 Češnje. Protočen, da boš mi došličil organizacji za imetnje. To je prišelje in veliko dobro, da se počekajte na ljubi. Prišel je master in režim. Češnje, da boš mi počekajte však glasba s Nishelgro. Však je tudi vsega spektrala. A da se došličim, da boš mi počekajte vsega spektrala. tako da sem tega počnega. kje vse pas všeč je vsečnja vsečnja kovoljšim vsečenjem karakteristiku 0, z vsečnjem alzibrajočnoh vsečnja vsečnja kovoljšim vsečnjem karakteristiku p. Zamovam z ok, vsečnju kovoljšim kovoljšim. Vsečnja prišelja vsečnja kovoljšim vsečnja kovoljšim kovoljšim vsečnja kovoljšim, Denout by OK bar, integral closure of OK in K bar. GK will be the Galois group, and OC will be the completion of OK bar, and see the fraction field. I will set S to be the spectrum of OK, S bar, the spectrum of OK bar, and I will denote by S the closed point of S, eta, it's generic point, and eta bar is the generic point of S bar. Then for any proper and smooth K-scheme, X, we know that there is a canonical, funktorial GK-equivariant spectral sequence called the hot state spectral sequence. It's any two spectral sequence, e to ij, this is hi, Xc omega j, Xc over c, twisted by minus j, which converges to hi plus j, eta of XK bar, qp, where we extend scalars from qp to c. This spectral sequence degenerates at e to, degenerates at e to, and even splits. This is the consequence of the result of Tate on the Galois homology. Tate proved that hi of GK c of j vanishes for i equals 0 and 1, and G not equal to 0. So hence, the spectral sequence in fact leads to a decomposition, which is called the hot state decomposition, hn of XK bar, qp, answer over qp c is canonically equal to the sum of hi Xc omega n minus hi Xc over c, twisted by i minus n. So this decomposition was proved by Falkings, conjectured by Tate, proved by Falkings and Tsuji independently, and extends by Schultz, to reached analytic varieties, and it was Schultz who formulated the spectral sequence, though it's implicit in the work of Falkings. So the generalization I will discuss is for morphisms, and it's slightly involved. So I will start to motivate it by a local version. Okay, so let f from X to S be smooth morphism, and g from X prime to X is a projective smooth morphism. So in our work in fact we work with toric singularities, but for simplicity I will discuss in this lecture just the smooth case. So first let's assume the base affine, let's also assume the geometric generic fiber connected, and let's choose a geometric point of X eta bar, and I will also assume that the special fiber is not empty. Okay. I will denote by V i, so first the fundamental group, so let me denote by delta, the geometric fundamental group in gamma, and let me denote by V i, the universal cover of X eta bar at Y bar. Okay, so I have my diagram like this X eta bar, it's contained in X bar, which is just the base change of X from S to S bar. I have here a finite etal morphism V i, and I can take its normalization, so all the schemes are affine, so this scheme will also affine, and this is just the normalization of X bar in V i. What do you mean by V i in i being the universal cover? So this is... Pro-represent the fiber functor, but you can normalize it in such a way that it's essentially unique. Okay. Okay, it's a process. Exactly. Yes. Okay, now I can introduce the algebra R bar, which is just the inductive limit of the algebras R i, and on this algebra, the fundamental group acts naturally. And in this context, the natural generalization of the spectral, the Hodge state spectral sequence is the following. So I put it as a question. So does there exist a canonical gamma-equivariant spectral sequence e to ij, so it starts from h i, x prime, omega j, x prime over x, tonsil from R to R bar, and basically, I inverse all of it. This is not Japanese shock. Inverse p, twist by minus j, and this converges to h i plus j et al of x prime, I take the fiber over the geometric point y bar, q p, and tensil over q p with R bar, tensil over g p, tensil over g p with R bar hat. Okay, so this is the question. Does there exist a spectral sequence like this? So what we know by a result of a brino and a hyodo based on the almost purity result of feltings is that h naught of gamma with coefficients in R bar hat where we inverse p twisted by j vanishes for all j different from 0. So if this spectral sequence exists, it will degenerate at e2. But for h1, we know that h1 of gamma with the same coefficients will vanish for j not equal to 0 or 1 in general. It may not vanish at 1, so it's not reasonable to expect that the spectral sequence splits. There should be no natural hot state decomposition. So are you in the case of nice reduction or any? So I ask the question like this. After that, definitely I will take small scheme, but this is just a question in some sense. We cannot answer this question, but we will see that we can answer it after localization at a geometric point at the special fiber. But before, let me mention that Schultz told me that he is able to answer this question definitely in the perfect case using hot state filtration he defined with Karjani for proper smooth morphisms of adic spaces, which I will comment slightly later. And that Bargav Had told me that he also has a strategy to construct this spectral sequence using the formalism of prismatic cohomology. So now let me tell you what we can say about this spectral sequence. So in the result of Ildo and Brignon, is it R is nice? Yes, R is nice. So let me just skip this. Definitely where we are able to compute the Galois cohomology. Ok, so what is our situation? So now I drop the condition that X is affine and nice. So drop the condition that X is affine and instead I will consider X bar, a geometric point of X. I will denote by X underline the strict localization of X at X bar. I will denote by X prime underline the best change. Ok. And I will give myself a specialization from the geometric point of the generic fiber, geometric generic fiber to the point X bar. By this I mean just an X morphism from Y bar to the strict localization of X at X. And then I can define an algebra which is R bar underline like this. So this will be the inductive limit over all affine etal neighborhood of X bar in X of the algebra R bar U. And this algebra is exactly the algebra I defined previously but for the scheme U. And this choice of specialization give me a geometric generic point of the geometric generic fiber of U. On this algebra we have a natural action of phi 1 X underline eta Y bar. And the statement is the following. U is an affine etal affine etal neighborhood etal neighborhood of X bar of X bar at 1 in X. So this is now X is a scheme over K. Yes, X is a scheme over S. S is now X is over S. Everything is integral. X is over S and the point is in the special fiber. In the special fiber, yes. OK, so let me now state the first result. So there exist the canonical gamma underline equivalent spectral sequence e to i j equal h i X prime underline omega j X underline tensor over O X underline R bar hat 1 over p listed by minus j which converges to h i plus j eta i X prime q p tensor over z p R bar underline 200 p OK So, this is in fact the first manifestation of this relative hot state spectral sequence. In fact, this statement could be globalized over natural topics whose points are collections of specialization points from geometric point of X eta bar to a geometric point of X. This stopper is nothing but just faulting stoppers, which is at the heart of the hot state spectral sequence even in the absolute case. So, this is what I will explain, but before I explain it, let me mention that this spectral sequence appeared already many years ago in the work of Yudu who discovered in the case of relative abelian scheme and where the localization of the special fiber. And here R is nice more. No, here there is no need for it to be nice, it's a localization. So, everything is over. When it's underlined, it means it's after geometric localization at this geometric point X bar. So, you can take neighborhood which abmers. OK, now let me introduce this stoppers, faulting stoppers. So, the situation is the following. X now is a smooth S scheme. Then I will consider E. This will be the category of morphisms V to U over X eta bar to X by which I mean just commutative diagram like this of schemes, such that U to X is eta and V to the best change, U eta bar is finite eta. OK. So, there is a nice way to see this category is by looking at it as a fiber category over the eta side of X by mapping V U to U. OK. So, if we take an object and take an etal scheme over X, call it U, then the fiber category over U is nothing but just the finite etal side of U eta bar. And you will equip it with etal topology. So, this is what I speak about side. Equipped with etal topology. So, I equip the fibers with etal topology and I will denote the associated topos with FET. So, this is the associated topos. OK. Now, we can put an actual topology on E, called the covanishing topology which was introduced first by the linear. And it's defined as the following. So, this is the topology generated by coverings of the following types of the types. We have two types. First, the vertical type by which I mean that Ui equal U for all I in I and Vi to V is a covering and we have also the Cartesian coverings by which I mean that Ui to U is a covering and Vi is just the base change. OK. So, this is a reasonable topology on E and we will work in the topos of sheves of sets on E for this topology that I will denote by E tilde. So, denote by E tilde the associated topos. In fact, it's not complicated to describe objects of E tilde. So, to give a sheve E tilde amounts to just give a collection of sheves for every etal scheme over X you should give yourself a sheve I will denote by FU which is a finite etal sheve over U eta bar and this collection of sheves should satisfy a co-cycle condition and a gluing condition. OK. So, it's a sheve of sheves in some sense. The transition map satisfies Yes. Exactly. So, this is the co-cycle. OK. So, moreover we can check that any specialization map from Y bar to X bar where Y bar is a geometric point of X eta bar and X bar is a geometric point of X defines a point that you will denote by rho of Y bar X bar of E tilde and the collection of these points is in fact conservative. OK. So, we are in a nice situation. Moreover this topos is related to the usual etal and finite etal topos in the following way so I will put it here because I need it later. First there are two morphisms etal topos etal topos of X call it sigma and to the finite etal topos of X eta bar call it beta. So, these two maps play a role of projection so this is defined in the following way for objects of the site the effect of the pullback is just take this Cartesian object in some sense. And here if you have a finite etal scheme over X eta bar you associate to it V to X which is well defined in E take the associated chief. Moreover, there is an important morphism from the etals topos of X eta bar to the felting stopos which is the cone nearby cycle morphism and it's defined the following way on objects of the sites so if you take V to U you just map it to V. And the first indication that this topos is relevant for the study of the etal topology of X eta bar is the following proposition which says that the functor psi is in fact locally acyclic for local systems which is due to feltings in the smooth case and it has been extended ashingar in the log smooth case so it says that for any locally constant constructible torsionabilian chief F on of X eta bar etal the R i psi lower star of F vanishes for all i bigger than one. Ok, so we will compute the Pia di Ketal etal topology of X eta bar by computing the Pia di Ketal of these topos and for this we will use Artin Schreyer theory so we need the ring and this ring is defined the following way so when I have an object of E I will denote by U bar V the integral closure of U bar in V and then I can define a pre-shef called V bar by setting that its its values on V U is just the global sections of this of the structural shef on this scheme and it's a fact that B bar is for the covanishing topology so like any shef as I said before it can be written as a collection of shefs and in fact these shefs are we have already seen them before so like any shef I can write it as a collection of shefs for any etal scheme over X I associate to it a shef I will denote it by B bar U which is now a finite shef finite etal shef on U eta bar and in fact if U is affine and if I take Y bar a geometric point of U eta bar then this the stock of this at Y bar is nothing but just the algebra R bar U I defined previously of the geometric fundamental group this one so it's really this representation and hence from this we easily deduce that the stock of this shef of the specialization of this point associated to the specialization is nothing but just the inductive limit of this algebras R bar U over all affine etal neighborhoods of X bar in X so I guess from this you can already guess how I will globalize the spectral sequence over the ring topos E tilde B bar so the spectral sequence I described previously will be the stock of the spectral sequence over this ring topos but since I need piedic topology which previously it appears to be piedic completions I have still just to introduce an extra notation I have to work with projective systems so let me give myself G amorphism from X prime to X I will say later what are the conditions but here just a notation I will denote by B bar N B bar modulo P to the N B bar algebra so the projective system of rings B bar N and I see this as a ring as a nuci algebra of the topos which is made of projective systems of objects of E so this is just the projective systems of objects of E tilde indexed the order set N in fact it's a topos ok, I define the same way ZP brevi so this will be the projective system of ZP Z mod PN and I see this in the etal topos of X prime eta bar and the last notation I need I can describe it here X prime eta bar which goes to X eta bar etal so this is the morphism G I gave myself so G eta bar and then from here I can go to faltings topos E tilde and I can extend all these to these projective systems topos of projective systems and I will put a brevi to denote these maps ok, so now I am ready to state the main theorem so the main theorem will take place in the category of B bar brevi modules up to isogenic so let me just give a name for this category so this will be the category of B bar brevi modules up to isogenic by which I mean the category with objects just B bar module B bar brevi modules and morphisms I tensor with Q ok, so this is in fact an analog of the category of R bar hat 1 over P representations of delta ok, so here now the statement so for any so let G from X prime to X smooth projective morphism so X over the discrete valuation ring it's a smooth it's always smooth from here ok X prime over X is a smooth and projective and X is a smooth in fact in our paper X is log smooth but here it's just smooth I wondered about the general question yes you did not the question about the spectral sequence yeah, the affine case you need X to be etal over gm in this context or if it has a nice chart like feltings the question which I did not answer but I would like to concentrate on this ok, so we can discuss the other question after so I assume that G is a smooth projective morphism then we have a canonical spectral sequence of b bar brevi q modules it starts like this it's a need to spectral sequence so first of all I take the hotch cohomology and then I pull it back over the feltings stopper so I have it here so it's something over X take the cohomology and pull it back to feltings stoppers I extend scalars over the pullback of OX to the ring b bar brevi q twisted by minus g and this converges for what converges to ri plus g g eta bar g eta bar brevi of zp brevi so this is the relative hotch state spectral sequence in fact these objects the sheaves that appear here are naturally equipped with gk equivalence structure in fact these objects the sheaves that appear here are naturally equipped with gk equivalence structure and we can see that it's not difficult to see that it's gk equivalent and from this we deduce easily that in fact it degenerates at it the tens of product is over what zp here here it's the pullback by sigma of OX so this is an OX module and you just pullback by OX and the first projection ok ok so it's like you did here in fact if you take the stroke of this spectral sequence over specialization map you will get the spectral sequence I described I stated before which means the result over the specialization over the localization so this is the first remark the second remark is that we assume here that it's projective and I will show slightly later where we use it it should be possible to replace it by proper and the last remark is that in fact it's here by a different methods Scholzso and Kariani have constructed hot state filtration for the periodic cohomology after extending to some period drain for a proper smooth morphism for the periodic spaces and this is in some sense the two analog results and this is the result I mentioned previously ok so in the remaining time I will sketch the proof when you think the vanishing result before you say that the differential is zero so this is slightly it's the same spirit but here it's just GK ok I don't use the previous thing so here I have just to compute the Galois cohomology the GK cohomology I don't need the gamma bar so this is even easier than previously ok so now to sketch the proof I will start by recalling the big lines of the construction of the absolute hot state spectral sequence and then I will explain how we can generalize them so let's first assume that let x be a proper smooth scheme ok so first result of feltings extended by Ashinger we know that h i of x beta bar etal z mod p and z is in fact isomorphic to h i of p tilde psi lower star of z mod p and z ok then the second point is that in fact it's not difficult to see that the natural map from z mod p and z to psi lower star z mod p and z is in fact anizomorphism hence enters now feltings main comparison theorem so feltings main comparison theorem will tell us that the natural map from x beta bar etal z mod p and z where I extend scalars over z p to oc go to h i p tilde b bar n, which means b bar mod p to the n this map is in fact now anilmost isomorphism and what Schultz extended exactly so this is so this was first proved by feltings in the context of scheme and Schultz extended in his context to his what he called primitive comparison morphism so which means that the kernel and the kernel are almost isomorphisms are almost zero ok so from this in fact this is the target of our hot state spectral sequence this almost isomorphic to this to we need to compute this cohomology group and we will compute it using a capon Lorem spectral sequence for sigma, which is the projection from e tilde to x etal namely we have a spectral sequence and e to spectral sequence h i x etal lj sigma lower star d bar n converges to h i plus j of e tilde d bar n ok, so to go further we need to express this term in terms of differentials and this is we can easily do by globalizing the local computation of feltings of Galois cohomology so we prove in this context there exist a canonical homomorphism of o x bar n, which means extension of scalars to s bar modulo p to the n algebras of the etaltopas of the special fiber so this is the algebra of differential forms but twisted by a la frontend so this is frontend twist which is up to a small power of p, just state twist so this is up to p to the 1 over p minus 1 is just state twist so we have a natural we have a canonical homomorphism of algebras into the direct sum of d bar n so into this cohomology algebra and the kernel and the co-kernel whose kernel and co-kernel so here they will not be killed by just the maximal ideal of o c but killed by a precise power of p so it's 2d plus 1 divided by p minus 1 times the maximal ideal of c and g of d is the dimension of x over x so anyway if we go to the projective limit and we invert p so all this will disappear and we find our statement which means that this carton lower spectral sequence will became the hot state spectral sequence so these three steps in fact extend in the relative situation Falking's main comparison theorem has been stated for morphisms by Falking's in his asterisk paper in 2002 and he sketched a proof in the appendix and in fact we gave a complete proof following what he suggested so this is the first step the last step is easy we will see it later but it's easy in fact the main difficulty is the second step namely the carton lower spectral sequence and the fact that it's difficult is that because there is a missing object in the picture which is in some sense the main novelty of our work so let me explain it ok so let me now consider x prime to x so this is a smooth morphism of smooth S scheme so we can associate to x prime also a falking stoppers and I will denote it with a prime and the construction is functorial so falking stoppers of x prime is naturally equipped with a morphism to falking stoppers of our x I will denote it by theta and moreover the projections are also functorial so I have here a projection to x prime etal I have here a projection to x etal sigma this is g and also the conier bicycle morphism is functorial so I have commidative diagrams like this and also the algebra I introduced is in some sense functorial so I have a natural homomorphism from b bar to theta lower star of b bar prime which is the analog algebra over e prime so in this context here is the generalization of falking's main comparison theorem so it says the following so as I said formulated by falking we prove it following his sketch so it says the following assume x prime to x is projective and smooth and let f prime the locally constant constructible sheaf of z mod p and z modules of x prime etal then for any integer i if I take the homology i g eta bar lower star of f prime then I push it by the conier bicycle functor to falking stoppers and I extend scalars over zp to b bar and I go to here I first push by the conier bicycle functor f prime extend scalars over zp to ring b bar prime and then take the higher direct image by theta so this morphism is in fact an almost isomorphism ok, so observe here that this sheaf is in fact locally constantly constructible by the proper and smooth base change so we are in the range where psi is locally acidic the other remark concerns this condition of projectivity and this is the origin that it appears so here in fact we use it to prove some almost finiteness result for almost coherent modules and we use for this SGA6 instead of tihel so it should be possible to replace projective by proper but we have not done it we use that SGA6 exactly and as I said it should be possible but it's already too technical ok, so let me go ahead so as we know previously from what I said previously z, the direct image by psi prime of z mod p and z is isomorphic to z mod p and z so this is means that here what appears here is that in argument of the spectral sequence we would like to construct so it means that we are reduced to computing this side which is in this case just the cohomology of b prime n so we need to compute this cohomology by spectral sequence ok, so let's look a little bit to the situation so we have e prime it maps to x prime etal here we have x etal here we have felting stopos and here our theta so to compute it by carton-laure spectral sequence clearly we need the factorization of this theta and the first thing which come to mind is to take just the fiber product of this stopos which exist so I will call this map 2 and this is gamma and to write the carton-laure spectral sequence for this composition in fact it turns out that this stopos has a very nice underlying site which is a generalization of felting site which we were able to define by inspiration from oriented products of the linear beyond the co-vanishing stopos which serves already to the definition of felting stopos so this is in fact the missing object that I will now describe and then I will explain how to perform the three steps so let me denote by now g the category of pairs of morphisms w to u v to u and this is over we have x prime to x and here we have x eta bar to x so I take commutative diagrams like this such that this is this etal this is etal and here I ask that v to u eta bar is finite etal as before and I will equip it with topology which is the following so I take plus the topology generated by coverings so ui of three types now of three types so the first type is just wi equal w and ui equal u for all i but vi to v is a covering the second type is ui equal u vi equal v for every i in i but wi to w is a covering and the most important type is the third one so this is the type c where the i is just one element so I can denote it with a prime instead of i so I have an object like this over w uv and in fact what I ask is that these two are equal and I have here a morphism and this diagram is Cartesian but this map could be any map so this is exactly copying the definition of the covanishing of the oriented product topology so I obtain a nice site and I will denote the associated topos with g tilde so if x prime is equal to x we immediately see that in fact this topos is conveniently equivalent to felting stopos so from this we see by functoriality of this construction that we have a factorization which I will draw here so I have e tilde prime it goes to this felting stopos goes to e tilde moreover the construction of the projections are factorial as I explained so let me put here the second projection fet beta prime beta beta and here I have x prime eta x eta and I have the first projection sigma prime and sigma and in fact we can extend this diagram by two morphisms of projections this one I will call pi and they are defined in a very natural way so if I have a w which is etal over x prime I associate to it w to x x eta bar by pi upper star and here if I have v which is finite etal I associate to it just x prime to x by lambda upper star ok so I call this map tau this is gamma so I can now ok and what else and in fact we prove and this is a fact that this diagram is Cartesian which means that this stopos is the fiber product ok so now we can go to our problem of spectral sequences so using Falking's main comparison theorem I know that psi direct image of l i eta bar of z mod p and z where I extend scalar from z p to b bar is in fact almost isomorphic to the direct image of b prime n so this is Falking's main comparison so as I said our target is to compute this one by spectral sequence and we use here the carton the ray spectral sequence for the composition theta is equal to gamma composite with tau so it looks like this is equal to r i gamma lower star per j lower star d bar prime n versus r i plus j theta lower star of b prime n so to find this spectral sequence we have to express this term in terms of differential and this is as I said we have we can globalize Falking's computation of local so first of all we can extend Falking's computation of Galois-Cohomolo g to a relative situation and then we can globalize it so to formulate the statement I will use pi but I need to enrich it a little bit so I will equip g tilde with a ring which is nothing but just the direct image of the ring b prime by tau and I map it to x prime eta and I equip it with the ring of x prime but x prime I have to push it down x bar prime so I have to push it down from x bar prime to x prime and I call this map h bar prime ok so I use this morphism of ring topos so the next theorem is that for any n bigger than 1 there is a canonical homomorphism of b bar upper schrik algebras over this Falking's topos from the exterior algebra of the pullback of the twisted shef of differentials x bar prime over n modulo x bar prime x bar n and this goes to the direct image of b prime n ok so this is and this has a kernel whose kernel and co kernel are killed by p to the 1 p to the 2 r plus 1 divided by p minus 1 times the maximal ideal of c and here the relative dimension ok so this is nice we find the relative differential but we are not completely done so what we find is we have the relative differential here and we pull them back here so we have to take direct image but we prefer to first take direct image and then pullback and this is what we wrote so we have we need base change and this is the last part of the lectures so very quickly I will state just the result so how we will prove this base change results in fact first we prove base change result for torsion coefficients this is just generalization of result due to offer in the oriented product case and it's just reduces just to the proper base change theorem for g so this is the first step and after that we need to use almost purity to prove proper base change for coherent coefficient so let me finish by stating this result assume that g is proper and always smooth then there exist an integer n bigger than 0 such that for any n bigger than 1 and any q bigger than 0 and any coherent o x prime n module I will call it f which is flat over xn xn flat ok kernel kernel of the natural base change morphism sigma upper star of rq g lower star of f going to rq gamma lower star of pi upper star of f so the kernel and kernel and p to the n and this pullback here are pullback for the natural morphism of ring top point and that's all so I think I should stop here so when you have the usual coverings which are flat and then the address is kind of almost the talen so you can control yes you can control co-home model so once you have the base change for the coefficients ital coefficients you can for instance one thing is that you have two rings here b and o x prime and if you take the exterior transfer product here it's almost isomorphic to the ring to lower star so you have o x prime here you have b bar here and I put a ring here which is just the direct image of b prime ok so this is upper shrink so this ring is almost isomorphic to the transfer product of these two this is already but you take the pullback and take the transfer product over o x ok and this is already uses already almost purity in your original question on the existence of the octet spectral sequence over in a fine situation so in which level of generality is it expected to work ok there are better results like I mean there is so before like in the time of haltings it was possible to calculate things but things with certain torric singularity but using the perfect theory and other things now with prismatic I mean you don't need all of it but still so what is the current so in some sense if you remain the spirit of what I said today what I expect is that if you work with the small skin for example in the smooth case in the sense that it's etal over gms or in the locked situation which means that you have a sharp a nice sharp as you know so in that case I expect that spectral sequence exists but in the context you are right we should expect this in the case where we have a perfect reader so it is true in general and not difficult to prove I don't know if it was the reference but faltings have been remarked using periodic semester that when you normalize in the maximal etal extension of the general fiber you get something perfect not called perfect this time but you can just write down sure but this is in the context of smallness I don't know without smallness faltings remarked that using certain equations I don't know where it is in the literature but what you do is suppose you want to take the piece root of some things and you write the equation x to the piece square minus this is to prove I see what you mean to get piece roots of functions of special fibers let me say at least when things I know this so in modern language faltings are perfect to it now of course you don't control but in any case the fact that faltings calculated what is now called this opla so I think we are getting a new proof of the statement I was saying by govern now so I don't know what you want to formulate in some sense you want to extend the expected situation for the local statement I gave in any case short support in the general rigid analytic so short support give a proof of local consistency of the entire local system so at least but I don't know if we for this you need this more refined calculation at least the way you do it you need the local calculation so you are limited to to the faltings okay I don't know thanks fully disappeared he is here look at the question so what is there you said that Delin defined the co vanishing what was was it in the oriented problem to redate this more generally for this no faltings defined his stopper in the beginning as you know and this but this one did not appear so this is in the new thing in our paper but when you mentioned Delin for the co vanishing what do you refer to so Delin in fact suggested from the beginning when Likro to Burbecki in Piedekar state decomposition he suggested to Lik that the oriented product should appear and it remained a little bit unused till we started to work with him no but this seemed with the horizontal and vertical that you call the co vanishing was it in Delin no like this it's a generalization of him but it's not far so it's in my work with Misha but it was in his letter to me I don't know if he used really the finite I think he not exactly so this is just the yeah so look you can so I don't know maybe we don't have much time but I think this intermediate factorization reminds me of relation between generalized psi when you have a morphism so then you get some oriented map which compares the two and so in this situations of an analogue for the co vanishing two of us for the first remark and the second is that I would like perhaps in a few words perhaps it's too complicated to explain how in theorem above on the middle board the map from pi per star of the psi minus 1 omega 1 to R1 total star how is it defined so some kind comparison of Galois with differential so how does it come so as you said it's complicated to explain in a few words but roughly speaking as you can expect there is an extension which lead this morphism will be the boundary of a long exact co homology sequence of that extension and that extension is in fact feltings extension so this is when you have a feltings extension but you have to globalize it in certain sense feltings extension exist you can define it like this and you can in general so our trick to define this without using feltings extension is to use kumar theory over these felting stoppers and to construct a map and to prove that locally it coincide with the other one then we see that it's caramel and co-camel armast it's the same as 1 it's like omega 1 ok, so if there is no further questions we will have a break and we will resume in 8 minutes ok, at 10 10.15