Using the internal language of toposes in algebraic geometry

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Using the internal language of toposes in algebraic geometry
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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 double-negation 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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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 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intrepid higher-order 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 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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
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
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
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
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
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