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Using the internal language of toposes in algebraic geometry
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Title  Using the internal language of toposes in algebraic geometry 
Title of Series  Topos à l'IHES 
Part Number  13 
Number of Parts  28 
Author 
Blechschmidt, Ingo

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  10.5446/20745 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  We describe how the internal language of certain toposes, the associated petit and gros Zariski toposes of a scheme, can be used to give simpler denitions and more conceptual proofs of the basic notions and observations in algebraic geometry. The starting point is that, from the internal point of view, sheaves of rings and sheaves of modules look just like plain rings and plain modules. In this way, some concepts and statements of scheme theory can be reduced to concepts and statements of intuitionistic linear algebra. Furthermore, modal operators can be used to model phrases such as \on a dense open subset it holds that\ or \on an open neighbourhood of a given point it holds that\. These operators dene certain subtoposes; a generalization of the doublenegation translation is useful in order to understand the internal universe of those subtoposes from the internal point of view of the ambient topos. A particularly interesting task is to internalize the construction of the relative spectrum, which, given a quasicoherent sheaf of algebras on a scheme X, yields a scheme over X. From the internal point of view, this construction should simply reduce to an intuitionistically sensible variant of the ordinary construction of the spectrum of a ring, but it turns out that this expectation is too naive and that a rened approach is necessary. 
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00:03
the and we would England and agreements for their introduction also thanks for organizing this conference I'm really having the time of my life he it's really good and thank you for giving me the opportunity to speak here so I want to start with the question even got Gutkin who posted on my home floor he said admitting it is convenient to reason about publicists and all internal logic has there been much done about them to unlock the logic of their goal was to skittle all of with the logic of the pope was required to much cumulative idea brought to feel like logic and this thing about this team has a very good point here and immediacy the basic objects of modified a branch trees out streets and schemes on local spaces which are locally isomorphic of the spectrum of a commuter Frank so it shouldn't come as any surprise that many notions and Niger Benjamin the unlike she few notions of globalized notions of what nanoscience in community fight about and also you know the business with the proves a major regulatory often they go like blablabla therefore without loss of generality we may assume that things are fine and then it's a simple consequence of what became of the privilege of algebra all approved store like blablabla benefits offices to check the condition of stocks they had stood because of intervention but can't I think topples logic has a contribution to make them the starting point then is to realize that any scheme has its purpose of all acts and that is of course of course she now she freaks but if you snatched away from article the producers could topple the publicist sheets on next this complicated cheaper offerings they look just like ordinary plain old greed months OK so I need to have a cheaper perplexed modules on your screen and from that time point of view this was simply looked like an ordinary module on that owning ring and this is the starting point as was already a noticed in the seventies when entire language was developed as several people and to so I quite to want to see a how will of power can bridge their hope of accomplishment .period so what and what won't do is like but indicative bids at both the dictionary between exforeign oceans and internal Shin but this dictionary once and then we can use it as often as you want and so I just want to give you a few of those items for instance receive a finite time seoul externally as chief demand space X of modules as final type it but only From the entire point of view it since that looks like an ordinary finitely generated moderate OK but a beacon for posted for and a sheet which is extraordinary and finally looking free this is precisely the same thing as from this point of view and ordinary matter which really is finally free not locally but that really the only from the you OK so bent on language a device which allows you to pretend that the base cumulative Obama is affected .period and to reduce the better situation .period promising plant Tokyo and that's to dust off you more examples so for instance may be when begin and end your learning and regimented then maybe your scale out of the diffusion of attends a product of ships and the border moderates and then there's nothing scary about them and that but then you're just learning then billing like he's got a problem recommended 5th mission only suppression then you have to she could fight and maybe the interface where where your there will you have insists feasible to fiction get away from that her point of view there has appointed of sheets of moderates is simply that ordinary tends a product which have long been like your 1st semester and in immunize about something and get em a particularly nice example is that it is that sheer operational functions K X saw as a prison functions and to end the safer knowing the conditions than you also know that it's littered the delicate in fact that there's a slot paper by Stephen Klein and private misconceptions about K X ready lists alike 3 divisions at about X which are very commonly found in digital image of abroad now which did not do not in she for instance that a case of the definitions litter the delicate but not from there husband of yours From the point of view no pain this year operational functions very simply namely Cagle X or exercise ordinary from them her party during OK and then take its toll reported ring the development of the work Was it yet ending you have to use the couple's natural sorry about your 2nd time Hubble's language and that all acts is sinking Audina ring and has effects of a regular elements and then you localize at that this executed beauty that it becomes so easy and then turning to the other 1 was the Kosovars opposes the use of resort to use the so some of the most I was so good the idea can look like a lot less than many sets up fact at any substantive wanted an investor will always be constant thing but if you look less of a set of regular elements than you will obtain the thing commonly called K X and because for instance and you can give and beautiful in telecoms truly intimate tone of the basics Of the fury of catchy devices and that and make a lot 1 1 March and so we had to end been abused selection then the fastest serve the situation that we had 3 1 theorem which folded if and only if that office was William we have a similar situation here namely there from the point of view you can ask yourself what is cruel dimension of the ring all acts any ring as a according mention now and then you have to be a little better
06:59
than in the user constructively sensible definition of the problem mentioned up to such a definition exists thanks several people some of which are the audience here or there and then you can visit this definition and ask what's the coordination of 6 dancers requirements of all acts is precisely the dimension of the scheme the base and then you have the following observation that I mention of a lacks the coronation of wax is 0 if and only a but I met with the Minister of the original if and only if then ,comma language Williams sunset some sense bearing all access to the site to shift control of the project on the top I want to take minutes to explain why in some sense of this talk is like a praise for my children this and son about having someone access on entire images of Publicis and most of the time geometrical will stress with which may find them but in fact they had publicist Kim to put more more Frank high fragments of income of logic they can intrepid automatic they can intrepid higherorder logic they can also intrepid dependent types which some of you might not know about but I promise you you're using dependent types all the time now and if you really want to import all of construct for constructive mathematics into the couple setting you also need a pennant types OK but the thing why I want to praise and because he made it possible to use unbounded quantification and then traveling so unbounded quantification is when you say for all it should hold that for all modules that support that now that if he did that investors unloaded confiscation of course needed all the time because you want properties OK and my so wrote letter paper which you can totally understand even if we don't know what status every justice small addition music producer not semantics of an of them of from To be able to speak about on for the occasion and no Clinton categorized In contrast to already interim categories what is over this small extension can really import all of constructive mathematics Internet upsetting OK and wanted a good example of I think the question is uppermost .period then let's have a basic itself so I faster less than University on that if you have a shot except sequence of ordinary nonteaching modules and you know that their father once all find generated themselves the middle 1 was the the hearing and the spread of hearing has an obvious proof and this obvious proof constructive that's what you can interact with this statement inventor language of any topples and then it will automatically rise for more advanced statement the statement you see on the border with sheets OK OK but so this is the 1st example where the couple's Magic allows you to like proof of your once and then interpreted in many many contacts and at once with out for the work 1 remark so you might object that the only proof of this statement is not difficult at all you might remark that it's simply routine and that it's very easy to do OK then you're right in fact you where you can but if you're familiar with indication than you would probably be able to move the statement at the at the bottom enlightenment but I argued but this is not the Midwest spent effeminate spent with like saying OK we have this generous on subsets of well subsidies to big I have to shrink it if necessary I have to shrink it again and win we there are it's routine but it's like 60 seconds they could have done something different again and with the couple's language a new independent you have the 6 of seconds for for free of a given to renew it because it simply look at the statement realize the because of the dictionary but it's about their interpretation of the way known statement of the knowledge about and then you have done OK you're going from 60 seconds to 1 2nd this is having a great idea also gain conceptual understanding because you know really know whether she's hearing comes from before that he only had a like a feeling but now we have like a formalized Negroes proof I'm on once examples of foreign salt shakers gimmicks assuming that its origins take achievable and what is often a title that the statement but this year will always automatic the locally free on and there's always something that was a statement so Evans and Robert Bach in his excellent lecture notes in the direction of PCs but this is an whole important hot exercise is an exercise with like agent which also would like to have a picture OK but in fact this is not a hot says it all it's a true the consequences of their constructive here often but the top In concept of algebra know that any kind of generated vector space it's not not free does not not possess a basis but as a concept if you can prove that anything of degenerate because there's really screen you can only improve this like to make a statement and they're not not translates to honor Denzel themselves I really like this is an example because otherwise you would have to like followthrough this hidden half a page and do not younger ones and economic twice and songs on an effect it's a tribute concert it goes some of the amendments flatness is being used in these we lose these that is what is the not usual usually so you have to like that and good relations with now let then at some point in time you will have to work but the point is that you can shift lots of this work and about which has already been done in constructively that injured impact can automatically halt these kind of things In all questions at this point again then and it's continued their to all the
13:58
parties was the top of his game I want to show you Icarus property which spent on universe Of the producers proposal and and always has so from that point of view I accessory and has foreign crews properties any elements which is not convertible this might portend and in a manner which is not in what was my so axis the following the future if you went to college by then I put elements he would have feud in this sense so this is the 2nd which awards and for generality honesty even if your based is base base ring for instance is nearing an undefeated but always sports so this was already noticed the special case by moving in the 70's and use it to make commenting on it he says this is surely important for its precise significance is still somewhat obscure as at the case of many such non geometric form of non geometric sense as explained in the so I am in their onehalf of 1 1 poppers office told this to convince you but then best exterior meaning behind the statement but it's not at all or obscure this year ,comma among "quotation mark
15:27
1st that South Korea top the to interlude that thing which I stumbled upon a proceeding that its main known but I've never written a scene of written references say publicist and you have perceptible given by local operator motor operator topology and at festivals like Cannes in himself again as the supplements and have you have to support and not take a formal and you might be questions so them what's the relation between the formula holding on the settop and the former holding about topics this any relationship guests this maybe the relationship was executed by the visible observation circle of high "quotation mark the subtopics if only if it's diamond translation invisible but I want translation is given recursively by these words that just as the for their allotted land on that obligation translation just with a diamond instead of an not but "quotation mark and this answer this is a very useful observation because it allows you to use entire language of the supermodel parts to speak about the subtopics and we'll see why this is very useful as a reference to the next the life of the nation he said that in the movies and know I mean the job damages translation assistant to a police tactical operational for given a former European standard translation by like putting diamonds everywhere in front of everything and then infected can prove that its officers for you to but the diamond in front of In front of the war was published in the Journal of the the death toll although maybe I should say that I am a little bit envious of Apollo's oppose you made you know every mathematicians only study what's true but the pilots oppose were already to the to study like things which should be true what do I know what does he know and what can be true but it's necessary to unsung sort of a person may have a study modern logic and so it 1st of the modernity is actually useful in like real blood type of applications and regulatory as well work and receive just various depending on and rich motor operator you choose you can incorporate like a whole host of that and 1 I just want to concentrate on for instance the first one circus cheese for Diamond said obligation that ability then Diamond high the high holds 1 events on substance the game consists with the crypt adjustment OK and then you can also wonder what to only dummies the diamond translation and if you do the calculation received but I'm I'm assuming best information would reduced but for the moment at that fight retirement means that this a form of high holds at the generic .period and so on and in direct encourage Eritrea's it's an important question rents properties for rent from the Germanic generic point to some dance opera subset this doesn't always equate the good thing about the "quotation mark OK and you can there analyzed this question a politically because you can put it in this form when despite the diamond implied diamond 5 when DOS 5 audiences didn't .period implied implies that important events of sorts OK and the good thing about it is that you then can import went on fears of a project for instance that this year and that therefore any geometric form enough this always holds OK and that way you can you have like 1 theory which gives you a whole host of renewable statements about spreading from points to open at once now and in fact if you bury them would operate beginning gaining more statements yeah His find industries 2 1 free variable which is 1 of the the season passes that he has no free variables in the state holding dozens of cleaning up as it may have fever was for instance it Natal and may refer to a given sheets of something that Seoul will article the States there yet are not the point of Essex views this is somebody's feelings ours .period buyers eliminated that these providers that may be at hand 1 for example that so and you might know that it the fund does not hold and that of moderates and you know that it's stalled manager but it does not hold in January that it would Arizona on dental and on and on they go the point this is because the Saudi now he began a concept understanding for this month period this is the cost of formulation from what would be 0 this monument it's for all acts in em exhibition 0 because of the condom plan this is not a dramatic fall OK but not presume that them as she was known to be a finite type it certainly didn't this means that the more the better model finally generated fierce generators X 1 X 2 and now you know know that you can read by the condition that any 0 simply S X minus your to assist user Iran's 1 extended 0 and this is manifestly adjournment to follow workers finale of conceptual understanding of life in general spreading does not approved for being 0 and why does the cool when the ship fine let's talk about Karzai
21:59
careerists their condition was chief of modest because I Corps in this very important regimented infect their meant but chief of modules which are not cause I careened should never be studied because they're like not geometric OK so cause you're wondering whether it's possible at all for that characterize Karzai Koreans and it turns also out that yes you can vary the conditions on the board and I quite like this condition the costs so a nominee you when on constructive advice but in this case you created great new constructive much about the because this is a condition which is not normally seen in photograph OK so they conditions but funny at all if you localized this module every from and then this new model should be achieved with respect to a sudden internally defined and model operator which is concerned about can sell and it's possible to give the hosts chief theory as conditions on its and entire language and that I do it for you this system has operated conditions and just south of the Britain's conditions and I quite like it because it's a far charisma to perform it says that if you're able to do to deduce that as neural given the assumption that about then you can unconditionally deduce but some poll have times and really is 0 Seoul and we can return to if you take it for you In a special case simply all all experts and it would take for S 1 then you accept a reduced the lowest formula and then you know you know the deeper meaning of this former this was just like little shadow Of the of the bevy of Greater Picture mainly at the picture of Class Koreans the
24:16
last year's lights up all fans all we talking and he on especially his writ really great ensemble of law on how to moderate states OK so and an abstract motivational fortify moderate schemes as reporting take a ring any who want to construct the free local ring all the saw this as the universe of property and it should hold that you like to have more officers from 8 to a prime than to be constructed grain and it should hold that for any new map Lokeren are and this match unique reflect all local not offerings from a price today and then you do a calculation and are disappointed because of this optimization problem has a solution said if and only if the ring at a has exactly 1 from India Candace Awards attrition again read this answer by the time they're not delighted to know that this optimism optimization problem does have always a solution you just have to allow yourself to chant the solution is the structure she all state aid in this scheme suspect now of course you have to define what it means for what what amorphous offerings living in different provinces that's what you can do it the series and decided he observers say to you that will not work look at and then of course you want to go 1 step followed because you are a fan of Kentucky now you want to construct a free operating all but any rain not necessarily a ring set again so what you do reduce them it's by the general theory of that this support published magic its officers to give a constructive account of the period of the use of spectrum inside the 1st attempt might be defined the spectrum as the topological space off the primary duty of a and I know that Primedia ideas I in constructive fanatics this will not have the right of property 28 saw that and my people my feature consider that topological space for the prime photos of a private as classic the same as a complement of Primedia only but it's directly accidents such as that for 1st accepted the duo notions OK this is much better but also privatizations of and constructive fanatics have their 2nd example by when he constructs a ring which is not trivial but doesn't have any of their profits what you are is that have to do with them but then OK off the prime suffering and in fact this depression was already the water remember leaders last lecture she at this long chain of equivalent says he was this she over spectator this has had an unconstructive the definition of work and then there was remodeled persists all of which you can constructively use and I just kind randomly decided for the final 1 Richeson OK b today and in this way not pain as and an atomic acquisition of talking stage and respect for the Spectrum Fund which I dissent etched onto to the forgetful thunder from clearing proboscis during purposes then if you had taken the decision does this like explicit site construction and resources you have to have the better side and the Wyoming deflectors sites are good but even better it's to not have to is a construct them by hand but was a general recovery top all y X and top assistant to X ever stop kind of instruction for you OK and
28:14
if you're familiar with directory trees then you also know the relative spectrum given back by the Korean Supervisor Ross of Boston next this relative spectrum which S & M scheme office and into acts OK and you might wonder how this can be described internally saw the obvious choice but this I would be to just take me talking Spectrum Fund but it's very general and applies to the situation but many do make the calculations and realize that and talking spectrum Fund budgets a different sites but the relative spectrum considered underachiever and thank you can check that we talk gives the car such if and only if this can be the best in the spot where this year dimension which of course it's not general itself so it's interesting to see the wholly in going after them like change affects but if and he is haunted and you again defined their relative spectrum as local so different frames but you don't take as a frame by frame of all a radical idea but only the Class Korean and you can difficult relations to arrive at this sort compensation of man a radical idea of a clause I Korean underwear is itself closet career again notice of this like true smoking behavior that's all decided he and I have to satisfy the condition that it an element s is an element of high under the assumption that and what then pastimes as really should be an element of life without any assumptions I can't say I just want to and we and our
30:07
mark on the points saw this location not spatial from turning point of view but you might still be interested in the points but that just sort of a better feeling that you have to know that fury which this spectrum testifies and it turns out it's not that fear of all time fighter fractured us but only of Mills which fulfills this additional cost and condition OK and to there then you can of course generalize at the X doesn't have to be a scheme that can be an unfruitful ring topless and then you can the they can understand the limits in the category of clearing publicists much more conceptually namely there simply given by the 94 minutes indicating a referring publicist and then given by a really localizing using this construction so and so and my last 0 minutes that turn to repeat on Sunset and that you can also like should differ the Brandeis account of the relative spectrum of the universe the property of the relative spectrum of the relative spectrum as Houston back to retreat and then you very clearly see the difference tell them a spectrum of encouraging namely the question In a directory tree it to construct the Freemen altering although a which is also free all was the basis of a although exon applications this except small but important difference I hope I convinced you about news and pop was a Mexican pop wasn't on language Brechtian prison fruitful thing that allows you to see if approved and allows you to gain a better conceptual understanding of emotions and of statements and directed the remedy for it and you have any questions now there's a view you know what that is collegiality directions to the nearest only heightens spectrum um this phase they're not this is always as such spaces hub located off me talking spectrum provisions of users face of losses who usually little they're not got anything you know that someone just
32:45
called the costs and the theory which thereunder expect intensifies is expulsion theory of the theory which that we have respect justifies the cost you have 1 extraction if the substance and it only at the base stealers urgent and then these motions coincide that you is the
33:14
most musicals yeah I S 7 people in the audience would disagree but disagree about that some less and less common notion that committed the resident constructed but in fact it's it's not so good faith and with the guys you have to have a great program rather than you Kansas taken on constructive proof of canned abided about placing them and the other element in what is so that it's like a tent and side by song that it's not a matter of enterprise idea OK you take this month constructive proof you put it in the machine developed by both guys and others and outcomes a effective find constructively acceptable prove which explicitly gives you bounce on the nite is a rule 1 out the classical and underbrush and he to cook all embodiment of producers legal I had no knowledge and research program of dynamical methods and Angela them I think we will have to find the In Europe few
00:00
Point (geometry)
Group action
Ring (mathematics)
Multiplication sign
Insertion loss
Regular graph
Power (physics)
Cumulant
Frequency
Operator (mathematics)
Interface (chemistry)
Selectivity (electronic)
Theorem
Condition number
Product (category theory)
Spacetime
Consistency
Prisoner's dilemma
Element (mathematics)
Division (mathematics)
Set (mathematics)
Ring (mathematics)
Functional (mathematics)
Entire function
Maxima and minima
Commutator
Logic
Hausdorff dimension
Network topology
Module (mathematics)
Ideal (ethics)
Object (grammar)
Spectrum (functional analysis)
Local ring
06:58
Point (geometry)
Polar coordinate system
Free group
Musical ensemble
Group action
Curvature
Set (mathematics)
Multiplication sign
Direction (geometry)
Basis (linear algebra)
Higherorder logic
Grothendieck topology
Subset
2 (number)
Frequency
Constructivism (mathematics)
Quantification
Aerodynamics
Contrast (vision)
Devolution (biology)
Free group
Addition
Theory of relativity
Structural load
Projective plane
Coordinate system
Price index
Density of states
Sequence
Open set
Entire function
Maxima and minima
Flow separation
Modulo (jargon)
Field extension
Proof theory
Category of being
Vector space
Logic
Hausdorff dimension
Universe (mathematics)
Module (mathematics)
13:56
State of matter
Ring (mathematics)
Mereology
Subset
Manysorted logic
Topostheorie
Circle
Rhombus
Theory of relativity
Process (computing)
Proper map
Generating set of a group
Point (geometry)
Moment (mathematics)
Translation (relic)
Bilinear form
Open set
Maxima and minima
Category of being
Arithmetic mean
Ideal (ethics)
Mathematician
Point (geometry)
Geometry
Observational study
Real number
Translation (relic)
Event horizon
Theory
Frequency
Wellformed formula
Operator (mathematics)
Gamma function
Renewal theory
Condition number
Element (mathematics)
Projective plane
Operator (mathematics)
Ring (mathematics)
Mathematical model
Cartesian coordinate system
Calculation
Logic
Network topology
Universe (mathematics)
Element (mathematics)
Free variables and bound variables
Game theory
Local ring
21:54
Musical ensemble
State of matter
Set (mathematics)
Ring (mathematics)
Multiplication sign
Decision theory
Water vapor
Discrete element method
Theory
Grothendieck topology
Prime ideal
Frequency
Duality (mathematics)
Wellformed formula
Topostheorie
Operator (mathematics)
Acoustic shadow
Free group
Condition number
Social class
Physical system
Series (mathematics)
Topologischer Raum
Optimization problem
Physical law
Algebraic structure
Mathematical model
Set (mathematics)
Ring (mathematics)
Hand fan
Mathematics
Calculation
Category of being
Arithmetic mean
Universe (mathematics)
Module (mathematics)
Right angle
Absolute value
Ideal (ethics)
Mathematical optimization
Spectrum (functional analysis)
Local ring
Matching (graph theory)
28:12
Axiom of choice
Point (geometry)
Logarithm
Direction (geometry)
Multiplication sign
Modulform
Insertion loss
Limit (category theory)
Goodness of fit
Grothendieck topology
Mathematics
Manysorted logic
Topostheorie
Free group
Subtraction
Condition number
Addition
Theory of relativity
Spacetime
Proper map
Prisoner's dilemma
Point (geometry)
Element (mathematics)
Basis (linear algebra)
Cartesian coordinate system
Limit (category theory)
Calculation
Category of being
Hausdorff dimension
Network topology
Phase transition
Universe (mathematics)
Spectrum (functional analysis)
32:44
Computer programming
Proof theory
Dynamical system
Musical ensemble
Point (geometry)
Element (mathematics)
Goodness of fit
Rule of inference
Theory