 Thank you for the introduction, and thanks to the organizer for the invitation, which is quite an honor for me. I'm sorry that my talk will consist only of a simple exercise, and I should say an exercise still in progress. I will discuss duality for vanishing cycles in ethylchromology for all torsion coefficients. Well, this is not really a new topic, but this is one, I think, in which there are still things to understand and even enjoy. Let me recall the story. So, I will work with strictly local tri-s with the usual mutations, s, the closed point, eta, the generic point, ethabar, several closure of eta, i, the inner shaco. I will take a prime number L different from the characteristic of s, and I will work with coefficients lambda, z to nu, z, for nu at least one. Now, if x is a scheme of finite type over s, the function of the nearby cycles is defined as follows. Here we have the special fiber ixl in x, the generic one, and here we have the pullback of x eta to eta bar, and here j bar. And for a complex L in the plot of x lambda of psi L, the definition is i for star j bar over star of L restricted to x eta bar. So, this is a complex on the special fiber axis of lambda of i not good. And for m in the plus lambda of phi of m is defined as the cone of i plus star m to the upside n is also in the plus xs lambda. In the mid-seventies, the need for that on the upside says dbc to dbc. So, dbc means the full category of complexes which are with the h i are bounded and consist of constructable community sheaths. In the early eighties, Gabber fooled the compatibility of psi with duality. To express the result, it's convenient here to work with a slight modification of the capital psi and capital phi. Namely, I put psi L as i lower star of psi L and I shift by minus 1. So, now this is an object in d plus of x and lambda i concentrated on xs. And similarly, I put phi L as i lower star of phi L, shifted by minus 1. And to discuss duality, I need the dualizing complex. I define ks to be lambda s of 1 of 2. I could take lambda s, but this is more convenient. And if ax is a map from x to s, I put kx is a approximate of ks. And I will use the dualizing complex d r home over x r home underline in kx. And if I have a complex over x eta, I still take the r home with value in kx restricted to x eta. Gabber's result is a canonical isomorphism between psi dL minus 1 and d psi 4L in dvc or x eta lambda. There is an account of this in the seminar on the periodic period. Now, so I think you put this around 1982 something like this. Combine with the fact that psi is right t exact, given that psi is t exact, in particular transforms perversives on x into perversives on x. Gabber also proved that phi sends perversives on x to perversives on x. It's not a trivial consequence of the first result. And that was the question whether phi is in fact t exact, or in other words if phi commutes with reality up to some twist perhaps. In 1986, no, no, yes, 1986, Bellinson sketched a proof of this, or at least a method which in principle could get a proof. Using the so-called maximal, what he called his maximal extension factor, details were never written up. A few years later, Morico-Saito transposed Bellinson's method in the context of topological spaces and duality, and also for regular autonomic d-modules with certain smoothness assumptions. It turns out that last year there was a conference in Montpellier and Bellinson was there, and I asked him about duality for phi. He said, oh, this is very easy, very simple. And he explained to me a simplified version of his maximal extension factor, which is maybe not so easy to understand in his paper. His paper is a how to glue perverse sheaves. And this is the method I will discuss here. With the further simplification, I obtained quite recently, actually, in collaboration with Wet-O-Tongue. So this was the first part, somehow some kind of a swift historical sketch. Let me turn now to duality. Consider the maximal elting factor. We have i to 0 of 1. In terms of p prime, which is the pro-group of order prime to n. If f is an i-module, I do it by ft for l-theme part, variant of f under p prime. So this is an ls0 of 1 sheaf. The factor taking invariance under p prime is exact. And in fact, this is the image of the projector, kappa, where kappa is short over order of p prime, or g in p prime, g. Of course, for any fixed action continuous i-action, you first pass to a finite quotient through which i p acts. And then we have a decomposition. f is f-theme, ft plus f-nt, which is the image of one of these kappa. And in particular, we have decomposition of psi. This is psi-t plus psi-nt. This pass will be my category. And similarly for phi. But you see, i upper star m is just this l-theme part, because of course p acts trivially and i acts trivially. So then we have phi is phi-theme, phi-t plus psi-nt. So now you want duality for phi. But you have duality for psi-nt because you have duality for psi by Gabler. So duality for phi is used to duality for phi-t. Now psi-t has a description which is similar to that of psi. Namely replacing eta-bar by eta-theme, the maximal l-theme extension, to xs in x. And we have here x-eta, g, i. And instead of x-eta-bar, I look at x-eta-theme here, which is obtained by pullback from eta-to-eta-theme, which is the limit of eta of phi to l-minus l, for phi, for my new parameter. So let me do by q the projection here. And psi-t for l in v plus of x-eta, psi-t of l, is i lower star of rj lower star. So in the sequel for brevity, we don't need the r before the, in front of the direct function, rj t lower star of l restricted to x-eta t, which is the minus one. So by pin s here for this Cartesian square, this is also i lower star of j lower star of j tensor l, where j is the direct image of lambda. So this j is, of course, a zl of one lambda module, a continuous one. So it's in fact a module over the ua server algebra, lambda double bracket zl of one, which is isomorphic lambda double bracket t, if t is one minus sigma, sigma, topological denominator of zero one. And this is in d plus of x and r. And similarly, phi is also in d plus of x, r. You want to, i upper star. I don't know, sorry. Sorry, it's fine. Okay. So phi t of m is also in d plus of r star. I upper star. No. I upper star. I upper star. Yes, thank you. Thank you. Yes. Thank you. Okay. Why the notation g? Because in fact, j will play the role of an infinite Jordan block. The torsion r module. And in fact, its torque at eta t is in fact r, t minus 1 over r. So this sheaf, this etal, this big etal sheaf with the big monotromy action is a key actor in this story. Now, in order to state the duality theorem for phi t, I need some notation. r is a lambda algebra, but it's an augmented lambda algebra. And following the balance of notation, I will denote by r of 1 to tau the augmentation ideal. So if I have chosen t, this is the principal ideal. In fact, it's isomorphic to t r. But I don't want to choose a topological generator, sigma, or the other one, and not t. But this is an invertible r module. And so I can consider its tensor power, which I will denote following Benenson again, r and tau, i.e. r of 1 tau, tau, tensor n, or n in z. Of course, if I choose t, then I have an isomorphism, sending 1, 2 into the n. Now, these are n form a sequence for n larger than m, r of n tau containing r of n tau. And in particular, we have r contained in r of minus 1 tau. Now, if f is an r module, I can define the Iwasawa twist, f of n tau, as f tensor over r of n tau. This construction passes through complexes. And in particular, here, this inclusion induces a morphism from f to f of minus 1 tau, identity tensor of this inclusion, which is in general no longer an inclusion, and which is convenient to denote by beta. This is again Benenson notation. And this beta is some kind of monotomy operator. Why? It's because if you compose here with the isomorphism with f given by multiplication by t, then the beta becomes multiplication by t. And t is 1 minus sigma, so it's some kind of monotomy. Now, the main theorem is the following. The theorem 1 for dvc lambda exists, which means we will construct a canonical monocontorial isomorphism, phi t dm. So there are two twists, Iwasawa twist and tape twist here. Now, the corollary of this, of course, is that you get a duality for phi. So you get d phi m isomorphic to phi t. So phi t d1 tau minus 1 plus psi n t dm restricted to eta of minus 1. And here, non-canonically, by which I need the choice of t, then you can trivialize this and then you get phi t dm of minus 1 plus psi n t dm eta. So in fact, you get phi dm of minus 1. So you get that phi is t exact and you recover gambler's result that phi preserves perversity. The proof of the theorem uses the instant description of phi by means of this maximum extension counter, which gives sort of a safe-jewel description of phi. Certainly the description of the cone of a higher per star m into psi is certainly asymmetric. When you dualize, the psi is dualized, but the higher per star is replaced by some higher per streak and you don't see anything non-duality. So that was the puzzle. And here is the solution, dm then some psi. We use a following notation. If a to b is a morphism of complexes, I denote like a dreamfell dozen some paper by a cone of a to b and a cone of a to b shifted by minus 1. Is it any morphism? Yes, morphism of complexes. Now, I will come to the beta here. If you look at beta for f equal to j, you get a suggestion of the kernel is lambda. You can see that I'm taking the stock and looking at the 1 minus sigma into t. Then if l is in b plus of x keta lambda, then from this you get j shrink l is the cocoon of j shrink l to j shrink l of minus 1 tau. You can tensor with l and you get a triangle. And similarly, j low star, by which I mean our j low star is the cocoon of j low star l to j low star l of minus 1 tau. You forgot some projection. Oh, sorry, I forgot the j. Oh, yes, sorry. You get j low shrink l is the cocoon of j low shake j tensor l to j low star to j low shake j tensor l of minus 1 tau. And similarly for j low star l, the same. What is the upside theme? The upside theme. So the definition is here. This is a higher percent of j low star, but we push by i low star, in the shape of minus 1. So actually this side theme can be written as just the cocoon of j low shake of j tensor l to j low star of j tensor l. There is no twist here. Now we have several formulas here and we can assemble them into some diagram. Consider j low star j tensor l to j low star j tensor l minus 1 tau. And the similar thing with j low shrink. Horizontal maps are better and vertical maps are canonical maps from j low shrink to j low star. And this is a community diagram and I can look at gamma, the diagonal map. This diagram can be in fact lifted as a community diagram of complexes by replacing l by, let's say, a good more resolution. And the Bayesian definition of x i counter x i l is the cocoon of gamma from j low shrink j tensor l to j low star j tensor l minus 1 tau. Here this diagram gives you two community triangles. The community triangle gives you an octahedron and the lower octahedron here gives you a triangle on the cocoons. So you get here a distinguished triangle. So what is the cocoon here? This is j low shrink of l. Goes to psi of l. So this is, I forgot to say, this is what we call the maximum extension center. So it goes, by the remark I made, it goes from d plus of x eta l to d plus of x tau. So we get j low shrink, psi. And here we get the cocoon of j low shrink to j low star, so it's a psi, but twisted by minus 1 tau. And the upper triangle gives you a sequence here. Here you have the psi, psi t l goes to psi, goes to here, this is the j low star. This triangle shows that psi sends dbc to dbc. And if l is perverse, then j low shrink and j low star are also perverse and then psi is perverse. Note that even for x equal to eta and equal to s, and l, the constantive lambda on the eta, the psi of lambda is not a trivial object. It is given by sequence by triangle 1, where psi t of lambda is just a lambda shifted by minus 1, concentrated on the close point. Then you see that psi lambda eta, in fact, is given by some element in x2 of this lambda concentrated on the close point and the j low shrink lambda. This is a class C in h2 with support in s of s to the lambda of 1. We just see the impact of the class from the class of the point. And compared to psi, which has the monodromy, this psi has the monodromy. In general, the image of beta from the psi l to 1 tau to psi l for l perverse C is given by the composition here. You go psi l or 1 tau to psi and then psi to psi l and you just get psi t of l. What is h2 as supported in s? Yes. Now, why is this interesting? It is because psi is related to 5. Consider the following diagram. Take n d plus of x lambda and consider n goes to j low star n restricted to eta and here j low shrink n restricted to eta. And j low shrink n eta goes into psi of n eta. In two ways, either psi n eta goes to j low star n eta. And the composition is a canonical map from j shrink to j star. So I have a community diagram that can be lifted to diagram, community diagram of complexes. Let me denote this by b of n. Community diagram of complexes, I can see it as a bi-complex of complexes concentrated in degree. So I will put n in degree 00 so it will be in degree 01 times minus 10. And of course I can take the simple associated complex to this and the result is that the simple associated complex is nothing but phi t1. And the proof is immediate, essentially. You replace this diagram by n j star j tensor m eta minus 1 tau j shrink here j tensor m eta. Somehow you display the psi in the j low star using this expression I gave before. So here is this. So you have the bi-complex. And of course here you can... So this is essentially equivalent to that. So this is of course a cyclic. So to calculate the simple associated complex you can ignore this. And you look at the corner here. So here you get i of star m. And here you get phi of psi tm. This is j low shrink with j low star. And this appears in degree 01 so this is the phi. So this is just an obvious observation. Now the main result is the following. The main point in the ingredient to prove theorem 1 is theorem 2. So this is just a theorem 2 for L in the DC of s eta lambda. There exists a canonical non-functorial isomorphism. Psi dL of 1 tau minus 1 2 dL. In the obvious sense with triangles 1, 2 and d and psi t dL isomorphic to d of psi t. The theorem 2 implies theorem 1 is easy. You see, look at the picture here. So dualize. Then psi is a dual up to twist. J star is transformed into j shrink. J shrink is transformed into j star and n transformed into dm. So in fact, if you define b minus of m by the square where you put here psi m eta here j low shrink m eta maybe I will take n notation for n in d plus of x lambda. You have a similar square here j low star of n and here you have psi of n eta. Then so this will be now in degree I will still put no sorry here. I will still put n in degree 0 0 so this will be now in degree minus 1 0 times 0 1 and you see that the d of b of m note before I do that note that again s b minus of n is in fact d of l. Now, dual exchanges between b and b minus d of b of m m is b minus of bm and then you get the result of d star. So in fact you get here is not exactly this I should put here 1 tau and minus 1. Here you might be surprised because at some point I put double twist and at other places I don't put any for example at m I take the dual and I don't put any twist. The reason is that if f has trivial dl of 1 action then take twist equals it was our twist. In fact you have that r of 1 tau but below of 2 tau is just under 1. So f of m tau is f of n if trivial 0 of 1 action. So then you get that and then you get the formula for x i d. Now observe that we have the canonical sequence psi t m eta 2 psi t m i lower star i star m and this map is sometimes called the canonical map and you have a similar sequence similar triangle here i lower star and upper streak of m going to sorry 1 going to psi t of m 1 tau going to psi t of m eta where here the map is a variation map. So the beta, the monodromy goes from side to side. Variation is a factorization of that which is the cocoon of 0 and beta. Now these are I don't have time to explain but it's very easy to recover these triangles in the description here of phi as a simple complex associated to b in fact you have a double complex so we have two filtrations 9 filtration by first degree and 9 filtration by second degree the 9 filtration by third degree will give you this sequence 2nd degree will give you this one exchange is b and b minus and exchange is first and second filtration so then b exchanges these two sequences these two triangles now let me give an idea of the proof of the 0 and 2 which is in fact quite easy the core of the matter is in this infinite jorgen block j so we call that j with q lower star of lambda where q is subjection from eta t into eta so j is an effective limit of jm where jm q and lower star lambda where here this is eta of phi projection qm also note that jm is canonically dual value in lambda by the trace pairing so the inductive system jm is also gives also a projective system where the map from jm plus 1 to jm is the trace map now the main lemma and the rest we still joined with weichel I should also acknowledge a very fruitful discussion with the author last week about these questions lemma which is for me it was an extraordinary surprise but this is very simple anyway so what is our lambda or j to lambda so j is an inductive limit an inductive limit so you think it will be a projective limit but no it's again an inductive limit so this is j of minus 1 minus 1 canonically so if you like these can also be written as our lemma of jm so this is a clue that projective limit is an inductive limit so the reason is that yes I get it first time how do you calculate this arm? you calculate the stock at eta t well that will be an inductive limit here comes an inductive limit of n of a projective limit of our lemma and jm now a simple calculation shows that the po object lemma m of h0 eta n and jm is in fact 0 so it's the projective system is essentially 0 and even it is a artinry 0 and for the how the guy so if 0 and h1 so eta n is the quotient by n to the n the other one so if 0 and h1 are invariant and co-invariant so h1 eta n and co-invariant with the shift there is a twist of minus 1 and in fact essentially it comes out that you are n minus 1 and then you apply the limit and that's finished so then you get a pairing, perfect pairing now if y is a scheme over eta of finite type and you take l in d plus of y lambda you get a pairing j tensor l tensor j tensor dl into k1 by star n or l and dl the last line last line the last line is the radial dual oh sorry, thank you you said it thank you very much, yes of course thank you and if x is a scheme over s in d plus of x eta and lambda you get a pairing j lower streak of j tensor l tensor j lower star of j tensor dl into k first in j lower streak of k then in fact in k of 1 of 1 let me call it double star now here I have 3 in that 4l in dvc or y or x eta then star and double star are perfect that is identify each of the factor of the dual of the other the proof is like this first of all you prove star is perfect well this is easy actually my simple dvc in fact you reduce to y equal to eta and then this is the result this is the dilemma now the double star says 2 things a that j lower star of j tensor dl of minus 1 j lower streak of j tensor l in myomorphism and b that j lower streak now changing l to dl that j tensor dl minus 1 minus 1 to dj lower star of j tensor l is in myomorphism now since star is perfect the trivial duality is that dj lower streak is j lower star d gives you a star plus dj lower streak is j lower star d gives you a to be a myomorphism for b gathers psi td psi atl in fact what is psi t the cocoon of j lower streak j tensor l of j lower star j tensor l when you take the deep psi t then already one of the d you know so the dj lower streak you know already then it remains this guy but you know the cocoon then you go for then you know it for for b so this gives you d you have to some compatibility reduced in compatibility with eta actually when you look at the where the map is defined and then reduced to eta but surely you have to check this compatibility so this finishes the proof of theorem 3 so once you get theorem 3 theorem 3 it implies the xi d 1 to minus 1 is called the dc because the the xi is also a cocoon cocoon of j lower streak to j lower star but with a twist so you know both terms and then so you know also the cocoon so this is immediate so this is finished now let me sorry I think I have learned two minutes there is one question that Le Mans raised about this key lemma how about replacing eta theme by eta bar would it still hold so it's conceivable but still we don't know the p prime is a little complicated and we don't know also Le Mans asked for j replaced by other theme so for example you you take a finite field and it's algebraic closure you take the projection you take a direct image of lambda you have that the jewel is the sheaf itself shifted by minus 1 we don't know in this case no twist no t-twitch should be algebraic so suppose you take now a q no I'm sorry you take a q bar you take a q take a j now to j or lambda so it's our home of j lambda j minus 1 no t-twitch because you take the zl quotient the maximal zl extension not 0 or 1 extension so then there is no twist it's plausible but we don't know I think it should be like this okay okay now also one remark on the proof you see this duality is very bizarre if you consider the poetal site of eta you have the map nu now you can consider the the projecting limit of the new caster of gm so this is the sheaf of arm modules on the poetal site which is a in fact an algebra and it's free of rank 1 so let me define the check so it is not 0 here but somehow the formula of lemma is equivalent to saying that our new star j shriek is torsion is our torsion and one can ask for j or shriek replaced maybe by constructable complexes of j shriek modules or j check modules now the applications are we just so 6 or 5 applications so the main application is that to local acyclicity so local acyclicity means the phi is 0 so phi is the dual so then dla is l a so then you get takashi site of stability of singular support and characteristic cycle by duality so ssd sdss and ccd dcc oh sorry ssd is ss it's too fast ssdf is ssf and ccdf and ccdf here it could be for x smooth over k and f in the ctf of x not by constructable and finite or dimension so you have this theory of singular support and characteristic cycle but these objects are controlled by phi and then since phi is a dual then you get that now I hear that offer can generalize all this very simply to general basis so dpsit is db and also for phi t or the s generally and then from this it should follow that La is also ula and dla is la so this is the cryptic so local acyclicity so la equal to ula means universal local acyclicity that is local acyclicity and dla equal la for the s regular excellent etc for the dual of the local acyclicity but well this is just maybe some sort of open at the moment at least in the smooth case but my time is over so I did it here so this was further studied I think thank you very much thank you thank you so this last result you mentioned what is this sub t well so I think offer should answer it I think he hopes some theory of theme team nearby cycles no anyway the so anyway la equal ula so using those results so there is an alteration or modification to which banishing cycle the lindon one banishing cycle will be as well and then you can use test curves so to speak to control the enough you have enough to control some constructible shifts when we entered the product of test curves so and then the so I'm not really claiming to well I did not study the so anyway because I did not actually I think the question was about the site so I think you told me that you were hoping for a theory of the team nearby cycles over a general basis is it right maybe this is a mixture of something from different times and so possibly but the problem with the oriental topos is that here you would have to take suitable certifications and well you have also probably to choose for the XI or certainly you have to do something like that there are notions of tenderness for shoes but I'm not really claiming at the moment I'm not no I'm not really claiming this what I'm saying is that there should be a good behavior for duality and perverse filtration so that the idea is to do it over any S by some comological descent there are some so it's kind of but not but of course if you have a situation with good behavioral vanishing cycles then because you know for test some results you should get something for the well-being but this is not probably you can get something but this is not really worked out sorry for I have a question on the name so this factor XI it's called maximal extension factor why it's called maximal extension factor so it takes eta so it takes lambda on eta so you have several extensions so so the j-loar star which is just lambda somehow and then you have also j-loar star which is just lambda but in fact the the XI as a part is not only as a part in degree 1 in degree 1 and degree 0 so it's not it's a complex with the non-trivial actinato it's bigger somehow than the j-loar star so in fact it's some kind of maximal say, dual extension of this lambda so j-loar so the j-loar star and XI are both same dual extension but one is bigger than the other and it seems that in a suitable sense maybe a sense which is not completely clear to me I should confess this is the biggest one maybe the demodule viewpoint it can be seen in other words offer do you have some explanation for this maximal extension other than it is somehow bigger it has more stuff there was this old paper of for instance you mentioned so I remember but I did not look at it recently so I don't remember the words that you are just saying so some of the bigger it has more stuff and more monodromes so the old question so you started with strictly Hansel in DVR Hansel in DVR well you asked me the question before yes so now that I have those twists right you see the 1, 2 and minus 1 then of course this is functorially if you have some eta 0 and eta and take an action by Galois by transportation structure you have to transport something into the other somehow you have a certain compatibility there but you have to transport from eta to eta 0 eta 0 eta 0 eta 0 eta 0 eta 0 then you transport the J twist and you have a twist but I didn't want to do so so then I kept track of this twist and then isomorphism I have is completely functorified isomorphism from something on eta 0 and the two things pull back on eta and I take an automorphism that is compatible so no problem so in fact that gives also in some arithmetic situation that then you can take the graded for monotomy filtration and then no twist it was a twist appear there and then you get that d of gray is gray of d for phi and for psi for phi thank you there is no more questions from fengen any questions in this just to precise what method you are using so in the case of this duality with this J large thing for scheme of finite type over x so in the technique do you use some no no no it is simply you reduce to something smooth and then you use trivial duality so you use a duality assumption like this and f over 3d is so this is a trivial thing so no no you don't need that no you don't need any alteration okay so you so this is a trivial divisage because you check your clarity you checked just for eta and then you have to no no no of course you have on y you reduce to a general you reduce to a general trick of something locally constant and then something smooth and then you reduce to in fact something smooth and then okay so when it's smooth and locally constant then you see the d of after star is smooth after star is after shriek and then you use a junction and so that's that's finished you reduce to eta no no that's there's nothing deep here but you're right that for the b maybe b here b of course is difficult and when in my previous approach I I proved b but I proved that I used the poet outside and then I could get that but assuming PsyT is constructible so actually it's almost as difficult to prove that PsyT is constructible as is compatible with d and so it's not so not so great and then if you the interesting thing would be really to prove for other as you suggested actually for torsion so for constructible complex the reason the reason for this is that you have a sequence 0 or 1 let me write c-1 so that you can understand what it is the dual so you have by using a little bit of water individuality you get this and so a thing which are filled by this gives you the isomorphism making this and this with a shift and so this gives you the torsion so but to do that then you will need alterations and the visage I suppose but also some non trivial results on the point outside with which not so comfortable or to prove generalization of this of this torsion thing other questions? yes I have a short one so I suppose the sort of the same proof works in the complex analytic case Moico PsyT gave a proof with with q coefficients first of all doesn't work with z coefficients and using it's not exactly the same proof you see Bennington original idea uses scattering method so it's a big word but it means that instead of j you take some truncation so you have this infinite Jordan block so you look at this infinite Jordan block and the dual but the dualized n there is a translation and so somehow you can ignore this translation so up to translation this is the so called scattering and this is what what Moico PsyT does so the length n Jordan block you look at its dual but then you will complex in degree minus n zero n and then using this but it's not completely clear how to so it has a size of course but not exactly what I defined here there is no it was our twist there is no such a thing so we'd like to have something over z but it's not at all clear how we can do it other questions? so if not let's take the speaker again