 Daj da me je zelo početno, da imam zelo početno, in da se početno sprednev, da so prejmalo Prof. Oguz. Tako povsem, da početno, da so prijemno tega kredite. Tako me početno sprem, da bo vse posledo v tukaj, As it is published in Astirisk and my work with Orgogozo on the p-dimensional field. And it is also related to question studies by Kato in the early 80s. So the first part of my talk will be application to commutative algebra of my results in et al ko homologis to teore of multiplicities. And so this uses, doesn't use the mix, je košnjenje krufijskel in vseh, ki je invertible, in nekaj je vseh, ki načinjete, in tudi nekaj zelo, da vseh znam, o kaj ga bo vseh hrveni do prečiha, da je pričočenja krufijskel, vseh nekaj vseh je nekaj način, in tega je početnja. V zelo, da nekaj nekaj, da imam tvoje 3 opročje vstavo, pa početnja vseh, ali bi ta srečenja, začo, kar je da se vse zelo izgleda, ki ne bilo izgleda, je pačno ta litera je zelo razpročila kompatibilitaj, zelo zelo počukovosti in taj všeč, da je zelo... In tako, da se prišli, ki sem predpravila za kredilje, so je zelo naštričko, da potrebno se počuva, tako... Našto, tako, primer tovore je o vsej multiplice, za rink, za lokalno-neterian ringo-momorofizem. Tako, pričo sem pripravljala, da je dimenzija D in da je zelo dimenzija Fiber, in da je zelo normalno in zelo vzelo. Zato nekaj imam vzelo, da je zelo vzelo vzelo. System. So in any case what I want to construct something which is equal to the degree of this extension divided by the residue degree, in the case of finite extension in thexi of buckOnly. Zelo, da sem malo spesivno, there is the classical notion of a multiplicity for an m primary ideal in the small ring. So I can look at the multiplicity of the extension divided by the multiplicity of q. In imam, da je to izgledaj, da je primari delq in je vzela v z1 over p, kaj p je karakteristik izponent. Vzela se, da je to izgleda, da je to izgleda s Susaninem Vojvockim red bojk, kako se načinili na nošnji sejkels, parametrizati v skim. Zelo se sejkels so fajnej taj, in sejkels je vsega geometrično lejonej branče. Kako spesualizuješ sejkel, nekaj vsejkel vsejkel, v zelo površenju vsejkel, vsejkel sejkel vsejkel vsejkel vsejkel vsejkel vsejkel vsejkel. tako, da dobro so tako početno tudi našli vsečo, tako, ki se nekaj roč, kot, ki je residu, tako, da početno, rečno so imeljimo, tako, ki residu, je odličen, tako, ki se nekaj roč, več, da se da se našli, tudi, da se odlično nekaj formali smuč. Vseh sebej pa je vsega inšla vsega? Vsega je to nekaj vsega vsega vsega inšla vsega vsega inšla vsega inšla vsega. Vsega je vsega vsega vsega inšla vsega inšla vsega vsega. Tako da vsega je vsega inšla vsega vsega inšla vsega, da se vsega je za Summer, da bi bilo vsega. V počke. Yes. In if q is generated by regular sequence, then it is the length of the ring mod q. OK. So now, in the case, I assume the ring to be normal to, otherwise there will be maybe several branches, it will complicate the formulas that and also I use results on a telecology which are stated in this case. files from the method, so in the cases of succinvavirty, they don't need an excellent, but things are fine type, you can even remove the Fellow habit почему je zemi zore guarantee for what they do, if you're careful enough. So in my cases anyway I have this formally smooth map, so by general results old fibers are geometrically regular, this is also normal, and I have a map that if it the same resident you feel, se drža, nekaj je finajt, je to nekaj degr, ki je vznik, in to je degr, je, tako, vznik vznik vznik v zelo. Zato način, da je resodiofil enemz, ne, vznik v zelo v zelo, jaz jaz jaz zelo, jeo je zelo, da jaz izvullo vznik vznik vznik, na residiju, v kompletnih terrenu, v lokalnih. Zato je tukaj materijal, da je unika. Ok, sem tukaj, da... Ok. Ok. Zelo, da K-prime ne je algebraik. Zelo... Tak. Zelo, da je K-prime finačen vsapravljen, in tukaj zelo, da je tukaj... tega vseče, da se razvijat, začel tega, zazve, da se tukaj dobilo, da se to bila, prvo jaz, da je pridivno, ki se res naredilo, bo se začelo, že je to je vseče. Zato, je to, in lega, razvijati ki se je zelo, zato ni počkala, da je vseče. Vseči ne lega, da je čas, da je način začin, ker so vseče, zelo, gdje je srednje. Zelo, da je tudi tudi tudi tudi, da je bila vseče, da je nekaj, tudi, ki jih se vse zbiji, na vse, kaj jih se vse zbiji vse zbiji, vse zbiji, nekaj spraveni, nekaj, nekaj, nekaj, nekaj. Eno, jih se je veliko zbišljeni, ta parametra je vse semetrična, nekaj, najdoš nekaj, doma, vse zbiji vse zbiji in boš vse zbiji diagonale. Vznikaj, da ztešljasko lahko. S kernerem diagonale pasvaj, da ztešljasko lahko. T eliminovaj nekaj pozostan predator, nekaj je vse dovolj vse, katakaj se počkaj če vse nekaj ne poten назад, je finajta, če so, če zelo, potrebno da bi se priletil v taziru in taziru in taziru, sa kompozitom, ko je finajta priletil, da sem se priletil in vem na mnega, in v particu del vsega izgleda, da je vzal tazir podok, je zrpravljena do pilove, da je vsega izgleda. Tako, čekaj, da je to početno včetno, moja najbolj oproštja je, da je moja rezulta in etalkoomologija, a potem sem videl, da je včetno početno z elementarijne metode in intersektionalne teori, da se početne. Tako, nekaj, da se početno zelo zelo... seems like a bit real. I here is a fact that we can see the complex and the truth in actually a few of the questions I have for you. So, now is it, did I hide something? Alright, so, now. So, I recall, so I assume in the, exp میں 17 of the book on my work. So there is the following result. x je nekočno vzniklja, nekočno vzniklja delarša. Preč njič, da obježim, kaj je zelo komologije. So, zelo komologije, da bi se nekočne zelo, tako komologično dimensičnega vsega je d in komologično dimensičnega vsega je x minus vsega vsega je 2d minus 1, kaj l je invertibel, tko ne je karakteristik vsega vsega. In potem, da je vsega kanonika vsega vsega, vsega komologično dimensičnega je, da se nekaj dobro, lambda d, kaj lambda je vsega vsega zmodulost vsega vsega vsega vsega vsega. Tko je vsega vsega vsega vsega. Vsega vsega vsega je zelo vsega vsega vsega. Zato sem vsega vsega vsega vsega vsega vsega. Vsega, da sem vsega vsega vsega vsega vsega vsega, zato sem vsega vsega. Zato sem vsega vsega, tko ne je karakteristik vsega, mater Jane, celosn Jezken, tko jasnja jezkenjefar, vsega bomo boj sigarnutaka,h delca vsega. Štačnobold noz, čekmačnev vsega. a danes je doznelo v lokal komoloče. Tako po vših vseh se srečen, which are studied carefully, I hope, then this will. Zrečen, so naredil je to, that appears here, and also you have to divide, you have to normalize it and divide by the residue field extension. And you normalize this branch to make it a DVR, you have some inseparable extension here and you have to divide by this degree. A potem, toga isomorfizim je kompatible. Včeo, je uniklj vzvečen za to. OK, nao. Zato, me ideje je, da, when you study my map of local rings with zero dimensional special fiber, both of them are degree D, you can assume they are strictly in zeljan, and excellent when you complete. The higher one I didn't assume, but you can easily reduce to this case, in je nekaj nekaj nekaj. Tako je. In potem je bilo, da je in dušnja mapa v lokalj kormologi. h2d x o spek o, h2d x o spek o prime, vzgledaj z zld. Vzgledaj z zld. Ok. To je, da sem zelo, da sem zelo, da je to nekaj na vsezatih, ali sem zelo, da sem daj, bo vsezatih je vsezatih vsezatih. In tudi, da je to nekaj na vsezatih, če in res je infinitivno, ta je in tegral in idejne teori, vsezatji, z vsezatih, z vsezatih parametričnih vsezatih. In v ovom različenju, možete kromologičnih klasa kratičnih kubandrov in minus kubandrov. Zato je vse FI zelo v H1, kaj FI v H1 zelo v opendu, kaj FI je zelo vzelo, v zelo vzelo v zelo vzelo. Zelo sem da sem dajte to, kaj je klas, kaj je klas, kaj je h2, kaj je kratko, kaj je kratko, kaj je kratko vzelo vzelo vzelo. Ok, zato. Zato sem dajte to, in sem dajte, da je to je nekaj, do vsej vzelo, do vsej, modul, do vsej, vs. do vsej kanonikov. To sem zelo. Ok, zato pa je to, If we move this, you have to look to do induction on D or look at the definition. If you know it, the point is that F2 and upto FD, okay, so now I follow, so the point is that you have a curve defined by F2 up to FD equal to 0 and at the all maximal point, generic point of irreducible components I know by induction that I get the right kako se vseh klases nekaj nekaj nekaj kronikov. Vseh je to nekaj kronikov. In tukaj da sem prejeljna vseh, da je je vseh vseh, da se je vseh, da je tega prišla, da je to nekaj nekaj nekaj, in potem je vseh zelo vseh. Zelo vseh, da je tukaj, tukaj vseh, da je tukaj, tako, čistim, da sem vse radočnega ljude in nesmah. To je tudi počasnje kromologijne, nekaj vse da srečemo vsega vsega. Ne je vsega vsega vsega na zelo. Zdaj je inštačno. Taj za vsega vsega vsega, vsem zelo da vsega vsega vsega vsega vsega z tudi vsega vsega vsega. Mi je modem drugi lokel, ma tudi možem prišle celu, to je kar naštri, tekur može je razskačiti in povržitati, tako kaj mi je, In čas sem zelo zelo zelo vso. Vse če sem pribejo, da je tudi, da sem... Zelo ne. Zelo da je zelo vそれで, in zelo, da je način način spektra zelo sodativ. da bo izgleda in je to dočen, da je vse genericoj nekaj tega, da je in taj, da bo izgleda poživljenje, in je začel poživljenje v katenarijah, da tega je in gleda, da je in druga, da je in z katenarij. Z sedim kljubovim kljubom izgleda, da je z njih kljubov, izgledクlja se obožaj mano niti najbolj z gruče. In neč tega reprejba nekih nekaj nekaj. In if there is one that misses the image then the class will be zero here. So, it must be subjective. OK, but now the idea is that you... if you have a regular leg and carving the regular locus and after some blowing up actually, you get some, so you take blow up of some primary idel as you get some normal birational model, even isomorphism in the punctual spectrum, with some exceptional divisor, which is in the regular law, but it's also regular, so in this curve cuts it transversally. And essentially, you are looking at something associated, that's the generic point of this component of the exceptional divisor. So the strategy for the geometric proof would be to work, to first define this invariant for generic points of the irreducible components of the special fiber, which is not so difficult, but still I don't know that everything is hit by the image. I got that, then I want to prove that they are equal. So I have some blowax, which is, I assume, normal, and I consider the corresponding normalized blowup of x prime. And then on the special fiber, I have a map, so actually I have residue field. So I have a projective k prime scheme mapping to a projective k scheme, and this map is, well, after extending this scheme to k prime, it becomes a finite morphism. So you can use usual degrees. Anyway, you can define between varitors, generic points. Then I want to prove it's equal. What the formula that I get is equal for. So the idea is that any two irreducible components of the special fiber are connected through co-dimension two points. Then I can use resolution of singularizes for surfaces as I reduce to the normal crossing situation. And then for dimension, in this case, it is easy to, again, to compare the invariant in the invariant there. Then this will give me the. So now the formula that I have is also the following, that if you look at this multiplicity, there is the following formula that I can, if I fix p prime ideal of O, so I can look at the sum of q over p, O prime over q over p, and e, so I can do the corresponding invariant for localizations. This is still normal. This is not normal, but in the strictly Inzelian case, it is geometrically only broadening. I can still define this invariant. So to prove this formula, you can reduce to curves, and then you use the telecomology approach. But then I can try to, so, the idea is to use this to use p in the curve touching the regular, whose generic point is the regular local, so define what this should be. Prove it is independent of this curve. Then after I do this, I can prove that it gives the ratio of multiplicities, because I can arrange that the generic point of F2FD are in the regular locals. Then I can prove this for, again, for any, for a surface, for a q of dimension 2, which lies in the regular locals using specialization to a curve. Then I, for a curve not in the regular locals, I can embed it in a surface and do this. And also, so, I can use the formulas in multiplicities interpreted in another way to handle curves of generic points in the singular locals. So, OK, so, I didn't, so, this was maybe, in any case, I didn't have detail notes, so I don't want to, OK. So, now, OK, so, in the mixed characteristic case, so, let me recall the following. So, I want to, so, there is a work of Cato on the Galois homology for a complete discrete valuation field. In particular, it computes the homologic, the p-comological dimension. So, it's natural to introduce a notion of p dimension of a field of characteristic p, which, so, you're assuming that it's fine, it's a finite p-base. So, OK. So, I have the p rank of k, which is the number of element in the p-base plus r plus 1. And for this, you use the Cartier. So, for any finite extension, if r is the p rank, OK. So, omega r is the same as the cycle. So, there is a classical Cartier operator whose kernel is the closed form. And then there is this 1 minus the Cartier whose kernel is the logarit, the chief of, et al, chief of logarithmic differentials. So, the, so, the condition is that if you look at this, which, in some sense, is like hr plus 1 morally, in some general, suitable generalization of the FPPF, so, for r equal 1, it's like FPPF-comology, some kind of. Can you read what you've just written? OK. So, you take something, which morally should be hr plus 1 k primus coefficient mu p tensor r. And this is given by, actually, this is not defined in general, but kato define it to be this, anyway. Recall some old things, which probably, OK. So, OK, k primus, what? So, suppose you have a finite extension of k, and, OK. So, you consider this invariant, which generalizes things, also related, so, classical things like the Brouwer group, the p part of the Brouwer group, OK, and, OK. So, the idea is that you look at this notion of comological dimension, where if all of those are zero, you declare, yes, if all of them are zero, you declare the, to be the prank, and, otherwise, you add one to take into account that this can appear, OK. And, what happens if the rank is in there? No, I don't consider it all this case, because all the work that I, it's based on the previous thing, where the prank is always finite. It's not, OK, OK. So, and then, for a Hensiljan discrete valuation field of mixed characteristic zero p, the p-comological dimension of k is the p-comological dimension of the residue field plus one. So, moreover, one can study the cohomology of k with coefficients in the basic, especially this kind of cohomology, which turns out to be the Milner-K group, model of p to the n by canonical map. So, this was, and so, isomorphic to, so this, first shown by kato, I mean kato up to questions on the characteristic p field, which will solve a block kato also by me, and then, they also, they studied, so one study is a filtration of this and associated, so the, so this is a canonical filtration coming from filtration of Milner-K groups, and I can describe the associated graded in good cases, so in particular in equal one. So, in particular, there is a quotient of this coming from the fact that the Milner-K group maps to a Milner-K group of the residue field, direct sum, so at least when you choose a uniformizer, you can do this, this is like the same symbol, okay, and this is the same symbol of the element times the uniformizer, next specialization map, and so, the, okay, so if the p rank of k is r, then actually h r plus 2 of k is coefficients in mu p to the n tensor turns out to be the h1, of, so via this, so no, actually you do this for the sep, so the, the galoacromology of the k, r, m, k modulo p to the n, which let's say is the new, new and r in Milner notation, so this is the co-carnel of, so this is also w omega r log, okay, for the deran-witt complex, which can be defined, the deran-witt complex can be defined for rings, which locally have a finite p basis, and then you get, so this again, so, so this gives the top-carnelogy in the case, where the p dimension is this plus 1, and in the case where it is k, the top-carnelogy is r plus 1, and it admits a map to, no, excuse me, excuse me, excuse me, this should be r plus 1, because it will use h r plus 1, ja, so, ja, so by this, this will map to omega r omega n log of k by a surjective map, okay, now I want to look at the, what could be, so, to prove the generalization of just, not of the precise structure, but just of the comological dimension, which, but then the description at least of this for, and then also some version of what I can say about this in the higher dimensional, local, excellent mixed characteristic local ring case. V n omega r, ja, w n omega r, ja, w n omega r log, okay, so, so in my work we saw goggles or so anyway, so using the technique that I use in the telecomology also, so I prove, so, let us now have o to be an excellent normal, of mixed characteristic zero p, and residue field k, with finite p rank, then the comological dimension of o, and also the comological dimension of o one over p of any open affine, is equal to dim pk plus dim o of the fraction field, excuse me. Now, notice however in this case you can consider not just o one over p, but there is also a characteristic p locus, now for the characteristic p locus, a telecomology is coefficient z mod p is quite different, in coefficient, I mean mod p coefficients is quite different because by, for affine zing, by art in Schreier theory you have only h zero and h one, so you get comological dimension statement quite easily, and so it doesn't matter, so when you want to consider the punctured spectrum, it doesn't, you can assume that the shift is zero on the characteristic p locus, on the other hand, the affine opens which contain pieces of the p locus can have slightly higher by one comological dimension, so, to compute the, now to describe now the comological dimension now for p torsion coefficient of the punctured spectrum, so there are several possible arguments, one argument in the case that I, so for the previous thing is using covering by several affine, but this doesn't work for the punctured spectrum, on the other hand you can deduce it from comological dimension of field of fractions in this, and also you can use in zalizations along affine opens in p equals zero, and the general fiber will be, so again by comological dimension result characteristic zero, there is an estimate for this, this can be also used to estimate the comological dimension, so in any case, I prove that the comological dimension of this is 2d plus one, well, it is less than or equal to, so the idea is that you look at the comology class and so it's zero the generic point, then you can try to look at the maximal open where it is zero, take the maximal point of the complement and then use the information there on the punctured spectrum and you can continue until you get you, so you prove that the, when you have a class in i degree it is zero and when it is in this degree it comes again from curves, contributions of curves, but the curves can be either of mixed characteristic or pure characteristic, okay. So, now on the other end you can have, now you can get a lower bound using a finite projection because you can take the completion by co-instructure theorem, this is finite over some standard complete local ring, where i is a coin ring, and then you have a trace map on etalc homology of the punctured spectrum, let us say it is constant with a constant coefficient which is surjective using comological dimension result and then you have to, so also there is a comparison to completion result, allow you to compare this comology with the one for the completion, okay, then for the punctured spectrum of this you can de-complete and so you have the closed point of sum, let us say p d minus one over i and then you can define a trace map from, so if I have some comology on this p as a closed point, so I have comology class on the risk coefficient in something on the punctured spectrum, so this is local comology on p, so this goes to comology of p, but if I take a shift which is zero on the special fibers, then I am in a situation where I can use a trace map for etalc homology, proper position in a trace map and actually I can compute it, so I can get for a suitable choice that come from a curve actually, so I get a map to the comology supported in, so this goes to the comology supported in the closed fibers and this goes to the comology of the finally of spec i supported at the closed point, which is essentially the comology of the fraction field, so in this case you get a map to the comology of the field of fraction of i and so you can detect that some classes are non-zero. Okay, so now, and so you can get equality here. Now, so now if I consider, so, anyway, so I have the following proposition that h to d minus one plus d m p k of spec o minus the closed point is with coefficients in some shift, p torsion etal shift is d minus one and the shift is generated by local comology with support in closed points of the punctured spectrum or any Zariski open, or any non-empty open in the punctured spectrum. Now, so first, using general comological dimension result, I get is generated by the contribution of all curves. Then I have to get only nice curves in particular, those generic point lines in the regular locus. So, okay, so I, so I need a moving technique, so roughly if I have my picture of the punctured spectrum and I have the generic point, let us say, of a curve, let us say in the characteristics, p locus where I don't, I have less information on the local comology, I put it in a surface whose put it in a surface. Okay, so, so suppose I have C in X, a curve, so I put it in a surface meeting the regular locus where p is not zero. Okay, then in the regular locus I have also the place where, so I have absolute comological purity, I have a map from lambda s. So, for s prime in s, some open dense, I have a, so maybe d minus 2, d minus 4. Yes. Okay, so, then what happens is that the comology class on the punctured spectrum, so I reduce, so first of all I reduce it to nice coefficients, so the coefficients are the statement to constant, so by usual the visage arguments to, and using information comological dimension to coefficient with the right height, then I can even make them to be extension by zero from some open, then I have a contribution from the generic point of the curve, but actually it comes as follows, I have a map, so actually I get a comology class for this extension by zero from s prime to s minus the closed point, so I have a comology class on the punctured spectrum of s which induces a comology class, so on s minus point to the comology on x minus the closed point, so if I want to prove the class coming from, so what happens is that by induction the class, the comology class that I have for the punctured spectrum of eta after strict initialization comes from contributions of nice curves, so in particular they correspond to such surfaces and then for each of those contributions I get actually a comology class on s which induces the one on x, so it goes to verify the result for the surface case and then I can normalize and then I have a comology class which is zero generically and it is zero, also at every generic point of the curve therefore on the initialization, then you have to prove that it is there, so you want to prove that it comes from contributions of all curves as in many specified ones, so you have to make it zero in some open which contains the generic point of those curves, so you have to, so anyway you have a comology class that is zero, generic is zero at this point, but then the point is that you have to approximate, anyway you use the fact that the comology of the fields are looking at generated by symbols and you can use easier approximation approximation method to so essentially I have something like so essentially I have to I am giving some so I have some comology class in the fraction field of the initialization of each of the xi and I have to extend it to some comology class in the fraction field of s by some approximation, so once you have enough roots of unity, so again by this information from the early 80s that for those kind of fields you don't need the block-cater conjecture, you know the comology generated by symbols and you can use this approximation so anyway, so this allows me to do this moving part and then so now I want to get an invariant associated to the top comology so a map like before for the complete discrete volatial field case so so so again I have the p rank is r so I want to get so I extend the coefficient by 0 to the locus where p equals 0 and so ok, so to define those maps I have the following so the top comology is generated by contributions of curves in even once with generic point of characteristic 0 so I can use the for the result of cut for each of those contributions I know what the so I use the result for complete discrete volatial field and of course the residue field becomes a finite separable extension but I also have no map which actually isomorphism on those things on the top so it doesn't change this so then I have to prove it is well defined and to prove it is well defined I use completion and some choice of coin ring and so the argument using projective space so this gives something depending on some choices but since this is generated by contributions of curves so it is independent of the extra choice and so this now this actually is defined without assuming that the comological dimension is this you can pass to the separable closure of the residue field and use Galo action ok and so it is also easy to see that it is surjective for separably closed and even when the p dimension is the is r of small k is r so I want then to prove that this is an isomorphism so so the theorem is that alpha is anisomorphism and beta is surjective for k equal to ks and in the non separably closed case the Galoa comology of k with coefficient kernel of beta is 0 and this actually gives this first statement and then there is a a more refined version of this which says that the kernel of beta is generated by the kernels by the u1 of the comology of the generic point of curves so so here this is defined by somino care group it has the canonical u1 servicers so once you know that since this is a filtration with essentially vector space quotient of vector space you can then control the Galoa action the Galoa comology on this is trivial but in fact I have some proofs that prove only 2 and then I prove also 3 so so I will try to sketch in the remaining time I will try to sketch the basic idea of the proofs so first there is a Birtini argument to reduce to a surface so I need some local Birtini theorem so roughly I want to I have a certain number of curves with generic points in the regular locus and I want to find a reducible surface which is regular at those points and which contains all of those branches so for this I need a variant of some Birtini statements in the literature there are several papers some paper of Felner as entry value corrected and then some recent work with some Japanese authors but I need a variant of this I can state what one can see show so for example in any case maybe because of the time I will not spend on this Birtini so in any case so suppose that you prove some Birtini theorem because it's there are several cumulative algebra techniques that go into this so then you reduce to the surface case and then so the also you can use finite covers in ecological dimension result so in dimension 2 I want to work like in again like my paper we saw suppose that I put a coin ring then I use it's over a coin ring then I can use and this also lies over some bit vectors for the maximum perfect sum field I can use Eps theorem to make the special reduced and then I can change the rest you feel in fact I can also variant where I don't change extension to make this kind of generically smooth and then this is algebraisable so it's a it comes from relative curve over I and then I can use by further extension I can I have the semi-stable reduction theorem which has several proof including the one of anyway this is the classical case of discrete valuation ring now so after some blowing up so making the semi-stable reduction I have some model so I have some external curves that is there is a part lying over the close point of my original strictly local thing then I have the sheaf essentially considering so this lies over spec I and essentially I consider extension by zero of some lambda from the general fiber to this so I consider in fact so I have some Y over spec I and I can consider I consider the chromology of Y with support let me say there is a part over the close point so all the components except so some external ones so corresponding to the original component of the locals were P equal to zero and this and then I have some vanishing cycles for this situation now in this case so the vanishing cycles were in the smooth case in the semi-stable reduction case but he assumes that the resufil is perfect which is not essential I think so he studied it for things which are semi-stable but also after extension of the resufil so essentially it works for things which are locally smooth over some things x1, xn equal to the power of the uniformizer and it works I claim that it works using the dry definition of the ramvit so he studies it using millenor k theory so actually one is to correct some point like instead of this you can consider the millenor k theory of the general fire of the strict generalization any case so using the techniques so one shows that the chromology shifts are generated by symbols they have a canonical filtration and essentially the other associate grade and there is a top in fact I would be interested just in the top so I have a chronological dimension result so I have to look just at h1 of the top rpsi so the top so one can show that there is only h1 we support in z I have to consider the top rpsi and then there is a quotient of this which is given by log log logarithmic differentials well actually it was not defined anyway defined something which should be the log the ramvit but I don't know how much anyway he has some definition by hand of this so this is a paper in 1988 so I have not studied if there are other references I mean I didn't know in any case this is only one of the approaches so one needs to improve upon because of some conditions and so on and also one has to prove compatibilities so the point is that essentially it's the coherence of the so I have to in any case I have information and the homology again it improves so one has the the ram so there is the fact that the top the ramvit is a dualizing complex on the scheme which has a local p basis and then it behaves well under f up per shriek so this is in a cadalti but under a security assumption working over a perfect field so I need it in a more general context and for doing anyway so this allows one to compute the top homology of the log differential of the of the mean shifts so essentially let's say if x is regular of dimension n is filled of constant k separately closed then I have to in any case I have an isomorphism given by suitable trace map so anyway so what I get is a certain map using this vanishing cycle description I have to prove it is behaves well on all curves so it's compatible with my map defined by curve contribution so then again I can make extensions I reduce a curve after an extension and changing the model to something that cuts transversally then I can see up to some sign check this so this will give the vanishing of h1k because of things which are essentially coherent and then there is another approach which is more rigid analytic avoiding the precise description of yodo so this is again after some extension essentially I can have a model where I now extend it to a proper thing a proper formal scheme and then I algebraize it where I have those a1 so the other curves will be it will be compactified so those will be p1s so then I have some rigid kind of I am looking at the cohomology modulo some rigid disks at infinity and then I look at my trace map and again so I will have some Galois cohomology so there will be some contribution from h1 of a rigid disk of an algebraically closed field which is not zero in the periodic world but it again involves some coherent information and so I can manage to get the idea of Galois cohomology this is a different argument because it really requires to have the semi-stable thing but it requires to use some rigid well actually one can algebraize this but one can use an initialization but anyway uses this idea and then by working carefully with the ideas of almost mathematics so one can also prove the third the most stronger thing by analyzing the last method so ok, so maybe ok Are there questions? Maybe so you mentioned two methods at the end one using the categorization cycle and one using properties and the but in some sense the question is are they really different? No, no, but because I say that I have this argument which uses the improvement of the stuff from the 80s about vanishing cycle and some compatibility I mean you have to do a lot of compatibility to do foundations for this kind of homology with law grid so you have to work on this language and prove compatibility with other map then the other idea is to attach disks at infinity to attach disks at infinity so this will be now what you are doing is the homology supported in this so this is like the relative homology of the world special fire module does a1 so this is like homology of a complete curve so it doesn't use at all the precise study here it uses more of the study at infinity so so union some closed disks with constant coefficients so the point is that on h2 so if you look at this h2 there is an algebraically closed field then you have this one you know but then you have h1 of u which is not zero but it is essentially for a disk it has to do with using so studying this module up to the end so then here I can have some valuation filtration and so again I have a kind of filtration by things which are linear over k and so this allows me to control well here I did not anyway I have also the Galois homology of small k acting on this so eventually I do anyway the Galois homology for the separate closure and then the residue field but the point is that because of it will act linearly on some vector spaces things which are essentially vector spaces of a residue field the Galois just by usual Galois it will be zero so this and then the ideas of almost mathematics so to speak is like what we are doing today so you can instead of using algebraic closure you can use just adding roots of unity and the piece root of representative P basis elements in the residue field and then again by some vanishing reza so you can reduce the the Galois homology computation that you need here because you are working things you can work with things which are valuation less than something and then you can use the felting I mean essentially the well the what is anyway the almost a tallness the fact that this is kind of deeply ramified in the sense that further extensions are almost a tall okay so the anyway in this case so I can read so the point is that I can get I have some I can represent the classes in each one homology of cave is h1d by certain using only this extension and then I can look at what it I can see that it gives me contributions of care I mean things in U1 because of the precise nature of the symbols which are involved but anyway this is recent it could involve some mistake so I didn't I have not I think that the other thing which has two proofs must be sufficiently stable and this one is only this proof and maybe I maybe I only imagine that I can prove it and so I don't know that I cannot say that but anyway the idea is to to improve the extension I mean to this kind of extension then look at which symbols you get from the kubanren then see that in U1 and apparently and I did not see a way to get it from the from the direct method of so I don't know it's