 אז תודה רבה for introducing me so I will speak about the work with Brian Conrad on So the title was given by Luzi I should write. Okay, so spreading out of rigid analytic families and observations on periodic Hodge theory. So this is a work which is still being written because there are some technical details which are not so clear. The motivation came from questions about rigid analytic spaces in particular results on periodic Hodge theory which were proved by Schultz general, I mean extending the approach of file things using perfect to its spaces So in particular if we have smooth proper rigid analytic space over non-archimedian field. So a non-archimedian field will be complete for a rank one valuation so I will suppose that it is a discreetly valued field with perfect residue field. So a usual local field as usually considered in the theory of Fontaine and others who looked at comparison using BDRAM, BGRIES and so on. So then there is so there was a conjecture of Tate on which was originally formulated as a Hodge state, the composition. So in any case one knows that by Schultz's work, so you have that the Z-mode P is finite dimension, the Z-mode P is finite dimensional, in particular the QP is finite dimensional, and then the comparison results give Well, in the sense of the, that is for dimensions it gives that this is the sum of dim like in the complex case HP X omega Q P plus Q equal N and this is also dim of H and DRAM X using so in particular the Hodge to DRAM spectral sequence that generates and then one can ask the same question for more general K. So a more general K could, so here the characteristic, the residue characteristic is the more general K could have zero residue characteristic could also be something like the completion of the algebraic closure of QP. So in order to handle this, the natural idea is to use something which is like what is used in usual algebraic geometry to set it a variety over the complex numbers, the sense to a variety over a finely generated Z-algebra. So for example, there is concerning the same question now in characteristic zero for algebraic varieties. So there is a work of Delini-Luzzi in suppose 86, something like this. So there were, they proved that the Hodge to DRAM, they proved without transcendental methods that is using reduction to characteristic P, the generation of the Hodge to DRAM spectral sequence using spreading out to smooth schemes over Z. So the idea is to, and this was also inspired by Fulton's Piadik-Hodge theory. So it is closely related to the material I'm discussing. So now the idea is that things that one can prove by Delini-Luzzi method for algebraic varieties, one should be able to prove using Piadik-Hodge theory for rigid analytic spaces. So there was a further progress. So a paper by Luzzi Carter and Nakayama in the log case. So for certain log-smooth map which are exact. So it's again a statement on the generation of the relative Hodge to DRAM spectral sequence and local freeness. So this again should extend to the rigid analytic as we hope using some spreading out technique. So of course here we have more structure with log structure. So we must then spread out some schemes with extra structure. So the idea is to, so for example when K is an extension of QP, we want to, so any X over K which is proper will be a fiber for a flat point. Proper map Y to Z of rigid analytic spaces over QP. So here rigid analytic spaces could be in the sense of Tate, but usually they are quasi-compact and quasi-separates and it's also in the sense of Addix spaces or Berkowitz spaces. So it's a, and then the fiber is over a K point. And also when you have some assumption like smoothness then this morphism should be smooth. When you have a diagram you should have a diagram and you have some query and shift, you have some query and shift morphism between them and so on. So for example we would like to extend log structures, but this is not yet covered by the technique. So on the other end this is, so this is more generally, you can have a discreetly valued subfield of your non-archimedian field and you want to spread out. So you can also try to, yes, you want to, yes. So you can also generalize it slightly to what I spoke about. Is it okay or like this? Okay, so when you have, this is discreetly valued and suppose that X to Y is proper flat map of rigid. Spaces over K, let's say quasi-compact and quasi-separate, and then by Renault flattening result. So this extends to, comes from a proper flat map of formal schemes, topologically a finite presentation over the ring of integer, which is also flat over the ring of integer. And now you can restrict to some affinity, so you can assume, let us assume that Y is affine. So in the original situation if this is affinity it doesn't mean that you have to blow up something. So then I want to say that this is a base change of a proper flat map of formal OF schemes with the same properties, topologically a finite type. And this will give by passage to the general fiber the result on rigid and elliptic spaces. So on the other end, yes, so I'm saying that to handle the problem, to handle also the RAM problem when the residue characteristic is zero, we need more. So suppose that I have a map from A to OK, where A is an Italian ring addict, so complete for an ideal and this is an addict map. So for example we can take some finitely generated Z algebra and some mapping to OK, an element that maps to non-zero element in a maximal ideal. And then I want to spread out to, so then the same spreading out result. So when I have this formal situation I want to get a proper flat map of topologically of finite type formal A schemes. So I recall that when I look at formal schemes, so I look at formal schemes which are coming from ring complete for a finitely generated ideal and then the locally something topologically of finite type is SPF of restricted formal series divided by an ideal. And there is a notion of formally of finite type where it's SPF of this with some formal series divided by an ideal. And in fact the proof gives something formally of finite type but it is known by certain blowing up procedure to get things topologically of finite type. So now, so in the proof I have to use the formation theory techniques but there is a problem to make things sufficiently canonical. So I have to prove the existence of certain versal deformations and this in the situation which I, of the formation theory that I want to look at so it's related to the notion introduced by Grottendijk of additive cofibre categories in his Springer lecture notes. So this was a work which before he loses work on the cotangent complex but his viewpoint, so I wanted to extract things related to his viewpoint from he loses result. So let me discuss general thing about Grottendijk additive cofibre categories. So suppose that I have, okay, so C is an additive category. So I will, I found a small modification of Grottendijk's formalism using pro-object which gives a general statement. So usually it is an abelian category and I want to consider a certain cofibre category over C. So let us say E over C is a cofibre category. Again all categories are small to avoid some saturated problems and also I want to take pro-objects so this will be a large category. Now the condition of additive cofibre category is the following that for the zero object I want to get a trivial category and for let's say a product of two objects I wanted this is an equivalence. Now once you put those conditions you can introduce an operation on, so this is the fiber of E over X. So now when you have a morphism from X to Y there is a push forward function from EX to EY. So now we can, if you are given two objects in EX you can using this equivalence get something in EX cos X and then use the sum up to go to EX. So this gives a kind of tensile product on EX and then, so one can verify that EX is what is called later a strict Picard category. So the tensile product comes with associativity and commutativity constraints and the zero elements satisfying all the rules including the strict ones that you need about flipping in EX cos X. So the strict Picard category is equivalent to a category of the following type. So I have a complex, so I will use a Gotendick notation which is, I think the linear is slightly different. So I have, let's say a complex concentrating in degrees zero and one and then I let the objects be K1 and the morphism is given by something in K zero. So if DX is Y minus, so a morphism from Z to Y is an element in K zero of which satisfying this. So any strict Picard category is something of this type. So essentially you have to give something in the derived category of a billion groups of lengths 2. Now there are several conditions on those kind of, So there is the condition of left exact when C is a billion and this means that when I have a short exact sequence in C I can map EX, EY, EZ. So I have a point actually. So EX maps to the so to speak kernel that is the two fiber products of EY and the point of EZ and I wanted this to be, this should be an equivalence and if this holds then more generally when I have a square which is both Cartesian and co-Cartesian ZEN EX is maps by an equivalence to the two fiber product. Now so in fact, so for every additive co-fiber category E to C So one can associate the left exact. So this is like R zero E. So this is like the construction of the derived where you try to take the limit over all exact sequences. So you don't assume there are enough injectives but roughly if you have an embedding in something injective then you take the EB, the kernel so to speak from EB to EC as a definition of R zero E of A and then you take a limit. Now the, and then Grottenlick also extends the co-fiber category to complexes but it is useful to extend it to the pro-objects. So E to C extends to maybe E prime over pro-C by the obvious rule that you take E of a formal inverse limit as the two inverse limit of EXI and this preserves left exactness. Now there is the standard construction of such additive co-fiber categories in terms of lengths to chain complexes. So suppose that I have an abelian category and the lengths to chain complexes. Okay, then I want to define for X in C I can consider home, so the home from these two. Okay, and then I take the associated, this was denoted in Grottenlick's notes by Q so something under, double underline of L. So this is not necessarily left exact so one can consider it's associated left R0. Now when we work with pro-objects we have the freedom, we can use projective resolution so the category of pro-objects satisfies the hypothesis well dual to AB 5 as in Grottenlick's to co-fiber so it has enough projectives and one can also construct them in some way directly and so I can find the quasi-asomorphism in pro-C and then the R0 will become the Q of P1 P0 X. So the fact that P0 is projectives allows you to verify the left exactness and also it turns out that the extension to pro-C switching the order. So let me, so if I am given A in the object of C and X in EA so I can look at the problem of the function of associated to B the morphisms from A to B with an isomorphism to zero of the push forward of X so by general consideration when you have a left exact thing this is pro-representable so omega X is pro-represent this so if omega X is actually representable then Grottenlick says that the typical complex exists now, yes? What does the notation omega X stand for? No I don't know I just looked at Grottenlick's book I don't know why he, this is a very old reference but I'm not sure if the generalities in particular what I'm going to explain I'm not sure if this is completely redone in more modern references but I'm not sure what the omega stands for I think it's just because of So it's just a symbol for the complex that you've written below? No, no, so what happens is no, no, the point is that given an element in ob C and an object in the confable category one looks at the problem of making this zero by mapping A to something which is more represented universal by something called omega X it could be represented so in which case it is said that the typical complex exists but it is easy to see that it is always represented by a pro-object okay? now so anyway so you have a notion anyway maximum and quasi-maximal object in the additive confable category so roughly a maximum one is where for any B Y over B you want a morphism so some object for which there is always at least one morphism to any other object and quasi-maximal means that after embedding something you have such a morphism and then the result so so for a left exact additive cofibered category E over C so it is equivalent to R0 QL if and only if there is a quasi-maximal object and the typical complex exists so this is part of other complicated thing so now this can be simplified so what is the what are the data okay so this is the data so suppose that I have an additive cofibered category and I give an example left exact, I give an example coming from an object a two-term complex and now I want to characterize those that come from a two-term complex so anyway this is one of the statements up to maybe some miss maybe slightly different conventions in gotten this book so the so the so the condition is that for any A and X there is omega X is represented and representable and that there is a quasi-maximal object in the cofibered category okay now so now what I can show is that for every left exact additive cofibered category E to C then E prime over pro C is equivalent to what is associated to a two-term complex with L zero projective in pro C okay so actually this is almost a tautological if I'm using so because you can instead of using a maximal object you just look at the all possible objects in the in the total space of the category we can take fiber products and get the pro-object which is in some sense maximum except it is a pro-object then this will play the role of L1 then the L zero will be the sum the singlish pro represents this omega just relative to C not pro C then I will get a complex which gives the cofibered category on C itself now L zero is not necessarily projective but I know that there is a projective envelope so I can replace it by a complex of P zero is projective and this is still maximal relative to C so this again represents the same thing then I know it is represented by this on C so this is a problem of inverse limit of those categories so it turns out that this has to do with vanishing of L1 from P zero of some system and this L1 is computed actually is the proof gives it in terms of some standard complex and this standard complex is exact in the category of pro-objects and since P zero is projective this is zero okay now so in some sense the maximal object is something which in the formation theory problems is like a reversal deformation so the property so again I have to move easily so maybe this is slightly different definition so I will say that X in EA is reversal if and only if it is maximal and the following holds that if I have a surjection from B to C and I have an object C in EB and I have a map Phi so suppose that I I have an isomorphism then this can be lifted to some map so so I want the lifting of phi and of disomorphism so this is like usual versality condition so it is easy to see that if now so the point is like this is that if so I assume left exact because otherwise I want to actually this surjective I want to pass to the fiber product so it's natural too so if A is maximal and omega A is projective then so A I mean AX then it is reversal so the reason is that the map actually is defined up to so the actually the category is given by this complex so the map is defined up to factorization by a projective and since it's projective one can lift so that's not so so in particular for if C has enough projectives then any so any good so let's say good means comes is of the form R0Q of a complex any good additive cofibered category has a reversal object so if you look at the proof one can also weaken it slightly so we can say something is weakly good if the yeah so if the the group of so the EA modular isomorphism is functorally isomorphic to X1 L A as above for some two term complex yes okay for some two term complex now so yeah so now in so this will be and then I still have a reversal object which is what I want so it means that I have to prove a little bit less so it depends on the formulation of the formation theory that I I use sometimes I just use the formulation I just control the set of isomorphism classes so so now so the problem that I want to look at is the following so I have X0 to T0 let us say I have an Italian affine base and this is affine and this is some let us say proper flat morphism and then I want to consider a first order the formation flat deformations over s so let us say this is a finite type also flat deformations and I want to construct a reversal the formation for this in this so this is defined similarly to what so now in every such a situation I have the ideal of T0 and T so actually I will usually well it depends on the case but usually so one can either assume or not assume but let us to be in the previous context let us just work so we have in any case a fiber category over the category of T0 modules was too slow okay so so I have actually I left exact additive cofibered category over OT0 modules possibly only finite type and the problem is the same as constructing a versatile object there and so for this I can use the deformation theory like in illusiz book so the idea is to so here I have I which is kernel OT so the fiber over a module M are also those diagrams over S or the kernel from OT0 as a given module so this so anyway to the diagram X0 to T0 I get a topos actually associated with this diagram and then I have a morphism of topos S to S let us say the risky so this is actually then a deformation problem for this so I will get it's classified so the solution is classified by something like a so I have the the pullback of I to the total space now so there is so to control this so there is a result which is based on growth and duality and which is the following so I have now let us say a proper flat map where Y is with a dualizing complex then I have R home I want to look at the problem like this R home to under some boundaries conditions suppose I get D minus square of X and L D plus D so anyway so it turns out that this so this is not like in the usual duality T or M which is well there are several duality there is a trivial duality but the duality we have upper shriek and some twisted statement where this is becomes R home of some operation on K L okay so this is actually so there are some generalization of this when you instead of flat you can twist by NF perfect bounded complex on X so so this statement is in some paper of Jack Holland I also found it connection with this so actually I need it not for in this form but for diagrams so for this stopos mapping to S but actually I have to construct a more because I want to look at all modules at once in order to identify that it depends on what one wants to do so if one wants to to calculate the isomorphism classes then one can just one needs some X to one one applies this and one gets home of something to the L so now this is proved by writing L as a double dual of something and using various operations in the drive category and one is to be careful about justifying things right boundedness but it is and of course it should hold a more generally without dualizing complex but I'm not sure what are the right hypothesis it's probably you need some approximation results but then one wouldn't have to be very fancy derived and higher context so I'm not but for our purposes usually the rings have a dualizing complex in any case the existence of reversal deformation can be easily reduced to this case so anyway so this gives a weekly good condition on the additive coefficient category slightly weaker but it's enough for reversal deformation this is one way now if I use a more complicated probably if I use all finite type modules kind of the category of all those make a topos associated to larger diagrams like in Iluzi's works I have morphism of such topos then if I do probably this is not yet worked out then if one works out the analog of this for that one will be able to to have some upgraded version where one proves directly that the additive category is given by this which is now so anyway so this is the first step in the construction then I have to to work I have to construct and order deformation but this is relatively easy once you have the first order one so again I want to okay so I am in the same situation as above X0 to T0 over S for simplicity in Italian ring since of my presentation this is proper and flat but I don't assume dualizing complex because this is just for an intermediate step then I can consider hence order deformations which are defined by the condition that the ideal to the power n plus 1 is 0 then I want to define a similar way a versatile hence order deformation and actually I want a compatible system of such versatile deformations which have the property that I have a kind of Xn to Tn such that Xm to Tm is obtained by modding by the kernel of O Tn 0 to the power m plus 1 where m is less than n so the idea is that I already so if I construct Xn to Tn then I already I can look at now again a versatile first order deformation of this now there the ideal is of square 0 but I can make it small in the sense of the ideal the kernel for the map to O T0 kills the ideal so then I have a small kind of first order deformation of this and then when I look at this and divide by the square of the ideal I get something which is possibly fatter than the original first order deformation so I have to contract something using the versality and after one contracts so one can check this has the right properties so now one can put all those things together get formal scheme and then one uses the operation really geometry to get what I said now I want to explain some in the few minutes to sketch very briefly the applications to the problem in about hodge to the ram and related things so so now what is the hodge to the ram let's say in the complex case complex case so X to S proper smooth of c analytic spaces okay but now we need some condition on the fibers I think that people know there is a class I think called fujiki or something anyway the bimeromorphic bimeromorphic to keller so anyway the result on hodge the composition extends to some well it's for keller but slightly so it includes a complete non-projective algebraic varieties so in particular I can have a proper smooth map of algebraic varieties so in any case so the lean proved in 69 that so using the absolute case so it proved that those sheaves are locally free and the hodge to the ram degenerates and the proof was by reducing to artinian local situation and then using the comparison with classical cohomology so here the natural idea is to use something like crystalline cohomology but in the rigid context but so the theory doesn't exist also in characteristic zero but one can one can mimic it using some check construction so the the strategy is like this so if I have now proper smooth let's say rigid case spaces but it will also work in the netarian case so I need to work in the context where I have basic operations on queer and sheaves and flatness notions so these are the classical rigid case only netarian case I and then in the netarian case I have enough what is called rig points or points where I can actually pass the usual rigid and electric spaces so the idea is that I want to prove the local fairness and the generation so and point by point the idea is to use the spreading out technique to reduce to finite extension of qp and then the idea is to follow the lens argument well the philosophy to go to artinian local things so and now so suppose s is artinian local and moreover we can assume that its residue field is k so k is characteristic zero one can but not residue characteristic zero possibly but it could be of residue characteristic zero so now the idea is that I have a constant family over s and the given family with the same special fiber so morally they should have the same crystalline cohomology so I can construct a substitute of crystalline cohomology so of course this is in the rigid case but locally in any case locally I have the affinoid in xs I can embed them in some smooth affinoids over the the base and I can consider the completed the ram complex so so v alpha completed along u alpha and also for fiber for intersection I take fiber product and complete along the intersection then I can consider a check construction from the completed the ram complex so let's say alternating check to make it fine and then I get some candidate for the crystalline cohomology of the special fiber so to speak so one is to prove but this is like a local one is to prove in the penance of the covering and that if you have a smooth lifting it coincides with the cohomology of the smooth lifting because you can take this is the only thing in the covering and then there are isomorphics so in particular the the ram cohomology is the ram cohomology is isomorphic and so they have the same length and for using this one can do the lean argument so and this is the proof so now so this problem was is related again to Piadikov's theory so in Schultz's work he has a statement like this but assuming the spaces are smooth and with some even with some twisted coefficients so here the spaces are not smooth so another question so this is the result is locally freeness and the degeneration yes this is one result and so the the so another statement which one wants to is about the the whole state spectral sequence so so now this is for X proper smooth over C algebraically closed a non-archimmediate field over QP okay and in this case the so what happens is that the etalcomology so this is a bit technical so the etalcom first first of all is five dimensional for Z mod P and for QP and so so so I'd reason etal in the sense of addyx spaces okay so it's not the usual but you have to define it in the okay now so this of course this is related so maybe I am is it minus Q yes so in any case what one what happens is that the periodic vanishing sequence is that it's related to vanishing cycles of so when you tens always so one result is that it's related to comology of the shift O plus mod P which also is the vanishing cycles of this or mod P to the N R give you something which is relative differential up to some torsion so one gets a spectral sequence but one doesn't have enough gallo action to control the generation so it could be that it so one doesn't know that it degenerates up to torsion yeah okay but now I think one was in several ways so I want to explain so one way if one knows it has the same the sum is of the same dimension as the Ram Cormology so one is to relate this to the Ram Cormology and this is not exactly shows this paper with one of the statement that he gave so essentially one spreads out X to a family over QP over a smooth QP original space and one has the Gauss Manning connection on the Ram Cormology the filter things and he construct some bid Ram plus local systems and you can pull it back and I think that one can probably this method gives a comparison result with this so it has the same dimension but I'm not saying that I didn't study this on so on the other end another approach is to specialize at some point so one wants to know that the dimension of this is the same as the dimension of the specialization as well one wants to prove that the Rn F1 star Z mod P is locally constant so for this the idea is to prove some kind of punk radality so in the rigid case so I want to look at this coefficient Z mod P so I want to show that if X is let's say connected then H2N X Z mod P is one dimensional with basis given by the fundamental class of any point and that the punk radality maps are perfect pairing if you accept this looking at the class of the diagonal in X cross X which can be defined one will one will find that this is given by something like the Kunis formula where alpha and beta are dual basis and then one can spread it out to a neighborhood in comparison and find that those must be a basis in a neighborhood so the main point once you know the punk radality you get local constancy and so to prove punk radality the idea is to tensor with O plus over P so we have to prove punk radality almost punk radality in fact so the idea is that the vanishing cycle for this, for suitable formal modules are complexes with almost coherent and one wants to prove almost square in duality for this and the starting point is the case of a nice variety where there are explicit calculations that can show the duality and then so one is to define the trace map so the point is that this so the in any case this is a longer this part is done in fact using the assumption that the model is not okay so but now without knowing that the model is nice the idea is like this so you have the smooth you can always assume that it is the model is such that the special fiber is geometrically reduced so it's generically smooth then you control the vanishing cycle shifts as being differentials up to torsion modelu to this modelu some small torsion they're differentials on the smooth locus you also know so anyway I have I have a model adapted to the perfect tier so these things which are locally composition of finite and rational domain anyways and I have only vanishing cycle in degrees zero up to n and the top one is a model of this small torsion is omega n on the smooth locus but in particular it maps to the double dual of omega n which is the dualizing complex so I get some starting point because this is the zero meaning of the dualizing complex so I get and I also have so I have a pairing from this this to the top shift going to the double dual going to the relative dualizing complex so I need a local theory of F upper shriek for this kind of situation of restricted like finite type and essentially I mean or topologically finite type composition of such things and so this is like the local theory like in archon Conrad's book or the Stacks project then and I need the residual complexes for this in any case so to construct trace maps then after one defines the pairing one is to prove it's a duality so the idea is to prove it in the nice situation then to prove it nice situation model refraction of a finite group and then to use Tamkin's work to say that locally you have many neighborhoods where which are have a finite tile cover so where which is successive a a nodal fibrations and also for the group action and then I have enough of those so I can the idea is that both sides are kind of morally have some kind of comological descent so I can I can so anyway so this is again some kind of almost mathematics it tends to be made more precise but but so anyway so this is one approach that I hope to speak about later okay so we will take two questions so first we will start from the Tokyo then Beijing then Tokyo any questions in Tokyo? okay I will question yeah in the pool of the hard generation first year so where does this cofiber category and co-tension complex and is it in the I have to construct I have to construct a first order deformation you look here now okay okay so I have to construct the first order reversal deformation and of x0 to t0 so I have x0 over t0 so the idea is that I have something over okay divided by some idea and I first use a standard reduction to finite type over z or finite type over the discrete valuation ring but then I have to construct successive reversal deformation so the problem is just the first order one and the problem there is to that the versatility condition that I need is not just in terms of the functor of isomorphism classes but with stronger things so I and I can compute there is the group of isomorphism classes so it's I need slightly finer information that is provided by even when I introduce the extra topos I need slightly finer information so one idea is that if one has this general thing on well it is easy to see it is left exactly cofiber category so if one has some general information on this and this can be used to to well in some case so it depends on the past which is used in some cases maybe one can use some cheap argument once one knows represent build by project and so for example I have this weekly good additive cofiber category which has a reversal object so this can be used to prove the existence of a reversal but it is depended originally but this was just a motivation but it turns out that the knowing the each fibers so the point if you know each fiber is equivalent to the category defined by a two-term complex and you know and then for a morphism you have so anyway suppose you just prove the thing in terms of canonical isomorphism in the derived category this doesn't carry enough higher information so to deal with the situation with three three objects in the base category so it will not give sufficiently rigorous proof of the reversal so then one has to find somewhere around it by passing so the but in any case if one works with a topos coming from diagrams one can probably get around it but one has to be careful if one is less careful about so maybe using this weekly good notion I can do so it depends I mean it has to be fixed in some way but there are several small technical alternatives how to do it thank you so are there questions from to you? okay some no that's it so are there any questions in the team? okay I have a question I am very confused on the last part on the hostage perspective sequence so do you want to say you can prove this degeneracy of hostage perspective sequence by using spreading out or do you prove something new? I don't know I thought about no but it turns out that I don't so I thought so some time ago I thought about this so just without knowing the so it will follow if one proves that the dimension is locally constant then I try to prove the dimension is locally constant by using almost puncture duality but I think that actually one can prove the degeneration without this by using by being looking at what this following the same problem let's say partings are short I think this is so in any case one of the statements some of the statements announced by Schultz was that Segment is known if you work over something like QP or finite extension of QP but we don't know it over C and the result is to go from QP to C so this is the new thing so this is the new thing so in any case so in any case without looking at so in any case so I think that so in any case one way is to accept so to just use as a black box the equality of the dimension for finite extension of QP and then try to prove that this gives a locally constant shift but using does the beginning of the Schultz's approach relating to Oplus without using all the period shifts and then there I want to relate it I want to use ideas on almost square duality to handle that so this is one way to long project okay so are there other questions in BG I think no are there questions in Paris so since we have short time let's thank the speaker again