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2/4 Singular support of coherent sheaves
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Title  2/4 Singular support of coherent sheaves 
Title of Series  Les cours de l'IHES  Singular support of coherent sheaves 
Part Number  02 
Number of Parts  04 
Author 
Gaitsgory, Dennis

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/16351 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  Singular support is an invariant that can be attached to a coherent sheaf on a derived scheme which is quasismooth (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 the
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plan this is a good move in and I think I could do that the the I want you the world direct
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so let me 1st um 3 capital bits of what happened last time so we introduced the notion of what it means for and I find scheme Over a grounds field came and was granted always assumed to be algebraic enclosed of characters to 0 and so Our when is a scheme I will mean and derived schemes and will almost always be talking about um right schemes of what almost a finer tied but not go into the definition that so we define what it means for such as Esther because this move and what it meant was that if we take it's tendon fibers so they would live normally again and not negative ,comma logical degrees and we want that so these colleges in the 0 0 for my great great so in this case we can define .period by support gregarious sheets so if there was incoherent chief and again when I say that and mean that involved contacts with coherent colleges so and as has a point so we're defining a single support at best about as follows so this has been our circumstances the algebra committed eligible since on the 1st colleges of our times fiber acts of the doesn't graded algebra were the generator sets and agree to act on his way to space and this is just the support this support all this great .period all were designated eligible for the yes and so the support lives In the vector spaces doll 2 but space which which the 7 and so it lives inside so 12 take H 1 contends fiber do what I it's the miners 1st home mortgages the contingent so that those are appointment definition and as Le Monde asked so how do you ever get a handle on this it's so theoretical but so in a minute and give you a kind of a couple of practical definitions of this thing the measures finishes this recovery capital aiding the story so we define a classical scheme saying occurs to me that trend Over the classical scheme underlying as secure all lower colleges suspect sameness of each 1 of the 1st this is the tendon sheath consider its H 1 it's a good year she's on the classical tests taken similar and so said theoretically that looks like the Union overall .period these guys of 1st colleges of the content fibers and then we said OK so we haven't been over .period twice supports so have weakened assigned a singleA Support Command which is some sense in assessing the sun the proposition is that it's closed is a risk equipment so In the next talk I will give you yet a different definition of single support which will make so we should make such claims obvious so for now understating it and so there was 1 essential result that a quota last time that said the following that sold single supporter of F is contained in the 0 sections if and only if staff is perfect so sailors aboard really detects the measure of imperfection of us OK so now consider an example hands a very kind of handson definition single support and then I would give yet another definition of single support focus here is an example so let you be a smooth the variety equipped with a function and say this function is nonzero solidstate is 0 setting function so this is your too hyper surface and is given as a complete interest sections of this as a classical scheme but it's also quite distant derived scheme was set last time that locally at intersections are exactly during those closets direct schemes that happened to the classical so let's say In this case of single best looks like well I gave a general criterium I want a general description of sink essence In the circumstances so will look quilting and the long cold differential of the snap it's so it's kind of India start course the "quotation mark tangent bundle to 81 is a 1 so this is really a map from the OS into that and so if the colonel of this matter I need it looks as
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follows so at a given point so that's a given .period it looks as follows it's 0 if protest is not equal to 0 and is onedimensional USA but some of the functions degenerative this point and so there is not much room for single supports well according to this theory would have is that so 12 all I can say just 3 status in this case that's on singular support at a given point a will be 0 if and only if if the test is perfect .period is a risk neighborhood of this point so In the case of harbor surfaces single supporters either 0 or everything you do all you have there was no reason OK let me say nonzero be nonzero and then if and only if it is not the of of the is London so no 0 means that income excluding excluding empty set and 0 I'm saying that you will hit the skate of if and only if you're actually part imperfect at that point in perfect 0 is perfect hour so now what I want to I Want to generalizes examples well 2 the general situation so we said that in general acquires a smooth scheme can always be fit into a Cartesian square where you need classical schemes so could use squares taken the right sense so and parallel to this so saying was the kernel of the maps the ground to star in the mystical Tennessee 20 star you restricted to ask so this is the door what the differential so I want to give a criterion its notice the following slid into disappears imitation that call the straightening the tangent space is currently at the point and so what you see is that this always sits inside the far times it's a very concretely the single supporter we live in the s times and you'll vector space what to the content and fiber demand via the point so now very concretely I want to say when an element here belongs to a single support so let me take sigh an element of the stock and I want to say when the pair In the past ,comma slide it belongs to support of Indian chief as saying he is a procedure and that is due to felled and you may so there's some vague analogy with vanishes cycles so its knowledge is really vague but well OK so here's what to do well here with me rewrite this again now choose the function on this whose differential at you're chosen .period is ISC side so this is my agent this was so did Indiana but my point is that OK and so on 2 vanishes of the point so it's meant to notice that the crimes and notice by the prematch that as prime this wasn't going to point and this is my house so if you wish that's what the set was the set of zeros of the map H and as prime is a much larger Close subskin it's the set of zeros of the composer a long history other than h none at a phenomenal year whose yet the 2 guys a smoother ages at everything yeah note that these guys this has yet to SNS primarily right well enough so as to what but in fact get tested both moved as a direct but notes that note however that s Brian is a hybrid surface because as primers as prime zeros of this composer map it may happen that this matter 0 but it's OK were still In fact when we discussed this type of service situation it's just as well applicable to the case where the function was 0 but this is more than sufficient additional options of course you know don't establish this is limited and there may be ineligible because they can file
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out get confusing rhetoric portents but so all it means just Craybas diagram and as prime is hyper surfaced that need to know disclosed embedding by I stand here is the theory is that so this belongs to the singular support if and only if the following happens so you had your leadership here take directed and you get a coherent chief here and what you want is that this guy the perfect so Bond In the of U.S. surveillance is if it is not perfect all this sprawling I want the risk neighborhood so if you regard perfection as an analog of the property of the balletic sheaves being leased but then I think you can detect single support by this property but not on the initial scheme but on kind of a bigger guys hyper surface finally I don't trust in the licenses None of it's no no complications here is just under close betting nothing happens this is a list of all the faces of the 2 you have there is a need this year it's absolutely there wouldn't recognize a direct images have a map of rings and adjust its forgetful under the map of ranks with the users use of the media and in the yet so this is the mostly functions not no publications here all right so so now I'm facing 12 it's not the dilemma so I want to give you get another definition of single supports which is very calculable it's it's complicated so please bear with me and if you don't like it just race the next 5 minutes to memory but in you just eat it may not be so nice the of the on most of the wealth of the Order of proves this is not as presented if you can't do this I'm not giving the stairs in the order in which they approved and to put all these have to start with the yet another definition which which of next week so that he can have the ultimate definitions the Hawks of go changed it and don't want to throw them at you right now that will do it next week all right so OK so here is that so let's call it gentle definition of the new Koizumi definition so we will we are still in the situation so this the installation of complete intersection and so large it is an object that has already appeared ecology V it's .period times .period over me which seemed as that before and um Weeden said that so called for this gene was modules all the following algebra so it's a perfect watches the mind you is the tendency space to this currently at the point OK so now I just noticed that well it's a good point over there .period and the definition in draw no group Voice over .period are groups so it's actually a group object in the category they're asking and moreover this good objects acts on our on our ass so mainly you can draw this diagram control want the students and yes so I have tacitly passed so our new world so we have this Infiniti category of derived schemes in that in this category the notion of a group object yes it as is our this is the center of the area yes I'm referring to this picture so if you wish this group or that acts on the points lifts to the group .period was both Gaza conclusions acts blistering report on us and in other words we say that his group acts on the group and even the scientists Ellis ultimate act only the action here's what you do and member I said that the single support is a film set in the Star tenderness you start moving objects when asked you listed the names of the action to here and then you apply this promises this equivalence is what I call crystal duality along the 2nd buried the 1st Variable said you go from here to a module and then you take its support some of rumors that singular support of equals the support of Whirlpool duality applied effect offers star as a subset consensus times so when you actually need to compute something use this definition
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such I were to see the 1st time I will be completely Buffalo how'd you could what is this action how would you compare this things so maybe don't worry about it right now I just wanted to give you perspective how to compute the OK we you have an example of the use of nonlethal nonfuel signal that there would the 0 yeah we we saw this example right yet so it was only at the end of last time and I'll come back to to the moment you so
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this could be used to service the futures all falling short of the number of years now but I'm going to call and undermined him in because there race is on the yes we saw the example of this particular I remember last type of this guy and so we said that this category's equivalent to this end so for example if you take the skyscraper he will get the model which equals the entire elevate supported everything there is no need for the quarter of on ball and then you can so that it got all the intermediate cases take any object here and there is no concern of course not so they're not take any any object take Baker :colon take any object supportive and apply the inverse equivalents in about OK so so In the next talk out loud and give the ultimate definition of single supported through various properties so the means to put this property is it's inconvenient to stay with coherent ships and so this is the next step but I want next procedure that I want to apply so 4 so we said that if the it was a complete triangulated category you can define a subcategory see little scene of compact objects I I sold the claim is that in most of the practical cases this subcategory retains basically the same information as seen in fact there's inverse procedure that I will explain right now the only problem for the interest procedure that well you really have to abandon the old world you cannot stay with in the world trying to other categories you need to Portland major categories so you could start within DG Category C not you can't produce and you want to call the INS completion so we basically trial characterized the uniquely in a moment basically what to do and all the direct songs and calls and direct sounds columns you want to do that a little circles to distinguish notationally C C so when this is a small and so on functorially it's characterized as follows so if you didn't see well this spring in the circle synod will become complete and when you talk about functions between Coke bit categories you can talk about those 100 that take direct sums of their times these the coal continues so continues Fontes from this and completion to an obdurate arbitary target Cook bit Category C are the same as monocled funk we used to have guests cocomplete local complete completeness is a property of the triangle at level just have directs the Department of direct these are is the best in the world yes again so these are yes this defunct former space and it's an equivalent of spaces there is again and this depends on the model I functorially in so so we have to operations among this going from a Coke bit category to the subcategory of compact objects and another from a small category not to the into completion so I won't explain what sense these 2 what procedures are inverses of each other the not quite canvases but almost 2 years of the world vision is going to have to do the materials of this year yes you would think that this as the as the latest of these you yes itself it's a filtered call it but the category of investors is a higher category I was just you needed to hire categories of misery index as the indices to create all this calamity I yes I think so I want to say what's hazardous operations are due mutual inverse secures a definition but some very very important 1 but cocomplete category said to be compactly generated if the following happens if you take this subcategory of compact objects and look at what is right orthogonal to it so all the objects such that home into them from compact guys on 0 compact generated means the survival is nothing save and if an object is right orthogonal it receives no Hobson compact option that the subject is is there or so this definition I believe Thomason and there was a beautiful paper of Thomas the words used OK so high so the there is as follows so you could start with our magical complete category you take the subcategory compact objects contained in he could do not back to see is fully faithful and that it's a crime is an equivalency if and only if C is compactly generated so most categories that you encounter practice Arkabutla generated and so tells you how to reconstruct C from compact objects and said the opposite direction the maps from seem not to the following suit that statement completion but cannot and must take compact objects that it's not an equivalence but this functor realizes that as see car To Reginald ,comma currently gets the Karrubi completion of this and all the images of all that important OK so however call Levesque of Islamist C 0 and I would really want to consider in scope all of us which is by definition in what called hold on the general consensus he to you know 1st the use since father for all of the interest right to Okolona
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given exercise Exercise Number One it's a it's a great exercise effective falls from something that's already written on the blackboard but you can describe it very very concretely as follows Nixon says the From takes 0 take it all these are the fact that is our the category vector spaces but the DG world so complex as a vector spaces so for some reason I want to possibly and completion so why so in fact I am interested incoherent objects why didn't bother to incomplete so it's well the reason is that it just so happens that it's much more convenient to work with this and completion analogies so imagine yourself from ancient Greek and 1 day someone came to you and said all its introduce negative numbers but why do you care electrolyte number both abstraction were interested to count our apples but so if you want to account for the fate of 5 597 it's minus something and you don't want to give to the keeps writing this all the time and it's the same kind of thing that the end goal by so you knew something was going on and a lot of just a small category soldier category all of the of this year on the whole thing is what the Lord Mayor's about well yes so it's a world of the categories of contras between them and they also permitted a category so some of the existing side by what is whole what do you know it's very close to Brown's representative is there so this this thing the the members yes it's fun our world so let me but we give an example of a situation so if scene is paying more and areas in algebra so is compactly generated it was what do you also the of in office we should this was the only yes he should do so for this go to uh it's Lowry's books 5 sigh of 5 . 4 5 . 5 with little to no more than 10 and fortunately it and that's what the apart from the use of gifts so this is an example of a good idea the category but when it's stood propoor knows proof so why is it the case that no objects here a writer funneled to combine guys this is Over associative algebra the U.S. market we have a good student here so here is a compact object namely a itself home from a is the forgetful functor so home From aid is the forgetful funk there's something want to this function is conservative and therefore nothing gets center 0 so no need to look at all compact object is among compact object is enough so particularly apply .period a primary listing around you obtain that at the end of the 1st defense maps isomorphic to classical and now introducing the new guy called in "quotation mark 1st which is if you wish to adjust definition in the midst not something that you know a priori just a new guy so let me play a little bit what was best also had the themes :colon can you have the old classical and 1st I claim there's a canonical a defined map a "quotation mark arrests so all the definitions of the word let's play a game so who will define the snap from yeah but you know it to somebody who doesn't hasn't seen this before would you be able to define it's not for me yes just using this stuff and above board that's all you have a loss is a loss of no that's not true code does not matter coop compact objects Corsica yeah is not contained coast Compaq UBS aquatic whisper Co is not contained in places to I don't I don't need compact object of apply this was seen not is cold and see another squad and sees clavicle we will defunct in that direction but we do have a functor from coal to quizzical just in that so this is your site 1 but yes but its construction so we have this that we just say that so this observation is that into cold has a 50 structure such that this fungicide is the exact and not only defined and in the finance and equivalents from P & Co S plus 2 classical dance for us so these are well utilities and things it's the bonded with bombs and below so the only difference between these 2 categories is somehow to minus infinity in it and wonder how can this be relevant so I said at the beginning it's all about divergences spectral sequence so it's really kind of the details of our categories a different and everything else is similar and it can ask how was it possible that this is of any use it was in the of the the government is we always used to say that we will use the power of the people is "quotation mark middle class means the following plus is I can say save it's it's equivalents on these guys for every on the volume of hours equivalent everything is equivalent to details of different offer sorry and I ask is that over the course of the year is limited by the people to know that I don't take and so plus means insisting that union Of these guys for all and that's a plus means where are you this is nominal into coal when defined go I don't refer to energy structure when their right call a set of last time coal means inbounded compasses with coherent colleges yet so when they defined referred to energy structure has been the cause of the EU's desire to call all I claim that it has a canonical defined in to
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structure which is in fact the uniquely extended from the destruction called it was on the way it's not not In the insults to the you usually really have to do some the more cause you can say that so INS makes sense not just kind of kept context in the makes sense injured general into infinity categorical context and it's completely true it's true that this Is the into completion of coal greater than they got in my innocence he said so in and makes sense always it's just a general procedure in inventor Qataris In makes sense always categories OK but the next question to the audience again for those who haven't seen before I want to fund in the opposite direction "quotation mark gave it to me don't be afraid well come on this is the 1st communicate Michael says 1st so Michael says this a is in Perth Ensco is in Co turf to call an end extend Zebedee saying yes sir no no no no no no no we have witnesses here OK I think was his said that so I'll call defined call Suntech signs and CSI is just well just in them call and then in extent of let's consider an example of the new danged the thing direct scheme as this speck carrying false generator signed side plans and degree negative too and even object here which is just the structure Schiphol and you want to send it to itself kind but it's not coherent and lives at the sentiment ecological degrees In the so it's it's it's not gonna work like this it's not going to S is not the structure she does not coherent so definition S is eventually go connected if the structure she has finally many colleges so for my large so example well example 0 anything which is classical is eventually go connected but what's important for us is quite who guys are this is exercised please remember quite smooth schemes can be locally written in the script products and using that should be able to exercise took this as a counterexample This is not counterexample this is a guy which is not to mention the connected so this notion of eventual Co connectedness is useful so state 1 thing about the methodological bitch that makes 1 appreciate more you have a map of derived schemes to the use the 2 of them on the other side yeah so you want to only only it's yet I won't go back to that in a moment I just wanted so let FBI morphism between this derived schemes again we're almost finer types then after years of Final Four dimensions if and only if it's fibers so you should think of the failure of an eventual connectivity is the failure of having final tore dimension "quotation mark the this crisis soon are eventually go connected so let me go back to the countries 4 so the claim is that there mutually joined former joint and b is that this CSI guide is fully faithful of just eventually go connective the condition that the CSI exist and so this is actually a very important point have therefore joint founders with this being fully faithful this is called the situation of coal localization so CSI allows you to view choirs Akos subcategory here but sigh Elodie attribute as a quotient category and you used this both for interview all the time so it's a sub but at the same time as some kind of course snakes of it's actually quotient of this bind something which is also very explicit and a lot of mistakes made by so when it is we have but it could the world of commanded the word quotient some of the stories residues of music fully fable means that a fun so fond of the fines amount between mapping spaces while homes before were sets now they are with the home between 2 objects now is a space the font of usefully faithful is a snap space is a necessary some isomorphism wealth isomorphic McAteer spaces which is home of equivalents they use in the U.S. eventually go connected off of yeah on Wednesday at his home of all the players and what his government in the world who made lawyer with the fact that even worse than that what would you say was the cause of death for the murder because yes so this is something so this is something we know and yes let us say 1 more thing so that but is a 1 more thing so I said that this is the quotient on this it's a very difficult me something with Bocelli Leonard tell you to quotient by what so the kernel of society is all those objects such that the memo had structure so the structure here is a
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degenerate so there also are objects that all of whose colleges a 0 Misty structure the objects and on 0 so for that reason this this Collison some kind of phantoms and inorganic and go back and ask how can these phantoms Varela without in a moment of give you in a concrete example for example here the question How do fewer phantoms like I want to I give you an object in the colonel I want to can produce a number out of it but something very tangible and will do it in a moment so it's actually possible to like really get a handle on this and the film by definition in this situation the kernel is right to funneled to the essential image of this a societies in the world is that that is note with a 70 this category has a T structure dead so it's a shy with respect to that of the 2 structure that has had existed only in scope all these things and I think it will be at the heart of the use of plastic that you would like to know this is because is 1st of the 4 years since the group was call all associated who moreover issues it's about actual growth over the issue of whether or not this Singh I'm just so that the the last of the you this year so I thought was the this is the home warranty physical moral issues these philosophers and lawyers were we're years of now is here is that these are the worst because of the division of it it did well in some sense this related to this this the structure In the in the extension of the structure cold you which is 1 is said to have used his power was all right here in the last match allegiance incentive in the world this year you want to in the last few years well yes it is and moreover you Danziger question is already contained here I said that so what Alaskans very good questions said he has can they think of this individual colleges really career achievement category here's how you do it you apply this font size you go here and take cohomology and it was said that size team exact so is the same thing as take on GMOs upstairs so advances the only thing that the objects there all of whose colleges in the sensor 0 of the objects on 0 so the structures was called non separated this was going USA so there you are you are based on the city's up so you you want to produce counterexample something tell me of what may be of help here too "quotation mark you some of the people of the last 2 years of work yes I'm saying that this functor is an equivalence every time you bounded from below I'll give it 1 more example you'll see maybe you'll see as the game so here's a question to ask if I were over I would ask can you give me an example of an object in the kernel is avidly asking a lot of the time because year of the growth of the orbital does not pathologist was infinity but maybe next wishes to produce an example and the 1 for his support and I'll give an example of a moment OK and let me do this example and will make a break so as I said there is 1 example that I find particularly illuminating and its as is my favorite .period kind .period over I mean it's from comes in new 1 and so here is the complete picture so into address is isomorphic now you take all modules no perfection conditions so Ensco is all of this thing classical as we said last time is modules was support and 0 so our sigh well just the tautological embedding and this fund the writer joint it's taking 100 called College was supports us something so writer a joint but if you have a but the Brighton sobriety closer prices of Fondren joint To the embedding of the full categories she said the radical supporters of sobriety To take Comal support the that's that's the wreckage and that's what it looks like so for example here is ending an example of an object that lives in the kernel of this site yes so we could take and need to be onedimensional then we are dealing with sand some because of its polynomials dignity and to not to be confused with that CSI would agree was negative 2 so typical module will be K of intimidating so this is a typical guy that will vanish under this Sumter on the basis of the Five or used to be of all this is our KwaZulu the other from last time Help Save the moment and so Koizumi reality and are home from the skyscraper the "quotation mark with costs more than of EUR by definition there did start life supporters of the periodic you apply Kasal gelatin used look weird to go to the western edge accountability gold not all modules but this kind of Wales this is similar to those in his home city of work yards of course could home that you and influences she found the image course we're all she's meaning um all not cause a career member of the crew of the other major cities in the front yard of a home they could use you as you schools serve chief support of the cover the use of computers in the market
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that but I want to finish this hour break something tangible but it was good so it's been this example but knowledge consider categories of what is it could very respected the action Jamba delegations so we can consider into cold of what it's our S and well you can take it either the quotient or its equivalent to the kernel of signing Jim McNerney so our phantoms and the claim is that this is equivalent to cause a clearer sheaves on the projected position of star it's something very very completed you wanted to number what about the number but it's a great and its actual cause a garish even something classical so Fiji if the dimension and means 1 then we think make love quotient so the colonel the considered view the categories to select so here if you wish to get enough numbers so it's really kind of tangible way to quantify the difference so that finish may giving the following definition so let's and B that SBA cause smooth and let as the conical glass of sentencing so that we have the following categories so we have in we have quietly coal which is used in Perth but you also have something in between that I did note into cold can and that is by definition in Co sub and when this was defined last time a school here she with support on so this is the very definition India call and this was fearing that it is the 1 in 1st and as before we have eased embeddings because the corresponding categories of compact objects as natural bandage to each other and not into general theory of lorries no this fund is always admits writer joints so so this Ensco is what it is the category of interest and it's connected to the more classical category but this pair of adjourned funders and to the bigger captive into Kobe a pair of adjourned fountains and again it's very easy to make mistakes so because you can think of in scope in 2 different while 4 different ways some quotient of quotient and you really have to make sure that you know I think it a summer a quotient is something OK so let's make a break with continue we are interested in and into cold understandable consistency and I thought that I should well say something about that 1st I should say what it means to me into of algebraic stack and so and also how is little it's all director the government right yes from now on everything is derived and that causes some trouble mean OK so into called the basic maybe a little bit boring but it didn't go through it nonetheless higher so 1st of all for this and for many of the things we don't want to restrict ourselves to let the 1st talk about Korzhakov so I will define what it means for classical revival to break stacked with it really makes no sense to restrict ourselves soldier breaks tax what we want to take why is an arbitrary priest AC and what that means it's a completely arbitrary conductor from I find schemes I mean derived defined schemes To avoid back of the what exactly I want representable frontrunners to be what I want my schemes to be among the stacks now home between schemes isn't facing reported because wearing inventor category so you've got to include unit intact "quotation mark spaces therefore market so what is the need to have cause a call on the priest as well what you want to say it is for every good enough for every S .period little life why you want attach so to specify specified an object of requires a on the please stack so to every S . little wife while you want to attach an object of classical and ask thought of as the pullback and every time you have cast someone goes to 2 so that let's call this little that's called a staff who is denoted by a Y 1 the composition to every situation like this you want to attach an isomorphic visits between an opera star of F S 2 White with what you've got on this 1 itself and of course every time we have a threefold called twofold composition so it's in the have and but the wife 3 what's it gonna have so it could be mean effort to restore order give the opera
1:08:54
star can when the but the landing rights for once and for all so on the 1 hand it will be so are adapting for this competition identifies it with half as 1 by 1 this is everpresent Star S S 2 where to send them so we have to be eases maps now which was so therefore the 2 circuits to go go from here to here these and naps between objects and higher category you want a space specify homotopy the man you want something close for objects and you give up and that's how and of his rights but so I I would like to use the idea that he is the holder of the launching of the new yes I do what I do I put my hands on Laura's book and I do and I say this that is the None None just 3 sex so what I say is less well at the right is amended saying so there is a functor From a fine schemes opposite to what did you can't respond is as those 2 classical have to believe in the existence of this fund and then we have the category of fine schemes over why he so Perez S ,comma little white there's a forgetful Fonterra from the map to so you've got Fonda from here to here and then you define classical of life was warranted limits what's called this fund to classical Leonard over this category of fantasies From this country innocent objects and so this is 1 of the things that the theory of higher categories good for I know what it was the precise images this is 1 of the reasons it's good to this theory exists and I have to think about it is right and never think about it um well but then they the questions of how to computers so for example you can take vitamin algebraic stack and you can do that a compared this definition with 1 in the Book of Vermont and will there be so what is true so remark that is why soldiers break what you know is that while there is always a T structure all kind of for a priest energy structure so this is definition on his bounded below party gets what always did on the entire category I have no idea and there's no reason to believe that the other side of the you have all so have all always is that could it was the 1st of the month you this was going to do a lot of people in the last year of the she decided it was time to learn how to do it when you were told you so what do you have a lot of school some offices How did you do this year's with them so I circumvent the question of how you homotopy good but writing officers forms the community during his this is it both as the same as what note the use easiertouse king of course to get the same thing but it even algebraic on the other breaks tax I don't know what the definition gimmicky minus infinity you go on to do something on his gesture of course is that could well you get along with his partner and you can say that he couldn't I I won't be able to so we will get there and what we knew and whether this is the 2nd the note it's 4 1 it's not each year you will see that the moment so it's truly the group Assem simple curve is isn't June is great and told them this time will be classical this case however we want to prove something about it you go through looks is Burrell and that has never classic of leases because of something that might not with any so you can give this definition even you can restrict itself the classical schemes and I will still be the same definition the problem going still persists so the division is you can do it in the draft setting on the classes of he said that the university and the resuscitation of only 2 lots this unity In the last 2 of the usual derived not union director artist he the vote yes yes did so by definition so when I say algebraic 2nd in 1 stack and those we are with you see the same it is covered by derived schemes In the same amendments is just exactly the same as in the classical theory the the rest of the world of the view of of the I will I will say in a moment OK maybe members of point said so definition so the nudists and only slightly more restrictive in general so algebraic stack Brady derived that is a priest With the following properties such that 1st of all it satisfies the tell descent and to that there
1:16:47
exists and the rides scheme equipped with a smooth map With a systematic so schematic means that if you base changes based scheme you get the right scheme please I was the world so I'm skipping so properly I should should started from other breaks basis so I don't this was the year subjective schematic reading students rejected this the reason is that like most of the rest of us in the world what is there this was the worst of the worst is over all you have to learn how to engage his assistant efficient alike and fine loaded up the stocks so 1st of all USA where the derived stack I impose descent condition say descent condition and then was overcame but let's call these guys derived stacks and now there's a notion of Kate Martin stack these guys are defined inductively as follows so for more money for cake was a negative 1 so I mean the they're very solid safe and became a negative 1 derived what artist case stacks are derived schemes and then perceived the induction so each time so why is a K artist stacking if it admits and map from the derived schemes such as this map is says that the fibers of this map arcane minus 1 artist acts that means that you can base changes by another derived scheme In this fiber products um what you want that to be so this yet is yes we want this to be a K minus 1 artist stack and they defend and then the induction it makes sense to require the snap beans smoothness rejected so it's kind of inducted definition it really isn't that reasonable scheme to the question about the value of the work that gets the most infinitum reports and then yeah so this is a reason was given definitions so what we'll be dealing will be algebraic guys under breakers is chaotic freaky equals 1 and here I gave well I don't like this sky motivation that has this for brevity I would normally should require this map not be schematic but with fibers build break spaces but that the requires something stronger about the work of the eye shields for esthetic musicians actually do Furcal race faces than than all the breaks steps but here just escaped How do you you you go out but I would like to work on Over the past buyers I hear but so that preceded think it's pretty clear now or you had this to have a question so the truce came under definition is as follows defined right schemes define a driver Gerais basis and then don't defined what derived 1 artist and then the induction and you can start induction from minus 1 already In the light of the new year by this form not yet it is half the size of a serious so drive to the right the race faces required enacted itself but that is something that is completely to the north this is a new it's the usual yes but the only thing that's so just to convey the spirit of things so that tell the family the following that it have this is on so there's a notion of a tall matter between a fine is going
1:21:49
direct schemes many form this check nerves and then you require that the wife so this is a .period infinitive reported many have this this is a Coast simplicial inflated reported you take its limits over the category Delta In a report of this matter limited the category of infinity reported cyberspace spaces and requirement has not been equipped and ISO morphism in the category of so that's the definition you it's a little frightening but any kind of future 1 has to follow one's nose meaning there all it is difficult to give a wrong definition here just proceed the group members of the community there was also longer for what here sheets or for these guys please you know I yes however I wanted to say 1 thing and it isn't going back to office questions so we've got this general definition quizzical so observations let's why the algebraic so initially for garbage at stack height took all assessments that says we want protesters have used to call all skews nothing to life but while break 1 can consider it 1 can replace bake With my smaller category so 1 can take the minutes fund can replace the minutes so here we took over the category of old defined schemes Over why what was it but so you can consider another category that Napster this index category they can consider and find schemes clinical it's smooth over why so it's a full subcategory here and haven't then it's over the category Kazakh Napster to this limit maps to the smaller limit and the Queen is of this limit will be it'll be an isomorphism so don't it's enough to consider all these schemes of maps smoothly but even more so you can take smooth over but you can't take at moment .period category where you require that the map between the schemes themselves recently she don't need so not only this asses maps similar why this estimate investors also sold and if it's going to be the same limit Navidad limbs wilderness amorphous this vision of the world strict meaning the result is that the U.S. and the New London as the U.S. economy no 1 was the use of the U.S. utilizing the funds yes overrated and then furthermore is that middle phase I love arbitral mass between the assets and out of ultimate think I'm recalled and between them and his services to the people who find it Of course we note in there is 0 soon singles he was going to be a lot of his it's a great question smoothed over .period we had the definition it means classical and smooth the woman was the move is that not so it's it like to that there 2 questions here he had a matter which artist accident makes sense to talk about the amount being smooth union cover enough so you can reduce the preciousness of maps this week artist question the contestant a that between schemes the 1 almost always skin doesn't mean schematic good luck to them even so you can say you can talk about that let's from Dragoljub divergent you can take and just classical to stack it makes sense for it to smooth over a point on his own there although yet so that's what I mean when I say smoother means and in this sense the move what is this 1 that was the yes but even if they stick to algebraic spaces Belgian unit this definition to any map from scheme they sooner this definition any map from a scheme to my stack is already schematic so that's why did definition so In this definition any map from a scheme is already schematic therefore it can and efficient would maps but it means for this not to be solicited is automatic it's already it's a schematic map you know you when I say scheme I mean derive scheme this is my view about the right schematic here so far this what you would not listen schematic a schematic if you want to know about year the year of our Lord on that's true so my bed you request a matinee idol that is not intended as a matter here so with less than proper definition Woods induction so in this case and algebraic space and then you can make sense it makes sense to say what it means for marveled race basis to the schematic to be small you know ,comma here who has skiing might be him that use of you received in manages to them is smoother is the grass that affairs when they as Eriksson's question what it means for a match between the right schemes to missing so the were so full we are in fine detail and definitions have smooth if and only if if it's fibers can derive sense FOR classical and the classical singing schemes notice that I'm taking fibers and Uribe sense so therefore flatness is absorbed already and as license as it means that after Tommy change adjust Intel locally smooth means that you're crossing with a the usual already I'm very glad that you didn't this didn't put you to sleep but on the contrary I thought of people disarming other racetracks um OK so knowledge means let me the final In well I will only decided for now 4 although the brakes steps or key artist tax the reason is that will enhance the formulas but for now we can only we can only do
1:31:19
this namely you could have a map are between schemes voters effort for star wanted to map from the end if the Nablus that this is easy to say what it is is just a take if the matter sold or more generally find it on dimension you can do at opera star it naps to cold it does if and only if the map is defined dimension and then he extended and this and that is not a factor dimension there is little film that tells you that it's actually possible to define it within a reasonable properties so but you there is this is this is the same as here so it you into Co is a functor out of the category of a fine schemes and smooth naps which then and then there is a lot of that was so limited use a preview so as star will actually failed for homage from office what will happen with rulings as ever streak so in coal is good for Everpurse streak and we'll talk about it in detail next time and as it is the only theory with a lot of content with of shrink so let me I want to test the level of alertness Al writer Theron which 1 of which points will the false and want to see if any will catch me Burton who called a doll in the sense that you look at result a doll in what has Laura sense start the other note Will you will too will talk about so I could have prevented it for the foot our current goal this this is good enough and will be looking so let me also say that the defined preferred same thing performance while following a phone call away so proposition again this is defined for all the breaks who is happy and who's not I defined separately and Co as that limits or I can define call the limit and then take the hint about his yet the scheme that's OK Saxons completely false all of OK so let me tell you when this is true so what you really need is still applies was a compact I find stabilizers this case it's known locally so this is the statement makes sense for arbitary price tax it's completely false trip tax so let me say what US troops the largest part so for this statement I mean this only makes sense in 4 in finer types I mean we all defined Cohen finite tight almost 1 attack this you can define in general and it'll be completely so let me say what is true so let's guide the was a compact with a fine stabilizes so in this case the gold the prime it is true that the 1st of wine is contained it was a cold so not much is true that this person dies as defined here are actually compact and I want to say that it's in the hands of funk .period well actually it's said it's the quality so that it can ask if it's true that in spite of the Firth of Clyde is an equivalent and I don't really know it certainly conjectured there's something lurking in the back of my mind that I recently saw a paper that proved that do as does anybody know so under some mild conditions so fertile conjectured that is true but it's not known it's true in many many cases soak in many cases again explicitly approve it in vertical folksiness and stuff like this the don't it's not in general so dark are some pitfalls so these are some of you will have more yet the geometric .period here it's kind of yeah it's really some of the year in which the "quotation mark where do you the flow of people get it through for schemes services with for all the race sex is just not the kind of thing it's not and I think that most of the people positions for these faults this is true so this is true this is true then this is unknown but you have a lot of people yet here this is true this is true this is unknown separated like this I just to say I have rolled in this way to kind of again to draw your attention to it's not such things enough for granted would you like to use the use of a result that I suppose is here was the go history of the development of the city it's a nontrivial theory if it is it is it is on the call was quite a separated yesterday by then it's too the want political this is the 1st time they have to note that in general meaning you will need to be here soon .period yet include this falls for arbiter priest acts so too have inclusion and yes yes so you need to have take the following skiing in detonated the judges into many points all will not be compact already there so quizzical comeback this is missus essential all a lot I'm really behind it but so finish with a few definitions
1:41:07
so let's Y algebraic status or if you feel comfortable with the artifacts continued KKR to stack a yes everything is derived and always have something almost a fine attack locally and all this means is that these schemes by which a smoothly cover are will almost a finer tape so definition why is cause if for every little life which is he noted .period we have the following H In the tension fiber is 0 4 by greater than to retrieve to just noticed that if you're dealing with gay artist tax and the cohomology will live From the degree negative came to infinity so this is if if um if White is a chaotic stacks I'm sorry so if algebraic stacks it'll be from minus want infinity so from minus 1 2 0 this is the school was responsible for 2 morphisms and from 0 to advance it like in the case of schemes for singularities so we're interested to bound the singularities character bound the automorphisms so this is quite smoothness equivalently and why his client assumed you've only for every ski mask that covers smoothly every such that s causes smooth so quite is 1 that is that if a cover by smoothskinned that scheme was missing so you define single in the same way classical K art and stack the defined as they waited can suspect relatives suspect of sin of attention shifted psychologically but 1 and you notice the following that if you have a scheme that smoothly In the whole of serene H 1 center just as before that you'll notice that in this case what is this about this is usually a well it's K or just an artist active why wasn't an artistic classical artist act so it's it's skin it is schematic over Over wide so if the situation were efforts alone you pull back the Single White as you recover silver maybe I should have said in general that the 2 had lessened maps and smooth out between causes small schemes that In general single of too for the afternoon With this 1 saying the first one issue a smooth map doesn't change the 1st call knowledge of the kind of attention for most of the stalls yes it's so we have a notion that have a K aorta stack may still make sense to for a map from from a scheme to that to be sort of a requirement that the state was which compatibility this was the I would say to all was this need for requires a smooth art stack the all the school the problem was that people will you hear the same thing some of the there is new follow up which is similar 4 F which is smooth we it From a scheme to an artist but do not Bloomberg said that is also true it's it's also true after you if you have a smooth map of artists Texas things just pulled back from Quaresma maps in which case that's the only case with things that defines so they the Saints fullback "quotation mark Swanson said you have follow yes so quite as soon as is a property local topology whenever possible sense the change qualities the image 1 each 1 of the tangent if you take the Tanjung downstairs tickets each 1 :colon back at the beach 1 of the tangent because each 1 sees only the so would meet in the remaining 1 minute From the give of our crucial definitions so far lipsticks and the risk closest contol single line and the cause of these as mortals upstairs to have a welldefined and insisting test for every has not to 1 then we defined in Co the proposed some band of y to be the full subcategory consigned Cowfly equal to the same limit index category but now you to take note of the entire into cos but in service you just limit well take limits of subcategories .period categories and .period category in the limit the move but we should not allow like that yeah I take the subcategories and 1 can again asked the same treacherous question Is it true that the Qataris compactly generated and the answer is being double triumph sometimes you always have the conclusion 1 direction and sometimes the categories compactor generated by that In particular told true Pollock's says basically every time you see in global complete intersection that will be but we know that this question is a particular case of that question taken to 0 so if we knew this was less responsive as from this year positive as well again we don't know but in most practical applications it's true all right so I ran over time and they still haven't finished my program for the 1st stock In
00:00
Classical physics
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07:34
Group action
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23:27
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33:28
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51:51
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1:08:53
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1:21:49
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1:40:49
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