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So mode km on dB km are the usual the abelian category of sheaves on its bounded derive category. And I introduce i km which is a category of in-object of the category of sheaves with compact support. Okay? And why with compact support? Because then you can prove that u gives i km is a stack. And I said it's very similar to the theory of distribution. Distribution are the dual of function with compact support and it is a sheave. If you take the dual of function is not a sheave. So it was my first part transport of this category of large sheaves in other words you don't because usually for your purposes is it enough with the in-object of things with some parameters conditions so you need in-objects of so you can't consider in-objects of like sheaves is compact support sufficiently constructive. Okay, that's after. Wait. No, that's the biggest the biggest category, the big thing. And after I define the sub-analytic side on the morphism of side and I explain I didn't prove, I just explain that the category of sheaves and the sub-analytic side is equivalent to the category of and object of the category of considerable sheaves with compact support. That was your question. Okay, so of course this category is contained in IKM. And I explain that we have to be careful because there is a lot of commutative diagrams but some are not commutative and it's the following we can send more sheaves on M we can send them to sheaves on the sub-analytic side by direct image okay, we have this function of side okay and we can send this in IKM because sheaves are and sheaves of course they are not with compact support but you take the inductive limit of sheaves with compact support on this it's natural but this does not commute but there are other functions which commute if you remember I construct also two other functions on these diagram commutes so maybe we work in the sub-analytic setting which is much more easy of course but it's not totally satisfactory because of this function and because of the diagram which does not commute so we will work mainly with hand sheaves okay but a finite result or we will work in the DRF category of hand sheaves of course sometimes we are obliged to quit bonded because some functions are not finite common logical dimension or we don't know if they are and we work here in I the sub-category of hand sheaves whose common logic is sub-analytic so this is isomorphic to the DRF category of sheaves on the sub-analytic side I have yeah it's a notation and are constructable it's a notation that works so for those who don't know there are notes on the IHS page or on my own page and everything is written because there are so many functions that it's a little confusing okay the videos also are the videos I don't want to see but it exists it exists okay so let's go further so we know this is general context and I want to apply it to construct sheaves the idea the fact is that working with sheaves there are not enough sheaves for an analyst like me many things that you encounter in nature are not sheaves like bonded oromorphic functions it's not a sheave our temporary distribution is not a sheave but hand sheaves is much bigger and with hand sheaves we can construct a lot of things which have no meanings in the usual setting so the first thing I did so so maybe I will repeat so now M is a real analytic manifold and I will defer so first I recall something of use you take a pre-sheave on the sub-analytic side then F is a sheave I didn't recall today what is the sub-analytic side so maybe I do it now for those who are not here what is this side the open set are open relatively compact on sub-analytic in M on the covering are the finite covering containing containing a finite sub-covering so it's a very nice site very useful I don't know if it existed before we use it but it's very natural so on to recognize that a pre-sheave is a sheave is very easy it has to satisfy the empty set so it's not too difficult and also for any U1 or U2 open set for this site that is relatively compact sub-analytic then the sequence is exact yeah from the beginning I work here yeah it's a billion yeah of course when I say pre-sheave I mean a billion pre-sheaves over K so it's a sheave if this is exact so it will be the tool to construct new sheaves so all these things I did it last last week so what is the pre-sheave of temperate sinfinity functions so it means that phi on all its derivative so in a local coordinate local coordinate system d alpha d alpha of phi of x is less or equal to some constant the distance of x to the complementary of U to some power minus n there exist c n what is the distance of the amplitude you asked me last week I told you and I don't know 0 or infinity I don't know anyway intuitively it means it means that the growth function is a polynomial function of the distance to the boundary like miromorphic function if you have a homomorphic function with singularity on the hypersurface it's miromorphic if and only if it grows near the boundary near the singularity is polynomial with respect to the distance so it's a very natural condition for an algebraist polynomial grows the distance to the empty set is infinite the notes is 0 I don't know you have to define in such a way you take a local chart and then you take a point you know I wrote two papers on this subject one with kashuwa is the other one with giyamu and we gave two different definitions of the distance to the empty set I don't want to make trouble with that I follow kashuwa when I write with kashuwa I follow giyamu giyamu ok so why when I was young when I went to seminars there was always Carter asking such question only if you take zero what happens only if ah yes thank you ok ok so why do we have a shift here we have a shift because of le jacévit inequality which is that if you have two opens you want on u2 maybe I take closed set for change if you have z1 on z2 which are closed on subaneitic then the distance of x to the intersection where is that to the power n is less or equal to distance x z1 plus distance x z2 so this is the qualities in local chart on the compact set yeah yeah yeah it's local of course I was unable to make a picture last week but now I give a picture where it's clear you take z1 on z2 so if you put some point here x you see if you take the graph exponential 1 over x this will not be satisfied the distance here can be very small on here very big ok on the subaneitic set it's polynomial ok so if you put together this on this you find that this is a shift ok so we have a first shift the infinity t is shift on the subaneitic site ok another shift is a shift of temporary distribution what is it's a distribution which is extendable it's it's defined by this exact sequence ok that's the definition distribution that you can extend ok so of course in the usual topology it's not a shift it's not a local property ok but with this subaneitic topology it is a shift so your distributions are like generalized functions generalized densities no distribution like in France people say schwarz distribution is it function distribution or subalife I don't know it contains functions it's not important for that ok no it will be important soon when I will prove a theorem but at this level you can take distribution or densities or it doesn't is it the same space db and t is those space to see infinity functions supported from you with compact support in you supported from you functions from the whole space which has zero outside of you no with compact support in you ah this one you mean ah I don't know I will speak soon of Whitney yeah maybe it's like that ok so it's a good it's a good transition to speak of Whitney functions so there is another in fact we define the shift of Whitney function but it's better a co-shift it's not a question suppose your varieties are p1 and you can see open space which is complement one point r yeah will be db of temperate the same as schwarz space ok so let's define another function so you for s a closed set in m define g infinite m s this is a sub-shift the shift of function which are which vanish up to all derivative on s ok and then we define a new shift no it's not a co-shift by c u with net answer with infinity so it's a pre-shift a pre-co-shift so so it's no it's a function so there is a mistake in my notes so to v I associate the section on v of the function which vanished on the complementary of v so this function which is co-variant you can extend it so it's a exact in the sense of my obituaries on the category of open set on you extend it to the category of shift so you can define f gives f with net answer with c infinity and since I want a shift not a co-shift is we can define the more general thing so this is just a convenient notation w w is for Whitney no but is it just in this context or it's more general is it notation just a dog for this or is it more general operation like that I don't follow what you don't understand your question yes ah it's a kind of tons of product because so it's like you know you know what this what this is but it goes because this is a function which are with compact support on you and this is a function which are it was your question which are supported by the closure of you if you is not too bad if you are as good if the dual of you of this shift is the closure ok so never ask keep the detail because I will not use it so much but we can define the c infinity w so gamma u c infinity w is where is that done gamma m you have to take the dual of the constant shift and you with infinity ok we can it's not important for what I will do after so but I just mention it because this was a push echo shift and I want a shift so I take the dual and then I have a shift ok so we have these three natural shifts and there is a another one but ok so this one will play a role for duality but I'm not sure I will what is the c no I explain I have a push shift and I want a shift ok so I extend this to constant shift and then I take the dual and then I have a shift but it's not so important the dual I say this is duality of shift the dual dual ok so now I want to prove a CRM I want to do something until now I gave a lot of definition but I want to state a important CRM and I will sketch a proof a direct image CRM for distribution so I have so again m is real analytic I denote by the m is dimension and I I denote by vm the space of real analytic densities that is differential form shifted by the orientation shift and I will use distribution temporary distribution but shifted by tensorized by these densities ok and what else and I use the language of d-modules so if I have I use a bimodule d a dm ok as a shift it is a m tensor f minus 1 a m are forms that is interpreted this is definition a m I mean a m whatever is in temperate forms yeah but it is not usual forms but in temperate forms it is temperate, temperate densities and a m also temperate no a m temperate I skip, I hope you will not see the problem but I skip the problem I should write beta here to be honest but I don't write it that is the problem of notation otherwise it has no meaning you are right because this is temperate if I take the tensor product and open set where this is not temperate then you lose everything so in fact it means beta when a m is the shift real analytic functions complex value so what is a theorem many distributions of function of functions anyway anyway it is not yeah now you are right I should have said exactly what it is but this is dual of infinity this is a dual of infinity functions okay with densities is dual of infinity functions okay okay so what is a theorem I will give two forms first the theorem is due to cashier so I give a a first formulation I say that there exist an isomorphism natural isomorphism okay so it is very abstract but you are not afraid by abstraction I guess I give a second formulation of the same result is the following I give you take our own f db t change tensorized by d m to n and it is isomorphic and I assume that f is proper on the support of f so in fact cashier as theorem in 84 was was better formulated as the second one you don't know what it is is it the one for sheaves not for the module for sheaves for hand sheaves but it is the same for hand sheaves magnetic sheaves as you want so before to explain the proof I want to show you that a corollary I don't know where I wrote it maybe it's here I want to show you that this is a generalization of a grotendic theorem in 66 why take f from m to one point okay and take f take u f the constant shift on u where u is relatively compact in m on subane etic so what is the left hand side this is a Durham complex up to a shift gamma u because this is this is r gamma u of the complex with it here is a mistake okay so this is the Durham complex with coefficient the distribution temporary distribution this is a translation of this okay on what is right hand side we find r of the direct image so it's proper of the constant shift on u with value in k in c okay and by adjunction this is the global section of the dualizing complex on u okay so this theorem tells you that you can calculate the up to a shift orientation you can calculate the convergy of the open set by taking temporary distribution the classical result for shift theory is that you can calculate the convergy of an open set by taking the soft resolution by distribution but here we take distribution with temporary growth which is totally different so you recognize if you take the particular case and I will come back to this particular case of a complex of the complex setting if u is x minus z on x is a projective manifold on z is a closed hypersurface then the the Durham the convergy of u can be calculated with temporary so here is distribution but after I will apply dolbo and it says that I can calculate this with the Durham with mirror morphic function with pole on z so it contains gottendic theorem of 66 you recognize no after there is one step I have not written you have to apply dolbo I will do it later but it is a real version of this theorem of gottendic which says that you can calculate Durham convergy by taking temporary things just there is maybe some notation there so you wrote you have t you define t with a check dbt and then you write it should be tv I forgot the t here so we will come back to this theorem of gottendic but it says roughly speaking that you can calculate Durham convergy by taking temporary growth ok I just say a few words on this theorem which is difficult it is a deep result so the sketch of proof I just say a few words first you replace dmn by its Spencer resolution then you find db calculate db tensor tsh tensor dm but this is free over dm so there is no derive product first you have to construct the morphism so we have to construct this morphism ok so of course this morphism is an integration of distribution by integration ok and after once the morphism is constructed we have to prove it is an isomorphism and then we can reduce to the second formula and then we decompose f as a immersion a closed embedding and a submersion and finally to close immersion is easy submersion you reduce to this situation m is n cross r to n and you reduce to the case where f is a constant shift on a closed set and move over so there is lot of work that I skip such that f minus 1 of x inter z is a closed interval ok so there is lot of work I don't give the details because it will be very boring but it's a proof ok so we will use I will give two important theorem today that I will use next week so one of them is this theorem and another one will come later ok so let's keep the proof to technical to be done here so now I pass to the complex situation so now x is a complex manifold and I will use my shift to construct new shift of a phonomorphic function so o x t will be the dolbo complex of c infinity t ok so what does it means o x t is r m over dx c so let me put some notation so maybe so I denote by x r the real underlying manifold on x c the complex conjugate manifold up or down depends depends on my mood no on my notes it's up I'm sorry ok so this is nothing but the dolbo complex and you can define o wittnay phonomorphic function the same thing so here is the difference between subanalytic on and shifts because you have two definitions so I should write you know as and shifts ok so o x is a direct image so I keep the same notation ok I look ok I recall this belongs to the derive db i cx ok and in fact so it's i dx ok and they are are constructable so it means that they come from db mod db excuse me cx or maybe d xs ok these are the image of things which are constructed in the subanalytic side but this one ok so it's a little confusing maybe I should skip it but remember that it should take db does not commute ok and if I take o x if I take a I will find direct image of o x and it's not the same because this one is not in degree 0 this one is in degree 0 ok so ok maybe I don't recall but I will give you example after it's better if you want I can recall but I prefer to give example because the construction is not easy so it is so the temple when I was it was a shape of subanalytic and the weekly one is you define something with some dualized dualized something ok wait one minute and I will give you examples and you will understand you will understand why it's useful I take examples and after I come back if you want I will take two kind of example take z a closed complex analytic excuse me I forgot I have another shift which is also important which is beta of o x sometimes I denoted o x ok this is w it's not and this is omega ok so I give you examples to make sure you understand things so if I calculate our arm I take the dual of the constant shift on z with value in o omega then I find o the restriction if I take the same thing with o witness I find the formal completion if I take no I don't take the dual but I take cz with o t I find algebraic homology supported by z and if I take o I find the analytic homology supported by z so you see the algebraic homology is recovered by o t and it's natural because o t means with polynomial growth so when you take the homology you find the algebraic homology so the witness gives you the formal completion and beta gives you this formula I don't want to insist on the witness because it's useful when we prove duality but maybe we will not prove them and I give you another bunch of examples so in the example on the left z was complex analytic now take m real analytic on x a complexification so if I take our arm on x of the content chief and m of o omega again I find the real analytic function if I take o witness I find the sinfinity function if I take o t I find distribution and if I take o I find satos hyperfunction so you see with this approach it's unified all these classical sheaves of analysis as obtained by applying our arm to some sheaf of homomorphic function and in particular distribution are constructed by comogical way but of course we have used distribution to construct o t maybe maybe it will be possible to have a construction of distribution without functional analysis so that's the end of one paragraph and now I will I will concentrate on o t and I will I will look at the operation but I think that the definition of particle particle local comology okay because the definition of hyperfunction one was local comology you took d prime x you take d prime x of c m is c m or not no it's the orientation shift shifted by the dimension so if you want it's c m tens of the this is h n supported by m o x okay and with this definition a m is injected in b m because e m is the same as arm d prime c m so you see the injection of real analytic function in hyperfunction is very clear with this definition of hyperfunction because a m the dual of the dual is c m okay and look at a m as this tensor product and by contraction you have this which is hyperfunction so this natural morphism is clear so I think it's a better definition that this one but you still have to prove that it's concentrated in a single degree why do you want to prove that no of course it's better the main difficulty is to prove it's concentrated in degree zero but if you want to do normal mathematics you can work with this okay okay so now I want to make operation an utility direct image, inverse image and things like that so I will systematically use the language of the module so now everything is complex analytic so there is a very very good book by Yeshiwara on D module at EMS I don't remember the title but it's a very short it's very concentrated but very good book with everything on D module it is in the bibliographies that I gave so in particular we use the transfer by module on the operation okay there are natural operations that you can guess that you know but there is another notation maybe not so usual the transfer product in the category of D module so this one which goes from Db dx transfer cross on in fact for an object also so Asheshif is a transfer product OX but with a structure of D module that I don't recall it's a basic D module theory you have to make to explain how a vector field operates so it depends if it's left left right right left right it's complicated but I don't give the details okay and also I use the duality for D module okay because duality interchange right and left okay so if we want to remain in the left D module we take this duality otherwise right duality and we use always equivalence of category between right and left D module okay so I will not I will not make any distinction when I write we have to take care but I will not be very precise okay so so maybe some generality on D module and after I will give the main CRM so so in a algebraic geometry always the hypothesis that you do is that the morphism are proper and you don't care about inverse image it's always good but for D module it's more in the macro local in the cotangent bundle direct image or inverse image behaves very similarly so if you have a morphism like that you can write the tangent morphism or you can write the dual diagram so I call this map fD on f pi because the projection I denoted by pi so there is a notion of non characteristic so if you have a subset let's say lambda here conic closed conic we say that f is non characteristic if fD is proper on f pi minus 1 of lambda and in practice lambda will be analytic until these are vector bundles to be proper is the same as to be finite so we have this notion of being non characteristic so when I take direct image I have that f is proper when I take inverse image very often not always I ask that f is non characteristic so I will not recall all formulas for d-modules there are too many and it's very confusing I just give a few one and after I enter my subject here I will take notation which are that I don't like very much but he will use this notation in his course so I prefer to prepare you to this notation so the direct image for example so f goes from x to y direct image of m is the direct image of for a left d-module on the inverse image that's casual notation I don't like them but he is the chief of d-module so I cannot say anything so it's that's the notation so for example I don't give all theorem but I give this one duality commutes with direct image if f is proper I forgot something also supporter of m excuse me I forgot to I don't work with cahiran d-module but cahiran d-module I have to say a little more I have to speak of good on kazi good say that m is quasi good it's like quasi cahiran more or less if so m is a d-module if for any relatively compact open set u in x m restricted to u is a filtrant inductive limit maybe small of cahiran ux so it means it has a filtration on m is good it means it's quasi good plus cahiran as a d-module in the algebraic case all modules are good but in the analytic case it's not true so you have to assume although it's good filtration so the theorem more precisely take f from x to y so if m is a good so it's not written but these are triangulated categories they are six of categories of the abelian categories and the dirav categories are triangulated then on f is proper on support of f then a direct image as a d-module direct image in the sense of d-module no the dual the dual of the direct image is the direct image of the dual and it's also good yeah that's that's abuse the dual is good the dual of a good module is good the coherence is okay but the filtration is something no I think it's okay but maybe it should be told without this global good filtration by some doing slightly better yeah maybe but then it's much more complicated you are right by hypercovering you mean things like that okay anyway it's enough for what we need and that's for direct image or for inverse image now coherent is enough because it's local so if f is non characteristic for n for n means for the characteristic to be non characteristic for a d-module means to be non characteristic for its characteristic variety of course then duality commutes with inverse image no global to be good is global not absolutely global but almost global for any so dual I explain it that's the definition it's dual for d-module no it's left to left here it's left to left but you have dual right to right right to left okay so it's not so important you want a break yeah so your f inverse is the same this convention is the same as tensor over oi is it so this f inverse of n is it there inverse as a d-module it's written here but is it does it correspond to the operation of tensoring over oi with ox yeah but for a structure yeah but as a d-module okay so it corresponds to what is sometimes in terms of yeah Bernstein notation is I don't know something like that something like that no I don't know Bernstein notation but anyway it's the inverse image for d-module there is only one definition of inverse image for d-module the point is it doesn't correspond to inverse image for constructive worksheets in other word under so when you go to the architecture constructive commology it corresponds to usually to something like f up or 3 I don't know anyway we assume it's non characteristic so in this case f-1 or f up or 3 are the same up to a shift but I think it's really up or 3 because this is the notion that up or 3 is good is the good f up or 3 behaves well when it's not in characteristic this is what he's claiming should be exact for d-module there is only one inverse image up to a shift and we apply it in the non characteristic case ok maybe it's time to have a short break ok after I will give two main theorem ok let's define the deram on solution from terms so in notation so deram of m is a omega x and so m and so so these are classical but they are the same with temperate deram temperate and ok so you see you go from here to here by applying the function alpha because alpha of ot is o x I forgot to say and of course if m is coherent then these two functions are related by not d like that ok so for coherent module we don't care we work with deram or solution is the same but of course otherwise it's not the same so there is a first result which is obvious is a direct image commute with usual deram maybe I write it here where is that no there is no hypothesis so a first result I recall f goes from x to y then if you take the direct image or proper direct image of deram it's isomorphic to the deram of the direct image as a demodule of m no hypothesis it's very formal and it's not difficult but what I will explain or maybe prove I don't know is a similar formula for temperate so to prove so so theorem is obvious but I call it theorem one one temperate the same thing maybe maybe it should be with double shriek but there is an hypothesis excuse me now there are hypothesis is a quasi good on f is proper on the support of m then the direct image of deram temperate of m is isomorphic to the deram temperate of the direct image as a demodule f star or f shriek as you want is proper ok so I want to give you not an idea of the proof but a very rough idea of the proof but to prove this theorem or maybe let's call it too this will be simple to prove theorem too we need a grauert theorem a temperate grauert theorem but even when we work with demodule and when we take direct image of demodule we use grauert's theorem so you use it in the case where there is a good filtration so is it known whether this is necessary as it is so the the final theorem for the module so I remember a long time ago this was a condition that we must have a global proof let's say it helps but it's not necessary but it's very very complicated if you don't make the hypothesis of good filtration ah so there are some references for this but is it? I will give you later because it's out of the scope ok but the final test result is it true I think I think in a paper I wrote with cashier in appendix we give a general tool in a memoir but 96 ok so let me give you the what is a temperate grauert's theorem is the following you take f which is O x coherent on f proper on the support of f then a grauert's theorem tells you that the direct image of f is coherent ok but a temperate grauert's theorem tells you that if you take the direct image as a subalytic shift and shift of the temperate version of x of f then it's isomorphic to O y t the direct image maybe I can put my symmetry here ok so I don't give any information about this result I regret that there is no that we are obliged to use such things which are horrible because there are a lot of functional analysis like grauert it is not deduced from grauert we have to do the proof again so again you have the condition no here it's very natural there is no demodule here ok but here it's proper you cannot change it here maybe you can take every time if you want you can do that if you want it doesn't change when it's proper the two functions are the same so the shift we see a very nice question it's not coherent as a ring of it's not in degree 0 it's not a shift it's an object of the direct category maybe maybe I say one word it's a parenthesis but we have a lot of time if you look what does it mean that f it's not wooded airflow it's not in dB the fiber dimension is not bounded on Y this one Y because if you have countless components it's different if you are looking for trouble you find trouble let's say everything is reasonable you are right for my day you are perfectly right okay assume X is connected okay now what I want to say just a parenthesis what does it mean on a site you have amorphism of amorphism of shifts on a site let's say on the subanitic site what does it mean amorphism it's a parenthesis it's out of the scope it means that for any open site U on any T any section on U there exists a covering on S I in F of U I such that U of S I equal T restricted to U I that's the definition of being an epimorphism for shifts on any site but here on the subanitic site we have only finite covering we have finite covering so for example if you take an open set like that the difference of two balls in C2 or in Cn and if you take as a dolbo complex this will be never exact after a finite covering so if we work with a subanitic subanitic site all is not in degree 0 but as a and shift it's different okay it was a parenthesis I don't know why in some sense can you characterize these shifts the things that are obtained as a coherent principle no we don't know anything on OT it's a we don't know if it is flat over O for example it has not so much meaning it's a complex yeah that's why I explained OT is not in degree 0 it has no meaning to be flat as an object no but maybe bounded the third dimension everything over O axis yeah but here it's not O axis it should be beta axis as I say I explained it's a difficulty I don't want to say too much okay so nevertheless you have this big result that you can deduce I don't give the proof but it's not difficult you deduce it from temperate grout it's a standard but temperate grout is not easy but after from temperate grout you can prove this one functional analysis like grout okay and now I will translate this result of cashew aura I will translate it in the complex setting where is it so here you have F okay now I take the complex situation then then there is a natural isomorphism omega X T transfer dx dx to Y but now there are some shift so it's a inverse image theorem for temperate I write it for differential form but you can translate it from right to left if you prefer so you deduce this from this by applying dolbo of course you have to be an expert in the module theory but it's for an expert it's obvious you apply dolbo on some formulas and you get this one so it's not very intuitive I agree but from this so if you transfer this one if you apply this function which function to this isomorphism then you find a new formula which will be useful if I give all this formula it's not for fun because we shall use them later then you find that maybe I can maybe I will write it here at the other side n is a n is a dy module then you find the following formula you find that the Durham temperate over X of the inverse image of n shifted by dx is isomorphic to the inverse image of the Durham temperate on Y of n shifted by dy so you see we have this so we have the temperate Durham commute to direct image on commute in some sense to inverse image but take care that this formula is no more true for non temperate there are formulas which are true in the temperate case and which are not true otherwise so what is the hypothesis you are right what is the hypothesis no hypothesis the half category of dy module but in the classical theorem they are all yours this is not true for Durham not non characteristic for non characteristic it's true but otherwise it's not true no no of course regular alonomic everything is true no no because no the idea is when you have some manifold the technology supported by the some manifold if you don't make temperate is a essential singularity is no more a demodule a coherent demodule ok so maybe I skip some many details so I think I have written the two main formulas inverse image on direct image on to to conclude today so now it's a new section or subsection ok so maybe I write ok no ok so I take a new section new subsection and I will look so it's a preparation for next week next week we will reduce the study of the alonomic demodule to the case of normal crossing divisor ok so I have to make some calculation in this case so I take y in x a closed hypersurface complex of course so o x star y you know it's a shift of momorphic functions with poles y and if m is a demodule I define m localization and y as m tensor ok this is the tensor for the demodule ok ok so I I will calculate ot tensor d o x star y maybe no I keep my I change my sorry I take I call it s it's better so now I need to stay in the hand shift ok not as a shift but as a hand shift then the result is that this is r i m I don't know how so we are in db i dx oh no if you apply alpha to this applying alpha you find o x star s equal r on cx minus s oxt so you go from here to here by applying alpha and maybe I will give a corollary a corollary maybe first I prove I prove the first formula I explain you the proof of the first formula because it's interesting so how can we prove this formula we prove it by coming back to distribution ok so yeah s was a hypersurface so how can we prove that I take oxt is a dolbo the dolbo complex of dbt so I will so let's denote by assume that f x to c is a real real analytic not 0 then denote dbt dbt bracket 1 over f maybe excuse me I need a notation let's denote by g j m minus s to m the embedding yeah m will be x r so I will prove a similar result but with distribution instead of allomorphic function and by dolbo it will be it will give the result so this is the localization of dbt by f if you want it's the inductive limit of db m or if you prefer is a set of u in db g star g minus 1 db m such that there exist m with f over m maybe it's not t here such that f over m u is in db m so here it's implicit that distribution which is 0 outside for ip of f is key locally for power yeah we shall use that of course yeah it's not I forgot to say thank you so finally we are reduced to prove that the section on u of db t no 1 over f is isomorphic to gamma u minus s dbt m so you see this is when I apply dolbo I will find the right hand side similarly and why is it true this is again it says that if if if you have a distribution then locally there exist m such that f over m u equals 0 that's what you said say a sinfinity function which satisfies lower service no last is distribution it means if you have a distribution which is in the localization by 1 over f on u it's the same thing as the distribution which is temperate on u minus s no it's maybe I didn't do that very clearly but it's not no I'm asking that's the analytic part that if you have let's say real or complex value function let's say that's the sinfinity distribution which is supported on the zero locus when you want to know it's killed by power all what you need is the service and the quality the point where f is small is close to the zero locus is it equivalent or enough so it just uses this quality and you get this or you need something I don't know the proof but I think if you work at the regular part of f it's obvious because if a distribution is supported by a smooth some manifold it's killed by some power okay I can do it with this regular part then it is okay it's actually smooth I don't know if you use it here or not the classical form of the service and the quality is the distance from the to the zero locus is equivalent to the f to f to some power okay but and this and you claim that this implies this kind of present no honestly I don't know the detail I know it's more or less obvious but the details anyway okay so we have this otx tensor ox star s and I say corollary we have this omega t okay applying okay I skip the details applying omega ox or taking okay because this will be the same as omega x tensor over dx oxt star s so it will be r cx minus s omega x temperate d ox and this is cx shifted by minus d so this formula again no it's clear I hope that we recover grotendic theorem on the homology of the complementary of a hypersurface he says that if you calculate the homology of the complementary of a hypersurface then you can calculate it by taking a homomorphic function with pole s so let me explain you what I will do next week because now today it was a little maybe a little boring because I gave all tools that I will use next week so I give a general result that I will repeat next week we want to prove we want to prove a property this is a property it's a little vague maybe for m on x so for all m on all x on m next week will be regular autonomic then there is a result according to cashier it's well known on abuse and the result is the following assume assume this property is local if x is a union of ui then px m is true if and only if p ui m is true for any i it's local it's invariant by shift px m is true if and only if px m shifted is true it's stable by direct image of direct factor by distinguish triangle if px mi is true for i equal 1, 2 then for 3 what else maybe I skip all these are abuse direct image direct factor so it's true for m1 m2 m1 plus m2 on m1 plus m2 then is true for each one maybe you are right ok anyway all these one are abuse ok in fact in my notes it was not written ok but now it becomes more serious now there are two other property so this one are ah what did I do ok on the two last one is direct image so if fx to y is a projective on m is a db good then if px m is true then direct image of m is true so the property have to be stable by direct image on the last one you have a local model which has to be true on the local model more or less I will be more precise you have d normal crossing divisor on x let's say x1, x r equal 0 then on m is d x for valenda then so what does it mean let's ah I will be more precise next time pxm is true ok so these are maybe I should maybe I I am more precise now m has regular normal form along lambda along d then pxm is true so what means to have regular normal form roughly speaking I will be more precise next week it means if d is defined by x1 equal x1 equal 0 then m is d over x1 power lambda 1 x1 power lambda r d r plus 1 d n no d d ok differential operator which annihilate this generator d x power lambda ok ok so you see this theorem is very nice because it reduce the proof to check proper direct image for good module on this local model and in the irregular case wait wait wait the assumptions are one six then the property holds if you have a property which is local stable by direct image unsatisfied for the local model then it is true for all the regular regular no I will say next week ok but this will not require to do all the theory no part of the theory that I will not give you have to know what I will not explain that the category of regular autonomic module is stable by some operations I will not do that next week what I want to say is that in the irregular case it is almost the same except the last where you have to replace x power lambda by exponential phi where phi is meromorphic and this is more or less the contents of Machizuki's theorem it is a translation by Kashiwara of Machizuki very useful now it is very clear to look at Machizuki or Kedlaya I don't know if it's so clear so I will explain the regular case next week on Kashiwara we will explain the irregular case three weeks after I think I stop