3/4 Singular support of coherent sheaves
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Title 
3/4 Singular support of coherent sheaves

Title of Series  
Part Number 
03

Number of Parts 
04

Author 

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. 
Identifiers 

Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

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Subject Area  
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.

00:00
Covering space
Statistical hypothesis testing
Complex (psychology)
Perfect group
Multiplication sign
Decision theory
Embedding
Algebraic structure
Complete metric space
Limit (category theory)
Mereology
Functional (mathematics)
Category of being
Field extension
Summation
Causality
Funktor
Large eddy simulation
Operator (mathematics)
Hadamard, Jacques
Object (grammar)
Resultant
Units of measurement
Directed graph
09:09
Greatest element
Multiplication sign
Direction (geometry)
Sheaf (mathematics)
Insertion loss
Complete metric space
Differential operator
Manysorted logic
Oval
Negative number
Descriptive statistics
Physical system
Area
Compact space
Spacetime
Counterexample
Process (computing)
Moment (mathematics)
Sampling (statistics)
Curve
Infinity
Staff (military)
Functional (mathematics)
Category of being
Proof theory
Hausdorff dimension
Module (mathematics)
Right angle
Cohomology
Directed graph
Geometry
Point (geometry)
Divisor
Variety (linguistics)
Lecture/Conference
Term (mathematics)
Operator (mathematics)
Theorem
Lie group
Statistical hypothesis testing
Forcing (mathematics)
Projective plane
Content (media)
Algebraic structure
Mathematical model
Set (mathematics)
Ring (mathematics)
Cartesian coordinate system
Limit (category theory)
Local Group
Quilt
Commutator
Logic
Quotient
Object (grammar)
26:24
Point (geometry)
Geometry
Spacetime
Moment (mathematics)
Sheaf (mathematics)
Curve
Propositional formula
Staff (military)
Differentiable manifold
Mortality rate
Parameter (computer programming)
Tendon
Number
Tangent
Tangent space
Calculation
Frequency
Causality
Lecture/Conference
Right angle
Fiber bundle
Descriptive statistics
Local ring
Physical system
31:17
Classical physics
Point (geometry)
Geometry
Computer programming
Slide rule
Flock (web browser)
Link (knot theory)
Direction (geometry)
Sheaf (mathematics)
Mereology
Explosion
Lecture/Conference
Theorem
Stochastic kernel estimation
Subtraction
Field (mathematics)
Position operator
Units of measurement
Physical system
Closed set
Element (mathematics)
Moment (mathematics)
Curve
Mathematical analysis
Total S.A.
Mathematical model
Bilinear form
Local Group
Automorphism
Tangent
Connected space
Field extension
Category of being
Cylinder (geometry)
Arithmetic mean
Duality (mathematics)
Right angle
Coefficient
Arithmetic progression
Family
Local ring
Inductive reasoning
43:49
Complex (psychology)
Sequel
Variety (linguistics)
State of matter
Correspondence (mathematics)
1 (number)
Mereology
Theory
Open set
Parabola
Binary quadratic form
Lecture/Conference
Term (mathematics)
Operator (mathematics)
Cuboid
Stochastic kernel estimation
Fiber (mathematics)
Area
Compact space
Fundamentalbereich
Compass (drafting)
Generating set of a group
Moment (mathematics)
Mathematical analysis
Content (media)
Mathematical model
Evolute
Limit (category theory)
Functional (mathematics)
Local Group
Equivalence relation
Category of being
Field extension
Summation
Arithmetic mean
Funktor
Duality (mathematics)
Universe (mathematics)
Module (mathematics)
Right angle
Object (grammar)
Mathematician
Identical particles
Funktionalgleichung
Resultant
59:50
Compact space
State of matter
Multiplication sign
Element (mathematics)
Propositional formula
Function (mathematics)
Mereology
Measurement
Parabola
Fisher information
Category of being
Positional notation
Funktor
Meeting/Interview
Lecture/Conference
Hausdorff dimension
Units of measurement
Amenable group
Local ring
Physical system
Social class
1:05:20
Point (geometry)
Area
Series (mathematics)
Group action
Product (category theory)
Link (knot theory)
Block (periodic table)
State of matter
Multiplication sign
Moment (mathematics)
Algebraic structure
Functional (mathematics)
Food energy
Parabola
Category of being
Arithmetic mean
Lecture/Conference
Meeting/Interview
Quotient
Theorem
Right angle
Sinc function
Physical system
1:10:49
Point (geometry)
Axiom of choice
Ocean current
Group action
Existence
Variety (linguistics)
Sheaf (mathematics)
Embedding
Mereology
Weight
Grothendieck topology
Power (physics)
Frequency
Mathematics
Pi
Lecture/Conference
Term (mathematics)
Energy level
Diagram
Antiderivative
Descriptive statistics
Fiber (mathematics)
Condition number
Physical system
Stability theory
Statistical hypothesis testing
Series (mathematics)
Conservation of energy
Closed set
Moment (mathematics)
Algebraic structure
Mathematical model
Dirac delta function
Equivalence relation
Local Group
Category of being
Field extension
Arithmetic mean
Summation
Right angle
Fiber bundle
Object (grammar)
Game theory
Cohomology
Local ring
1:24:30
Classical physics
Point (geometry)
Free group
Link (knot theory)
Differential (mechanical device)
Multiplication sign
Correspondence (mathematics)
Direction (geometry)
Mathematical analysis
Line (geometry)
Complete metric space
Mereology
Dirac delta function
Theory
Entire function
Category of being
Frequency
Goodness of fit
Commutator
Funktor
Lecture/Conference
Diagram
Theorem
Matching (graph theory)
1:31:21
Group action
Presentation of a group
State of matter
Confidence interval
Multiplication sign
Graded algebra
Propositional formula
Food energy
Frequency
Lecture/Conference
Theorem
Diagram
Position operator
Compact space
Addition
Element (mathematics)
Volume (thermodynamics)
Mathematical model
Estimator
Field extension
Category of being
Funktor
Commutator
Triangle
Object (grammar)
1:36:51
Classical physics
Group action
Variety (linguistics)
Real number
Multiplication sign
Sheaf (mathematics)
Open set
Theory
Mathematics
Causality
Lecture/Conference
Natural number
Subtraction
Moment (mathematics)
Algebraic structure
Limit (category theory)
Functional (mathematics)
Local Group
Tangent
Category of being
Symmetric algebra
Field extension
Exact sequence
Isomorphism
Chain
Quotient
Module (mathematics)
Right angle
Object (grammar)
Cohomology
Local ring
Resultant
00:02
and while do this is the approach of the world and rooms a and communicating directions In in so recap what happens
00:33
around last time so if as is and I find that the right scheme and we are working under the almost find assumption so we said that there is this category was Corvallis and it's been recovery can be recovered from the subcategory of its its compact objects and the latter is perfect so then we also introduced this category yeah right here the last time there was no word on the then there is the use of the and repeat what's relevant so this wasn't in the completion of the category of coherent complexes and we always have a perfunctory I denotes it's the fungus that in the extension of the embedding of coherent acquires a coherent and so on so if test is what we called eventually go connective "quotation mark connected and that means just that the structure she has medical knowledge is in this case the functor so sigh has left you and this left a joint is um just the embedding of perfectly into cold so this assumption tells you that the structure she is coherent and so can bed prefer to cook so that was 1 thing that we said so additionally we said that if 1 of the of you know this is completely evident from so 1 thing is the it's easy to see that perfect is indeed compact and the only thing to check is that Korzhakov was compactly generated but it is decisions should generated by all "quotation mark yet this was a fine yeah submissive visitors most of people will know there is no this is the result of 2 of the also we don't need any of this so will would do it for fine and then leave cover everything that I find so we don't need any of the nontrivial facts about undefined schemes so not here the traded shares of the so if test is eventually go connected sorry causes so so we can introduce intermediate categories namely for every 10 which is a chronicles a risky close except inside the Senate recalls saying we can define the categories In cochairman of tests which was great definition of the end of coal and and the sits in between so it's related to bad paraffin John functions like so and we're gonna use this fungus that was 1 thing we did there was reviewed roughly the 1st part of the lecture in the 2nd part of the lecture we talked about priest accents stacks slipping give you digest of what happened the city's yes there are 3 but the 3rd is a bad guy it's not continuous wait we don't considers these things unless they really necessary so In funded adult commuted arbiter big sums are for reasons that I will actually mentioned in detail elects lecture on a bad guys for the kind of operations of the Buddha performance I'm trying to stay away from them and look at least at least unless they really have to OK so the 2nd part of leisure we that lies a priest AC we can attached to its category requirement .period use of life by taking the limits over and finds that map to and this category we don't have much control over it we can define perfect complexes as the exact same way limit of perfect but small true that in this case the perfect will be contact here it's not true anymore for arbitary priest tax so um Wilson said that if wise elder break instead of taking all of finds it not to white it's enough to match those of finance to take those of finds it maps smoothly and also if Wieseltier break you can also assigned to wide in scope but the same procedure is at its limit over I find that a fine scheme that map to why smoothly of of them and moreover if if assume so I know that mentioned so it's 1st question but it is it's isomorphic actually 2 units of calls go wide is cause a compact compact with a fine stabilizes and if benefits crisis moves for every um an inside a single twice you can begin attach category in the end of white again with the same procedure limit over a finds skaters skaters map smoothly to widen in all these schemes are quite scene themselves because why was crisis moved so and to comment on what over was asked me about Co and is compact there into gold but I not known to generate the general but you will this is year with all right yet so it's not if and as old sink vented equals lists well now the searching pollster here but if it is 0 0 this is recently this that here is the world's most it's easy to easier to construct the name of the people of the world over these are the use of this
09:10
sort is 1 of the largest in the world where the use of force in the area yes the is the on the use of the term yes yes there is 1 essential thing 1 needs to prove I don't want to edit wasn't planning to go here in the stock but just say once it's cool theorem that if under this assumption it's what you have to prove is that the funk from global sections of is continuous so it's seems I ball is what U.S. and it's not obvious there is no yet but we are stacked in it's it's not obvious because cohomology is a limit In general and it's you have to commute to limit for the call limited it's not grant it's not it's not coming about yet so the problem is that the negative infinity say have to bound so but Morris true is is actually unbounded ,comma logical dimension and in view of the world for last was you to Hi I'm forgetting its it is in our paper Denfeld I'm forgetting if there's a counterexample it's a paper of gentle them myself I'm just going to there is there is a counterexample when this stabilize another fine I'm forgetting what the success of the country's sample too this is my pleasure locates a it is so there's a bit technical discussion with me just say that this is what we know but for why being equal to Beauxis that it will be true that's for any and all that means cold and of y compactly generated In view of the world there you just wired just in general yes you will always know that these on the compacts is maybe that contacts generated by see I can't imagine for many people the discussion about cutbacks compact is boring but for somebody who's lived in it for a long time it's very kind of very wake up at nite and think about combat support projects he 2 the London just 2 of the interests of the policies of the world kind of and that's why I did this in and will seize more evidence of this as I said last email previous lecture you for compactly generated categories the datum of is equivalent to the bottom of if compact soldiers on the 1 hand compact object to retain all the information but wants wants to start considering functions like presented have left to join that sounds Quebec subcompacts that writer joint will not normally subcompacts a compact and it becomes very inconvenient to restrict itself just so well this categories compact objects so for that reason I really kind of wanting and complete so that it's technical but it's remember I said this about negative numbers so it's like 5 minus 7 and you don't want to be saying 5 71 to say minus 2 and its passengers in completion is like saying OK I have a negative numbers and notice for them all right OK so but now the time has come to discuss this looks system so what is it that a solid may give a definition so before I do that give additional looks is that the 1st defined bungee what else they now is not at all surprising it's kind of routine but let me just go for it not nonetheless more generally suppose have a target stack why in the source stacked next In this case I can default considerably priests of maps from extrawide and by definition is defined so that home Front St To the capping stack is hauled from times next life no surprises here but so this is partly 1 Y 1 wants priest stacks in general these guys will not be stacks of anything that is just very convenient to have them because even tho they're just arbiter Bruce tax still can calculate tangent and contenders spaces to them you don't to discard them their convenient to have so biology is by definition the mapping staff from Europe With every taking bungee on 2 the classifying stack of the group but invisible bungee is knowledge misstate what hoaxes so for this it's could it's convenient introduced yet another guy again no surprises here but it's very convenient language said you have any please check X in this case might actually the curve of wood which are working to it's out attach another priest called X in the wrong mainly hauled from s from a test scheme 2 eggs the wrong is the definitions on from S introduced so this realizes wrote and big city of crystals so 1 applications of this definition is the following will not really go into it right now but let me just mention since look you can we defined classical for an arbitrary takin particularly can take it for eggs the they claim is that this is nothing else but the models OK it's it becomes a fear of content with content if you know what the models are we know what the models are assumed schemes in which case it becomes a theorem so taxes a singular skin so how do you define the models you go to this nightmare of the 1st defining for a find the Kashiwa going don't do it defined in this and then Kashiwa steering In this context so this gives a can of a oneshot definition he said that the so
17:47
as of the world you it was always going to be among the lowest in the world what is the use of worldclass using a lot of you want singular or non singular the have OK so let me let's let them so here the statement assume that X is a fine that will In this case so just answer the 1st question so if X is after is a fine so the model is is the following is the full subcategory of the model on some text still With supports on access you will see that that the I'll say that in a moment so where X is closely Benedict still there next to the so that's how you define it theorem so now the question is what is the devoid sonic still that this is defined as well Extel is our finest move the differential operators it's the derived category of all models oversight yeah so we began in find case it take the ring and take the use of us all I I don't I don't do it that's that's how I define it just In my personal experience In my lifetime I've never seen the USA notably geometry of non well of none was a career and things but was all of you that you can do it in different ways it said OK so that's my that's my official definition yeah I go I don't know if it's a quilt is indebted to its on deep losses like that things might go wrong and the miners hedges don't know Over the members of JI just you and then and then and luminous and so on so this is Drumcree stacks of this is 1 application can now book Sears on anything so Luxor's G Is this this mapping stock from Exeter on 2 points Monday so this is the official definition Fluxus you along on the side which is this is not the sort of 1 of the factors which is or isn't worth the cost of his humor is 1 of the 1st Williams 1st all personal give a definition that too but it so that the 2 definition but they're all all they're very topologically equivalent so there's 1 which is answering phones question so if lies stack so we already have 2 definitions of work on the 1 hand and the modules on why can be defined as requires a career she on wider on this this mission 1 war you can glue so defined it as the limit over the schemes of the models on as and those 2 are very ontological equivalent so when wait here why does it have to be a stack again why can be Prestatyn ,comma work so why can't an arbitary in this case it you could take arbitrary maps if Weiss it's enough to consider small guys so the the to definitions now in some particular cases if y is the quotient of a variety of a scheme you want to give a slightly more if this description of what you would call and a chicklit variant direct category on Wednesday OK so this is my looks this and all but we all know ring the the operations there is no timing there's no such thing that this was a good deed is of racial tensions rather go overalls quizzical and anything Is this same Tanzer category that over the something else that's Duncan knowledgeable go there but it's it's not it's not it's not it's not infrastructure all right so then so 1 proves that he Texas proper then look sister is algebraic stuck by living not to go to the polls I just cannot do everything so it's 1 of those things you can there is a section of my paper with the region when would do these things in detail and this is because there's a lot of people who were among Bloomberg well it's a stack what it the only in the points of albeit anything the points of what to think they are yet it's yeah I'm just saying it in this way just to see how to send set of definitions so that that makes sense with the director Djibril geometry so I want that looks is founder of the war property rights scheme it in the doesn't matter because you see it once had taken eggs wrong it the wrong doesn't feel any of the direct stuff this is going to be yes I work encourages 0 and world yes so this is it's might data with the region uh I think it's either section and forget about them remembering some 10 or 11 somewhere so who knows where it is the right word yet so this thing and it was woman's question earlier it produces an object of dry you did have to bring geometry you prove that it's the soldier breaks tack but you can ask is classical not something we went smoothly covered and a Feinstein does this defiance scheme have lower as the structure if those destructor she'd have lower colleges the answer is yes in general sometimes it does not like Group a summer simple annexes a curve is rated G is greater than 1 it happens to be classical in general it's the right and you will need this guys because in the process of proof you use the stack for a more relevant or you'd be Maximina portend when it's
26:25
necessarily the right he has a lot of to learn about the it's always causes move which is what I'm going to say now so it's it's it's that will see that and in a moment the causes movement is derived you cannot get away with classical style yes so now let me say proposition that if X is a smooth curve profit um looks Jean is cause so it may give you a kind of a token calculation how you do this things so well for this to make sense you have to 1st prove that it's uh soldier breaks that but that approved by some very general arguments this to progress every concrete arguments appear Yukon compute something so what do computer and while they have to complete the tendons basic points and see what colleges you you're getting so we actually gave the definition of Riga's official within the rate geometry of the tangent space so let me 1st to do it and armature case maps work X Y just anything even here anything but we are within this local finite type setting in here can In the setting you can compute completely Fagin spaces so if you fix it .period signal that the tangent will do this news will be global sections of text From quarterback of the tangent wide if you think about it this is exactly what you expect and that's what this is what you get the director Gerard geometry so again no surprises suggested that makes rigorous sense said in particular and it's a it's a great exercise such as if you if you want to acquire practical knowledge of how the writer rated G 2 regiment works just to do it so if wise . 1 g we obtained the following description of bungee whether it's gamble OK but now so let's remember what is the detention space so I take this point of this staff so therefore it makes sense to talk about that tangent space to the stack at this point who knows what it is apart from those of you who I know what it is yeah we mixing euro but who knows shifted by 1 and if you look what what that is so you can be intake is the algebra you twisted by a bundle and should logically by 1 and similarly now we take extra beds thereon so we take .period what gene but effective the on and the tangent to look and again for any acts To me what's known as the oncology so in particular so it may prove this proposition so you've got this complex and all you have to see is that it doesn't have colleges higher that number 1 but here we are dealing with a local system on a curve local system shifted chronologically by 1 while McCurry Hakim up to 2 Venetian they won't have to up to 1 period so
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that's that's crisis movements so I just just the fact inched down and around local systems the indignities categories degrees from 0 2 2 I just did it because the demonstration cylindrical debris geometry something people would just be waving your hand such as this has something to prove that something you've got something to prove very concrete handson way so the end of the day we have to complete summer completed comes down to something very very concrete evidence that the Buran easy way to see that the point of allusion recover 1 of the virtually hoping constitutional but note that he also proved will In the there's give that doesn't give a classic ality right through the fact is classically yet that progress in some other way I think using kitchen something Minnesota in the book it Hi OK tho in some countries including the United States from the United News has reported that 1 the few on the wanted to see more precisely what means the eye yet even so classic holiday of prospective usually hard To yet so if you want to see you got something as a priest you got we've got something like cloaksis asked Is it classical and these questions are harder but because at the end of the year your corneas to still ended the year with the endurance burn GE and they're all year long and exercise program but is also critical of the None of these ingredients of something the this flight no I other than in the form of the merger of 2 OK let's see I mean OK In particular I cannot describe what seemed of this is OK so what did we say it was only 80 priest tax so it's an element signal of your prostatic and then an element both each minus 1 of the core tension so let's understand what this age minus 1 of the Crittenden so it would take H 1 of the tangent and that was what that age 2 unlike curve with sufficient local systems garden as the twister Billy algebra by my local system so now I want to do all eyes but missed this little Of the call attention to well I have the generality of the curve so what this would be a big H 0 0 With coefficients algebra twisted by the same signal it now it is no sorry thereon but now I'm going to use the killing formed his this you're very right this is where I stopped in the group another 7 reductive but in the reductive case and will actually see will be dealing with problems it's in the moment that's where I stopped the group is not productive in the group is reductive use the killing form to identified with GE thank you mentioned the butt it's fine because for what you want to build it'll be up to quantities after the elections OK and why do they perform this for the following reasons OK so now you please do not have to use with it as it it doesn't matter it's the note for what I'll say it's not that there's really matter so I just want to draw attention to what this thing is take a look at it you have a local systems it has a group of automorphisms and this is it's Lee algebra in the very quick in the classical sense so I want to hold Sigma ,comma a were 80 let me just make slight abusive language call them and amorphous and Sigma Mary Cheney an element of the Lee Elder broke the group of autumn offices but is infinitesimal to mortgages and just "quotation mark this agenda I think it was received you to so this was this is the doll this and this and that all of this the passage from here part of death yet the passage duality so the passage from the slide of his life as a globalization and the these adulterated utility work this is by definition of boulders and this happens to be that all of the lead investigator in the this your you can put that in fact millions somewhere in the
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middle of of this when you say that there are any of you from the beginning of correct indeed so yet to the right of the top colleges that needed to put in the family but not the units of the race foreign Minister what the totality of this is my thing England OK and all I can give the crucial definition of quotations submitted is a closed subsets in this scene of locks since where we require me to be important and while enlisted men in different ways so for example a section of this thing it has a value at every point and it's enough to require that it be able to just 1 point because it's it's horizontal respected connection so wards globally but however you want to think about it 2 I don't give his official will I mean will see nondeductible to the moment just give a definition for the purposes of migrants and finally so the genetic London's Is the following conjecture that the mold on bungee hot we are actually defined at the models so we actually know what they're talking about is supposed to be equivalent to 12 in scope many people flock system OK so I'll give you an example of how this works whether this is actually a theorem Bloomberg said that there were no other Caribbean Islington's language has to do with herbs and get them to live in their own positions on characterizes the failure of course so the view that the whole yet so I think that's what I started from his 1st stock yet so wrote 100 page paper were adjusted is listed all the things you want from this a sewer and my 1st off I said I'll only mention 1 thing namely compatibility with very parabolic induction that this would it's only 1 thing so the New American so yet so as we work in the context of the cocomplete categories and continues founders in this context anything is given by corporate Colonel and so it is given a colonel we dont going To understand this kernel is as difficult as to understanding the conjecture itself so would you expect the review and a film about me and with nice means is there is definitely Colonel I don't know for example you can rest refused to open the way the the soul this kernel is not chronologically bounded so I don't know so I don't know in what sense this kernel is nice it's a colonel in field has thought much more about this kernel he what he wanted so he was um expecting can't have very nice properties and just use them to prove it I don't know to what extent is from exhibiting work bring worthy of the Hall DNA and then at the instance of the laser camera yesterday I mean it would be nice to have a bigger house not catching can be characterized differently so that link it is Mary is Whittaker compatibility 1 the facilities at the say that this is a bunch of things that you want us to to to satisfy that and again so since I can I want to go and direction single support that we just not go any this thing that is really relevant the single support is this parabolic induction somehow all the understaffed In a sense it's more interesting because when dealing with a customer part of this category but what happens is because part doesn't see this single supra problem so if the camp according meet language the it said that the saying I think it is the 1st in my 1st lecture that like without amorphous forms Castel but more reports are the most mysterious but they don't presented political problems and here we're dealing with analysis so that's why we can't win dealing here with as seriously as wanted these guys to converge somehow OK so but Live
43:50
8 mention a few things remember inside here we have classical sitting there so therefore to it's the corresponds something a Classico is in school with a 0 so here the single supporters have the perfect complexes so here single support is required to be nil but I can't take the the subcategory missing was supporters 0 so it's our classical yes remember we had this to whom because of the into cold remember we had this problem we have this sum considering this is betting on so therefore it corresponds to something call something 10 but tempered so what we can do what assuming the conjecture you can intrinsically characterize what the subcategory so the conductor story can say what Was this will go to you do it in terms of hake a fundraiser I will not do it right now I will I'm a very happy to do it question time but that does not look good right now just kind of old that With the geometrical say tempered his means bird with Nunavut homegrown ones I know what you mean it is up so that so that I don't know I don't yeah I don't know what like Huracan restrictive um the single supporting read model sense you can try to see what his group I don't know if it's it's it's a great question just I mean that it's so let me just say another thing school of the story so as I said this kind of theory holds for looks so this Ensco is in need of coal so so compact objects are cold there are now as we shall see in the sequel from his official state sequel you could take on an arbitrary buys smokestack if you take that as a yes no had taken the category of coherent compasses with given single support they'll be Sara the ality Alt equivalents so I wrote the same thing but he wasn't with op yeah well before believe this conjecture it's supposed to go to the modern and gene so there must be an equivalence between the category of contacted the modules and its opposite cooking gets what this equivalency can you make a wild guess take a compact objects of the Category D models and try to make make it contravariant lead to itself and also you volatility the letter was yesterday we were talking about the models of the notion that the problem of things not so but it's limited to take some answer because I think what you watch just has a name yes attempt has it has a name attached mathematicians name who knows what I mean nobody wants consider taking D models what so if it takes d models on but other great variety take those which bonded with Goodyear collagen acclaimed as Qatar has an evolution I love you all here yet it's colder duality diversity of views on the wing Of the Dingell there's something intrinsically defined the so 1 way to the final fine cases as are home to the Vatican so we can do In the context in which it defined the models 38 realities of With yes Burton yes hello Of the very tempted let me just making a 2nd 1 OK here's how to find so this is useless agreed on the floor by floor nonanimal you don't need any of that just give give me 1 moment and just trying to think if I can do it I really easily because of the area but it's so that is so just aside from home from India 1 to 2 years the oncology Indian on why so this this with this winter weather generalities look but that's not the problem so you want to say that it's again but this is this happens to be completely wrong and but let me just say 1 word is considered yet top model a very cool correction that was invented felt so what happens is this so described how compact objects and Benjy look like so here we really getting into analysis of the problem is that gene is not quite a compact well we know from other more performers the fundamental domain so compact objects In the Bungei look as follows all it takes acquires a compact open college you you take a compact and I want you can and should be extended so now there's a problem the shriek extension of the models in the accepting is not always defined so it may or may not be defined so there is a fear
51:52
that there is enough of such user nephews such as these guys are defined and the compactly generated category so it's a it's the result of some content this means that all of them more useful I don't know whether they know the meaning of the word yet because the extension was Europe from you so it is what it is to show that 1 ever have a monk was a compact other breaks stacked compact objects must be of the sport you the moves from the store fibers outside star fibers outside of quite a compact must manage you necessarily have that it's it's not true that there are between opposite the badge of extend the catalytic about generated you might not have enough of these guys but it so happens a specifically for bungee reduction theory tells you that there are many of these guys to the categories actual compactly generated but he said the university yes as you know from noncelebrity models due to a freak may not be defined you this city the world view is yes so it is usually the glow of a thousand exactly Bill left joined may not be defined Lo Afeaki still left a joint EU restrictions In this letter jointly not be defined the court will begin its I'm I'm in my setting but it's yet in the closet .period antiJewish right yet so I'm saying it now not for every you if you will be defined but sometimes defined let me let me give an equivalent and it we have hospitals forecast it'll be fine so India when Ltd comment on this a limit of say 1 thing and local back to constables so L of rail and answer questions as follows at parallel phenomenon for this not always be defined as the folding that it contained lower and Matthew was Compaq this guy make may no longer be compact In fact vehicle made no longer be coherent so it was this was coherent as will become classic coherence used this fine a generation obviously jail lost are used by the generation it just so happens that bungee structured in such a way that there may many for which you will not lose defined a generation but whatever happens if you apply the generality to this guy guys what you will get is jail or star of the Virgin dueled with you and this is a problem if you if you believed into anything like this because OK this was compacted letter to gelatin but this is no longer compact Japanese duo fees combined guys along parts of this just has no no chance of being true where did reality does not some combined guys provide so would filled invented the he introduced another explicit internal operator she called a miraculous duality that's um but if you do it in some himself yet in the functions of security setting so he deduced the funded given by explicit kernel that sounds store extension to freak extensions so it's a very cool stuff um that said let me go back to custard so forecast lower improves the following there exists a particular open subsets such as anything which is Constable is what's called clean it's both foreign streak extends tensions yeah also this miraculous reality acts as identity and cost but also so he doesn't do anything to the customer also but does something very weird to Eisenstein's namely this miraculous little mayor it sounds there are 2 versions advisers funk is 1 with a treatment star it sends eyes with star to ties with a chic but for the opposite parabolic so this is mere "quotation mark so he introduced this magic tricks all right about it until the conclusion is the we conduct was formed come come come composition of the gelatin in near the I so so the tubers advisers says founders because you can pull back with this for forward with the sheik or pull back with shriek and performed with a star Samira terms my eyes 1 version to another version but for the opposite parabolic it's some very weird version of a functional equation if you wish OK so I guess I'm still running behind us and I still haven't covered the material my 1st stock I don't always agree on an islet celebrated his criticism no it was supposed to contain everything if they covered all that they wanted to cover because of yes the next hour as speak twice as fast OK joking single support started because there was embodied in the name formulation me in this bug was most visible but with is that we try to explain how this bug get corrected via this theory I will give exactly half of explanation that Havel who will be next week so we are considering bloke since the mapping tool boxes and "quotation mark accused that this is the problem so was so it is following functor so the specializes in serious we fall back From and
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performed became so here there isn't dangerous notation to set up a streak and it shouldn't be taken for granted but please do for now and will withstand sometime next week explaining what is so extreme pullbacks ha it's a theory with content so we'll talk about it but so what we want we want that descends into cold New Look this and check and 2 well to do this I have to talk little bit above the ceiling of this guy so how does a State of the look since she look like well it's a local system with respect to peace output signal the well it's supposed to be age 0 of people twisted but of course is extreme noted and there is no cure informant people but there's a cure for 1 G so P. Doolin identifies with Angie Margalo the unit for the radical of the the killing 4 1 gene does best so and so they it will be an element although this did listen missing and notice that inside GE was in about a radical there is key what do we do but radical of people and that is M delivery so inside here there is an twisted by and inside here there is you can take new Bond elements in em and so this is what I call for the parabolic This is what I called Newt the original movies have to on shares of the parabolic you made in a contributor class but not the end of the season due to problems at all with the same levied OEA depends yeah it's indefinite detention there's a beautiful story but that's just part again I'll be happy to talk about it so name is these guys so it's important that the lady of lady just saying No expected next time that this functor which in itself is suspicious sounds into cold that need to look since the new closest so proposition Proposition 8 amenity is that this long and not only intercollegiate and preserves compactness these are different statements 1 preserve compact doesn't have to do with single supported it can be measured in the entire category and hold it says that this Fonda maps coherent to coherent and that has to do with the fact that this matter is itself causes smooth so in vertical Final Four dimension for falls back clearing to clear the is that peace that lost stands now this category you could have a and preserves the again preservation of coherence has nothing to do with single support its candor measure all all all genes called in all it says that sends cold to call and is just because it's more from his proper is just a profit this week 1 of the interesting thing is that it does this the single supports and Al actually show where this comes from so you don't run out of time will explain this Clinton Anderson
1:05:37
the state also more so I wanted to explain that so why do the why do we think that this is the right formulation of something else I wanted to explain that this is kind of the minimal fix for the conjecture if you wanted to the North selfcontradictory there a sense series so that if you wish the main theorem of our paper with writing contains the following that do the following to do this Eisenstein's funk but applying them not too To the smaller category don't apply them so the enlargement they do them evenly the flight quizzical looks so you get something smaller you will land into Ensco New exists due for all parable X and then see what generates inside called the claim is that it actually generates into Colonial you know the although analysts including G obviously so insensitive minimal fixed yeah I think so early members of the future they don't know and I'm going through colleges across a parabolic when was year when you say you you have to know if if there always the quotient for me yeah insights here it's modern the area but the link between the news of a year or 2 years probably corporate there is tremendous white wine as defined this as a dramatically improved will mean like of I ask you where does this come from this is the view of likely because the new and I will explain them yet look so now I wanted an example of 1 holdings it looks like in which case this at this statement is is a theorem on look question to the audience howled as blocks since G on the 1 will play knows yeah you know that it's someone else that is a derives reader reductive I don't matter any connected with action so how many local systems are the key 1 yes so what hitting the sack and how what what about amorphous there will well that's not so the claim is the point the looks is that's it taking my favorites derived schemes .period ties .period over G divided G8 has nontrivial DG is right structure again um I'd be very happy if you ask me how to prove this but after in the question time it's lovely exercise handson exercise how you prove such a thing directed region yeah it's acting it's actually a joint energy and of course acting to the end of the year the year of the use of the U.S. retail so that there are people take the right product it's acted on Beijing and then it started to suspect quotient but it was only when I can the meaningful for the available functions what we no I'll elder of the various president moment so let me just stated this looks like this and again I'll be happy to provide for it's very nice Ladies formulation no I'll and there is a reason but it so Langlands anything to legalize meaning that is likable you can you yes so the gentleman who Bloomberg News home yet that status is just take a local systems were attributed to the
1:10:50
layers of fabric 1 . at 1 point but the way this is not completely canonical this action depends which of appointment yet it's that it's not it's independent choices on the 1 but many case Langlands so that's what it says in call commitment but we have been through this crucial reality game already so this category's the following year taking algebra series of GE shifted ,comma logical eBay to to the right to taking models it taking single support on political the sense of primitive algebra and you taking the aquarium category OK so unfortunately I placed it but may temporarily was because of what put something here and I want to prove that my favorite into without support connected by this clarification from tourists and this will be sustained no support condition again this is without obvious embedding just this things will support sits inside all things and this is part of the world where here this and this is the right the joints namely take sections supported and said here who have twas a coat this is missus stocks supported 0 with so in a way that some reason that such an equivalence exists is while the war's experimental evidence of his republic of think American independently when on and then so if you look closely with arrive exactly to this formulation begin now here comes the question I want to say what this equivalents so in the system land was going to do so here comes the question they want understand what this equivalence actually does so on the genetic side I think the following very concrete object you know that bungee on 1 contains .period mortgagee as an open subsector are the summer stable locals yeah the triple bundle is an open conditions take the following things on the on the left Jane lower shriek the Bureau of Constantine model know what this deserves a better place than I what other races and I'll tell you where it goes to underline once so as the diagonal North as well the .period after all before you divide the G it's derived scheme bubble Blunkett has 1 point just Mathis .period in clinical it's incidental but the way I am :colon testing the level of alertness something on the left and of what you know about what makes no sense can use can you tell me what makes no sense to get clean this Is there a video OK because you can actually understand many different ways and this is what I said early on that you can really get confused when you use is focusing on 1 side and embedded into go cause inside and called viewing of the caution so here is 1 way so I want this to live here if that's what's supposed to live .period but let's see what meanings I can assigned to the subject on the 1 hand after all it said Korzhakov year Schaefer to the direct image of 1 prosecutor sheaf under another on the 1 hand I can't understand it as an object here and this is the wrong answer His can of this would have corresponded to something tempered and this guy's not tempered so that's not what we want so this subject does not belong to this category His other possibility you would agree that this guy is a coherent chief coherent short lives here therefore but it doesn't belong to the subcategory so the weight of standard ' you take interpreted as clear she can apply this writer joint the weight as a coherent sheet it corresponds to the structure ship it's not
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supported on the new political have to to apply this writer joined with a little weird only thing they could on well this also was on the left side of the life lifestyle of the exactly how this assumption is that this yes and I'll draw a diagram in a moment where such a thing will be generally explained I will right in the 2nd I'm saying that when their weight I lost and I'll do go there a moment it's ambiguous I don't want to understand and here I do want to understand each here and then apply the writer John so the answer GAO chic of the constant goes to this thing Taylor Taylor skyscraper apply the John so now will destroy my conjecture will arrive to conviction Informatics not emit management with my conjecture great OK so how does this thing look like as said let me go to conceal side it took the structure she and they to cohomology was supports on the close variety and you're not getting something coherent right like imagine exude variety take a motor was supported a point you get that kind of delta function not the skyscraper the whole delta function it's not coherent again takes take a smooth variety take .period burning the only no no no coherent models New Site Finder generated cohomology so the colleges not find a generator so we've got the good war here it will become a compact once applied this writer jointly it stops being comeback and so this is no longer contact but the the conjecture is true well I had earlier described how compact or just look like so I took this guy can't achieve what what's the problem yes on the classifying stack the constitute most compact classify sectors has you know go where income knowledge of the group is infotainment degrees the constant she is no longer go back so this guy's not back here so no contradiction now let me modified God instead of doing this the Nicholas J. till the matter from the point of view no I wasn't it was a joke I was saying that we ran into a contradiction mathematics because I said that this guy and the right that is not compact but notes that the guy on the left is not compact either because of the classifying stack this constituted not compact instead they can modify consider the matter from the point to the classifying stock and take the constant should there and that it made it to the classified so it's something close fibers article knowledge of the group and this is compact enough so and ends at home what it will correspond to the on the other side so they Jane lower chic of pie lower streak of cake so it will correspond to something which is supposed to be compact I don't know know how to describe it here but I know how described here this is the structure sheep of the new political another surprisingly Kenneth and killing off the everywhere in college of the group so in terms of this equivalent sums of this Kazuo realities will be the destruction of an important call all right this is the 1st to for each current nursing conservatives there's a guy yes I unblinking now on the descriptions on the spectral sites In the the work it so here is my piece of ignorance but they did not just mine as far as I know the capitalistic Mills on bond you 51 are not known terror in the system seems to be the only goes so let me say that I I think this is not well investigator so we understand this equivalents but there is a bunch of things that are that remain open for example Intrepid the kind of mystical animals in terms of Lang was over I don't think this is not that the kind of thing so I'm 1 and half an hour behind my schedule so that there were 2 things that I was planning to do but I think I can only do 1 1 was the actual definition of single supporting knew that game this .period ways definition but there's a more powerful definition terms of a toxicology tests Option 1 option to that would be to discuss this star business and explain explain this so you have to accept the definitions singles the existence of single support and I'll explain words such things come from
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the direct emissions of so what are the preferences that means you there is a link between the removal of U.S. professors from the University of Minnesota famous 1st look at this this yeah OK so part of the ignored Laura's book and we're pretty much done reasonably it's like it's always it's only beginning of any kind for the side and you suppose it have a match between just schemes acquires Osinski schemes to local finer what they want to define on the 1 hand and have a lower start from classical so I claim there is another from those from other finalist entrees it's kind of stupid for some reason it works and it makes the entire thing work before I defined as longterm Al tell me I'll tell you "quotation mark properties it has more property does not have the only thing is that I should have written this line the walls so this will be confronted that I will define and this is the 1st time that we already have look at the property that the will have is that this diagonal commuter remember this writer joint the property that it will not have Is the visit diagram while missing these schemes were eventually go connected wills have this fungus is in the opposite direction and this dagger will not commute In just across the South and so that's 1 point 1 1 can get to really confusing so we see there are that 2 ways to go from Kwai secure sheets on a swan to Insko years she's honest still you can take the quasiperiodic image and then in bed or you the embedded than it takes in Gujarat image and these are different and that's what we saw With this delta function and can on the more already sold knowledge until you how this funk as defined the on its very stupid unfortunate so to define a member in good cheese was into completion of something I have to define just as functor from here to here and it's defined as follows I embedded into causing the years she's been remember that goes to the passport then I applied the usually push forward and going to cause acute issues and to plus and then they remember all right these guys were actually equivalent and invented the book is no sanitation right yet inbounded producers bounded from below OK so knowledge the comment of inward such things come from so you can have anatomists want to well there's a code differential there is a diagram just taking do all of the differential that look now a in particular it could take "quotation mark 1 and then defines amount correspondence he was like this and we call this month saying book so here is the theorems so there is this 1 theory .period will backs and it will be at peril theorem for shriek full stop forwards and it'll be peril the repertory pullbacks at all discussed next time it says this that 6 and once considered press 1 and 6 and 2 in thing has to analysis using the following cleverly headaches the free image on purpose often 1 then it will be contained in the list can estimate from
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below like the and 1 thing
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is smaller that something has to women to then proclaim is that this means cold it sends In and 1 promise 1 2 into cold and to say to 2 but that on the way no uh see my conventions so yeah it's this new funky of lower stock you you also need the support and war yet yet in Florida yesterday .period Getty received maps yes vote sorry no no no that's thing he doesn't act compact objects to compact objects but it maps and given subcategory to you know you're in my application sexual proper so in addition to that effect schnapps coal to call you get more than half of but any that anyways for many different things so against England December again is Maxine just remarked this fund define a priori from into cold also support of us wanting to go all supporters to the cleaners of this pompous has this subcategory of this subcategory scant estimate from problem but this exactly the same volume of you you will listen to you always play well yeah yeah so the parties can similarities with don't I do I don't I don't know how to make more more precise now it's a great exercise at life this theorem to gets proposition . 8 the leaders will oppostion sorry .period B Wendy 4 .period able do it next time the so I just couldn't exercise Proposition B from here because of the cost of the of the core of with confidence in you get if you take it take something of a parabolic which is unique pulled into modeling for the radical but it wasn't it wouldn't then there's the can In the end it's all very concrete all right so now it's a room which remains sometimes I can try to save How action the financing support so this .period wasting is not really manageable so 1 has to do more robust definitions so do people still have energy forfeited minutes we can call it a day note that I just feel free to go to sleep I know I know I'll survive Lawrence so framework is best for them for this framework we actually don't need higher higher categories so this discussion triangulated so the triangle category and aid is a positive liberated algebra and committed cauldron that maps to center of state of tea yeah that what I mean by this is that for every element homogenous elements stadium degree and you get a functor from team dish to shipped to end and these guys commute with each other for different the purpose of the present yesterday and Camilla matches history snaps to the center so natural so it's a matter of 82 d prided diagram creates so so consider speck and for each element homogenous element they would
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wouldn't make twice and made is so this kind locus of zeros the group will grow even we kind of figured he immigrated secede yet so and ghosts near the shipment to em think of it as even degraded it's not of convention so what I'll do well defined the category for each a Alden noted let's called what minus why these worldly things supported in Wayne not supported on open subsets away from the zeros of a namely team belongs to this subcategory if this map is an isomorphism it's a subcategory I rolled it in such a weird way that its stance on the right I because this Italy admit left the joint and we get this what you can call it the exact sequence of categories namely define things with support on white 8 to be those guys that they get get killed by this and you really should think of these objects supported said theoretically unlike any and this is this is it's right funnel so in fact you can write down this memory laughter joined explicitly "quotation mark localization had and yet so many categories complete so this will be the defined as follows the installment of limits which are usually not defined and trying good categories would defined taking its or natural numbers so words it taking the lid 2 these animals not and it with House miles a so is this means that this is a subcategory of objects defined Wong 0 so they now use if you have an it's a risk closed inspector you define Tiana as the intersection of all the data is such that and is contained in the White to a certain way so that's how define support and there is so what you want to see that it really works like support for Corsica issues and there 1 thing to check their so it's not completed logical there's 1 thing that we need to check that is that this containment he is an the quality you don't need any extra assumptions it as follows as it is but we it's the 1st thing where we stopped completely automatic into you know something to prove it and then so this gives you a welldefined In the notion of support so now what do we applied to so as his
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cause this smooth and but he was in use for those of you who will work with me in it so they are the the and is 1 of team and it was a very the moments that come with them that the issues included only after another sorry this is a really bad time so this is the quotient in this this is included it is a real things of support so team is computer screen triangulated category corresponding to think :colon screens Cole was this DG things we don't care about it anymore we're taking the corresponding tried triangulated category know who is 8 before a defined a L define what comes before a indeed In the 1st time the joking so for this definition would wouldn't you need Horschel Co change which is estate Koksal cohomology as as it is now so let's notice of what we have in the so cleaned the colleges this summer that acts so into 0 toxicology we have to take 1 thing you know that functions on Europe variety maps into while it Hartford could change take this 0 College of that have the zeros quality of your functions like map to offer could change a toxicology another thing that you know what you need to know a little bit about half of the wholehearted contains look like so he could take sections of this shifted tangents well that's actually a maps to talk shows sorry I should write a whole should go so this moves steps to halt should go chains of us In particular um H 1 maps to the 2nd half colleges and it's snaps as a module over this and as a result if you have with this and analysts it take the symmetric algebra lists over this Lisbon officially algebra functions and my saying so we'll take a look at thing was this classical scheme maps to this is my page so In this story you really haven't used harsh chains but you will use them and see their different degrees of heaviness with which you can use them for most applications like if you want to compare with my pointwise definition you all need Hartford coach aims as a Sociedad refrigerated algebra in fact contributions have more structure there are was called the algebra and you can get away without using it it's my work in the into use it for some obligations but the action don't need to to know what 2 algebras is what indulge a to to develop the theory all right few