4/4 Singular support of coherent sheaves

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4/4 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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my own and I think I could do the around the last time we talk assignment of this start to the scheme and locally almost a finer types attached to this category of configurations so in fact that the reason this was invented 1st of all this was invented simultaneously of closet simultaneously many different groups for different reasons and the way I came across was not the cause of so it and know that this was fixed the problem handling this correspondence my reasons were different and explained what they were are the following so yesterday In the last week we said that if you have a morphism next why there is an Iceland had flunked director image in this context direct image and its direct image and defined is the game of tennis and definition namely you take an object of you regardless of the cause a coherent the wide use of direct image and then you notice are high actually is In the blasts and therefore can be lifted to into that was for so is designed so that this diagram commutes more and I reminded so I was a natural puncture going from being called to classical you embed coding to question machines and then in extent so this compromise if you behave nicely with the start of something it the trade mainly to remembered that the in he eventually collective case that was also left the Giants so closet where she is in good shape and this upward diagram will not commute I wanted a careful look at that In addition to pull back to Florida action was to have pulled back so if there proper you want to create a joint proper and of course you can you can do it with cause acute receives about something bad will happen so so effortless chic so there it was just usual classically Rashid who used the acknowledges as it's like a giant it's exists for some general however so what's called not continue and I remind you what continuously so continuous means the computer with In direct sums on the last day of the year yet but but D-plus is not close and direct you have to do that but I don't know what I wanted I wanted the whole thing and then the Anatolia why does include director is
director of the so we have the following letter I actually stated on the one-half of it already supposed have a pair of giant hunters and as is usual wondered about his left the joint and always in the world of categories with arbiter directs so it is continued with them on the compacts the compact and it is compact generated this is and so in this situation it's easy to say that if you have a proper map there is no reason that it would send compact guys made the perfect guys next to perfect guys and wife take the inclusion of a single point in life and then you get the images skyscraper will not the perfect the question of the years you can going somewhere and not on the phone with the program of the long so that he could take care of you this is what is going to work ultimately the most valuable and that is impossible because this functor is obvious even in homage the length of equivalence is unbounded Yeltsin is a disease of unbounded attitude it yeah well please someone the bad side so it take the constant chief bungee it will go on the spectral side to something which is goes in from the left to the minors so therefore you don't want to cut the because he believed 1 of the main players in the constant she said a lot about that so we want to work with ,comma category is big categories look so therefore this will not be continued but in the analogous situation here so again the right joined will exist for topological reasons and it will be continues because appointed the overlap so knowledge may save the words why do insist so much having continuous functions by wider care so that 2 reasons 1 is very practical and wonder the other is more practical my continues so as to what is in front this continues it's difficult to compute anything so sustains the affirmed the principle generated by a bunch of objects generated means that every other objects can be obtained as a iteration of procedures to take indirect something calls if ponders continues to have the algorithm of how to extend you if you know what a function does on given collection of others in what but extended everywhere well just because we have had control it's manageable in the this continues even if I know what my father doesn't generate except I really cannot say and also so it was a much in the way of the world that the people of the law in the office of the disease in the most of it in the also note that that's that's still finds these things are so fine we will get there actually innocent so just say that have more control but he was really in batteries so you can consider worlds of these categories and continuous function the main observations that these guys can up so this is actually a matter .period ordered categories so you have seen you can create a product which is another category of the same kind so it we give an example of such a thing as a soldier were a associative algebra and then the answer would be able to be thing and I'm talking about all old models and so and so it turns out that this operation of tens of product gives you a lot of power so you can a lot of things and later on today's topic some of things so for for example typical examples have and algebra object inside the right said associative algebra 1 of the categories it have left model and write what you can tens of them up over a and get yet another category so this gives power to produce a lot of new categories something construction but in this case you knew what you want to get to the intensive models but sometimes we don't have expressed support and some categories that you wanted to exist will exist because of this thing and yes it's not it's a well-defined so I'm saying that this category has as a matter of when Nadal structure and it's really functorial only Specter continuous maps you have seen 1 acting pursuant to that the want answers will map to see to intensity if the fund funds he wanted to do was continues it will not matter if the fund was discontinued In St. thinks so this is and when little category if you have a less and haven't algebra object you have left to model the right model just may attempt to take this as a product is well-defined operations among other categories if you will yes it is sir it's algebra objects inside the danger caps continued well a 1 little category 1 category of and if I asked to define what it means by went Category I will refer to her as definition of the there's a product will say it's algebra object in this in this category so when ordered categories are eligible objects in here although he was of on the other categories of news in the morning I'm saying that this is
1 of the categories the cause of the separation and also on the yes it's here that we live in an enchanted world yes yeah I was if you think the world and then again there so the operation a product is all can functorial suspected continues bunkers you have the Montecito he will not be able to sense this morphism unless a founder was continues OK so going back to so we still want to define it and that we wanted to find out what the sheik From Co and cold wife and collects well in the following way so you want if have an open you want to the left and joined the fountain image and if there is proper is a proper you want have a chic to be the right to join in general I mean when you want to attempt the construction of course what you do when I got people factory morphism like you will save for Bush's 1st mark 3 times just shake and you know what they are in each case course you can't do this in infinite setting I just can't define functions but they all had allocated to the factorization there is infinite number of comment of consistency that they have to check yet you can set up a machinery using the speculations of organized big big things and make it more so In this has actually been done in the world of constructive machines by so there's there's 2 papers but before you I'm forgetting his 1st wages so the constructors have a bit of defeat so if it is a functor the almost a finite Frankfurt Yukon yes sir yes according to my conventions don't even say that the so I'm currently writing a book with because words that we a larger framework for the construction so I'll be very happy if you ask me a question about this but maybe in question time so In this way this looks like a construction but in fact you can state and pull off but because this is a unique history so it won't an arbitrary thing so you can make a statement that such a functor exists and is unique give even some more structure and they can be very happy to talk about it but not in in the meantime the lecture already so now I want to go a a few more steps in this direction we know what up all on schemes so manically that we actually extended automatically to priest acts so we technical so I'll talk about priests text cold locally most defined term laughed so these are my definition is a contrary fountains and taking fine the right that are what we call eventually got connected the have item in ecology and these arbitrary function from this to reporters so this is kind of these before we defined just priest tax to be arbitrary founders from all find schemes to billboards in Huribgah considering these guys I didn't want to live so I'm leaving this former definition all 3 tax all are actually .period category while the mobile priest so there is a fully faithful embedding however it's not either Allied writer John to left a joint anything so it takes a little thought to say what appears again I would be happy to say but again in question time to adjust don't think this lecture just want to say that this notion notion are connected extent you go for all from from schemes to priest cracks do you can't say do anything as how we extended classical from schemes to stacks name of the priest wife "quotation mark well it's the same procedure so now we can talk about go on precepts give examples at the test of alertness to see the so for example stated .period on the sky are in court the last time introduced scheme would affect the wrong it was a priest at such a reminder home from the wrong was definition From this reduced to acts of Mississippi State and No say that the models next arbiter conditions into about arbitrary Our people happier what right now I'm not it's supposed to tested if here we yet this is a really supposed to provoke some reaction In a so I came here to see what was in out arouses suspicions so last week we gave a definite different efficient mode so this today yes last week week with get a different
officials said classical OK so you can as was going on and why Wyoming changing from here to here and you can ask at the same actually 4 OK so explain what's going on here so the 2 more pieces of that data so that on and find scheme is considered perfect this is 1 of the categories contends of any contends coherent she made perfect she yeah just answer up and can be extended and you have an action of this 1 1 category and this can and the way definitions work Korzhakov in the same intruded any please tax Classic will act and called on the 1 hand on the other hand so if you have any scheme staff members a canonical maps to the point and you define the deal rising to the pullback back of forces and so using these 2 pieces of structure you define functor :colon for anyone to function goes from classical of wife you go action rising Napster instrument for any priest act you have amassed from classical to into and the theory is that why is due on anything for respect within our final at work and then this year "quotation mark but the shriek from the point we now have the formalism soaking up Soekarnoputri why is this so why is obtained From so completely arbitrary time by taking the wrong there are is some drama small but put it like this and if why is the scheme man who was equivalents if and only if the scheme and so forth he visits was equipped only small guys but the the 1 thing is clear some means of classical and classical why why is the right scheme and this fund will be because only if is given only wise classical and singled in the usual sense so Abramoff as any priest acts so I'm extended so I had my function defined on Sky and that's what I do is technically I gave a formal wouldn't and go isn't but this former has a name is called the right kind extension I extends from schemes to priest tax in some automatic way and obtain functor promptly stacks to major categories the the do Sheikh yet everything with that trickle back with you all later the phone call I think I understand the question explained so I think here's what offers protesting against this has nothing to do with the fact you can ask this question about schemes so they have more schemes in any of the following guys there is classical of X there is quite because of why there is being cut off effects and there is no call for life so then the following operations here there is usually bobbled back here there is a function of 10 straight up by the globalizing here the residents of Hawaii and here's the sheik back the claim is this diagram commutes this is the world of schemes but then by extension in a priest acts the only thing that many years you in this time of year you could have used over the last years yeah establishing that said that I erase this but what said that after she is but the work of Liu and Zhang opera she is defined as a functor from from cat category opposite of a fine schemes almost a finer type 2 categories of data the frenetic shoes but the same construction works for him to Co the more you know the sitting on the floor of the correct and all comments in 2nd this is the 1st to know they are equivalent I'll I'll I'll I'll explain a moment so that's where we're going in the 2nd half of the year you use the word of the more than I can think again that is among the more I will explain a moment it is good enough the Americans what you think it is so that's because it happened so so that's another moment of test detested weakness so let's live in the world of schemes I just said that it had a scheme that you have a sponsor Omega x from classical so this is the founder and its cancer might like he are you in the sense
that we are in no what what good will it have a bunch of functions between these 2 categories before so what was that all you have spun in this direction and if X was eventually go connected you also have a functor political side and here I come and say that in addition to all of this you also have this and what is going on well indeed you have all of these guys and so well you have all these guys and their different from there was 1 of the for example the effects so the event with state of the world the way we set up it's going to be just the same as classical there's no difference so so X equals there is the identity and nothing is going on in this case this prompter is censoring up baby doll lies in line coming about should the and shifted to remove minus the dimension of the effects Over actually ask another question he detected a discrepancy in how I set up into cold on all in the case of breaks stacks let me suppressed for a moment because it'll it'll take it too far if you again is the question of very happy to answer can the question so we have to we have always that just say that it's all only this this from that actually makes sense for arbitrary priest acts so these guys are actually feature of schemes and it was guided it's extends arbitary priest acts so that if this is the more fundamental and more fundamental function and hear state of affairs this functor is equivalent of anything which is the union really yes so in this case yes actually should OK so we have these 2 categories and other 2 font the side and there is strong it can ask how are they different and they differ by tensing bylined bond so this local complete intersection this analyzing still makes sense it's lying and these are 2 functions from 1 category to another and they are not the same with the differ but answered by lying so that's how it looks like we have been around for years and now know he you know could have called what any prostatic and talk about the content and complex and and that there anything which is the wrong this potential conflicts 0 because the content complexes test Ban mapping infinitesimal thickening to your stock and the wrong set up in such a way that there a I call on yet so could have accomplished as defined in the way we talked about in the 1st lecture at taking the squares the extensions and you mapping I knew that I knew that this would be the all Duran Duran Duran Duran just summarized what has happened and what has happened was a crash course in Upper Street now I coming back to the notion of single support and I want to comment on what how it behaves in respect of publishing in the reason I want to do it again because we're looking at this diagram looks parabolic looks at what level looks for this is called the spectral Basescu's victory and our Eisenstein's function spectral was the composition of acute back up street and last time I said that we have to talk about it and now we have it With peace pact in Koloa stuff and what I wanted to explain is that the this font maps the category In on looks the looks at what we did explain last time was following with half of that we introduced come all officers and explain that direct that sent to you in court looks due to want explain the 2nd part I want to explain that he backed she maps could looks In a looks just just the other half of what it we did last time OK let me 1st remind you where this came from this was a general assertions of specific to calls it something general about single support so reminder so let
x twice the amount between classic so we have strong Of course director hinge on the 1 hand and also had his correspondents Over map to sing X projects down to sing wife and called snap think of this year from last time with the following so fixed on subjects and the next and in 1 instance was and assume that but you take this singular goal differential saying of signatory image of it would be contained in the while so In in this case claimed was this direct image the next 1 to and why when 1 so he naps in this subcategory to this that was here from last time so now I want to state the corresponding results 4 of the sheik asked students who volunteers so the circumstances and accidents like the the opposition P and Q 1 1 2 you could Apple shake OK so here was to give an answer please and so it has a singular no answer it happened In fact the finale but in a sufficient condition on that as reserves it will be some other place you'll see will gather in the 2nd just Environment Minister such as result is containment it's an estimate from about anybody yet we want this to be small compared to this like the bigger and access the easier it is for this to hold so what's estimate that there quite smooth schemes In view of the universe yes there will be something that will there is a parallel assertion about the model it estimated the characteristic variety of upstairs in terms of the downstairs to give analogous thing here so what if you have the downstairs if you want it's a single support of its fall back to the contain some work what's the condition no no notices containment the chances of being the yet it's the same thing in the opposite direction so clear so where can provide that containment just the opposite direction that and why 1 is small enough as compared to the sum is contained in the Senate something else if half of what is contained in Annex saying conditions just rip invert the containment it's exercise applied to the morphism Q and obtain this containment From the but he wanted more so we wanted a stronger ties not only to respecting the support we actually wanted to send coherent things to good things we wanted to preserve compactness 4 this piece back or start it happens because the morphism was proper nouns doing the kill upper chic up at chic will not general Pervez preserved coherence you up a chic like you have a single variety an appointment included if the after 12 years you get all the extra bit the infinitely many of them it's not doesn't preserve coherence so so you may want in these circumstances ask so well is effortless chic of coherence it was good and this is well pure Worthington's herself before the transfer salad so we can actually give a criterion when creators would preserve in terms of single support this we'll have In mathematics because its schemes but I know that this is what you're saying is too weak We can give you can give a final condition let me do something more general what after all laugh at the chic is and version of tens of product like just say something more general voters abroad so here's a theory Texas classes and as 1 of 2 or coherent the following are equal so what is that he should take the usual turns a product of course derived 1 his career I mean could hear music has signed many colleges the you can take
their shrieked as a product that this means so much as a product of opera star under the diagonal map of the external transceiver that this will be up a pushy good home in either direction "quotation mark internal home in the opposite direction Sheikh you take half what times of 2 on extends acts and you shriek pullback respect that argument and of course this is all boring but all this is equivalent to the following maybe take the single support of F 1 it's the subset inside saying that take singular support of of 2 it's something inside saying intersecting you wanted to be there there was as usual this year because of the art of it was the wrong war you you the group has groups you have to do it you it is not it's not trivial so I said I think something more general instead of taking up Bush back I talked about as a product of sheets so you can this book that can be expressed in terms of the tens of product Bloomberg Sheikh pullback can be expressed in terms of this kind of tens product and stop and expressed in terms of this kind intensified all of something else and currently and and blanking and what I think next on X times why something like this on X times wife because yes so you'll be that you've been doing and after answer the structure Schaefer you censoring the with the structure should for the diagonal something like that watching the example and I'm saying that if you know that you can give a criterion for when clearances preserved the thing is I don't know if there is there is there is duality do you think about you this year yes it's as important and Deif 110 served 1 of the key points of the yes yes and that's what I'm saying that if you want you talk about that the Union using .period view of the theory of normalizing for for quite as smooth and as a matter because the polarizing is aligned bond the 2 of you In view of the yes but because the other thing is it is just the globalizing its single support the 0 time single support of the original if you're in your view of the you know the question is do you think you'll see a lot more what it's the same as did you regard the usual commuter section is derived 1 and created its thing and a subset but you cases a particular case of so many meet maybe even if it is given you the right to so you know what I this volatility Dalton dudes while I'm standing out during the break so I think that you can get criteria did use this data from here out of but with number just standing up all right OK so well these 2 theorems Oregon basic basic from 3 properties of of single supports that's how they behave with respect to push forwards and example all right so I think the way they were this is part of his life and his work at the on the losses because of the computers she bore you with the image you ask this question before at least on the grounds that of something of artist in the years to come to in the form of the minister the question again so this is my friend said that because of what he OK so we want to computations years starting with same it's a point in looks and the looks the Taurus yet practical .period skyscraper what let's be careful now a category that is 0 for the Torres Strait so this is actually in classical look system could but that was 0 but we start with an Eisenstein's the yet so and this is the object of intercom look at you it's a particular object that so so let's ask a question about so what kind of question would like to ask if there is a very interesting question 1 can ask but it's of a letter asking questions we get we get a well-defined object so you just have to get it to stop stacked yes indeed so and it's easy to say what it is it's either overseas reducible or when it is reducible this too one-dimensional guys and honors this opened up on that opens of stacked this it's going to be just classical and then what would you get what you think you'll get the division's direct image well I mean that there can't be anything because it just didn't have the usual classical 1 interesting subsets work this is not 0 you're getting some and yes this question before so you ask budget years ago this was mystery of and its that's where it lies that's exactly mystifies time has to do with remember grow this treacherous diagram the 1 1 died and the not Korea this has to do with us all right so the cost of the war on work on the C there will be a phenomenon so it for open part where
0 you that you will not get anything interesting about something will happen so I think it would make a break so they just say what I had in mind for the last hour there are 2 things that I wanted to cover and I think realistically only I will have time for 1 of them so let's just say what these are the number 1 is the following so with that effort she is the line that saying support which is something it's it's really a subset there's no comparative structure can ask is there comparative structure came into something slightly better so and let me tell you what kind what kind of better things 1 can do so let consider object not an goal but the quotient in Kobe classical it's called the Qatargas singularities so in in general it have held a notice with the circuit in court what a quizzical so well by the same amount of same principle to it you can attach an object in the projected a of the same both remember so we have this injunction so so you can this quotient is high that the kernel of the functor or the cautioned that the image of the country the same and this is fully faithful you can take this model the image of the long-term auspices was the colonel about functor form but the whole world is all so climb because gets supported this year's section there for taken object of the quotient it'll have a well-defined supported the group projected positions so say to this is a sign that state's singular support with circle of which is a subset of this project position but what you do have to have more so here have not exactly skin 30 structure but had a falling piece of structure so beautiful in the area if consider the category models on this project positions and this category if someone little category and now we remember what talked about the beginnings of someone on the category in the world of the 2 categories as such it acts on this category of securities so not only do you have in this dump the dead single support something act something and so this is actually of central technical importance for and I can explain that so it can of the central results that allows you to do things so this is the number 1 and the number 2 is the following it has to localize local and this is still very highly conjectural we don't like mobile lines we don't have a statement that we can know what it's supposed to be so it's supposed to compare the following so the alliance was equal itself a tariff categories this category was supposed to because this 1 local analyst is supposed to be equivalence between a pair of 2 categories this 2 categories posted according to this so what I want is category "quotation mark With the action of the group yes yes I mean it's DG categories equipped with the system pieces of structure With this academy in this category these guys together for 2 category Instituto category the totality of these forms an infinite at a category it was the 2nd no no no no no no it's different the word to me that I allow non-convertible from office morphism there is no longer there there book and in the process of the use lurid did in the 81 categories and these guys are the to categories don't want more of year so this maybe not so that OK
so this is the genetic side here in the categories the same again I don't understand it's technically yet sheaves of categories Over local systems and respect the upon former function and and it's a mess so I've been working for many years trying to make sense of what it is but I am able to make sense of what it is this kind of hole for the same reason as the global guy was wrong namely here there's more stuff in here so there are there's a phenomenal Whittaker generous G check :colon was due to yet I would so that it if I'll go in either the direction his direction the next hour and then I'll will explain whatever I can explain so but what I want to mention is that even then managed to the it feels wrong because we have to account for the same phenomenon in the global case we have to replace classical flocks we have to make this replacement of code so we enlarge the spectral side and that it had a chance to be equivalent to the geometric so here are so shades of categories over a priest act you can think of it as some kind of categorical To categorical classical so then we should be able to also somehow In the numbers being categorical In the only problem is that no no 1 knows how to do it so in the case of classical cages taken here she is incomplete and that your here is just completed not clear complete I discovered there so I see this as the 2 category and then you enlarging the 2 categories and the worst
is that but relax for a moment so sure you guys have seen 2 categories analyzed all categories I've seen are of the following source its catch at each to Qatar has the following shape it consist of categories equipped for next with action with excess structure its categories with With extra structure so they need to category that I have seen in my life comes equipped With forgetful puncture to the 2 category of category well if you can't think of any other please let me know that I haven't seen 1 so seems we've seen categories that are not equipped with national from 2 2 sets that's why I'm so so we know categories that on not that structure like Susan on a manifold it's not set with structures and plenty of other examples do you want to the French the only thing you know you have you know I don't even if it did he do the same problem I can do it but I mean it's so these on the the technical problems of this but even if all these people still do not enough for the following reasons I'm creating 2 categories without and this can this being the category is really doesn't have a forgetful funded To the 2 candidate status if you don't live in the UK and the people of the things you can say museums and on all of on on it this was the "quotation mark according to the government and the people of the city the Of all I would say that the 2 options for the next hour and I would like to know which ones which 1 would you like to me to go to war you what will the so we can take a boat and then I can during question time I can do the other but there is this kind of he let's let's think about who wants 1 but the look of this who wants to like there is parity the technology you know so what do you think 5 minutes what is its content of another paper withering it's called the category of singularities and global Springer fibers OK so I said that I will talk about the following the modules on the single project position saying act on the Qatargas singularities but I will start not from here also motivated the lanterns and I'll say what problem where we were something and then how that that thing came up so let me
remind you something that the stated at the end of last time this sentenced take all spectral sensors founders in you apply them to Quantico don't even modifications so this classical sits inside car you get a collection of objects inside in cold and see what these guys don't generate so the theory was that was the generator is exactly please call a looks look now let's take a look what it what it means to generate suppose as usual you have a pair of John founders some of them might is that the image of generates G including chemical to join the bloc so generator the phenomenon if perfunctory his conservative conservatives In the world of DG categories means just that doesn't send anything to 0 except 0 but so you can imagine that to prove something this conservative and it will not be enough so I kind of what we want but we want turn this kind of statement that some farm produce actually fully faith so this is so deep Q so we want to map this category fully faithfully it does something has to do with classical but parabolic and so that is this is what this is where we are going do so OK a scary creatures will appear so don't be afraid they have morphism that from next life I want to talk about so when members would have to have to a finding on next but what is wrong you go Corsica so I it but say will define relative to monitor what kind of classical ships or in conditions that are no connection with the connection along the fibers how to find out what the definition so vertical D models on exit specter that are well then figure out if you want according to convene geometry machines on but the fibers nesting you can think like getting .period net into why use some yes so considering this creature eggs and around all over wide around with wife to this kind of thing look at what do you want to consider classical and go well there is the actual but I see here so you see if X equal to wipe this is just Ex so we used canonical and while you control that this case is fully faithful betting but what happens is that the category that we actually does in his between it is genuinely in between and either note that there is a map from here From wife to this not that it would be the story is about from next to here so therefore now will so that's why I talked about the streak pullbacks and that's why I preferred the 1 because we can use what talked with the developer triple so she back defines a map from here to into collects triple back with aspect this that cannot think of it from getting the connection forget the
revenue was almost now let's assume that X and Y are quizzes again assume X and Y are causes In this case what also known as "quotation mark negotiations things with 0 single support analysts define as the pre-match so the those objects here such forget the connections they have Zinc 0 single support and it's very easy to see that the image of this actually lies here so there is classical there is into :colon there's something in between and it turns out same thing moment that this is the beast Winterthur and let me give it a name because it's too much notation recalled 1 OK so what can be considered more hasn't locks since he locks we're interested claim this so you can ask why which this and not in some of the reason is that so we're
still trying to prove language and well would you want to do you want to translate things from the spectral site of the genetic science and it turns out that this guy if you assume Langlands book it is an induction this guy can be translated to leave what can be fully faithfully embedded into something explicit on the genetics it's just by induction on them and rank this category you to put people generate Whittaker category you have to understand what this provision Whittaker categories I'm just saying this because explain why I'm interested in this category OK so these are like building blocks and so the state the diphtheria and then I'll explain With its meaning so the the theory goes on to say that certain functor from into court on looks G 2 0 will want to take all of them together in a way way that following simple all right when he announced policy for some so there is a faithful so would have to explain is that what do you mean by this guy and what is defunct so well and so it's kind of easy to see that for the last joint what the essential was generated by the essential image of this is the same as was generated by these guys to the generation is the same but I'm getting some relief and so the theory is central to England's because all yes that's with what explained on the geometric side for Jill and we have an analogous in what do you want onto could well have a map in general and we will degenerate into them we have perfunctory general known to be fully phased bidding for Gillette and if you believe what I said you get fully faithful functor in 1 direction so that's kind of the mechanics of the proofs of of the length so that's why we're interested in this assertion you it's just part of the yeah I will I will not be talking about the genetics I'm saying that I'm just giving motivational why I'm interested in this kind of assertions I'm into the assertion because I have parallel assertion a on the genetic side have found between these categories which I can prove that he is fully faithful that's what I mentioned by induction in the semis simple rank and then I can show that fight on the generators if I go this way the generators are in the image of a strong and this news Newfoundland direction which is automatically if we fully faithful to this this this and this is how the approval language fraudulent goes we don't know little known work these 2 of this is this is our land concerned with trying to compare this and this difficult life so we are trying to prove that this categories equivalent to this I wouldn't try to do and trying to stop the function 1 direction and moreover wanted fully faithful and we aren't doing it and embedding both into something larger categories and unconstructive fully faithful functor between the Florida categories but the impetus to these countries finally equivalents but the various stages of the people and the state will show that the fountain is actually fully faithful to the whole you to pieces I have to say what In moment you insisted to on work well it's well courses when is the man on this and I hope he will be for people do what will happen is that I'm dealing with classical and so classical ecology book of the same thing at all is a scary words and not what happened it goes to the fully faithfully into the wrong version of representation petition and this is what it would have to be so it's wrong it's wrong version versions so you putting again I don't think it's the category wrong as the person they belong to so 2 amass as as rewards question why this is so with it's useful because this this embeds into something local that's the way you do it you you can exhibit generators union all of this is completed geometric so you then you here in Europe and the U.S. is the opposition yes this season with professions so there's a functor Whittaker model that takes goes from here it's a paper dedicated to you so and so he was a motivation wanted this so that it has been just go ahead with this and of course the the final 3 very interesting the touring station erased the so what is the he setting they have some index category a and have functor from ancient Egypt Qatar so to objects of have some major categories see and have been having arrow from a to B. We have found that have to be be so in this case the couple things if 1 is something we can all along you can take so this is a category whose objects is to think of the soul of these objects little CA for every and for every hour of the Union the limit is it's really kind of a compatible system of objects in each of the categories but there is also another thing it's called Blaxland and I'll explain the example in a moment this also collections of objects like now we no longer require some offices we only require maps in 1 direction In this case the direction and there to laxatives and the collection In 1 yes at this autumn it and this is what I call you and Alex well so
and let me let me give an example exactly but he example what anyone so consider the following example so is this category has to objects 0 and 1 1 arrow and consider the following example of this seat is that why beautiful logical space and why 0 being open and white 1 will be a complementary closed so the euro will be keen on 1 0 again derived category 1 will be shipped 1 by 1 and the 0 2 1 will the she G 3 so when a claim that the good she was I would shoot on way she is and in precisely this sense in terms from of the strike it's not a limited selection so this is what I mean by the right-hand side of the US and then it'll be the arrows in the opposite direction it'll be collected I ruled this example because of wanted in this direction no no no shift but it's as as written but I went and direct categories fund-raisers fund-raiser as written 1 of the 2 work you the company take all the both set the standard parable excluding is standards economies account for about was yeah find post-Sept or it so the the visibility so so this is what I mean the glue that misstate what this functor is so if you look at the definition of what it means to what I have to do have to produce the family of funds founders you To these categories that's are not compatible under pullbacks but their lacks compatible and only if a go from unprovoked another about 100 to trigger composer 1 and needed to propose up to set up the national transformation to say what the sponges are so the front following go from being scope looks Jean I forget that I'm back into the whole thing then I do not wish to ease Co look the looks suspicious they're wrong g day look I've got here I wanted to land subcategory but Llandudno and not here and we land here what do John so you it looks like the G so if you have a joint I will land in the strict limits but because it has been a joint stake system broken and I only have natural frustrations so these are my Funcas interference says that this this the fund into the categories of food from OK so the the remaining 15 minutes of tried to explain why what is said about the models helps support this if it exists at the writer John the medical exists the General Re the continues because a stronger sense compacts the compact and yet all of these categories a compact generated many of the people of the yet so there is a map from here the stack maps to wine yet the production of this book the writer John right at you that left so this is the right joint it is continues because the initial functions has back to conduct OK so far I would like to explain not but the deal's approval is there and is the following and mentioned the following fears that there the canonical action the deal was projected stations and the see things was on the course with circles by life which is been definition the because what's called the category of singularities and again here now I'm going used this before but this procedure 10 ring up to here on the future In terms of this action I can see I can expect the same support 4 that the a closed subset In this project station well I can take him to cope with the circle With supports in why so before to subset think itself we attach the subcategory but now take a subset of the projected station in this subcategory will live in the canton of singularities so this category turns out we can express it in terms of this action namely you take the entire in tents the on the projected station we did not on them so there is no restriction on from and sits insiders predicted nations resist fiction that tenser up and you recover with the category with given supporters this is .period the fear and point being with him pertains to that situation will have not just my X over wide we about 4 2nd we will introduce annotation by what it will be on the line work kill the boring parts and kill the classical so .period the says the
following this category I'm not can also be expressed so again and now the question to the audience so it will be the School of the model of the same things what with what can you get so the the difference between this and this is acclaimed also gotten this procedure of 10 strain this procedure product with something so let me say that it's the more projected as a something so I want you to guess what this 1 that something is the prison so that OK just call it's not potential political relative thing to say what it is here is well equipped to answer this question let's look at our picture again so would have liked thanks it maps to sink while it is too soon and here the think the colonel recalled that the guys here the this so is what I want to so this is a map the fiber wrestling a map and I'm taking its current and take the pre-merger 0 section a series acts away so this is what this category looks like so I want to say How is this kind of thing helps to prove the theory that the blood work well you wouldn't several 1st it's quite easy to see that it's enough to prove want out everything by classical so it's really enough to prove this more if the core differentials I like feel like Look you differential geometry in differential geometry I have put tangent of what times contingent of wide over wide banks have diagram the familiar this is the derived I think so I want to explain how this kind of thing helps to prove this kind of thing soap so now you look at the left side of what you see is that so now both sides are taken from the could g by history all looks at all over the world looks something is the whole in all cases and killing the Korzhakov and that's what the circle signifies you the so a claim both sides are obtained with something so that may explain what these something are so on the 1 hand have this project in positions of so have you want on the project mentioned here the category from the following things would be if the projected Nations relatives relatives things on a lot of the books so so we replaced the original statement that some fond of it has to do with his car I found that only has to do with the models the advantage of the letters letters addressed to Portugal so if the statement is that certain types are captured let me say very explicitly so so statement it is supposed to 2 fable about on the types have knowledge and in the remaining 5 minutes I'll see what the somewhat of a types are CS now the usual sent it out to you know nothing just somewhat of a types as we know them so you know this precisely what they are so 6 . com In this sort of thing is local system for the signature is appointed boxes and its
actions in the wrong section of joint it's important amorphous so now let's that he the parabolic he attached following which are called Global Springer fiber the the following things it'll be reductions a single piece of things G 2 people such as belongs not just of the parabolic and I think the most important radical so that they conform somewhat of a type by going and there is this reductions In the claim is that the somewhat of a type so so L we don't know what hepatitis contracted but we know the whole adjusted a similar thing yes so there will be a diagonal spaces at the back of house In this case the diagnosis but in this case the diagonal district yesterday the Holding Co Ltd but ,comma :colon it's something and some sort of push-ups and the value of the month due to the true yes yes yes it could hold knowledge of a colon is the calling of colleges so this it's not a great idea to do it because the way the when you said you would to be on the according to the law of the land is about to go to the earlier in the day of the people of the 2 you see that's what it called laksa limit you can equally call it lacks calamity and then it really has universal property from the beginning and called and it has been a problem for mapping out so lax limits and call limits they have watched Universal properties and they have both so you have relaxed limit you can say what it takes to map out of it in the same way as what it takes not into it Is just the same Madison's formations so when you talk about lax things in this context I couldn't call these guys just the same way lacks :colon the the theory you go it is not that you want to go over the use of the name and and so there you have it in for you and what you think Maria of yes yes I know that and so I don't want to keep people but I I'll tell you exactly which part of it is it's this you this year you can call it that you do not want to but in the end of the day missile ,comma to Tigers very it is well it's no no it's not something that we had seen before but but in the end of the day after going so we certainly conjecture this summer to protect the estradiol which proved its home logical triviality so you've got a bunch of Springer fibers different parable exceed them somewhere in the claim is that in the end of the day know so but what is involved is the following so you have a spring of fiber In a like this so haven't that the local system that the system can think of worker is just at the bench if home-office from from the fundamental group to To do so sir so you have a so the sprinters fibers and talking about on the intersection of this spring of these bigger fibers you know it's plus the fixed-point locus for these and amorphous acting the fine since these kind of things occur only appears the fundamental groups namely the six-point sight of the generators on the so that in this all yet this part is you take strict diagram of topological spaces the maps that are involved I'm not inventing so we have to win taking this calamity have taken home a typical innit this but the claim is take hold of a government at the end you get we conjecture it's a point we have improved but 2 of the Rules of the the you open until 2 this kind of thing for you to trivialities but while well after we started up to here in became a triviality 1 got so the real stuff but it has to do with the oldest theorems about action on models this is Ontario to produce too but this kind of statement already went it's because we only have to parabolic it becomes very very easy but you again tempted to say where the diagram looks like 4 jails 3 it looks as follows you consider the flag variety the completes and writer-producer 3 and then you see something like that To be ones meeting at a point and this diagram follows the proposed that will be the Oakland 0 going to 1 . 2 2 and the and 1 that collapses 1 component of .period another that the collapse component to the point of with together to the left of the point it's symptoms something like that so again I didn't want to be very precise here I just want to say how the statement allows you to go from you career shoes to actually topology and then you attack the problem topology explicitly using a building something In terms of them


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