 So I will speak, as the title said, on some ideas related to questions in Piatic-Hodge theory. So I consider Piatic-Hodge theory for rigid analytic varieties over complete algebraically closed, varied field of mixed characteristic zero P. And in this case, for when X is smooth, the cohomology, so the total cohomology with coefficient Z mod P is finite dimensional by result of Schultz, and this is done by using results on the cohomology of the sheaf O plus, so mod P, so it's an almost module, almost coherence of this. And then, so in fact, this tends to O plus mod P, so O plus is the ring of integer in K, and O plus, let us say K plus the ring of integer in K, and O plus is the sheaf on the arctic space or the rigid analytic space of functions bounded with absolute values S not equal to 1. So now, so in fact, one has a canonical map, which is an almost isomorphism. Okay, so this is when X is proper and smooth. In fact, smooth is not necessary, as I will explain later. But now in the smooth case, so what I want to prove is also the prankaraduality, as I mentioned in the end of my previous talk, so we have this, so there are results of homological dimension that show that this HI is 0 for I0 in 0,2D, where simplicity, the dimension is D and the variety is irreducible. So the prankaraduality should predict that there are perfect pairings. The dimension of H2D is 1, and the generator is given by a fundamental class of any closed point. So if P is in XK, there is a class of P in H2D PX, maybe Z mod P to the LD for any L. So a priority that is not clear, it is independent of P in our case, or even if it is not 0. So in any case, so the statement is that this is generated, so this gives a basis, in particular it's independent of P. Now, so once one has this, then there is an argument that allows one to prove that for a proper smooth map, the higher direct, so that the RNF low star Z mod P are locally constant sheaves, and in fact, Schultz reported that he proved it without using prankaraduality in a certain way. Now, and this is useful for comparison results, so even for, so between DRAM and chromology, and pediatric et al. chromology which are shown under such an assumption. So now, so I will, now this duality is reduced to some kind of duality for this, so we have to prove, and then we also have to worry about the top chromology and to show that we, the duality that we get is compatible with this class of points. So there are also some... At the end of today, there are six operations for medicine. Okay, so this I will talk in the end, so there are technical difficulties to define a trace map, so I will speak in the end, so I'm not sure, so what I will do is that I divided it into several parts so the problem is that first of all, I prepared less than I thought because of various emails. Then it turns out that there are some notations that are needed in some parts, so I try to order it in such a way to, and also I will give just from memory various lists of steps, and I think that essentially everything is okay now. Then in the end, the point is that the trace map is constructed by hand by starting from, I will explain how to do it, and then I don't have the, so what happens is that I will explain what one can do, assuming that one can consider suitable trace maps and I have, in principle, one can prove what we want for proper maps of regenerative spaces, but it's not, I mean, still there are some points and in particular what I say is a kind of hand waving about which sign do you use and so on. I mean, this is after, after, okay. So the basic idea, one of the important, so one important idea comes from the paper of Falkings where he said it seems which have nice reduction and he has a certain statement of duality for, well, what I call nearby cycle, it's not really, I mean, it's also in Falkings. So this is, one considers the map from a, for a formal model, one can consider the projection to the Italo Zariski site of the special fiber and then what I call vanishing cycle is total direct image, so it's not nearby cycles, but it's just for the title of the talk. I mean, by any case, in Falkings case, he has, he didn't use rigid Italo-comology, but he has some Galo-comology, which then we know now with the perfect theory is equivalent to doing this essentially up to almost zero sync. So in any case, so he has some local, some duality result, so I will show how this extends to other things and I will also comment on how you prove it because there are some points which are not clear in this. So now, so the first section was, so I will, yes, and also I will need some geometric presentation in terms of successive semi-stable curves following Temkin and also Reno explained several years ago in a lecture on permanence following Temkin some rigid analytic ideas, so we'll follow those ideas and they will give a kind of rigid analytic treatment with a refined sufficient, some refined version of the semi-stable reduction result, so in Temkin paper it's just for schemes in any case, so I think it's, and then I will show how to go back to the original problem. So the first section is, I will try to indicate what, so one is to study certain kind of schemes which are not orings, which are not netarian, but still they have good property because so, and some of this is done in my book with Ramero on foundations of, or now it's called foundations for almost ring theory, but in any case it's, I will need somewhat more general class of rings because I have to consider algebras topologically of finite type, not necessarily of finite type, but the general factor are the same, but still for the proofs one needs a little more, so I will not comment on, if there are questions I will comment how you prove things, but it's not there because there are many points to discuss here, so I will consider algebras of the form restricted, so I have a rank one valuation ring possibly, so usually not discreet, but the discreet case is the usual netarian story, which is not new. but now I can consider algebras C is algebras, which are questions of, so finite type of some restricted formal series, and the morphisms are abstract morphisms of algebras, now then one can, in fact those are good enough, so this of course is a topology, so you want to know that there are no pathological things, we don't respect the topology, so if A is in C, so we have this subring visit topology, which is the image of this, so in fact any two choices of A0, for any two choices A0, A0 prime, the topological tensor product maps in economical way to A, is a third choice, and one can go from A0 to A1 by adding torsion and taking a finite extension, and also any morphism of such in C from A to B, you can choose those kind of, well not let's say called, it's like rings of definition, the topology, you can choose such topologically finite type things over which those are finite types, and the map sends A0 to B0, so then we can, now so those things have the following properties, so how do I, okay, okay, so those rings are coherent in the sense, like infrasurge brickware on the kernel of a map of finitely presented models, finitely presented, so in fact any finitely generated V torsion free module is finitely presented, and then there are also almost notarian in the sense that a sub-module of a finite type module is almost a finite type, where they're almost, is relatively the maximum ideal of V when V is not discreet, and if V is discreet, it's already in the 10, so we don't, and also they have nice properties relative to completion, so you have Artyn Ries lemma for exact signal finitely presented model and finite type ideal, which allow them to get flatness of completion, but also they have nice dimension theory, so there is a dimension function, which is universally, so it's, and also the cruel principle ideal theorem holds, so the way to introduce a dimension function like in the general fiber, you essentially take the dimension of the closure plus one in the special fiber, you take the dimension of the closure, one is to show several, so also, so what is important for me is to have a formalism of groten-dig duality, so there is a, I need the local theory of F-Aper-Schrieck, which is developing Archon Residues and Duality, and in the book of Konrad on duality and Bayesian in the Stax project for, in the Terian case, and in my book with Sramero, this is overvaluation ring, but this is just a small generalization of, and so then we can, so in particular, so this map something like D plus, so actually it's just in a fine case, but because it uses the factorization through an affine space, or well, in a generalized sense, you're using restricted formal series if needed, so you'll do it for smooth morphism or closed immersion where the usual way of to show independence or the factorization composition law, so, and then one can construct the dualizing complex, so to speak, a dualizing complex like F-Aper-Schrieck of the module V of respect V, but then one can represent it by residual complex as in Hartzhon's book, and this is canonical up to unical, and therefore this glues to the scheme case, and therefore one can then define trace maps, so I will use the formalism of duality using the approach with residual complexes which works in this, in our case, so the residual complex is, well, it depends on dimensional convention, but it's, so essentially it involves the sum of our points and given the grid something about points whose dimension function is i and you have some module, something which is like the injective hull of the residue field, but in this non-iterranean case one can define what it means using glum-local cohomology, so, now also there are depths in variance, also the underlying topological space of those things is an eterian, simplifies slightly, then there are also depths in variance which are denoted by delta and delta prime in my book with Rameiro, so essentially here they are the same because the underlying space is an eterian, essentially you just look for when you have a quasi-coherent sheaf, you look at every point where the lowest n such that this is non-zero, so in any case, and you have compatibility, one can define some, do some kind of local duality things, in any case, so one can, so one can then think about Cohen-McCauley and Gorinstein properties, although the things are not in the eterian in this context because you can just look at things for which the algebra itself is up to shift and it's dualizing, Cohen-McCauley just some depths properties that one can formulate. So here I am speaking not almost story but actual coherent, because the ring is coherent, so there is no almost yet, but of course later it will be necessary to worry about too, so now, so in studying the, originally for the study of Felting-Spurity theorem, so Felting-Spurity theorem originally was, when the base was actually coming from discrete valuation ring, where you tap past this algebraic closure of the, let's say, in zillion one or completely, so this was generalized in my work with Ramero to more general valuation ring, so using some model algebras and then it turns out the choices both covers all, I mean, is much more general, I will still have to work with the nice local model, so in a special, so one can consider nice models of toric varieties or of course fine toric varieties over the field of fraction of V, so for simplicity let us just look at is a nice model of a torus, so for example when you have GM you can have something like semi-stable reduction XY equals C, where C is in V, okay, so I want to generalize this, especially I will need just things which are successive semi-stable curves, so in any case I have a torus whose coordinate ring is direct sum of K Chi where Chi is in the character group and so I want to have an integral model where for every Chi I have a free rank 1 V module of course this should be a finitely generated algebra and it should satisfy that for a power of N this is the generator to the power N gives the generator, so now when you have such a thing then you can, so should I, okay, so how did you, ah, you took the one, okay, so in any case so if I have such an A then you could extract P-sroots, you can extract P-sroots so instead of, so let us say that K is algebraically closed and of mixed characteristic 0P and so I can look at inside Chi T1 over P so the corresponding, I can extend by the rule that I said because this rule tell me what to choose for so now this kind of thing when periodically completed is perfect with algebra originally the purity theorem was for this so let us say that I have A0 is the original algebra, AAM is direct sum over Chi in 1 over PN Chi T and I also have graded pieces for, which are A0 modules so, and then I have this A Chi so now what happens is this A0 is an example of of something which is morally like a Gorenstein so it turns out that unless I made some bad thing in the definition so in any case, it's discussed in detail in any case that this can be descended it's not, actually it lives of evolution but it can be descended to a saturated log-saturated, log-smooth saturated morphism in the sense of a Tsuji I think or so and then there are cohen-McCauley properties and flatness of the map so the fibers are cohen-McCauley in any case, we can also see the Gorenstein property in this, so essentially the dualizing sheaf on the smooth locus is the module of differentials which is actually free so it's on the and the dualizing cohen-McCauley must be the reflexive extension of this so and then those are a kind of maximal cohen-McCauley modules and this is a duality even in the derived sense and then I want to consider like in Fulton's the co-homology of so there is a group delta which is home maybe from this module this to the roots of unity so this acts so one can consider the co-homology on maybe let us, we can do it module of p to avoid considering continuous considering continuous co-homology and now so this group is essentially a power of zp we have certain generators and the co-homology is like co-homology on the top was b delta but for p torsion modules the r gamma delta m is represented by the causal complex of the generator minus one acting on m so this starts with m then the gamma i minus one to m to the dimension m to the dimension over two binomial coefficient and so on now we have duality so I will need to this okay maybe I will try but do I have to move this you decide what you have to do do you want to put it on top? yes it does not raise so well this one it's okay so we have then a pairing actually so there is a in fact on quite generally for top process so there is a kind of car product pairing and here I want to to have a duality coming from so let us say that m and n are dual with values in let us say walk modulus some element to avoid continuous cohomology and then I don't want to consider a derive category of representation just simplicity on actual module so you no so suppose now I have a not modules but maybe coin McCauley maximum well so a not modular a let us say so I want to avoid continuous cohomology and let us say that they are dual to each other in the naive sense because I said that this is kind of and then suppose that I have an action of Delta so of course continuous but then because of the torsion property some it factors to a finite quotient so and then the description is a causal complex works and then so there is a way to write it down in terms of causal complexes and then one can see explicitly that so we have our gamma Delta a zero and then this goes to the top one placed in degree minus D which is a zero up to some one-dimensional normalization so you want a duality with values in this when so in any case so this is again set in following's paper but without doing the but anyway this is elementary is apparently so the causal cohomology mean the complex computing causal cohomology when you do it as you get some shift up to science of the same thing but then you have to show it's compatible with this map but now what happens is that if I I can apply it now to each of the A each of the graded pieces and then I have actual duality of but now if I pass the full A then I have almost duality but the graded pieces are not in accordance so what I do is that I excuse me I made the mistake that I have to divide by the character of chi I have to excuse me I have A chi bar so this was a mistake so I have chi bar in 1 over p over character group of t and then I have A chi bar as grouping everything that goes and this is an A0 module so I don't I had okay so then I have for cohomology of A chi bar and A chi bar inverse I have a duality but as the order of chi bar grows one can see easily that it gets annihilated by something close and close so you get almost duality for R gamma delta A so you get that they are almost finitely generated which is as I said the same is almost finitely presented and almost well module of some A and if you use continuous cohomology by hand it's almost finitely presented by the same algorithm yes yes yes okay but you don't need to know that anything which is if you don't need to use it in general yeah yeah it's yeah it is just by yes then there is a more general now more generally let us give one formulation which is like in feltings so M is an almost now almost finitely generated let's say projective A module with a continuous action of delta or A module and then it is continuous in the so one is to define what it means but when you have A module just essentially a limit of things on which finite quotient act so it's then we want again a duality between so this is actually the statement in feltings up to the changes in some conventions I mean he has one over Pi he works with anyway it's not on the other end the proof was so in my viewpoint essentially to actually have a proof of this one is to approximate by actual representations on finitely presented things and then one is to use the kind of infinite resolution at least in my way of thinking I'm not sure maybe there is a way to somehow handle it directly but the idea is that it can be you can have an almost well up to any degree of precision you want in terms of almost you can resolve by by things which come from free module just there are there are enough elements fixed by subgroups open subgroups and they form an AN module you can cover it you can approximate it by finitely generated even free AN module consider the induced module so you get some resolution which just are very simple and then you the point is that once you know the duality for each of those or the almost duality then you you can get it for so the dual will have an almost resolution in terms of the dual you can do work with the estimate the point is when you go far enough it does not affect the co-homology in a given degree so I but I am so actually I will use such an infinite resolution argument in the geometric it's already present when you analyze this it seems to me from this fact no no no he said something no what he says what he says is that we have to make some some small well he said something which doesn't include no so okay anyway so there is some only some minor details need to be checked I don't know okay anyway so okay so in any case now I will pass to the geometric part which is a semi-stable reduction following Temkin and Renault and me and other okay so so this is so first there is the so called there is a reduced fiber theorem in rigid geometry so you decide with you so I will so now there are many types of rigid space are attic space but that is mistake original thing so this was in two kinds of rigid geometry so called the criterion case and the classical rigid case and for simplicity every region analytics space I consider will be separated quasi-compact so they are always represented by a formal scheme separated and formally of finite type over SPF of A raised in a terrain ring relative to complete respect to an ideal or rank one valuation ring complete respect to an ideal and I think one can easily extend it to so called microbial not so high rank valuation ring but this is maybe needed in considering fibers but I think one can avoid it in what I say but in any case it should extend to that case and then I have a map of so of formal schemes like I said such that the rigid general fiber well is a flat with geometrically reduced geometric fibers so here the flatness is well actually the naive flatness in the sense of commutative algebra but it's also equivalent to analytic flatness in some situations people are distinguishing analytic in any case here there are so called enough rig points in the there are enough or closed points in the classical rigid case so one can test only with those and the fibers are in the sense of points is values in algebraically closed complete valued field where you have some rigid analytic fibers supposed to be reduced and it's not enough to look at just finite extension of the field in some case you need infinitely inseparable extension in any case so in fact there is a previous theorem which is called a flattening theorem so this is in the papers of Borsch, Ludke, Bomer and Renault well there are several parts of this so I think this one is in the last part so in any case a flattening theorem says that you can arrange the formal model to be flat and then the reduced fiber theorem will say that now and this is after an admissible blow up but for the reduced fiber theorem we need a rig or rigid attack in fact base change and then there is something finite over this which doesn't change the general fiber and this one is flat with geometrically reduced fibers and now so rigid attack well I assume people know about rigid attack but it's it can be generated well one can instead of rigid attack one can consider a more special thing which is a finite attack cover of a neighborhood and then some a formal model of this so here usually usually one attacks here the normalization so it's not just because certain things could be non-reduced you need this high prime but you can think that the y prime is you make it closer and closer to the normalization the general fiber and then this will the distinction here will disappear oh is it some kind of where we find the X okay so what happens is the reduced fiber sometimes there is an app but over DVR then historically Grawert and Reimert proved using certain method in the 60s they reduced just over a complete rank one valuation ring and then this was generally in the theory of stable field so some more general statement successively but now this is a geometric thing so it seems to be difficult to directly reduce the general case to the app to the discrete valuation ring case maybe one can do it with some stock machine anyway but and also there are several approaches to this so I'm not going to review the historical I also didn't read all the but of course the semi-stable reduction theorem is deeper than this this can be done but some of the proof of this uses if I know what Temkin does I can prove also but in any case and also it's not hard to actually reduce it to the case of Grawert and Reimert that is maybe this is the way they proved so once you know it for but except that here we need higher rank because of the points of the addicts but we need higher rank cases and if you know it for those cases and in fact the higher rank case can be reduced to the rank one case so essentially by so it so it is essentially some souped up version of the Grawert and Reimert thing which was proved in somehow in a global way using projectives but there are some as I said there are several approaches to this so now what I am interested in is also doing it better for the the topological part of this but do I I don't find my ah yeah okay in any case I recall that for rigid analytic spaces there are several approaches there is a classical approach of Tate there is only close point but this is now very old and there is the addict space or the Zariski-Reiman space of X and then the Berkovich space which is like the the kind of house-doorfication house-doorfication of this so there is a continuous map so this is essentially compact house-door space is a spectral space and you have analytic so this is like so called analytic case of analytic addict space for analytic addict space at least there is such a thing and then there is a segment non-continuous section so essentially you have a height rank one which corresponds to points in the Berkovich space and then you have so called secondary specializations any case of those which so and then this map is closed with quasi-compact kind of proper in the weak sense that without separated fibers were just and then we can speak about ideas of wide neighborhoods of things so this occurs in several contexts in rigid geometry so one sometimes uses notation compactly contained for certain things so the idea is that certain things let us say over a base so let us say you have something open in V all quasi-compact so it compactly contains means that on formal models the image of U bar let us say the special fiber closure inside V bar is proper over the special fiber of the formal model of X and this is independent of the choices and in particular there is for affinity domains in a given X so basically you have for rational domains you have certain domains of the form f0 like f0 up to fn-1 less than or equal to fn where the functions generate the unit ideal and then you this is compactly contained in what you get by putting 1 plus epsilon and this generates this kind of thing so now you can also look at wide neighborhoods let's call it of a point well in different senses so in zRx by which I mean that it contains the pray image of a neighborhood in the I don't know the Berkowitz of the corresponding points so of course this is not in general not quasi-compact another way is that there is a neighborhood so you have one quasi-compact neighborhoods but you have a phenoid containing another and this is supposed to be so a V of this type such that there is a smaller one so this is you so a smaller neighborhood that is compactly contained and this is if and only if there is a wide neighborhood in this context so the point should be does it have any valuation no because I just look at the neighborhood in this so then it contains all the fiber automatically so it's not but here I take let's say a point in the risk agreement space so the U could contain only it doesn't have to contain all the others but once I enlarge you to something in the wide then it contains all the other points in the fiber so the the refinement so convenient formulation of this instead of taking rigid et al I say the following that for every c in y the risk agreement or the Berkovich space there is a wide neighborhood v then a finite et al cover okay and then a formal model of v prime over of course the original thing and then I can take the pullback and I claim that this is enough yes I can do it with such a thing not with the general rigid et al in fact it's almost the understanding what rigid et al topology means because over the Berkovich point you have some a finite number of pullbacks which give finite separable extensions and those can be lifted to finite et al covers of a wide neighborhood so you can have something so you can construct such a thing such that this now instead of being a rigid et al covering will be covering for the usual but kind of get split into into a finite domain and then you can then so now I want to describe the formulation of Tiamkin's result in the same terms so I have so so what should I I don't know which I should use but in any case I can move this and then so in Tiamkin's paper on relative curves the situation is that I have some relative care well actually there are assumptions there are different kind of assumptions one can put over but let us say that everything is separated okay so in any case so let us say that we have a smooth a smooth general fiber but it's not so the point is that after normalization in the finite well it depends on hypothesis well usually it's a normal scheme it's a scheme but it will be more like I will work in the rigid setup and it will be the so but in the case of schemes of normal so you want to so of course if you since you work with scheme you can reduce to the netarian case if you want then you want to to to have a semi-stable model after normalization which is generically etal alteration and then you want something which is flat with kind of so this is a smooth curve so let us say pure dimension one and here I want flat pure dimension one and with semi-stable okay fibers mapping by a proper map which is generically birational to that of course as in the usual semi-stable reduction theorem this is there are many semi-stable models but there is a relatively stable one which is better so this is the usual condition about some smooth P1 in the fibers which have to cut in at least three points other things and if there are bad components you can blow them down so actually for the blowing down in general one is to work with algebraic spaces but this doesn't matter much in our case in fact there is the relative dualizing shift of this becomes ample for X prime over X so if X is a scheme we must get a scheme moreover if we are working with so anyway so Temkin shows existence as uniqueness but in the scheme context of course the proof uses valuations and then of course Reynolds well actually I don't know exactly what Reynolds observed because the notes are not very unique so essentially one can do the same thing in rigid analytic view point now I also want to notice that when you already have the reduced fiber theorem so you have only many non-smooth points then the X prime will only modify the stable one will also modify over these points and it can be viewed as some kind of generalized blow up because some power of the relative dualizing shift so you have this thing where sections actually give a projective embedding so you take the direct image of this which give you on the smooth look with some line bundle for non-trivial line bundle and this is like some kind of sub-shift of the extent of the direct image of this line bundle to the whole thing so it's a kind of a blow up so to speak of this sub-sink so in this way now so the idea is that in the rigid case now I need the for the analysis of course one has to work with the points of the attic space so I need the high rank microbial valuations and I have now formal scheme situation with let us say with reduced fiber curve with generically smooth okay but then the previous thing was for skins but here it is easy to see that there is an algebraization but I'll kick type results so I'll kick this is for but anyway this is known how to do it so essentially I can the singularities are uniquely well up to anyway can be algebraized and then there is and then once you do it you you have this blow up that you can do and then the resulting semi-stable or stable model can be shown to be independent of the of course two different algebraizations are in any case they are closed to so up to an automorphism sufficiently closed identity they are the same so in any case one can show that so one can show uniqueness and so on so you you get it over those and then the idea is roughly so the idea should be to in the case of now a map of rigid analytic spaces with given a formal model so once let us say that on the rigid analytic spaces it is smooth relative dimension one then for the formal model you make it first flat and then geometrically reduce fibers then for any point you look at the way to so you need some finite extension actually you have to maybe hand the lies and take a finite extension of the corresponding for every so the point is that if you if you make this stable modification at a given over so anyway so it seems to me that the proof of I mean of course it's not but it's essentially a variant of the technique so it should and essentially it may be what we consider better but let me just say that the as that you for each such eta you find you find an extension of k eta plus over which there is a semi stable modification or even a stable modification and this is given by some kind of generalized blow-up so the idea is that this generalized blow-up will work also can be spread out to a rigid etal neighborhood okay so a generalized blow-up means that I have let's say on a scheme I have on an open I have a line bundle and I have a finite type module extending this line bundle on the whole scheme and what I want to do is to find a morphism together with a quotient of this finite generative thing which is aligned by extending what I have and I want the also the open well it's retro compact I want it to be schematically dense so there is a universal as you expect so you just take growth index the quotient rank one quotient the scheme representing rank one quotient the schematic closure of what you have on the u okay so and then but the point is that this is a very so the so the point that when you are working on the Zariski Riemann space when you look at the local rings they turn out to be certain sickenings of K-up such things but it does not what you it is such that well one can see that for centers of usually blow-up the possible centers are the same and then I claim it's kind of the same for those generalized blow-ups so that because I'm working with so the well maybe we'll discuss some kind of commutative algebra that one can so maybe we will okay okay okay okay so I will continue to state the kind of local uniformization result and what it gives us in our context so we work over a rank one complete valued field it doesn't have to be of the mixed characteristic and as I said it could be also more general types of rigid spaces maybe now one can avoid all hypotheses if one uses the right the right machinery but let us but we need some any case I don't have the so I start with something well let us have a succession of map, smooth well we're of formal models where this is smooth all of them are smooth relative dimension one and for simplicity quasi compact and separated then I want to have an equivalent nice model of a neighborhood in fact it will be a wide neighborhood actually in the application that I have I don't quite need wide neighborhoods but this is also seems to be related to some shrinking techniques島 איביה מאוד לא לא אבל לאBLE look צ pin סמוסופינויד, ואני רוצה לעבוד איזה פעם, ולוקחי אני יכול להכנראה שאני זוכר שיש סמוס, סמוס ומופסים בלתי-במשימה, ובאינת שאתה יכולה להכנראה את היחסים או את דיסקים. אז לא, אז אני, אז לך לראות שאני already כבר כך כתוריזציה, שבאינת שיש לי, אז זה לא הרבה רשת, ובסדר, אני רוצה להשיג את השגר של תמכין הרסל של איזה ללב, איזה שאתה... כן, אז מה שעשינו לתמכין הרסל של איזה ללב, שאני אני אוהב את זה לנסת כתל, וואלה שאני אוהב שם קיבל, שם כתל, על פינוויז, ובסוף, יש לי איזה סמיסטיבל מודול, אבל ביוניסטי הטעה, זה חדשת את כל זה. So I actually have it over some formal neighborhood of the final detail cover. Now so the idea is that I'm starting with a point here or a point here, then I have a certain wide neighborhood and then a final detail cover over which there is a nice thing, but then you can, this is smooth of relative dimension run over that. So you can take the image point, find the finite et al, you can even make it Galois, so I will do actually. So then the situation is like in the Young's theorem and things like this. You have some finite group and so let's say XC in the Zariske Riemann space of Chi1, then there are wide neighborhoods of XC1 and its image up to CN and finite Galois et al covers. So in any case, the idea is that the, so the, yes I have let us say, yeah no, it is just applying the same thing but stupid induction without any fine technique, so if there is no, so I have, so essentially the fiber products of those, so anyway once you go to, yeah so in any case, so over Vick C2 for example, there will be a finite Galois et al cover of the pullback of Chi with group G1 cross G2, which is a successive semi-stable vibration and so on. So I actually, I will need it when Chi N is M and K is algebraically closed, in which case you don't need the last thing. And then in particular, it will imply that any point of Chi has a wide neighborhood with a finite Galois cover and yes, and also it's important that you can see from the proof that the formal model and the key variant, a key variant formal model, which is a successive semi-stable curve, which is smooth on the general fiber. Now I claim that such things can be, so locally one can look at the equations that are introduced and see that they come from relative toric varieties, maybe some log, so as I said since it comes from log smooth saturated maps, so in any case you just have to know something about the units in, so this involves some approximation to compare again, so those things are formal, but they can be, they are resilient, well essentially finite type models and those, so essentially I need in the case of log smooth and saturated seeing for over a normal scheme, so there is a general fact actually that if I take, so invertible function, you have an open contents in the open set of triviality, suppose I have a log structure, which is trivial, so I recall that for log regular thing we know that the shift of monoid is just direct image of O star from the trivial log, so here you have that the shift, if you take direct image of O star you get essentially when the taltopology you get, well you have some log structures, so you can take some, so roughly the only invertible function that appear come from invertible functions on the base and things in the log structure, so but one is to anyway, so there is some push out that gives J log star O star and this is used successively to describe this, so to see that this is, so it comes from a, in particular from a nice, for a nice model of a, in the taltopology for a nice integral model of a torus, like what I explained before, so in particular the previous results apply, and now, who, if you don't consider any action or go action, no, you see, so of course anyone can form this extract P to dance roots of things and then you get and, so, but, so in any case I said that I did not, so I have some points that I didn't prepare precisely, but let me continue on, so we need the fact, we need that now to work with, so in some sense this is to obtain on smooth rigid varieties, but without imposing any model because you can get a nice model by this local picture, so locally, so locally I have this, this is what you, but you need that the variety is still smooth, the rigid variety, for this, yes, yes, okay, so now as usual, so there is the pro etal topology, actually there are several versions, it seems, of the pro etal topology, either using finite etal, and it shows this paper, more general things, there are several ways to put a cardinality bound, but it seems that they are not, for commology of usual things they give the same results, and in particular I have, one can do, so this is in the mixed characteristic case, so O plus is the usual etal shift, and then when you take the modular P to dance, you can either do Janssen style continuous commology or pass to the pro etal shift O plus roof, and then the, actually this is also true in the derived sense, so without almost, just on the nose, in all versions of a pro etal, in one of them it's almost by construction, and other things you have to prove something, a little bit. Now, then the point is to construct, to study also in faultings approaches interpreted by Schultz, you consider the x pro etal, yes, and also I want to mention some general facts since we are discussing this, so there is, so in general you can consider shifts like for an element in the value group, you consider functions with absolute value less than or equal to a, or less than a as shifts, so you can consider quotients of such things as shifts on the etal, for example, but then it turns out that this is commological descent, for surjective map of quasi-comparing, quasi-separated, so this is closely related to the so-called faithful topology in Schultz's lectures, and so again, so this is reduced to looking at what happens over microbial valuation rings, since you get some situation where, when you pass a limit of things, which admits sections, so it is, but then this is, then we can use this resolution of singularities in characteristic zero, which is known, actually, with a canonical one, so it should apply to the analytic case, so then when you study these shifts on a possible, the shift O plus roof on a possibly singular thing, you can use a hypercover like in the Linoge theory by smooth things, then you get the finiteness result on any singular thing up to almost, almost finiteness, I think, and then by the same method you can compare to Z mod p etalc homologies, so you get this comparison even for singular proper things, and when, even for non-proper things, of course, the Z mod p homology is related to the O plus by various Arte and Schreyer things, so one of them is O plus mod p, O plus mod p, this one one is Frobenius, but this is not, this is sensitive to almost zero things, but another way which removes there is to use something like O over less than A to the P, O less than 1 where A is close to 1, and this is Arte and Schreyer, and then this somehow recovers the Z mod p etalc homology form, anyways, so this is also infalting, so now, yes, so what I am saying is that then there is a calculation for nice integral models of tori, I can calculate the, so I have my A0, so the point is that, or something etal over A0 and periodically completed and I have the An, so the essentially you, when you go to A infinity, yes, there you have some perfect, well in the completion you have some perfect things, so the etalc homology is the usual, anyways, and there are cyclicity results, so essentially it resues, so up to almost zero things, the looking at the direct images of this, let's say from X etal to chi, so then one can look, yes, so first of all the interpret, it shows its finiteness proof, it uses the shrinking argument, but instead one can observe that what it does, in fact, shows that when you look at such shifts and you take their direct image, the etal of the risky side of some model, then this is almost coherent, so in fact, you get essentially the same thing on the etal, I mean, the one you get here is the pullback of the one you get zero after almost zero things, and on the fine pieces you just calculate the, essentially what I explained before of A, well when you do the completion, you have some continuous homology with values in A roof, but now, oh yes, there is an important thing, yes, to recall about the, which is defaulting, extension, so, but let us say that, so the relation is differentials, so so, in fact, the indicator, so in this case, so there is a well-known calculation of this, where you get a lot of torsion, which is something called junk torsion, which is killed by Zeta minus one, and in fact, there is also a canonical map, which is important now to observe, so let us have a function in OKI star, so there is a kumar class in h1, well, a corresponding thing on x, mu p to the n, so actually, and then you, so actually I want to get something in the, let us, OK, so I have x at all to kai, so I have r1, Taylor star, maybe mu p to the n, so I want to send differential df over f to this, yes, and then the claim, then I want to extend it to a map of omega 1 of the formal scheme to r1, Taylor star, o plus mu p to the n, up to something, so you have, it's generated by such things locally, so you have to check that some relations hold, so the relation is df and d1 minus f, essentially, but this is not, so this can be done by looking at the way the feltings described it. Yes, but first of all, you have a twist here. Yes, with a one and modulo torsion, modulo kernel of one minus zeta, OK, so in any case, so there is this thing explained in feltings paper, but so essentially the same thing, but without, yeah, no, no, OK, but as usual, I omit the log and I don't have almost, I mean, I observed it in our case, many exact sequences are exactly, without almost, so one can, of course, maybe you get, if you depend on what you use, maybe you get at first that this is well defined up to almost zero, but so there are several things that one can observe here, actually, I have, I think I was able to check that this kind of R1 or this kind of H1 just does not have almost zero elements, or almost zero, but also I can just check directly from some version of the felting extension that this holds on the nose, it seems. You will not ignore it, it's OK, you will work just with this model. No, no, no, this is a global thing on any formal scheme, any flaw. This is problematic, I think. No, no, this is well defined, because for any, the point is, I define it by generators and relations. Once I show the DF over F over, once I show the basic relation with 1 over F and F when both are invertible, then it follows by trivial, if the map on generator satisfies the relation that I want for F and 1 minus F, then I have such a map. Now, to check this, then I go to a smooth situation, and in the smooth situation, I have the felting, so the smooth situation in question is over ZP, I take the completion of the fine line minus zero and one, and I add the piece roots of unity to ZP and take piece roots of F and this, and normalize, and I can see that this is enough, I don't need to do more, and then I can just work with this felting extension for this and check that everything works, and so I get even a kind of, so what I claim? My thing is that there are two possible twists. One is by one, and one is by the xi of the front end, OK? And the difference is P2S. No. So there are two possible twists. I think the one which is, which glue is not the felting's one is the one with the xi of the front end. No, because of torsions, we are not working over QP. No, no, no, what happens is the following. When you calculate in nice models, there is a quotient of actually the H1, the full H1 will, modulo torsion will give the omega 1, and somehow the point is this is a canonical description, because you calculate it using some toric presentation, but actually this becomes more canonical, but if you look at the proof, you can see that what it actually proves is the following, that you can define map, maybe this is not exactly what people do, but this is very close, so you can just try to define by hand such a map on generators and check if the relations hold, and you can check it, that the relations hold up to kernel of one minus zeta, and in fact it holds on the nose up to kernel of one minus zeta, because either I prove that there are almost zero elements, anyway, I can work out the felting extension just in a, just in a simple case where you take zp, I mean in fact I don't need the full, it seems that I don't need the full thing, I just had to add certain kind of functions and normalize, check that the argument works, but maybe there is something there which, but it indicates, since I have two ways either proving that there are no almost zero elements, I mean so it's very stable because anyway, so I can, and even if I have almost zeroes and I still know it doesn't matter much for the rest, so it's just to state, so in any case, then we can go to higher differentials and define the same thing module again kernel one minus zeta, because you have the same relation, you also have the relation df which df is zero, which holds, well in general actually you have the symbol relation f minus f equals zero, but here the minus one is doesn't, well, because there are roots of unity, so you have, yes, okay, so now the point is that those things are almost, as I said, they become almost coherent when you go to a particular nice, yes, and they're also almost coherent for not just those things, but for now using the perfectity, in other words, you use a formal model where the affinoids are well, they are obtained by rational domains and finite detail cover succession from things in, from the product of annuli, or you can even generalize it slightly to allowing a tile mass is quasi finite formal model, but anyway, so there are those things for which one can control, so, so then one has the finiteness, in any case, almost finiteness of this, and also the existence of this map, so the, the situation, okay, maybe I will need another blueprint, because I said too much in, so, oh, maybe I will, I don't know how to push it, okay, now for any formal model chi of x, so I have a map from x-tile or x-pro et al to chi זריסקי או רטל, then I can consider R kilo star of O plus over A, so, so actually, so this is, so there is some homological dimension result, maybe it is zero in degree R i equal, so anyway, it is zero when i is big enough, maybe bigger than twice the dimension, by some results on addix spaces, okay, on the other end, for nice, for nice in the weaker sense, for perfectly nice that is, no, no, no, no, there is a homological dimension result for p-tortian shift on certain chi of addix spaces, like in Uber's book, there are several ways to indicate, so it is based on the dimension of the space and the residual dimension, homological, then phycomological dimension, so anyway, so there is things which are zero on the nose for large, large enough, this is one thing, then you have almost zero and bound, so in order to study פיידי גולג'ט'ירי, one is to work with, so anyway, this is a little bit common, so, so, so there, so in the Schultz's thing, it's even, you have some more conditions than is actually needed, but in any case, we need a phynoid for which one can, which admit in any case, so you, you want, because usually one can control fractional domains, finite et al, infinite et al cover, and not any et al map, so you need a composition of such things, or more generally of et al with quasi-phynite formula, between a phyno, a finite composition, so those will be kind of nice a phynoid, so if you have enough of those, then you know that this is concentrated, almost concentrated in degree between zero and d, and it is almost finite, almost coherent, okay, and moreover, you can refine any model by a nice one, this is again a general, not so difficult, okay, so then you can use, of course, well, I didn't say, but there are finite nest theorems for coherent homology for those things, because they are coherent schemes, so you have finite for coherent homology of coherent sheaves, for almost coherent sheaves, so you get it for any chi, you get that this is almost coherent, where almost coherence means, you tensor it with m, and then you get actual quasi-coherent sheave, which is almost coherent, in general, this is not necessarily quasi-coherent, but up to almost zero, okay, it's quasi-coherent, and almost coherent, which is, in this case, the same thing as almost finite presentation with your own, yeah, yeah, so the module, yes, now what happens is that when you look at rd, so now let us do it with the pro thing to simplify, because otherwise I will, I will need this to avoid torch, okay, but then this is, this was a pro etal, or Janssen's continuous homology, okay, so again, this is again almost coherent, yes, and what they say here is that, in fact, we have a map from omega i chi to ri t-loure star o-plus, o-plus roof even, module or some small torsion, killed by z minus one, when a twist, okay, which has the properties that local sections, which are unit on the formal model, go to df over f, but then, in fact, you can also extend it to local sections, then you can view it as a rigid, yeah, then you can extend it to df's, which are just invertible on the general fiber, because you reduce to some universal setup in one-dimensional smooth thing, like some kind of differences, so like some annulus, and there you have the, well, if you prove somehow, you have a calculation of this h1 here, and in any case, but maybe what they said is, So, here, what is the assumption on the formal model x? No, here there is general, maybe I will not, yes, in any case, so you can also, so there is also the nu from the pro etal to the, so actually you can tensor with the rationals, and then, yeah, and then you have, okay, and then you have, and there, any local invertible function as this property, you don't? You can start defining it by koumere, and then the question, so you define it on the mono it, and then the question is, does it factor through, but you don't like to, yeah, you are in the max, you can start from koumere, say, man. No, no, no, df over f goes to the koumere chi f, the problem is to show that it, it factors omega i, and then I have to, so the way to do it is that it does not factor if you take this, but okay. So anyway, so according to this, so what I understood that working with this folding extension, you see that it factors, and also when f is 1 minus f of a unit, and you could also extend this calculation to get something when they are just units on the general fiber, and then you get some possibly larger torsion, in any case, which one can, and this is what I believe, and it seems to be, I mean, okay, it doesn't seem to, okay, so yeah, so in any case, so this one is, the point it coincides with the analytic omega d on the smooth locus, so actually, yes, so what happens is that you can do it as a map, then it becomes actually an isomorphism on the, when you have the etal, the pro etal, and when you work on the rigid general fiber, so this is, so, and also in the smooth, so it's a morphogenism, often on the general fiber on the smooth locus of the special fiber, and now the dualizing shift, so we recall that our kind of things have dualizing complexes, so there are a start, so we normalize the, so actually the omega, which is kind of our parallel sheet from S, so this usually it's for a smoothing of relative dimension D, it's omega D concentrated in degree minus D, I think, so in any case, so I, let's say that the first, the first commology shift, we'll call it the dualizing shift, and this must be the, so the dualizing shift is omega D, well, is some extension, the unique reflexive extension of omega D, and since this is almost coherent, so we find that, let's say the maximum ideal tensor this, which is actually quasi-coherent, will map to the dualizing shift, and in fact this will be, so the maximum ideal also will map the dualizing shift, and then we find that when you go module A, because the A plus one is almost zero, again you have a map from R D till star O plus maybe module A to dualizing shift module A, because actually the dualizing shift, yeah, so you have like H zero of the dualizing complex module A, and H zero of the dualizing complex module A is included there, so you have a map to dualizing shift of chi, module A, chi, so this is under the weak assumption of just nice relative to the perfect theory, and then one can extend this to, yes, and of course the dualizing shift maps to the zero-scoromology, so it maps to the residual complex, which is a canonical representative of the dualizing complex, okay? okay, so there is something, yeah, so if you look at the talcite of a smooth rigid space to the, yeah, the pro etal side to the talcite, then I believe, yeah, so this was that, excuse me, R I new lowest star of O plus roof, not, or actually O roof, well, you have to, it's, is it O plus or O? I think you have to, in terms of rescue. Yes, we have to get yeah, okay, so you, you, it is a graded thing, yes, multiplicative, yeah, so this is the usual color differential on the rigid space, and this is, but not the dual complex, no, no, no, no, no, no, no, no, here it will be just, not the dual complex, yes, it's just exactly that, yeah, so and I think you have to, to the tensor with the rationales here also or not, I forgot, otherwise you have already the x-1 as you said, the z-1, no, but when you go to, yeah, of course, yes, okay, so this is, so in any case, so I will, okay, so I will try to, so this is actually the crux of the proof, but I will try to, maybe I can, because there are several, no, no, but there are several two important steps in each of them, okay, let, let me, so, okay, so for any formal model chi, so you dominate it by a weekly nice one, and then you can say that rt lower star of o plus mod a, here is rpi lower star rt prime lower star plus mod a, and now this maps to the residual complex, and then the residual, direct image of the residual complex goes to residual complex, so you get a map to the residual complex with a suitable shift and twist that I omit, and so, and then the basic thing is to, to prove an autoduality for this, so this tensor itself, so again, this is a general fact on the opposites, which actually, in most general form, maybe it's treated in the STAX project, but in any case, so you have a, so anyway, so this goes to a residual complex, and those things are almost queried, so I want that this is a perfect, this is a almost duality, this will be the theorem, and here we assume x problem, no, no, no, this is a local theorem, so just a local theorem on smooth rigid space over a complete 0p algebraically closed valued field, yes, so and a is anything in the maximal ideal nonzero of the valuation ring, so you can see that in fact the dualities on chi and chi prime are related, and so if you know it for chi prime, you know it for chi, in fact, they are direct, what you have here is a direct image, and the dual is also the direct image, so if you have a duality here, you have a duality here, so the first step is to prove it for quotients of nice in the previous sense formal models by finite groups, where of course the group acts freely, so on the general fiber it's a finite תלמה, but not on the special fiber, so here roughly the idea is the following, so I have chi prime over chi x prime over x, this is a Galois cover, and so actually I use some kind of גודמה resolution using some family of fiber functions, I have to work on things, on diagrams which are commutative on the nose rather than in the derived category because of the nature of the argument, but this, so what happens is z, so I will have something like this on chi prime, this is a גודמה resolution on chi, I will have actually the group G which acts, and this will be both the invariance and the coin variance of the group action, well, so actually, and also there are several statements that are, like the taking the direct image under pi is also the direct, so, yeah, so roughly I have some duality here, and I want to get the duality here, and this is the idea, this is obtained both by group homology and group core homology from this, and the group homology is compatible with, is dual kind of the group homology, but to do it technically, I, what I, so I have, I have written, but this will be take a long time to, to, yeah, so in any case the idea is to, you, you take this, anyway, this maps to itself, to this, and then to this module, the coin variance under the group, so I get, after taking direct image to a chi, I get a pairing of this one, and this one with values in this, but this coincides with what you get here, so you get a pairing, but now on the level of complexes without any derived, okay, also a pairing here, and those pairings are compatible with going up and kind of trace in this, moreover, I can look at the canonical complexes coming from group homology and core homology, so I, I have, like, when I have something on which the group acts, I can put, like, the complex calculating the, the, the homology, so, so there is a complex calculating the homology, and a complex, the standard complex calculating the core homology, so I use trancations of those complexes for this, the one for homology and one for core homology, and so I have actually a pairing on the nose, I claim, between those with values in this, and on each term I have a duality, or almost duality, so I have it on the full complex, but then the full complex is compared to this, up to things which are far away, and they don't matter in a given degree, so I get a duality here, this is the idea, but I want us to be very careful on doing this, and then then I want to discuss how I prove for any formal model duality theorem, so let me take another blackboard, so anyway, CHI is a given formal model, so the, the previous thing, the tanking-like presentation means that you can cover it, so there is some rigid cover, a usual topology of pieces which are, which are quotients of nice by finite groups, okay, but those are, those are just models of some opens, of wide neighborhoods of some things, but then, all the affinoids that appear there, I can then find a formal, a kind of admissible blow-up, which somehow dominates all of this, which means that there are pieces there which are the pre-image of this, correspond to the pre-image of this, and they cover CHI prime, so I have to prove something, so I know it for CHI I, this sandwiched in CHI and CHI prime, but the duality, I said that the duality that I need for CHI is kind of the direct image of the duality for CHI prime, so the idea is to study the cone from my object to, so I have the, actually since I'm working actually with a, so I have RT prime, lower star of opera plus, actually M times this, and this is calculated by some Godmore resolution in economical way, and this goes to, well, actually the top cohomology goes, then I can extend it up to homotopy, to a map to residual complex, upstairs, and then I get a corresponding map downstairs, so I just fix those maps, then then I have actually a pairing of the nodes on this, because this is valescent residual complex, so I can consider the cone of, let's say, the cone from K, using this pairing to its dual, to our home, our home is just represented by the home from this, and then I can do it for intermediate models as well, or even partial, like the CHI, so the point is that this cone, the point is to prove, to control this, that it actually maps to this, that this cone maps to this, to other one here, and maps to the one here, yes, so, and I, so I have, oh no, okay, in any case, so the idea is that if, yes, and also those things can be thought of as part of a big, actually, something in between, but this has to be, so with the type of, yes, so I think that, so in any case, I have a certain cone here, and the point is that its direct image to chi-i from this to that is, if I control the models properly, I have some cone that I want to prove is zero, and its direct image to chi-i is exactly zero, up to almost zero things, but then I look, let us say, I have to prove something by induction, either on the first homology, chief, or on the dimension of the support, there are two versions of the argument, and the version of the homology of support is actually like the proof of my finiteness theorem and exposes 13 of the, but then I will do the one with the, so let us say that you have the zero, the first homology, if it can appear, and actually it turns out that it's not really necessary to control, once you control maps in the drive category, instead of working with the homology of the cones, you can use kernels and co- kernels of the maps on homologies, so you can do this less strictness in the drive categories than I say, so in any case, then the zero, the first shift here is up to, well it's an almost shift, it's a direct image of this shift here, but then this morphism shows a direct image from this to this of this shift is actually zero, so there are no sections here, and therefore there are no, since they cover there are no sections, but any section that comes from, goes to there, so my shift must be zero, so this is one version, then there is another version, is the dimension of the support, and then there are some very, there are some subtle way to control things precisely, and to be sure it's honest proof, so I am not, but I think that, yeah so, maybe it's time to, okay.