 אני thank the organizers for inviting me, it's a great pleasure to speak in this conference in honor of Ilu Zee. I knew Ilu Zee for a long time since I came to the IHES in the early 80's, and before this I knew about his work, when I was a student at Harvard, I wanted to learn about fancy things, and Mumford suggested to me to look at Ilu Zee's thesis. ובסמרתיים בישראל אני רואה, אני רואה את הלקטרנות, ואני יכול לראה את הרגע הרבה, ובסמרתי, אני רואה את הרגע הרבה, זה היה הרבה טקניקל, לעשות פוגרס, ולא anyone else took it from the library. זה היה... אבל בכלל, אז זה היה מאוד ילבד, לצער עבור, בבקשה, בבקשה, ובסמרתי, כשבאה, בגלל, אני עושה על פלטינג' אורנוסנותימטיקס, ובסמרתי, אני עושה את הרגע הרבה, לנסות שוברדים, ובסמרתי, אז, בפרח, קונפרנטים, בנוראו פילוזי, 2005, אני רואה, על... פינטנסטיורי, לנסה, לנסה ביליון, לנסה של קונפרנטים בפינטנסטיורי, לתלכומלוגיה, לסקות כתוביל, So, in fact, he wrote, he wrote in 2006, a manuscript called Vanishing Cycles of a General Basis after Deline, let me see. Anyway, after Deline, or Blumon, or Gogo Zogaber. So, I spoke about a question that he asked in the end of this paper. And so, this is, in fact, a subject I already spoke about in Tokyo in 2007 and 2008 and 2009. And then, in 2018, I gave another lecture in Tokyo about things, other results on Vanishing Cycles on a general basis. And this also led to some results I spoke about in Oberwollfach last year. So now, I will reconstruct the first lectures for the one about the question of Illusi. So, this will be about comparison of oriented products. So, this already appeared in the talk of Abes and Rigid et al. And, in fact, this, so there is, so in the case where I have a sub-scheme of a scheme X, let's say defined locally by a finite type ideal, there is a map, so there is some kind of, well, something that Fujiwara denotes as a tubular neighborhood topos. So, this is a, so you've, you'd take the formal completion kind of minus Y, which doesn't exist, but it exists as some kind of attic space or reno types, rigid space. And then, you take the et al topos of this. So, there is a comparison map, which also occurred more recently, there are some lectures in the seminar of algebraic geometry of Ousei. And this, so the result was that for, in the quasi-excellent case, for a constructible shift, a torsion constructible shift, torsion prime to the residue characteristic, the natural map is an isomorphism. And this is a kind of a version of comparison between algebraic and rigid analytic et alco-homology. So, in any case, so we, So, the oriented product is a solution of two universal problem. So, when you, so, so you want to process this maps, T to TI, T to T1, T to T2, so that the composition, on the composition you have a map like this. And the, the, so the definition of the two categories of maps between, I mean, the definition of the two categories of torsion is like in the ZA4, in particular the points, there is a notion of specialization of points, so that the, the amorphism between, amorphism between points is amorphism of the corresponding fiber factors in the opposite direction. So, and there is a defining site for the oriented product assuming that you have defining sites for the, for the TI where, let us say with, with, I consider only sites with finite projective limits, and the morphisms are compatible with the limits to simplify, at least it is used in some references. And then the, the coverings like in the previous talk you consider, so you consider like, like U, U over F indoors W, V over G. So you have coverings of the form, coverings in U, coverings in V, and then the, like in the previous talk for any morphism for, I mean, the same as in the, so this is the asymmetric part where you, as soon as this is, well, this is the, the, so if you do it on both sides, you get a site that defines the fiber product of toposes, not the oriented product, what? You have the map from above, from U to U. Change notation, you have switches. No, no, so the, the, I don't remember. You need a map, you have the three things, three things, and the map, so one of the three things is both of you as the original, I think, so you need a map from U to D, I think. Hello, here I have sites, so I have sites, there is a site 41, site 43, site 42, and then there are morphisms of sites which are functors preserving finite limits, and then the, the, I have a map from U, well, I write it, I have a map from U to the, to what I call inverse of W from V to, so this is like U over W and V over W. What I mean D, the upper object is U to V and D. So here, here I have a, So this is just type C? Yes, so type C, then you have a, some kind of localization, so it's like a section of a number, so U to V. Anyway, I think you, you, you have, no, in any case, this is a standard, I just recall what was already. Both triple, the first term is U. Yeah, both triple, the first term is U and this changes by a pullback. Okay, so this, in any case, so if both are, so this, this was, yeah, so this was something introduced by Deline to, and then studied by Lomon to define vanishing cycle of a general basis using this canonical map, and then I need some properties of this relative to change, so for example, in the visage arguments, so if Y prime to Y, let us say, so this is exchange in the first, so actually this is an equivalence, if this is let us say integral, so it's like finite, and this is an equivalence for et al. And if instead it is proper, then it is not, but you can, you can still prove that at least for torsion, that it doesn't change the cohomology of sheaves, so now, so those toposes are coherent, but they also have the property, so if I have, let us say, a coherent topos, I can look at the so-called localic reflection, so I want to map it to a topological space, or a topos that comes from a topological space, so I use the sub-object of the final object, and here it becomes a spectral space, which, and then every point of this can be lifted to a point of X, this is a small extension of Delin's theorem, and then I have a special type of toposes, or let's call them type A-quarium toposes, where for every point of the corresponding topological space, the corresponding fiber-product topos is equivalent to representations of some to G-sets, so the topos of G-sets were G-sets of a finite group, so for example, a Taltopoi of schemes, or of attic spaces are like this, and also this is stable under this construction where all morphisms are coherent, and in fact you can do this construction also when one morphism is not coherent, and now, so one important thing is a base change, or in fact a universal base change result, which is like what Abess was speaking about, so in fact I wrote an email to Illusia, I think in 2006 discussing such things, so we have a general situation, so we recall that I have, okay, so in any case, so there is this general construction that Abess called co-vanishing, so in any case, if suppose that I have a fibered site where the fibers are sites with finite limit and finite covering families, and you have a base which is a topology and you consider on the total space the horizontal and vertical covering families and corresponding topos, and in this case, it maps to the topos downstairs, and if s is coherent, that this is coherent, in general it doesn't have to be coherent, but the higher direct images are computed by sheafy-fying on the level of pre-sheafs, the cohomology of the fibers, and moreover it has the property that for any map of toposes, the corresponding fibered product topos can be described in the same way, and actually there is a kind of universal kind of formal type of universal base-change theorem, so we can say that the map of toposes satisfying universal base-change theorem if for any, you can give two toposes over T and you consider the corresponding fibered product and you want that, let's say for a sheaf of a billion groups that you have a corresponding base-change map like in the proper base-change theorem or isomorphism similarly for a sheaf of sets, sheaf of groups, so this kind of situation satisfies a universal base-change theorem and the construction of the oriented product is a special case of this construction. Okay, can you record your horizontal and vertical morphism? So I have a, so suppose I have a pseudo-factor from the category S to Sites, I mean to, and then if I have a covering family here and I pull it back by a Cartesian thing and I have a covering family on the fiber, so at least, okay, so at least I'm, so this can be defined quite generally, I mean you can also relax on my hypothesis like this finite limit, but this I don't, I mean it can come technically, any case you can, the point is once you assume that the covering families are finite, you can control sheafification by just sheafifying, I mean, well it is done in this paper of Abess and also I wrote to you an email and okay, so this, and then the universal bench-change theorem is kind of just looking at this and seeing that you can, you know how to describe, by choosing to define inside, you know how to describe the, It was in your email 15 years ago Yes, yes. Yes, in any case, as a sketch at least, so in any case the, yes, so in any case, so these toposes where I have this, in this situation I have the, one can define the notion of constructability for sheaf of sets and sheaf of groups and I mean the usual conditions are equivalent like there is a stratification which is locally constant or is a finely presented object categorically and so it's a, and so how do I get another, so in any case the, so one can consider also morphisms between, so first of all, there is a notion of constructible sheaf that they say with coefficients in the finite ring and then you can also look at netarianity, so in the case of local netarian schemes the ascending sequence of sub-sheaf of a constructible sheaf is stationary, so something similar is true here, so for example if y is netarian and x1 and x2 let us say a finite type over y, then this is not, is also netarian topos, coherent of type A which means that any, so any quasi-compact object is also netarian and this is reduced to netarianity, local netarianity of the underlying topological space, so here one maps to this, to the corresponding fiber-product space, so there is a local reflection which maps to this and so objectively, and so you have to look at the map and so any point here is actually lift, there are finally many lifting and you need some generic generization property, so there is a criterion giving netarianity of the corresponding space which slightly stronger than the fibers being finite in this space being netarian. Now, so using this, so the basic points are given by geometric points, so x1, x2 and a point such that and a morphism between y, the images in y, excuse me, this is again, so for any such point, the corresponding automorphism group of like Galois group is contained in the Galois groups, the product of the Galois group, so the idea is to reduce statement on sheaves, to statements of sheaves of particular kind, like sheaves of the form tensor product of sheaves on each factor, tensor product of sheaves come let's say z-model sheaves come from each factor and this can be done, so one can prove that if I consider such sheaves and you consider the subcategory which is closed under extensions and direct summands and there's a morphism, the smallest one containing this, and you get all constructible sheaves, the z-model sheaves on the oriented product, so I can reduce for certain comological statements to such things and so yes, I want to speak about the rigid analytic so I will, I have to use does the middle one or the tubular neighborhood so Fujiwara denoted by tf over y but this is also the etal topos of the attic space so I have let's say y in x let's say they are quasi-separated y is defined by a finite type ideal then I have the formal completion so I kind of, I want to view this as a rigid analytic space and take the corresponding etal topos so first let us consider the usual rigid, the usual topology which gives you the Zarisky-Riemann space so you have the so you have the blow-ups of a relative admissible blow-ups to the inverse limit and I have chi let's say this chi and let us have the part over y then I can so of course there is some foundational issue in defining the in non-aterian case but at least one can define an algebraic version of the also it is in Fujiwara's paper but one can define a kind of a zillion version of the rigid etal so let us consider the following site I have x prime to x which is a finite type and etal outside y and the coverings are those families such that the corresponding Zarisky-Riemann space so this chi relative to y you have surjectivity on the corresponding Zarisky-Riemann level so in fact you can so think about it, after you blow up everything you get a situation essentially you have evaluative rings and then you can also consider the topology of the special fiber and then you have to take you have to take some what remains is just a finite etal cover of the general fiber so in fact this can be described like in the previous talk as a kind of feltings to a post so I have the so I have the let us take on the limit space I have the etal topology so I can consider O chi so let us say O chi locally you invert something and so you can consider U going to finite etal O chi O U 1 over phi algebra like in the previous talk but you you restrict it to just to chi bar so you have well this is a kind of a fiber side over chi but you have to pull it back to chi bar so at every point of chi bar you get in the limit you get what like in the previous talk the corresponding thing for the stock so the point is that the co-vanishing or the horizontal vertical topos for this one can see that this is equivalent to the ridgid etal topos and in particular one is some kind of universal bench change property it can be used to give variants of some proofs in but so this is for analytic adic spaces now so just to so in any case so this rigid analytic topos has a map to the etal topos of X minus Y this is a non-coherent map between coherent topos and it tells also a map to Y and then there is an arrow like this so this means that for in etal X minus Y you can define the corresponding shift so this comes by just look evaluating it on this and the shift on such things and so this gives a map from this to the oriented product of etal topoi which is so here I have coherent maps to this is a non-coherent map in general and then I want to prove a comparison theorem for epsilon so for constructible shifts I want to prove that this is an isomorphism so so roughly there are the visage processes where I can restrict the shifts which come from the factors and then using behavior for finite or proper maps and I also need my local uniformization theorem which was used in the asterisk book so at least so you reduce to you reduce the case where the shift on F on this is let us say reduced to F2 to be extension so I have to evaluate this on some object so actually I have let us say I have to to evaluate on an object of this site but I will reduce so essentially the main point is to reduce to reduce to cohomology on sinks of the form this where V is open in X minus Y and using the structure of shift and the visage I reduce to shifts coming from Y so this isomorphism implies the computation of vanishing cycles so we know that if you compute vanishing cycles like this or if you pull back and push forward you get the same thing but your isomorphism there I think it implies this the isomorphism you wrote under a strong condition of quasi excellence quasi excellence and constructibility and what you are speaking about is it's just for torsion you mean that on the diagram above exactly you start from there you can take the vanishing cycles the main vanishing cycles means higher direct image pullback and you can also take the pullback and higher direct image and we know this is isomorphism for torsion okay this comparison is kind of an easy comparison here it is a comparison for kind of open non-quasi compact domain in the rigid space like comparison on aphinoid minus F equals zero what you are saying is comparison on aphinoid so of course the point is that all the previous things would enter into steps in the proofs so I will not but essentially the so I am so maybe my notation was omega before so the idea is that I have omega to X so I have some direct images of as I said Z mod L from just in the schematic sense restricted to I and I want to compare it to the rigid analytic kind of rigid analytic direct images so going to going to Y so I want to of course so this is a non-quasi compact map and when omega is everything you get a previous comparison result so the idea is to reduce the any case using weak local and also I have a twist by some shift on Y so the idea is that using certain universal base so omega will be exhausted by domains in some chronicle using certain blow up by some aphino I mean omega I by quasi-quasi-separated so essentially I will prove it for a large enough index I have comparison in the quasi-excellent case and the twist by a shift on Y will be ended using a universal bench change theorem so and so the so essentially it reduces to calculation in the constant case now the in the constant case I will reduce to things which come from logarithmic geometry so so let us say you have suppose you have log regular and this is let us say X- some union of strata and let us say you have a fine open immersion actually what then one can prove that it is principle so locally it is given by so U is like locally X1-P and then there is a purity absolute comological purity which implies the stability of the regular case and for the inclusion of the strata where the log structure is trivial but you can generalize it to such thing so you have that r j lower star let us say z mod n is wedge I of r1 and r1 comes from the Kummer sequence so you have j OU star over O star with a T twist tends to z mod n so one can check this and then this is classical what? yeah classical but I will reduce so I will reduce the calculation yeah so I have my omega let us say so I have my y X let us say and then maybe bigger one is z so omega is the complement of X-z sorry r i and j lower star I mean I was in immersion yes no no r this one is j for example so omega is an open of what? no no ok so suppose that I have so this is defined by an ideal i and this is defined by an ideal j and then I can construct the blow up of j plus i to the n X and the open where i to the n generates so this so you can think about the case where let us say those are principle ideals let us say f and phi so you want to consider the domain with absolute value of f is at least phi to the n this is obtained by certain blow up so essentially add f for f phi to the n so and then so I have my kind of omega n which has a nice model using this so the idea is to take this so in the I work I reduced to the log regular the normal crossing situation but in any case then I will need to to make some so at least I will have after normalized blow up I will have a log regular situation and I blow up some idea like f phi to two elements so the point is that I will have a locus where when I look at this x prime and normalize 2x and restrict to i so there will be points where at least when n is large the fiber of the blow up at least the full blow up is a p1 and a fine part is a1 or just the origin when I don't like when so I will have a closed locus where the blow up gives a1 in the fibers and this will be independent of n and then when I normalize I just add some possibly some roots or on the reduced the point is on the reduced fiber on the reduced locus there will be still an action of gm at least after some cover so all the shifts that I get from like x-z going to x prime n so all the direct image shifts that I get on the will be calculated using this and so there will be so in fact relative to the gm action there will be monodromic in the sense of dn and then one knows that the homology on the kind of the special fiber that is over y can be computed just by restricting restricting to the zero section so and then when I look at this calculation like one can compare the zero sec I mean see that well this also gives the direct image from x-z to x so one can compare the two commutations and see that you have a comparison for large n so this actually will be so I will prove the statement in the strong form that for large n so omega n ridge to y so for large n r rf low star z mod n n is r i there is a morphic to r i f n well z mod l so at least for a given degree of homology since I use a weak local uniformization I will have this and also tensoring with any shift on y and I think that one can do it also uniformly in i even when the dimension is unbounded using the technique of X for z 13 and that's the risk book and so now so I want to first I use structure of constructible shifts to reduce to a case where I have well I have to also to evaluate it on some object but suppose I already know that the object on which I evaluate comes from an open in the in x-y but of course this requires to make somewhere but then the shift is again a tensor product but you can assume the shift is like an extension by zero something locally constant on some and then by passing to a finite cover I say extension by zero of a constant and then constant so I but then I want to make it a situation nice to use the calculation in the log regular case so I need a covering for the H topology like like I use in the finance theorems to to do this and of course I need all the the psychological descent thing that goes into that and the similar thing for the rigid a a homology but it's a a well there are a so here a since I it looks like formally I need a since I replaced by a it looks like yes yes yes so now when by the way so when the same situation in the non there are failures in the when you are in the non-ateria non quasi excellent case for example it is easy to see that for for valuation ring of rank two this should fail so because you have situation where the rigid simply you have two points somehow when you look rigid analytically or completely along that you don't see the generic part so you can see that this comparison in homology of omega omega could be a generic point for rank two valuation ring and this would be empty so this cannot hold if you work over rank one valuation rings and the same even non-ateria and then one can reduce to let's say those with algebraically closed field of fraction and then there are there one can have an analog of the weak local uniformization in terms of a log smooth saturated morphisms and one can generalize this approach and also over one dimensional so one can have a generic version over an open let's say and then you at finally many point you can compare to the completion of the normalization you have so the the quasi action is not so important for one dimensional DVR because you and when you have a in the case of Nagata's example of normal non-analytically normal thing so one can see that it gives a situation where this fails in the dimension two Nagata type example of non-analytically normal you will have a situation where when you do some completion so before the so simply the when you do completion along why the analytic the algebraic etalcomology will change so it cannot and after completion it will be because of the accident so you will not have comparison in general on the terrain case so I am sorry so I I tried to read some assorted sketchy notes but I was yes so anyway there are more results about the questions of O'gogos but this was the no no ok so there are questions about but this is not related to this I mean like which blowing up you need to make the to make the vanishing cycle nice but what? no when did I start oh no ok I thought that I am ok so in any case the example that I mentioned in the end about the non so when you have let's say I is a DVR and I have something in the completion which is transcendental over I then one can construct in the you can add to I so you can consider things like y minus phi to the n minus some approximation divided by power so the uniformity of the uniformizer so you have some extension of the local ring so the idea is that it will be isomorphism on the completion even the pi added completion and you will have so you will have something which is irreducible you will have a regular local ring and irreducible seeing which in the completion becomes an end's power you can even make it strictly and zillion and so this and then when you take an end's root you get this so I say n prime to p so so this is an example of Nagat also discussed in the hysterisk book and so here I will take omega to be the complement of z so I can also complete a long z without changing the properties and so I will have a situation where so also I strictly visualize so when I complete okay so when I so now over so when you complete along the special fiber the since already this is like completing the local ring so then this will be this will have n components so omega will have n components so in any case I have a situation where just when I do the direct image of the constant shift in the original situation is z mod l and let's say in the completely long special fiber situation is z mod l to the m and after after I do this coalition I mean the quasi excellent case so I have comparisons of the rigid analytics so the rigid analytics space will not change so this shows the comparison phase now let me maybe about this twist by things coming from the base is like this some rigid analytics space some tubular neighborhood maybe to y so I have some quasi compact and quasi separated map and then I claim that the actually this is a general fact so I claim that r i f star for all shifts in the analytics space commutes with tensor products with a shift on y so this must be, this is related to statements of Bateson theorems of Hoover but I didn't find it exactly in his book but it is, so this comes from the universal Bateson phenomenon that I explained because I hope it's let's see so no I no this excuse me so I have, I need to introduce the Zariski Riemann space here though it will be the map the direct image of this Zariski Riemann space can be computed using Galois homologies locally at stock so it then I need I need this monodromic shift argument because I have a situation which is non-proper but I control the non-properness using the monodromic shift so then in all the cases when I compute I get monodromic shifts I can restrict the zero section so this is used also for twists by constant shifts by arbitrary shifts on the base so I think, so this is not okay so you said monodromic shifts are they the story? yes so verdiere at least in a finite type situation over a field but one can generate on a more basis and also here it's not exactly finite type because possibly an inverse well in any blow up yeah I work in yeah so you have a kind of a con situation or a spec of a graded algebra I thought that your business with the phi n or something like reduction with normal cone so monologue with that yeah so in any case I have a situation where I want to understand this rigid analytic so I have some something defined by f equals zero this is defined by phi equals zero I want to understand this by blowing up and after I blow up I can what was the question? oh it's similar to reduction to normal cone somehow that reduction to monodromic shifts but I'm not sure no in verdiere there is this normal I mean he uses it for his specialization of shifts but here what happens is that when I when I do it at least with n large enough the point is that f I want to exclude cases where f like f could be locally phi to the n times a non zero constant I mean times a unit and this I don't I don't want so so there is there are points which are in the closure of f equals zero outside y and the other points so at those points I get just the zero section at least on the reduced local at those points I get the full a1 but after I normalize I get possibly another a1 but up to isogenic GM still act on this one so I can see that of course this requires a commutative algebra at least when I take the reduced local after some isogenic on the reduced local the normalized blow up and since I can calculate the stocks using logarithmic geometry you get something which is constant or essentially on the orbits so you get the monodromic shift and then you use very thin actions of this business well this here I just need the to yeah I just need this kind of hyper cover and commological descent for both the rigid and the oriented flow situation to reduce nice situations so I very time appears in the proofs but not in the so this is an instance where oriented commogical descent really enters because in the book eventually you can get rid of it yeah there was an appendix of shank that you could do it with yeah yeah in fact the appendix of shank is like the argument I had in mind which was when I lecture in Princeton it was something like I mean like to look at the structure of the proof of the structure of the argument commogical descent still it gives what one needs but then it will it turned into oriented commogical descent in the seminar but the original idea was like what Jank did in this and which also works in the complex case without having oriented without having a theory of oriented products in that case so my last question is somehow philosophical what kind of applications of this mean theorem that to the effect that f is our epsilon f is our epsilon f star f so does it simplify and improve in analytic region analytic geometry or does it simply no this includes so the main point here is to prove a comparison between algebraic and analytic commology and this is like a comparison and it is proved in the expected way by kind of the resolution singularity if you have enough resolution singularity you reduce to nice calculation so it it cannot but then I thought about more refined so the thing I talked in in 2018 was also things about like the properties of vanishing like so maybe we put it for 6.20 maybe I I will talk in the next online conference for I will do the I will continue about the other topic so in particular I can prove I think in any case so if on an open you have I can also do a truncated version so if the vanishing cycle of a general but it's up to some degree are well behaved on an open and it's enough to blow up something in the complement so it's a question for you to get okay and also you have also the local situation so in the case where there is a good theory I can prove that the map of balls is a so you know there's another proof for this using another possible modification but I can prove that the goodness already implies a for me no balls and also there is a question of the Japanese on good and very good I mean it's transitivity and and so it yes, I met for my next for Takashi first let's keep this in mind for Takashi okay you're almost okay that's just for the driver for the ball okay