On the vanishing of negative Ktheory
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Title 
On the vanishing of negative Ktheory

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Part Number 
3

Number of Parts 
5

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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 
2016

Language 
English

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Subject Area  
Abstract 
Weibel's conjecture predicts that negative algebraic Ktheory vanishes in degrees less than minus the dimension of the ring. The conjecture is known in characteristic zero. In the talk I will explain an approach which reduces the general conjecture to a very weak form of resolution of singularities.

00:00
Statistical hypothesis testing
Group action
Exact sequence
Matching (graph theory)
Lecture/Conference
Modulform
Object (grammar)
Sequence
Subgroup
Resultant
Dimensional analysis
Element (mathematics)
01:50
Point (geometry)
Group action
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Multiplication sign
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Characteristic polynomial
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Parameter (computer programming)
Mereology
Rule of inference
Dimensional analysis
Goodness of fit
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Agreeableness
Negative number
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Ext functor
Nominal number
Line (geometry)
Measurement
Numerical analysis
Residual (numerical analysis)
Inclusion map
Mathematical singularity
Arithmetic progression
Spectrum (functional analysis)
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Local ring
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11:40
Point (geometry)
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Perfect group
Logarithm
Image resolution
Characteristic polynomial
Maxima and minima
Mereology
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Theory
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Reduction of order
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Lucas sequence
Commutator
Maxima and minima
Mortality rate
Complete metric space
Sequence
Arithmetic mean
Ring (mathematics)
Vertex (graph theory)
Game theory
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Resultant
Local ring
Gradient descent
Inductive reasoning
19:43
Group action
State of matter
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Sheaf (mathematics)
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Uniformer Raum
Manysorted logic
Different (Kate Ryan album)
Homotopie
Square number
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Fiber (mathematics)
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Block (periodic table)
Closed set
Moment (mathematics)
Stress (mechanics)
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36:00
Group action
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Sheaf (mathematics)
Maxima and minima
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Student's ttest
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Element (mathematics)
Prime ideal
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Sign (mathematics)
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Theorem
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49:13
Group action
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Prime ideal
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Forcing (mathematics)
Algebraic structure
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Oval
Network topology
Order (biology)
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Spectrum (functional analysis)
Directed graph
Gradient descent
54:18
Presentation of a group
Group action
Multiplication sign
Direction (geometry)
Propositional formula
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Mereology
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Meeting/Interview
Different (Kate Ryan album)
Homotopie
Oval
Square number
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Fiber (mathematics)
Position operator
Descriptive statistics
Area
Covering space
Block (periodic table)
Nominal number
Price index
String theory
Discounts and allowances
Category of being
Proof theory
Ring (mathematics)
Order (biology)
Mathematical singularity
Resultant
Point (geometry)
Computer programming
Image resolution
Similarity (geometry)
Regular graph
Rule of inference
Theory
Element (mathematics)
Lecture/Conference
Term (mathematics)
Reduction of order
Modulform
Addition
Multiplication
Focus (optics)
Surface
Commutator
Algebraic structure
Division (mathematics)
Game theory
Local ring
Gradient descent
Maß <Mathematik>
00:06
the government has backed the bill yeah I think you are a match for the invitation to speak here and should make 2 preliminary remarks because of all the results of to speak about today are not as complete as I would like and 2nd I mean that I should say that most of them somehow came around discussions with nephew moral OK so in this talk by a scheme makes I always mean this scheme which is classical objective of 1 of the rendering of financial dimension for simplicity walking and 1st of all we want to study K groups so let's start with Kizzie walked out of the the next and of course classically you can just define this test the free European group generated by the picked up on ingredient on next what you will the exact sequence somewhat you the subgroup generated by elements of the form the prime that's the diamond prime minus convenient there you haven't shot except sequence of this fall OK and then best
01:53
tried to work defined not only Kasey will but all of us to define a negative cake and that his ideas as follows 6 so he made some on motorbikes antidoping process so he defined came minus of X to be some sort of deal version London of his evil in this number is a construction which reads as follows NEC find at the pool all of the men from his of DEC's times anyone that we just right extends a as 60 cities upon it 1 last Kasey takes times a oneputt note the parameter maternity to minus 1 and then disposed to X times With the natural met and then you take a call coowner of this and colleges a minus 1 and now you can go on and define came under strong effects similar just by the same construction land construction that apply to remind us what it does to send coconut but just replacing Casey will pay my muscles and so now best suggested to study these groups and that this is motivated by the following fundamental theorems due to fast and it fall include I also think quantity and assess the following In the 1st of all it if you applied this number construction tool positive fury defined as defined by couldn't then you get you mean you could just get K I minus 1 song for hiding began and evil and under of Islamic construction like to play of Texas just hire minus 1 and the 2nd statement but it is also some old part of this fundamental theorem is it if you applied this review make look at this construction of negative hate groups to to reckless scheme then you this manages to for extra it is true that OK I of Texas finishes for life president talking to somebody's negative gay groups are measure for the singularities uh and but it's a very complicated amount to :colon and the most important conjecture about these negative cake groups of schemes is the conjecture of violence that I want to discuss in this talk this source fulminated is a question by by 19 inches 1988 and it says the following 2 things many even forcing the schemes of final mentioned these negative K groups finally vanished what precisely is KIO fixed and the old if Chai less than minus the "quotation mark dimension of and there is also the 2nd part of this conjecture namely that's OK prior then time is equal to the dimensions minus the dimension of text then this is homotopyinvariant so OK I'll fix that thing toward Pat I of fakes times maintained a fine line it is Nizam office all this is true actually on I'm innocent people to minus the amateur so this is the conjecture usually Italy the final version of the abided defines it the yeah let me come to the assistance may just proved as conductor but let me come to the cinema is brutal so let them let's call this conjecture W C X for this gimmicks and just 1 more small indentations so let us call W indeed the statement that you know this conjecture for such schemes of dimension at most the do so for all 6 of dimension most and then um physically or in in the last 2 years the following theorem has been shown 1st of all is the 1st major progress is that why the short this conjecture up to dimension to it so we have statement W C 2 1 which is due to wider then we have important resides didn't it always follow the writers of captives agreeable so what next although some few decay of character 6 local finance time and this is a result due to continuous his Amaya cheating invited and then also be if he assumed very strong form of was emotional something that isn't positive characteristics of a perfect feed he also get the right conjecture therefore the subjects OK so perfect shield envious assume get exactly the same statement as proved by Hamanaka in still I was writers elected this good strong resolution of singer what so this is what is in the literature mourners now that you can come to work so too to grab us question and you can also ask the same conjecture for what became a just mentioned this shortly so modification you the following that gave you defined by by the 1st of all I considered the standout under by Jayson Blair Blake's over the integer just to just make you take when a Marine I was to the old TJ I you this idea and then homotopy Kseries defined as follows namely on what appear capable of of the scheme X is defined as 5 from atop the Group of the homotopy calling at the nominal bright K of scheme by this I mean there None nonconnected caterer spectral and to a secular model comic of equanimity holidays simplicial spectrum and then you take and then the new fear my 1 2 0 presented today at squalid is the following namely 1st of all if you replace in the formulation of fibers conjecture the cake group Idaho motorcade became group then this is true soul congestion on the interest so I think this was gonna ask your question and 2nd so what can you say for K through this so that the results are not as complete so we can't rule by the conjecture for this scheme X and the following assumptions 1st of all you need some sort of fine and this assumption namely if all the residue fears of positive characteristic of Aubenas final it thinks it's a point of heaven itself so it was a federal state dinner for being used as a fine at an office in London and if that this is only a very
11:38
small assumption which depends on some technical problems of
11:41
topological cyclical monitored and if the very weak
11:44
form of resolutions singer that his horse which I want to call best endeavors the and explained later so this is just an extremely weak form and I'm quite optimistic that we are able to prove this fall you could also assume just that the dimension of X this lesson could for it so on and in this situation we can prove this kind of resolution problem so
12:10
that's fitting resides in over there that statement the question the nominal and I come to this and there's some Mato reduce it to vise extends gains in particular so the sale of a special point of this Part 2 of this part 3 of the immediate their nomadic conjecture was for schemes of characteristic zeal could find even containing Jew so but is I mean this is mostly due to Mecham also with some contributions premises OK so not a young already mentioned you you have to reduce some holiday this kind of either convicted 2 7 4 with better schemes like excellent schemes and so on so let me know I spend his reduction and although these reductions are really elementary I mean I couldn't find them and even the literature so it's quite interesting but chose to use for the U.S. is onerous nonaligned it's always assume office for yeah and close calls for Syria to fuel that is meant as a scheme known as means such a good scheme you can generalize it of course to more general schemes but then you must not require this kind of Cape Fear but Casey of perfect complex or something that involved his duty yet the reductions I'm coming now so I uh that that been what at mean reductions the soul 1st of all if the following kind of production namely if I know mistakes as any scheme if I now provide conjecture for all the stocks of the substructure she then I know the wider conjecture for x so maybe the ideas presented at expanded quickly sold out for this kind of game theory have tells the descent spectral sequence of avoiding form the tool Turner's just because the Coleman Aegean of that sells the occasion of this None None connective the when this convergence and paying clients he accused of bricks an hour if you assume that you have this by the conductor you get some managing of this chief at certain stocks depending on the dimension of this point IN ANY futility ,comma and then there's this sort of finishing result of Courtney shares added political munching ministers followed mentioned schemes but you can John marginalize it or given version of this system it's also for such any any she's which have so damage support the roughly speaking so then you see that for miners P minus Q was smaller than the dimension of X assuming this year the vanishes and then you use "quotation mark make brandishing foot the necessary example the next reduction in there and also surpassed tool an excellent brings an excellent schemes and particularly so let's say someone negative 6 indigenous then we have the following reduction if you know the wider conjectured for certain rings ,comma which compete local rates of dimension and most of the for such rings then Fiat Uno divided conjecture for schemes of dimension up to the McCain sought to put divided condition and was adjusted to prove prove it for a complete local In particularly are excellent and we have no for the new later the whole the strength of your knuckles emotional singlet AlQaeda idea again it's not very complicated so we proved this by induction on the dimension of X Of course from the old up to the them so the 1st of all by the 1st position I can assume that X there is an ogre scheme then IT knowledge by accepting bids the completion of the respect to the maximum idea and I do not buy smaller takes the closest point of fixed no we have the following commutative diagram with XX roles roles are just that localization sequences of gave their support of Thomas employable take care plus 1 of 6 0 at the close .period and we take the eyes of its support and applause .period when you take care bye of texts and then we take pay of 6 of the close .period and then we can write the same for the completed rank the fact is an many of course vertical markets like this and the roles I accept and what has been on the note this year's Nizam offers the excision theorem of Thomas and football for Cape theorist supports because its members fled you also know now if he assume that fire is less than minus the dimension of then these schemes
19:45
here they have dimensions 1 less than the dimension of X sold by action assumptions we know that picture these groups the Greenwich as far as his groups so we know that in effect here we have also isomorphism so we see that he may have an isomorphism and we assume that here the school brandishes the or assumption or cancel this simple people so now we have produced some simpler situation but he can't get his example have resolution of similarities because he'll knock approved is for closet excellent local rinks now so the basic idea now In all it proves so that I know of that of fighters conjecture is somehow 1st of all you have to know you need some sort of finishing reside finishing like for example Texas regular than on negative cables menace and you need some sort of the sent results so you somehow result thing letters of X then you applied astonishing and any apply some sort of dissent this is all very small is the same structure in pool so 1 definitely need some sort of dissent under it said so this is something I want to explain now and this is motivated especially can by some work of Matthew Mark so let's consider what kind of 50 Cent 3 1 to consider his consider and the pullbacks failed schemes of the 4 former exprime MEPs to X Y and that she then we have a closed set scheme here while fixed and we take Quebec and quality while Prime so then this is a public "quotation mark a class and we assume there a P E is projected and we assume that Peterson isomorphism outside wife so in other words uses Nizam his From exprime white prime toward takes too long life because such a thing in abstract blobs clear Of course of course if you block X and widen this there would be special kits note from elected the city center so let's say why it is defined by the coherent idea chief then let us set y tend to be equal to the such grows up scheme defined by the coherent ideas chief their fight to the end and it does defined take a group dedicated furor spectrum former schemes associated from scheme Whitehead to be beat the homotopy limit wall and offered pictures taken of white and then the cake groups itself had just crossed the defined the ,comma Topeka this spectrum can now I can't consider the following apply just a placate fury to mice Graham on this accord could flying scale fact the few spectrum to X then I consider cases victim of this former scheme Whitehead and consider carefully spectrum of the UN and OAS former scheme by Prime head and all sorts of directions you quantitative exprime and I ask that this square is homotopy contusions so this is some sort of dissent property property and X as stated the Expos if this is homotopy cut for all abstract blobs Grasmere sits in this corner then I say that the 6 points OK so why should we consider this song some examples where this kind of dissent under stress drops holds following which are known for example if the mothers and P from exprime 2 exes finite then follows from the vocals of skier on accession fucking fury and sunblock of methanol now the example is having this so In this case we know it and the case is if exprime then a abroad of X and closeups scheme y and if y 2 x is a regular closing emotion so then we also know this for this special case we know them however I should have consistently pressured the electors of planned order for orders Phoenix I'm dentists 1 and this was shown by Thomason it will usually go away you have the loser year's over I up but in the completion it's a whole Amin was policies are not regularly most but I mean if he she and findings the consent of the policy can just take a different there powers of menu does a different system of ideas which are I mean IndyCar London all of the the um so but this is just a special case so what we really need some more general statement and this is the 2nd main fear and other staff and it says that in all this send conjecture for Cape fear for the scheme X 4 makes classical projected over ring are artists compete local and again have to win posters there for being spine at Nestle it's a M is the next Medea and the 1 that is feared his whole being is finite and we also have to impose certain very reform offers me something that is that I want to spend now just for what it would take schemes over this ring are to be precise so now I really have
27:55
to expend this unpleasant assumption here about so to say resolution it's rather than resolution thing that is in some sort of principally as a nation of ideas she so it started this recall the following definition of it will not come a close immersion of schemes wine X say that this immersion this moment pseudo fled I played in nation and it's a talking excessive intervals scheme Christmas tree to this case if 1st of all why has not been and secondly if I consider block of X and Y then I ask that all the fibers Of this global why have the same dimensions In the dimension necessarily has to be the core mention of wine X I mean I don't know laughter but do you allow dollar the well a form of viral you know I don't care I don't care so this is Hamanaka's definition maybe this can happen I don't have an example this will be the weather you None of this is that this is a part of from above so they the dimension could survive as it always has to be at least this could imagine but I don't want it to be that demand imaging can go out anywhere I don't say anything about that all components of and mentions so this is of course is the conversion than Nomar flatness knowledge flatness that would mean that then somehow the exemption devices fled Monticello roughly speaking from camp and there are no I can form this was motion assumption so this is a statement followed by the protective integral schemes for what this fixed it's a complete Lokeren power and therefore any Monday closest subskin zeal of X but there exists some composition of blowouts and now meets Sudafed determined "quotation mark P from external critics that NPS means no flat and a factorization sort and section and the pullback half the year 2 exterior is an locally principle is defined by locally principal idea she I don't have a lot of people deserve to hear the views of all In the end evidence today at any money then center so the point is that this is exactly the minute he will knock approaches was emotional similarities he states he can't principle lies sounds closeups scheme by a certain sequence of up mainly in regular only that sent us that's the important result of fuel market but she it's much weaker in so I'm quiet optimistic that actually does this kind of the resolution process and can be achieved so unfortunate have no final result it's just state some partial results this what you do you have your the conclusion is that this dissent property for K period was for any the abstract blobs Fairbanks sits in 1 of the corners if X is closet projected over although this complete lock ring which is probing is finite and as you have discussed emotions options so this kind of resolution problem is actually much closer to a process "quotation mark according according the cases which was the developed by cuttings so I cannot say very much about this and just 1 remark 1st of all it is Our contends the Russian numbers then this dissolution problem it follows from the Book of fuel market so if the residue Patterson said I was always a complete local rainfall could be any excellent Lokeren this is the 1 article but the following thing I can prove a modern IT namely this kind of the resolution problem now for some scheme X the provided like this and any closer Norman subskin zeal there can be proved up to dimension free it can also be approved for their close substitutes Xeon X is the dimension prophesy is listening to what In TI some I think and this form let off cover 2 million annual visitors dismissed a similar question all so notice was just preliminary so we want come back toward the of fighters conjectures under certain assumptions so now what we want to do this we want to work did you say term fear style I want to be more precise and just want to restrict it to the 2nd part namely this year the year of this resolution assumptions and this being as finalists using this for and other stuff achieved
36:05
so this is quite nice so as I said to me it sounds menacing something like wrenching for reckless schemes that we need some dissent OK so many firms expand we cannot really resulting really was a Louis in that history cannot stick to address the fundamental theorem about the vanishing of Cape Fear of reckless schemes so what is the replacement of the squad useful contradictory proposition so consider reduce scheme and it's consider um case class size in Korea minus Jr off makes J as any positive integer then I claim that I can after some modification I can make this CSI managed print need exists modification pleas from Texas prime protects so by Irish a projected by national such that the pullback .period of CSI in minus J both fixed fixedprice this venture and so if you if you have a arisen was something lattice of then you can just take take exprime ministers was emotional Zanetti because it can of course all the class because then the school said but so this is a replacement for already 1 replacement for resolution of saying that and the key thing is that uses innocuous that if you someplace that more so 1 trillion preliminary Marcus the following namely the existing natural ejection 2 1 the group came minus J of so explained to you that this comes from the Steagall looking might exercise and G. M repeat I might apply by J. copies of GM inanimate out something of this case the local this scheme so at least there exists no surjection compares your off ex Thomas G. and for detail put utility image in this case the old world of his yourself X times pages at least this is modeled maybe more vivid on care for the moment he just considers ejection the latest come out pools this proposition he proposes to 1st of all that we can assume so size not represented by an element in this group so without loss of generality can assume that is represented by vector by times GMT since then as a sign on can extend the specter of 1 on this scheme is the arrests came as a coherent chief at least so we can extend this to withstand thing called on to was called the bomb which is a coherent she's on its in jail now because X is reduced I can find an open bins such scheme of X session the this fled or so the urgent the by restricted to you would times a day the to you on this I mean that I mean what I mean is of course it is chief inspector of pumpkin but now they note was almost ready because of like at mall visitor following namely assigned confined blowout 2 for the next body on which makes some holidays the baffling more precisely what they tell following need exists some projected by IrishAmerican .period from exprime text but it is an isomorphism hallway you'll end such if I consider the socalled strict transform all of the and let's squanders the GOP primary which is now the coherent chief on exPrime times a jail Mendoza's fled although takes pride now what is the students home just needs to take the usual Quebec of cohesion she and then UK the torture the can all sections which menace or when the prematch of you this just a caution of the initial pullback so they get a gift to give you this but then it follows that automatically if this coherent chief has finally told mention this is an easy check the reason reasons just that it's flat or what is exprime and of course this scheme is this move over others by now so then it's the difficult to OK but any coherent she is financed .period engine gives rise To some cars Kasey all this show make will be but just wasn't resolving it by the governmentfunded resolution Manitoba and so then this element is the cohesion she gives rise to a class in Kitzbuehel X times a day and of course if I restrict this the school him chief here to x times GE To the jet GM to the Jr I get the class of the pullback of my origin and CSI song so then this restricts opinion promised staff of CSI this induces the sentiment and then
44:08
of course any such element managers in negative K groups as I explained 1 is that foreign policy a few of the storms of CSI and sent in important observation so now I can come that which approval of theorem stance the 1st applied by my reductions does explained I can now restrict to schemes which are closet protective of all compete local rinks so from now on in this proof the by scheme all this mean this scheme which is the project all but it wasn't thinks complete lowering and for such schemes he won't approve them this vanishing conjecture of either by induction the dimension of X so let's consider some element sigh and care 5 of fixed their kind assistance minus dimension of X so we want to show that CSI vanishes so I want to stick out to where the 1st part of why congestion at the homotopy immense was simplicity now the key propositions tells us I mean 1st of all he can easily reduced to X reduced escape this weekend and find some modification which carries this CSI so choose a modification people Friday from breaks trying to X such at the that both disks on a team bench of the price King now because the is projected by Russian Diffley find the closest nonIndians subskin y of X such this piece an isomorphism wife now we can consider the following the background rules on this or not and I don't think that's what roles but they just use ally descent property namely the users fear and double standards because of some sort of dissent relations and immediately get an exact sequence of the foyer informed KIO of X receives paid higher classes on the from the scheme why Prime head Mendes goes toward the of exprime last I of former schemes Whitehead so this is exact and what can we say and now we have Alderman CSI and why and by Prime or them Italy's former schemes have been mentioned less than the dimension of X so by our induction assumption we can actually apply applied we can deduce that minus I plus 1 of Wyatt said finishes this is the induction assumptions but then it's not difficult to see that this summit is not reduced activists image this sitting in this former scheme doesn't change fury this some easy option that just based on the observation that in that case the hope of luring does not depend on you put and elements can just move of the tenements with or changing Kasey overrate you can insist but you get a mention here and by the same argument this is just the case of
49:14
what Prime and then was a violent action assumptions this venture fear and but this means of course that can forget about this match here and say this is injected but I already know this psychosis evil here so I I deduced that the OK so this is the basic idea there has usually just before he was just what he is going to be things here has so you realize like this he has to find prime I have to use it so classically a there there was a real something that is was used twice in all these approaches you can't old they say they want the next thing just hear this formula might just for those close to you recently None of you need this of this condition is resolution in order to prove 1st a 40 cent focus fury and then classically it there instead of consolidated people used because use of the resulting lettuce here so classical solutions yet this is proved the EU's encompassing zeal for example in the book of fighter quirkiness and sewn into a different base 1 tool to obtain dissent and wants to walk a pulled for this kind of people position so let me know um sustained just a few words about this whole weekend all sent fodder so this year and other stuff no Saul they ingredients this an idea going back to a cooking as is my violence on this group of people namely these spilled up a fury into something apart and some parts which doesn't see the number reduced structure talking to this works as follows so instead of the Cape Fear of X he considered the socalled "quotation mark infinitesimal K fearing Of the 6 which is just defined as the homotopy 5 all the picture fixed meditating by the tool topological cyclical commodities spectrum of and for establishing sinking commodity this kind of of this kind of dissent property located .period that was proved by Dundas and so here we know this kind of dissent here we want to prove only and is enough to prove this descent for this quite so good in print as McAfee few and important property of infinitesimal K fury Is that OK in and all X this the same thing as came into force be taken to the of the Associated reduced scheme this is the result of could lead in a coffee so this is ridiculous so that's the most important thing to do so what he knows With for this case invite can apply some very general kind of dissent reside which generalizes biographies mine so I want to explain our some about some commodity .period schemes which guaranteed at this ,comma fears satisfies something that descended on extra abstract blobs clears the storage of Asmussen has amassed users and he really use the strength of he'll knock us was elusive singularities and a new thing is that I make this work so to say this very weak form of resolution singlet of social principally session of ideas source now we've got a lesson fakes complete altering this is fixed and consider a commodity furious of schemes many consider schemes which are closet projected Obama tool the Kentaro spectra the political spectrum and this college age and
54:18
then the odds of a relative age namely if you have closed aggression while sitting inside 6 D consider the relative age .period age of excitement too wide just just even to the homotopy fiber of page of AIX met him to within each of what so I can say some properties which are usually easier to check the trade guarantees that this age theory satisfies dissent under abstract blobs many so is 1st of all be assumed all extremely weak form of wasn't of singularities although this ring on and if page satisfies the following properties namely 1st of all an extension by schemes which means that page of its 80 consider finally generated and about what our which according to the age of spectators plummeted include loss scheme defined by some idea I Jr meps Azevedo clues to H of the 1st thing they prime directive toward the close subskin defined by either fail paid To private summering extension and with the property of higher some visible some idea not aided by the assumes that I as also idea In a prime so then we assume that this society concluded this is extreme and that it holds for and article the next thing we need is Miami us finances the coverings and then we need this kind of dissent for regular blobs In this sense of muscle come can expand the minute we also need this in my hands and not invented some extensions on page of any scheme next maps as a meager couldn't stretch of the regular scheme and then these properties implied that's 1 abstract North Korea exprime met into and containing duties .period schemes wide by time the footings career is on Wikipedia confusion in the age of eggs the question of why that the flight time met and talked .period turned the TV don't read anything about the former skiing because this theory doesn't see it None reduced structure and came to the should be the "quotation mark people that the subject of the division and the media Of those parts of the hero the Yahia I mean In a sense it is meant to noticed pollution in Qatar thing like they used to establish the growth of both of whom described we used have the because of all of this is the thing that this assumption is hidden in publishing thinking :colon this is a in order to prove discount of dissent In such a situation quadriplegic cyclical Meiji unit is hoping this it had I don't want to say anything about the sisters I cannot free and not really an expert on missile I cannot say much about what OK so the point is that now descend on abstract blobs was HQ means that this is the homotopy pullbacks square and by for regular blobs of course native exprime is a block of Texans inside this is why their wives regularly must then you have have blocks and 2 air by his alliance approach to a result like this is going very involved and he makes really for this use of the strings of bosoms housing that is US uh I find the sophistication of his approach and somehow you just if you have any abstract he also explained to an extent somehow you just dominated by this resolution assumption by it blowups in knowledge to reflect sent us so then by some simple trick you're reduced to the case the exprime is a in and knowledge to deflect center 0 just have to prove descend for this end the idea is to reduce descend for this to dissent in regular because if you have some knowledge to deflect blocked it saying you're scheme XSL Asoka then you can't just consider this idea defining this why despite you find the socalled reduction of this idea defined constructed by knockout agrees it has generated just order mentioned both y and X many elements and if you make some the apply some additional tricks From commuter divided by you can make sure that actually this idea is generated by reckless look the area haven't stopped he also visited the world leader in Beirut in losing to all the political groups close to the governor of the People's because of what they have to listen to of the U.S. global the leaders during the visit the solution of the soldiers and the focus on the point about how much a became furious that I mean that in principle approved works very similar that approach and say this is pop music you can find it on the archive prove that reside on home what indicators published the rest of this the idea is humming again you need some sort of they need to kill negative cake groups this is made in mourners assembly as for the key proposition people position can also you can His leave program for would pick few than they next thing is you need some sort of dissent result from a K fury but full multiplication area we know what exactly descent along abstract blobs present dissidents because it was bought by Dzerzhinsky using computer different techniques so on formidably carefully have descended on subplots and you don't need To avoid this complicated things that fits inside this opposition I think this is information users of this in city of these fucking Young has also for services you have true possible nomination will disappear minus 2 and cut commands 1 so for example if then you take a minus 12 of enormous affairs it's a Dennis just isolated Singletary and end up in Jan that say you take just a from local ring and then it just Collinson animal flew in those emotion and make a minus 1 is more complicated but the thing 1 can say what it means that if you have a dimension DC scheme and you consider case minus D office came then somehow more unless it just depends on the common at Horrocks of some resolution similarities of the schemes will commit likes of exceptionally vise mayor of his his 1st heard it's area but it died when this is conjecture because of know want it was a lot of the and the damage was his name yeah I understand this is the future of the the rules of the game to lose the castle exist some description of of enormous surface similarities Bye by bite of Biscay minus 1 in terms of some life the new kid on the side they also has a free amino and catch a signal from the assuming resolution similarities he said facilities solution is years the I'm not sure I didn't think about it but I suppose you could forward if you somewhat interesting the the we to the move