1/4 Singular support of coherent sheaves

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1/4 Singular support of coherent sheaves
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Singular support is an invariant that can be attached to a coherent sheaf on a derived scheme which is quasi-smooth (a.k.a. derived locally complete intersection). This invariant measures how far a given coherent sheaf is from being perfect. We will explain how the subtle difference between "coherent" and "perfect" is responsible for the appearance of Arthur parameters in the context of geometric Langlands correspondence.
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if they In a world and there a high to we have to be you In thank you very much it's really an honor to give this course obviously um so thank you very much for the invitation to help well I'm sure will disappoint you the best to minimize the damage the so OK so 1 could say that this the whole idea of single supporters technical so it emerges from a rather fine point of logical algebra and so on and let me tell you while the source the origin so there are these things called spectral sequences and we used to them converging and but sometimes they diverge so aspect of sequels doesn't compute what you wanted to compute and so this whole idea single support is a measure to which a spectral sequence does not converge and you may think it well this speck of signal than commercial carriers is boring but in fact some the Manning's does emerge from there and I should say that it's not the 1st time so the exact framework that over this discussing has appeared before what so 1 is mentioned not temples in succession 1 is and was called matrix factorization these things were discovered with physicists so these people are smart guys and another context was much much earlier so this state ecology so the they're also smart guys um answer but it's it's the same idea of that too will be dealing with to date has so has said we measuring the divergences spectral sequence in it's a kind of functional analysis will be done functional analysis within Category Theory so now has to the framework of the lectures today well is the sort of introduction so Gail fund used to say that if you write a paper the introduction was contained at all so what you must say everything you want election than the few technical details left left for the remaining 100 pages of paper sell basic afraid to say it all in 1 or 2 days and look for them but what will happen OK so let me begin so a lot the elevators grown-up problems no Idaho lifted this year yeah but it was not often seen in greater efficiency in public on you yes so far yes all of this is a joy work with the mining can but so for us the motivation came Al from Jeanette abundance facility begin from there so X is a curve assumed complete overground field paid to Georgia briefly closed characteristic 0 this is because Mailey just because I'm a coward and freighter adjusted I N G is reductive group so genetic when begin begins as follows you consider bungee This is the moduli a space of Principal Jim a curves and you consider the category of the models of that derived category well when you talk about genetic Langlands 90 per cent of the time to talk about this but in this course we won't so it with gears motivations of all steps that much time talk about the models and whether to direct category means so on the other side you have the stack of local systems on your curve with respected you check and this will be of interest for us and we have actually spent some time on it the fund more precisely In the next lecture and so we consider the derived category of Korzhakov years using the incident and the main form of genetic Lennon says that there must be an equivalent and so this name for takes place as is when Wenger's commutative II Taurus so that is given by the Freedom Kyle on transformed and he fails when Jews noncompetitive and this this failure is exactly well Our interest in this in this course when you think that the Government of course you don't just throw it like this you want to specify what properties this equivalent has as usual while we're doing Langlands the most interesting part is the comfortable part here and the irreducible part here but again for us this that it happens not to be of central interest for Leeson who bullies lectures because well as you know from offiical mortar functions as the functions presented no while the mysterious that presumption all analytical difficulties they have rapid decay and analytical they're fine and as they said will be dealing with function analysis so it's the Eisenstein series that cause analytical troubles that's our interest so I guess will be doing function analysis unit In algebra geometry so I will now stay the property that this equivalence is supposed to have with spectators and serious and that's in it and from there will see where the trouble is so Securities properties compatibility of Langlands with Eisenstein's sold its peace the at parabolic sits inside gene projected onto its lead and have the corresponding parabolic inside Langlands long and so we have the corresponding diagram of moduli spaces here but but
but but but but but but but but but so with the call mismatch P and Q and there's a cold peace spectral because we call the spectral side and his spectral while corresponding to the diagrams with a fine functor is let's say going in 1 direction from the levy to GE so here it's the Eisenstein's functor so we pull back that means of this map Q and performed means of key and here we called spectral signed with pullback was the same thing so if you all all where governors you would ask why this is defined because we're dealing with non-holiday models so this function is not defined in general it's a left a joint in my might or might not be defined it happens to be defined and essential image here all of this so again I'm just saying this for honesty now that it's exactly and sigh applying it to long-haul imaging models but once so not all guys here and 1 was the comeback the news of this come as fullbacks news of the snap and you can show that to these guys the letter joint is defined on them the job of 5 p up streak Kilpatrick is always defined on older models but PLO treatment under defined you know what you know that you yeah yeah electors of pursue our online editions of standard but this is like I'm using the most under the tuition the upper streak is the fund is the pullback for the models the toll was defined in the view of the idol of Joe Europe stock Eurex secure prefer you may ask why defined it's this in the D model world it's not always defined by the supporters accuse smooth such as defined the question you have to go His name was the I just home rare coming from the fire and his and he correct and so you have a loss from complicated the futures when the fire so the reason yes there is a map there's another Contra here and you can also described what happens here but number of from people the show was articles because there was no submitted on Monday for pitcher and I'm going from here to here I'm not even trying to go from pitcher from your side of the University of Houston yes it's true so I go from here pullback before so long as you were when you do it with them all the conviction the year the National is usually in the form of grants and concessional loans and so you have to choose which 1 year ahead of the I the don't soldiers do you decide which 1 of them so you just thinking that they can be awarded musical compositions as well as to what you what Europe what you're referring to the local fearing he was a good friend of the family and what would you do you have to all this cost home yes but you see it correct but I'm not trying to decompose what automorphic representations in December presentations on the water so that even representation thinking the functions I start with the kind of function on city's space and I consider that function so I'm not interested here and the questions of breaking it up into the world we don't know what is theirs was the featured most Jesus yesterday said so and so There are those of the offices of all of our here among troubled cheers and tears to obtaining a visa to the University of Missouri the virus US-led lives lives about 4 stated the other otherwise there OK so where more than 1 government ministers and great so will we'll talk about it a lot "quotation mark so In the morphism so there is a theory about fee for good cause requires Qureshi this convicts on indirect image and it will be paid for morphisms of fine at 4 dimensions such as this 1 yeah this it's up a streak that ended up shaken appointed issues it's this kind of other streak and and will return to this point but I so far it read that people are paying attention not just symbols symbolically pleased this is a problem by walking off court of law that's exactly my point so so far in the conjectures falsely stated I work it's it's acquires a coherent category I'll have to say is that all it is it's proper is the 2nd stop of this guy's not probable his dies from these good it preserves clearance you jumping ahead we will get there in 5 minutes it preserves coherence but it and preserve something else which is crucial for us In the eyes of the preserves abound coherence but that's not enough for what else 18 soon the defendants of it so the promised compatibility is the following year so let's write the same diagram for Bolivian so here we have this Eisenstein prompted and have the spectral Eisenstein from 3rd and we want this diagram to commute so this is the compatibility of with between the group and its leader subgroups so what I'll do now I'll explain why this thing cannot hold just this is impossible claim again and this has been with us and the emergence of spectral sequence so let me throw you into the cold water OK so I'll
start talking about the on the subject so me the plot is becoming technical so 1st of all what kind of categories due I was working with is so as over asked these arms well when they're right quizzical and in the unbounded derived category no finer his conditions a definition triangulated category is said to be cocomplete if it contains whole direct sums you go there and was not exactly I was expecting this question so let's meet please ignore set theory so using up to half the loser of the day and still lower than the amount of leave look Montenegro there yes can all practical purposes accountable is the word OK so now we give the following definition so CB category and objects is contacts yeah Holmes From seeing commutes with direct sun so they give you a familiar example let's 1st did not invade derived category was due to rebellion category so let's take instead of for a moment I will take the Ring pain and consider the category of a modules so this will be my notation for the unbounded giraffes category like to write D but so when they needed and category How could do little heart it's solicitation and invented so many really see hot it means um the heart of 2 structure some deviating from this context so let ourselves ask ourselves the following question when it does and object and I mean the usual immortal where is the logic such home from Panama to something commuter director I don't think the you have to shareholders of the day so hard in the heart of the 2 structure civilian category for a moment signed deviated from this context I want to ask the same question indeed in category so let me it should turn to members so what you say both of these is the this because of the ongoing violence because I know but I am asking I'm asking American I'm asking very concrete questions when there's this commuted the at times he's just OK bye and generated so question and answer if and only if a missile degenerative mold look at but let me modify this question now I'll take indeed the unbounded derived category but plan will still be in the heart but I now ask so when is a compact His compact and indeed if and only it's a perfect complex if it admits as fine at resolution and projected resolution whose terms are I find generated projected models so compact question and and then become silly that they not in right regular the heart so when an object complexes contact if and only if that it's perfect so II by definition admits well equivalent to and finally complex of we will find generator projected watches OK so now let's a specialized to the case when there is a fine degenerated algebra so I do know better as speculating on a committed and so when their right quizzical unless it's a definition means with the devoted in involved to unbounded Drive category so from here we see that the following so popular with Lamont an object compact given only if it satisfies the following 2 conditions so conditions 1 Is that after belongs to what I do know call so call means I find it conflicts with coherent
colleges again this is something that's more classical would be denoted by Deby called a dental again electorate deep so find competent with Korea colleges and there is an extra conditions so that is that it has a fine toward dimension so for every point 1st so with my case unless the size skyscraper With for this derived fiber has clinical knowledge so the "quotation mark the forest was and yeah but this this is enough the always so we do know note this ,comma backed objects by turf so perfect best sits inside called that and their inclusion is inequality only if necessary we are over the radicals OK and what is the use of the erect so knowledge may explain where why this is trouble so we have the following a very basic level I suppose you have a functor between cocomplete categories that we have the a pair of a joint ventures so will have a lot of punters in the upcoming lecturers abusing the topologist imitation the 1 on top is the left to join and 1 of the bottom is the like John that's limitation and I became convinced that it's actually found the convenient rotation so let me very good so let me also to a piece of terminology a functor called continues it could commutes with arbitrary direct if takes direct sounds direct sums the yes I mean we're talking with strong chip onto the defendant categories Amin what do you recall exact bunkers the center except trying to say wrangles I'm not referring to any kind of 2 structure takes a direct challenge to the victims so I want to just add a word of warning it's that it's not guaranteed not every functor takes direct sums to exams namely should take some kind of infinite product that would fail to take direct sums to direct OK here comes the little among the so this is stock yeah so was missing I didn't want to know so then what is the following is acceptable to give exercises of the unions his lectures with and so I left the joint is almost continuous if you found peers laughter joined is automatic effect as saying the exams indirect substances so it's an exercise in the East Jean is continuous and then on takes contracts to compacts OK so now I'll explain why this is trouble so the claim is that this Eisenstein's functor only automorphic side is actually a left joint so is good that we have enjoined to another fundamental constants terms and well just go the other way and as diagrams so you back with respect up chic and that's a fungus always defined on the models and you push forward with communes star with specter so hands on his Texas combat compact but now let's look at what's happening on both sides of of you will be high yes so visas standard functors on the captain models there my construction continues the other essentials says opponents of something the use of Johnson terms it's I'm just mimicking the definition of the level for dimorphic functions I saw well I discussed compactness for and find schemes I will return in March much more detail 4 in the case of algebraic stocks such as looks it's so believe me for now that things work In the same way so that a compacts are perfect so the claim is that the the functor please provide inspector so 1st of all as never remarked In dozens coherence to
coherence so there is no trouble with that the reason preserves coherence is because the morphism the spec is proper so this is OK but that does not send to the and those with stacks and I will return I'll go by there are artists do you this will not be in the novel none of the other .period among Georgians who would you lose their use of force the most to define this other L other turned to that of term that so these categories can be defined using any kind of topology Wikinews smooth coverings of flat coverings you know all of you always get the same we will get there in the next lecture so to answer McCain's question No looks is actually not in its it's as close a compact because you can become very unstable he won't admit interconnections and the reason for this is that this fails to define tore dimensions it's easy to write an example for P 1 this morphism immediately producers fear for you objects you started with structure she can get something which is not perfect and officials story so so this cannot hold as stated and so my role over this lecture the next I will correct the right side 2 make the conjecture while at least not self-contradictory you have to get it over with in huge exactly as you can call no I 1 can yes it's more difficult to work with because evil corrected by some kind of phantoms and these phantoms of the Korean side are more or are easier to work with the stands on some of the model site but you can't OK but now comes the next piece of bad news is that you can't stay with algebra geometry if you want to do any of this you have to go to the other bridge on 3 so but there are 2 ways to learn the rival debris geometry side you take it lorries books or hearing 4 of the Concord that you read them all spent 3 years as an example to me and you still not knowing what locals this is a dynamic because it yet so the only way to learn that I think is just just billion that is start using it how do you just go to use this thing and finds that we also know that the world see it was the day of judgment and you were God you would ask me What do you mean that bedeviled the regulatory al-Sadr I don't know and I don't care I just yet seriously and tell of elderly wanted me we have to go on and on and on and on and on about you the idea was to to the development of the people of the people he is the home of the users so take because we are are of the rate what because interest 6 2 0 you can it's enough to do committed DG synergies a soul because of this it's not to stay there I want to explain so when missing well I still make a lot of mistakes in the directives or bring geometry and in higher category theory lots 99 per cent of his mistakes happen to be and the level of feudal categories and make the state for falsely minus for higher categories happens falls on the 10th somehow if somehow you you know the key Udall category theory you and that make incorrect statements about the higher categories that's that's my experience of it so modern the rattled regulatory so if for no other region noted had committed ringing you can attach to its expecting some kind of topological space with the cheaper functions so what you have to know bedraggled agent is the same thing but aid is now :colon senior G and commuted to DG algebra which lives in not positive ,comma logic of agrees the latter is important but to start from 0 goes to the left so so you can ask what is the topological space so although suspecting is exactly the the same as the topological space of its zeroes in on the top cohomology so that and it's exactly the like skin versus variety is the new Bolton's don't matter for topological space and you really should think of this ending as some enhanced importance you can play in the I'm used to you said that the White House yes I will know yes yes it's absolutely so you with the kind you know have you been Will you see I say I have no I mean know how to know how did notice and army but how do you know the way in
which the only I think this is the reason I innovative minds because I actually to seize I just saw these abbreviations as as at the muscle I can handle it and commit itself incoming differential graded algebra the will of the people who live in the city of Neapolitan rates of sits in degrees 0 and below you the usual practice so beautiful logical space is the same as that of its top ecology and you really should think of as a as some version of people you with so we are used to having algebras when importance this elements of a cells would luck while the give rise to functions by these functions may be another elementary algebra but the function is 0 it's important element so the same thing you should think of as a it's going to have a richer fluff on important OK so here are so in order to do algebraic geometry you really 1 thing they need to be able to localize so it ordinarily bring geometry of the haven't element f you mean reduced so killed importance in this case you will be able to localize the local I'm a to local a expected to localize not really respect elements in a cell but actually respect elements in the 2 groups reduced and the same happens here if elements in Asia's year-old and even take reduced and that you will be able to compete I could produce a new algebra new CD G 8 and so we have to risk a localization out of that in building new build your ordering geometry a "quotation mark alone this sort of the I'll say something about so saying you have to and derides schemes so their wide schemes former category they do so you have this 1 is to conformist set so and it's fine however this mainly you to a dangerous place so it may save what else you have and why you want to consider that something else so what you really have is but have is a homotopy type of naps s Montas too such this whole mind-set is defined not of this type so derives schemes the what's called a higher category so users of exactly you know what to do with it was a little yes and if so what I do I have to say have Laura's book on my desk I put my left hand on on the on the book and the type with my right hand and that's the best I can do origin believe the existence of higher categories where of users and that eluded them on the the I by the time of the year exactly exactly so I can of the in honest answer I don't know I don't care so I believe that there is month I believe that there is someone just 1 fear of infinity categories the 2 people whom there is just let them do it so if you want a cake civility I use lorries model for not far from igniting the use that but but I honestly don't even use it all right so let me the same award why you want to consider the space instead of this instead of this the set so here is a typical no way and so can receive higher can work with academies maybe this is the point of entry so the export Plano gives you motivation why you want to think about the categories so bad there was no fun listen to have products so say I have with the final derives schemes 9 1 form this so this is let's say the suspect 1 the suspect to the suspect 3 so on the 1 hand I know what I want to get this is going to be that the suspect of 1 derived sensor product into a 3 but I mean as well worth and taught us it's not good to just this new definition of explicit to win the final did you want to say what fund which represents so in order to bring geometry we know that Hong from the international by the product is so the piston apparatus so this is what happens on underage image analysis falls in the wreckage revenge on what is true and it is kind of of business when isn't like it's easy to add custom get itself used to thinking and higher categories I never raised this right and this is true it's fiber products all homotopy types well now you can take
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you think Is this something did so and its 5 brothers ,comma besides the fact it's fiber product in the category of spaces "quotation mark ,comma deliver products so In the end of the mood of the this is yet again so I know how to computer there is no need to be exact means contracted visitors addresses them in this higher continued fear produces such things so any STO petitioner you you won't even bother about me when a computer after the party lost really find such a solution you all you care about how the thing looks like an exacting to candidates look at so this is the set up a devil debris geometry so let me that they give you 1 example would advise you you have to have a lot to do with the use of and that's what it turns out that you 1 very unlikely to miss 2 2 To make mistakes in if you just by yourself into the ideology of infinity categories you can knowing how this series set up his really unhelpful the all right so let me give an example so for the future of the misstated so as the is well here's a kind of homered in the rugged region toward almost a finely tied yes the minutes your a series insurance almost so if so age 0 win over case he is a finite type in the usual sense and all the other H. Chinese car find it generated as models Over Edition so this is kind of the derived algebraic geometry and analog being fined attack that's where things like this is what it is and yet so without almost eventually it's actually find OK so in this case you can't define the tangent complex safe suffered a tidy have some political tension complexes but only classical has the same meaning it's a model so while Uzi of course has a definition but let me given the finish like this in the world of durables revenge on the tree it is a rather well defined but the tendon fibers so let me take a point of US troops demeaning .period the following day few losers yeah yeah it's a it's a man from suspected K list thank it's a map from 80 to when that aid to carry the automatic that killed all the law colleges appointees point underlying the missing OK so it can classical mainly fastest suspect saying this is suspect it's not classical is this completed that and complete analogous to passing could be reduced but not killing all and killing all the law colleges as the point of a scheme is the same as the point of of the underlying variety the same thing the point of rights came as that is the point of underlying classical scheme that uses of the apology .period we want to come out of the news of the release of hostages in the United States of America and the topological so that there will be closed on them because from the beginning of the month right but I'm saying that if you think of .period Azmat suspect spectator ATU recovered the same thing mainly just a couple of holes and Kate K points so I want to define detention space well the Tangency Drive tangent space so it'll be a complex so where the way my notation for the category of chain complexes vector spaces is that if a lighthearted means I mean just Rector spaces as usual when the right less story ,comma 0 of greater than legal 2 0 and means complex is so situated lower degree degrees below 0 and above 0 so the tangent is a complex that His above 0 so h year-old Of that is the tension space on of the underlying classical as we know it who knows how to define a tangent space of a scheme at point what would you do if you do all of that but I want functorially our our people have the right to yes so it's home From suspect epsilon Kenny 2 U.S. such that well at the close .period Acehnese sense to your point does anybody have any idea how acknowledges ,comma who came to the highest in Europe so in that direction home from a classical to the rind is the same thing as a former home from classical to the underlying classical the same thing as map from a variety to skiing is the map to the underlying variety saying the rise in look no notice so that's Hall from classical to derived discreet in that direction you know I yet so here topographical drivers map spaces actual discreet so who has already of what that is supposed to be part of it you have that but now change the meaning of Absalon so it would be no CD G 8 where the degree of excellence is negative and that recovers so what like movies tangent vectors have a genetic meaning it's also the maximum virtual doll numbers but it did alter the degree you know how the a is home I wrote home which is which is by 0 maps and of course again recover the entire thing so that they can give another exercise the audience recovered not the individual colleges by the entire thing like in me a formula that gives just the content of fiber as so if you really want exercise of the very happy to talk to you all right but these so this is
our values and the Government of the Union is so in almost find type at least to be said informations the information are equivalent so I'm not losing any information I would not do the same in infinite type and it's in general it's better to do it Is Better talk about the contention really talk about the Tanjug news because it hazardous he needed dramatic interpretation OK so I'm proceeding apace can you are significantly slower than I thought I won and I'm really glad that town because it carrier like that people are active so Jews we're at the it's been going on for an hour shall make a break did you think it's a good idea the absolute make a break and then will 10 minutes something new as I said we're interested in measuring the difference between 1st and cold will return to these In the direct sitting in a moment I will not be able to really measure the difference in the general case but there will be a a specific class of derived scheme for schemes for which shall be able these schemes a caulk was smooth to let me give a definition 2 offers the the following 11 so so all my schemes from now on will be assumed locally almost to find it so as he is a smooth classical scheme classical mean that always 0 exists H I below 0 them as 0 if and only if it's tendon fibers only the existing degrees 0 so this is something you know for classical schemes even take a classical scheme you know it's smooth if and only if 2 the tendon fibers are take a disease could complex take fibers table indoor license I was always his ears ,comma logic but it's also true in the direct setting with the same crew so now comes the definition as is likewise a smooth give these guys admonished for private greater include so smooth difference from causes moves but in that you allow 1 extra College so this is a very convenient definition as a definition that the reformulated in maybe more comprehensible terms so 1st of all let me just say that's this is equivalent to the following so that you can equivalent To the following you could take the court tangent complex events so there locally the scandal-ridden conveys a local so he can be represented by length to complex consisting of 3 guys locally now let me do it you know over the yeah so you do the but you know what to do I want to no free the premium to direct some of Sept several copies of itself no shifts take consumer copies of hearing put degree negative 1 stake several copies of the degree but degrees 0 have a map take the call and we know that's what I want this to me what do you
think about that locally OK but melody the geometric reformulation no yes this is this is derived so here is the land which is parallel to this land so fast is cause if locally it can be written as the fiber product linguists when their write and their find space and really to spectrum of the deployment algebra classical on a map of Fernandez as we said a map from classical to derived the same of underlying classical this is real given by honey and political than variables and this is a point of like this side of the Staggers completely classical but new forms With fiber product in the deride settings informed consent derived from metal rings surprise move means that it is derived close so derived locally complete intersection In the past the the government other adults remember that it is but in the In any case doesn't matter so I think the president but not so and let me actually prove this thing in 1 direction it's because we the use it much of times supposing have in general supposed to have the fiber products like this ,comma From why 1 way to anything so let me right the tendon sheath of S tender should draw the content of the necessary whether there's so it's going to be the following to take well it's it is what it is what it is supposed to be a Soviet you're taking the tendon sheath of Y 1 In restricted to S & differential about this India collapses shift away tourists with the tests and it takes a while colonel the drop of a very geometry could cocoa Coppola just like to "quotation mark fiber so but In this setting you will see that the Clanton chief is the tender should foot a and is just a free module Frank and so we'll see OS alternative nothing to worse problems fiber so 6 explicitly complex of length to set foot in degrees this year and 1 was content and this is degrees negative 1 0 so if you wish this is the proof in 1 direction and the growth in the opposite direction it's pretty much the same as the so and who remembers the definition of locally complete intersection the classical leverage between of scheme give the classical skiing when runners at a local competitor section all of the funds have already you have on the stroke but great but differently from what used to be the of the the growth of just 1 reason news of the yes or you can say like this little bits gold good classical classical LCI is equivalent to the following concern choirs assumed that last classical so he could take a classical scheme during regarded dried scheme so just had doesn't happen to have lower colleges it's LCI if and only if its causes it's so that partly explains why LCI is so nice so basically can of classical schemes that arise archivists move or
they obtained from smooth schemes by the taking of the fiber of such I'm so we think the fibers such as with the fiber fiber products and when it really gets in the classical setting while you can think of it as follows you taking the drive setting and then you're throwing away the law colleges so you're doing something horrible were er um so L the eyes of those classical schemes that I I mean do you didn't do anything horrible to the von von below "quotation mark on over the years has gone to the world I'm forgetting its good conjecture complete I'm forgetting I'm forgetting the status of this thing other so in the other Jimmy geometry can the it's well you can have the contention complex of any length and yes so when this matter exactly kind of passage the classical you know you something very nontrivial from the political point of view truncated "quotation mark so not now will play another game which is what we needed for our definitions but it's I think it's found in any case I find it funny on the Blair is being here so let's do this take the tangent fiber and shifted chronological to the right by 1 June no the people who you know that this has a canonical structure for the algebra why do you know that most of the time needed to talk you don't the avoided some somewhere in the building :colon writers musicians basic rates it's going the other 2 within the during World War Two but I'd we didn't we didn't say would yet yes so "quotation mark Greg so Quillen tells us this but let me give a different interpretation but not equipped done different safe but its candidates it's very intuitive and that In the but the work on the list he says you can be anything but take any scheme ascertained Redskin mistakenly .period that is the kind of but so far it has a competent vector spaces in event of an and effective realize it's like this in this way is that the question of whether you this this is the 1st of his duties he use told the parliament this of the a outlines of some images images in 1st something that he could take this particular ,comma selector spaces shifted they won it it will have a canonical structure of the algebra that's where this is going on here because this is the same year what I mean the Can I realized any any public spaces in the form of loans through the yearly In Army warrant the French admittedly algebra canonical objectivity Infiniti category different admittedly algebras so well constructed experts yes we have and I'm doing it .period ways so here the construction you have your point consider the fiber products .period times .period over my skin In the dry tense but no well because it's a point it's actually a good object in the Redskins it has a name political inertia so now indirectly big geometry it had been a group Tankersley algebra and if you look at this calculation like of this what happens on the fiber squares you'll see that the algebra of inertia underlined while the algebra of while attendance faces a complex so we algebra and this will Will this so you want to think of the of the structure it's really eligible for the group's name this 1 innovation again and I'm not saying anything different from Quillan adjusted reinterpreted in the Al quit original definition what was the result of that so quickly peak it's was almost of Qatargas between that's a commutative reigns committed Belgium Italy algebras contravariant something the listed on 1st 2nd and this is my definition OK so
here's another fun fact about this so consider member with an old break case best the skyscraper let's consider our home so I usually I don't like I don't write are the the right and just if I were to write home it will look at me and what I mean is derived and the morphisms of my skyscraper so what structure does not have its own Sociedad different from that of and Secretary of State this 1 and committed of question why do I mentioned this afternoon With work on the bombing would be the year yet again there is commitment from skyscrapers skyscraper to ourselves to the G algebra I started talking about inertia then escaped to this White groups "quotation mark of no support and I'm an actor .period OK so and now we have written the financial single support you have to conclude the monitoring group statement on is beyond that there may be no less than the cost of those the sense that it is a classical thinkers so he can you know you can take it to the French King realizes by the different really algebra think it's universal delving into very classical sense take take its cohomology is it'll be just ingredient associative algebra can the accountable faxed such a true statement there was also a lot of of people in this city this is the good things that I said to admit that we just of Union what would only make sign canonical accrued canonical isomorphic the worlds of gangs have infinity category of associative is the GLG breasts and it's a conical isomorphic and that's In the set of his devotees alongside other overseas available to members of the foreign and said intractable said yes choices for them but final other ones invited to use your like this union linking the brackish the music of the long the coverage and on knowledge so I need the cost you can constructed you will yet so yet so we'll use the same form 1 of the definition single support it it's good for theoretical definition so like it's in my experience this is not the kind of object that you actually want to realize base that could change some of them some judges will cover you want to write down explicitly like you want to write down really model from the readily algebra for some of them you don't injure experiments which in theory Kelly this is not the kind of thing that I want to write on the New year movies there only a single losses up to you will see complete statement comes to you and that is where we must be aware that none of you will and again of out I make sure the things a computable single support because there so this is good for a theoretical definition and for actual competition into something else so I mean I men next lecture and give a bunch of different definitions some of them are good for proving things some for computing things and so this is 1 from which is this 1 is good for troops for defining things all right wherein the defendant Newfoundland region as is the final yet so I really want to get into an example and I hope it's so realistic so far that I might not it will I where you can effect if I don't have time to do it now livestock from extreme example next time have all right can I wanted to and now
wanted to find single support them so so this is almost the final time and so it a coldness so these are projects in quizzical such that the individual colleges are finding the time generated I over page 0 and and bound sidelining 0 colleges for but financial mainly so let's start Starting from this S defined for you a subset somewhere and that's what I'm going to we would take an object and we take it .period and we consider it's acted on just as they man amorphous stores the 4th In particular so there is danger lay differential greatly algebra In particular let's just now it's past individual colleges so to which it is agreed In greatly algebra after all at and more the so now I assume that as his question so then this differential rated algebra will have only 2 pieces if you have a piece and degree 1 that corresponds to well it is 0 tangent space and Peace and agreed to while his shift that corresponds to each 1 of us is particularly house In to have this Andean Lee algebra which is just H 1 so could you consider H 1 yet notes this shift so that can the placed individually told Amelie algebra acts on this module in other words the symmetric algebra exodus fanatic algebra but the generators of placed and agreed to now you can take the supporting the move to the center just supportive
of crude algebra have agreed to model over the ingredients polynomial rain so we defined single-A support at canceled left to be with us for forgiveness if people are still in their wake this support will be a subset of what which great .period yet conical because it sees a greater the support of the directive as a module Over more losing his by
happens to be friendly generated the group also it's hard not to know that it's not it's it's justified generated model this algebra I'll states series to this effect next time so where there is you could see no is difficult there has but it acquires this is a list of the it maps next access it axed from next ital cyclists this is here now so that's assignment so it a risky closed conical subset OK now we want to take all of those answers together where do they live these days so when you 1st come to the 1st course of differential geometry defined the tangent space to tangent bundle how do you define it for metaphor no the tangent bundle is the unions over all parts of the manifold of tangent actress the latest blockade saying of best using union or points it's been brutal looking from the definition of a very clear why this is another point of literary writing the look at the way it is isomorphic to relatives suspect it's suspect who still awake that's something wrong but there is a policy analyst with the union that is something not as good as model but cannot because of the money from the Union obstructive .period intended regimental music no 1 other than anything that's going on here so this a classical middle of the road games here in the fall of the Soviet take tendon sheath actually ,comma logically they wanted lead degrees during 1 shifted take ages 0 and it's no longer vector combined a coherent chief take it said Over all Western technical back know the well it will be the same because I took each 0 2 many of them you want for our purposes but don't care because we will be a only interested in supports and nobody who worried that the 3rd case information may have you may have importance so I is eroded skin 0 I could replace invited religious part for my purposes all I care about is that it's appointed defined in this way the OK and look ahead to sacrifice some of so finally so definition for commitment all the singular support while the Union for in as of this point where single supports such each of these fellows sits inside each of these fellows so therefore it makes sense as a subset of the so proposition that I'll explain while fiberwise it's theirs he closed as rotates not only is there is a close but have not difficult to prove it is most of the year want we all no that's not it so will get there to the force lecture which it can get you absolutely cannot get anything on this scheme it will only have support along with these guys I will talk about this in detail so badly that it's impossible to really cannot produce anything lives his scandal-ridden here like this very scary all have security so this is something I don't know so we only know single supporter don't know anything about cycles this question has been investigated by officers might have been investigating the people just have the knowledge that it might be interesting solutions that a endurance
but I'm interested cider doughnuts 1 of the reasons to something called the pointwise yeah I have to generalizes and we will do I just wanted to give definition of single support which would did Will next lecture will be devoted to 5 different definitions this will be 1 of them if you can imagine that hold 2 of the characters you have to In view of this I think we have the means and the will of of the people in the morning and of year meanwhile not exactly banishes cycles yet so outlook will choose a function and then something will happen along this function yet there is essential in this dividend filled but so states a single support does indeed never something Singh 1 directions very easy from here to here but it just happens that thing was supported 0 Europe thing was perfect so sport does measure the degree of imperfection of your object just the way he look at this thing this variety measures to what extent your best was not so that state of Islamabad on that land see if these guys are all 0 Dennis was smooth so busy measures how much as is far from being sold and have a single support measures how much effort is far from being perfect of the I so it has no it has a meaning in effect that it only makes sense for smooth so for quite a small schemes so the advances that have accompanied the allegiance of LCI that it makes perfect sense but otherwise for just classical the movement usually 1 of the up but so the mold this example is coming up in it's not yes but it's not classical that's the thing the mold this example I will do it the knows all year's section please the Americans contains a section foreign direct so earlier this year he called the semiconductor and it is eroding support in the form of those that wish to do so and I know what is known In and what it so select looking now we're ready for an example now if you want to keep that OK this is the of the year yes and so here is the most league Mosbase example requires Osinski defection happens not to be classical and it's the kind of scheme that's used all the time so named to take .period times .period over electricity the suspect from the specter of seeing it from the start OK so what so this is my personal best OK so further before we talk about things but at the point of this thing points that have we used to have great result 104 . like K points of harmony the so what's the classical scheme underlying this 1 I had to get to the year it was at 1st asking ,comma What are the point of this guy about 1 . so OK now what's the single what is the office but rate so now will go nowhere analyzes example in detail it's likely that coalesced has a very explicit descriptions these are all this is Kozol duality and its scores of reality usually 1 breaks down because some find this conditions don't hold back the exactly bring down because of the difference between Cohen per so as written is exactly true and respondent 1
direction is given take sorry have goes to what hauled from Torin unique .period I should have said this "quotation mark this a seen I'm considering the modules for this such that the colleges of find In the classical sense the value of these players was lost all this not that the ability the world is the correct so this is not a legit object and rattled a rare John tree and I'm not regarding it as such I'm not trying to take the spectrum of most of the it was the he that under so here you again taking the of these I'm taking models could take the cohomology is they become integrated model over the greater drained and I want that to be generated a lot of the if you said want to warrant the same as perfect models so by the way inside here and we have 1st somebody gets that corresponds to yeah it does not supported 0 and the the right kind of 0 would you support colleges are supported his is an 0 you want to know the problem is that yes and so on well and basic and just applying the definition but did this gives you a way to think of a single support so you start with object Shearier transferred here look at the colleges it's not the greatest module over soon think you take its support so like this because duality what in this particular case just exhibits you by definition what the good news is that the weapons all its module over this differential minutes Algebra I take its individual colleges were my mother is is my guess is as I I I always that's limitation I called any it's different rented elder when I write a modern Koch ,comma logically graded so this is if you wish well In this case this this crucial reality gives a single support explicitly that's what it does and it turns out that all the fears improve and the end of the day he reduced to this case so this is the most basic the Qantas assistance scheme to give you a lot of what used to be a good yes well know that example this along with its it's the same because to tell local it's the same as a vector space so kind if you if you want to have an example you won't have this or it may be a L and now In Perfect is just you know whatever can be at whatever can be constructed in as Findley many operations on the free guys so that's that's a definition what I mean about so these are the guys what I can tell you what they are in the active fine dimensional so supported 0 means is 13 so that's what happens to perfect its on because other goes 13 found but then you can save and hitting the support that's supported measures how much longer Tinian your are OK so I think did maybe the Missy 2 thirds of what I wanted to do but it's fine Indiana for Bush thank
my mine


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