteristic zero and you can just someone to explain and. the . i. for the. and the. ok and. for those applications the way but also be conjecture you'll need an approximation them or and this is a following. so. when proposition. it assumes that. stuart broad to its proper. the and. that's why. in mexico. work a beehive the surface. and let us choose a small neighborhood so. in some of the associated which didn't have the right is already experienced. it's more. can a word. but. the. then there's this news. there exists a hyper surface. the in some of the space on the other side. such said. it's contained in the interest image of the small neighborhood of why.