 really much. So it's a great honor. To spik for. Professor Kashiv, as Holmes birthday. So the title is so application of non commutative geometry in that I sense to arithmetic geometry, so these are the first step of. Of this program so, so this is what I just said. So he is derived geometry, but most mostly non in tudi v dobro teži vzataj, da se se načinj upcoming coup dve vse vse zvukaj. Če zvukaj, načinj, ne zvukaj, in tudi z vzutaj, da se počke počke počke počke. Nistaj, ne zvukaj, da se počke počke. Sve, da se počke počke. Zvukaj, ne zvukaj. Ne zvukaj, ne zvukaj. Ne zvukaj. in komoloži in in komutativne motivi. I will try to give, to espljene, o ko je tudi podeljno strategij, for approval blocks conductor conjecture, in z njiha, to razvajte, was espljene z Marko, vzelo. This is joint work in progress with Bertrand Tohen. So the first three things, the first three items there would be very quick, because I'm just recalling, kako Marko Robala včešlja taj morač. Zdaj sem koncentrujno koncentrujno na 4. in 5. pače. 4. pače je taj vse formula for digit kategorij, taj vse formula for non-commutative spasje. Zdaj sem zelo, kako je vzelo, in vzelo, da se vzelo, da se vzelo, da se vzelo, da se vzelo, da se vzelo. V skup omrečno Marketovskim spilje? following the ideas are clear to Kurnovič, and then taken by some of the people there and other people as well is just that you want to study algebra a variety, say over a field K you can study using the derived category associated to two x. So, the perfect derived category of the queried bound derived category, Kaj amputnih kategoriji, kjeljjjži kategoriji, zato jaz se je amputnih, kjeljjži kategorijiinden. Ko sem spetka, da spetka neče, k je nekaj izpelj. In izprončno je vse vizivna, da tudi jah bo k jaze. Zad načine, kaj je zmičaj občaj, v monojda kategoriji, kaj je kaj jaze, jaz da vse je amputni občaj. Kaj so kaže? Vse mjelite res prejševaj na teraklju. Prejševaj na kalitse, koliko te ne zelo, da bo se umeril, več čutimo je več ni bo. Ćudne objezke proj. Iložite. O Slovenijo je nezela v izgledu, na razredom vojpa, na razredom veče. Več nekaz jaz je. Kaj je to, načerš creation? Da weče? The wood is damaged. A and A dual are not isomorphic. So this led to a more general program in non-commutative algebraic geometry, which is study non-commutative spaces on their own. So where a non-commutative space is, by definition, a digit category, perhaps with some hypothesis. This depends on the context. So a non-commutative space would be a digit category. And these are several applications already. And mirror symmetry is something in bi-rational geometry. And non-commutative motifs and k theory. OK, let me give some interesting examples of non-commutative spaces, which are not too far from commutative ones. Commutative means of the form d of x, where x is a variety. But they are still not of the form d of x. So for example, when you have a semi orthogonal to the composition of a variety, or the derived category of a variety, each b1, b2, bn is non-commutative space. But it's not, in general, it's not of the kind d of something. For example, in any cubic form, there is always something which looks like a k3, but it's not always a k3. Or you can take the equivalent derived category, g is, say, in algebraic group, acting on x. And you can also take deformations coming, for example, from the formation quantization of dx. And these deformations are not usually of the type d of a variety. And of course, you have all the focaia categories. This is an example. So what information you want to extract from a non-commutative space? So mainly, I mean, you can speak about topology points, but this is very poor. I mean, this is not good. I mean, for this kind of things, the dj categories are not very interesting. But for homology, they are interesting. And you have a list of homology theories, which we already studied. So actually, homology and variance, studied by these people and many others. And the question that we posed at the beginning of the project is that there are new interesting homology theories for non-commutative spaces. So the answer is yes, and this is eladic homology, which was the content of Marko Robalo's talk of this morning. And probably what is at the end of these slides is probably the most interesting thing I'm going to say, though it's rather vague. So what we realize, somehow vaguely maybe, is that it's very useful to realize when you have a complex coming from a usual algebraic geometry or arithmetic geometry, it's very useful to realize this complex as the homology complex of a non-commutative space. So it's a kind of philosophy or intuition. I cannot substantiate this too much, but I think this is something that should be taken into account. And we saw this in arithmetic geometry, so application to arithmetic geometry, so this is why I put this further specification. So expect also periodic realizations, is periodic homology? I don't know what ends. I mean, I haven't talked about that, but I think that should be something like that. Yes. I mean, as soon as you have, I mean, this machine, as you saw in Marko Rualo's talk already, as soon as you have a, as soon as you have a periodic realization from commutative motives, as soon as you have that, you compose it with the, and so you have something then, of course, you want to say, prove some theorems, yes, yes, yes. I think you have it, yeah, because you can take crystalline or think periodic, yeah. As soon as you have something from realization, from Vojbocki category to, yeah, you compose with the machine that Marko proposed, and it gives you something. Then, so very quickly this is, I'm just recalling super quickly the thing that Marko was explaining today. So we fixel, which is different from the characteristics of A, and the theorem that there is, there exists an erladic homology, so an erladic realization for any ADG category, which is this HT with QL coefficients, which is a Q erladic shifts on S. And for example, if you have a proper map, finite type, and the source is goods, it's quasi-compact, quasi-separated, then you take this recover periodic, two-periodic etarchomology of X over S. This is we already saw. So the notation is the same that Marko used. QL beta will be this sum here, QL twisted by N, shifted by 2N. And this is, I view it as a commutative ring or infinity ring in erladic shields in the monoidal category of erladic shields over S. And not this homology as erladic homology or erladic realization, as Marko explained, is two-periodic up to a T-twist. So there is H, is shifted by 2, is H, but not quite because there is a T-shift by 1 on the left. So in other words, this is a QL beta module. So how do we construct erladic realization for these categories? So you already explained. So there is a first step where you go from DG category. So you want to go from DG categories to erladic shields. So you break the construction in two. You go from DG categories to commutative motifs. And this is the construction that Marko explained. I just use a different notation because I put a do up there. So basically you send a T, a DG category, to the functor on smooth schemes over S, which send the smooth scheme Y to K of the K theory, which I mean the non-connective, a motoping variant algebraic K theory of the DG category T tensored over A with the DG category of perfect complexes on Y. But of course, I mean, you need, this is just a naive definition. To make it infinity functor, you need a lot of techniques, but Marko already explained this a bit. And also said that this functor is laxmonoidal, so it sends DG category S to modules over the realization MS dual of S itself, which is BUS by definition, which is the spectrum that represents non-connective or motoping variant algebraic K theory. And then you are, now you are in commutative models, then you can apply any realization you want. For example, here we are interested in eladic realization, so we use this eladic realization, which is based, actually define, we only give an infinity categorical enhancement of construction by Siziski Degliz and IU. And then you compose these two things and you get a functor from DG categories to modules over the, these commutative realization small R of BUS, which is QL beta, as Marko showed. So you go from DG categories of S to modules over QL beta in eladic sheets of S. And this is just the restainment of the previous, the previous thing that you compute relative eladic homology when you, when you evaluate these unpaired for vex. Okay, and DG categories of singularities was already defined, so I will give quickly, will be quick on this. So a Landau-Gisburg pair is just a pair of a scheme X over S plus a function on X, global function with values on OS. Then you take the derived fullback, so the derived zero locus of this map. And for any scheme, quasi-compact, quasi-separated, you define the absolute DG categories of singularities, which is the quotient of coherent, bounded, divided by perfect. This has nothing to do with the base S, this is absolute. And then you define the relative DG categories of singularities, so the DG categories of singularities of the pair, which is, you take the kernel of the singularity categories of the derived zero locus, push forward to the singular singular category of the ambient space and you take the kernel. And you can also describe this as a quotient, as there were coherent bounded over the derived zero locus divided by coherent bounded, which become perfect when you push forward to X. As Marko recall, I is LCI in the derived sense, so this definition makes sense, so it's true that this sense perfect object, perfect object, the push forward I lower star. And if X is regular, there is nothing, I mean the relative singularity category is just the absolute singularity category of the derived zero locus. And also if X is regular and the map F is non-zero divisor, then this pullback is just, is not the derived pullback, it's an ordinary pullback, I mean they coincide, so X zero is just the usual scheme theoretic zero locus of F. So you can do a computation, so the singularity category of S with the map zero, so of the Landau-Wiesbel model where you take the base S and the function is zero, is just the perfect modules over A, A with U, U minus one adjojnj, where U is a variable in degree two. So this is periodicity, this is why these sing are periodic in this sense, two periodic. Marko described a version of Orlov's theorem that you have an equality, I mean equivalence between singularity categories and matrix factorizations. And okay, matrix factorization, you already defined what was this category, so I will skip this, but you can read it. And X is still regular, but F is allowed in this comparison, it's allowed to be a zero divisor. And as Marko pointed out, I mean these two funtors, matrix factorization and singularity categories are monoidal and the Orlov equivalence are with appropriate hypothesis, it's an equivalence as a lux and monoidal natural transformation. Okay, so we are in the setting of this morning, so you fix an incident rate, fix a new firmizer, so BK is a fraction field, small k is the residue field and you have these open and closed immersions. So whenever you have a scheme over S you base change along these and you get what I call X sigma is a special fiber, X eta is a generic fiber and you note that if you have X zero is derived zero locus of the composition, then X zero is the same thing as X sigma, so as a special fiber. So for any sheaves on X, erratic sheaves on X, you have the vanishing cycles with respect to the map P, which is a Galois eta bar equivariant sheaves, erratic sheaves on a special fiber. I recall that this is the definition of inertia, so inertia is the kernel of the map from the Galois of eta bar to Galois sigma bar and the theorem that Marco explained this morning is that if you take X regular, P is proper and flat finite type and the base is strictly encelian and excellent, then if you denote this map, so sigma i is the closed points inside the base, P sigma is the projection from the base change to the closed point, then there is a canonical equivalence between, so on the left hand side there is a vanishing cycles but two periodized, shifted by one with the invariance under the inertia and on the right hand side you have the erratic realization and erratic homology of the singularity category. Remarkz is that strictly encelian is just because it's easier to state the theorem but you don't need it. Just that has to be excellent and encelian and both sides are naturally modules over these, so the invariant of QLS beta, homotomy invariants with respect to the inertia which by computation Marco showed is the same thing as this algebra written there and this is an infinity algebra in erratic shifts over S and the equivalence of the theorem is actually an equivalence of modules over this. I think this came up in the discussion after the Marco's talk. We think actually that this, you see I mean this is an equivalence of vanishing cycles and singularity categories after taking erratic realization but we think that actually it's also true before taking erratic realization so it's true that the motive, the commutative motive associated to sing X of F so m dual of sing X of F in previous notations should be equivalent to iub's motivic tame vanishing cycles. And we just have to work a bit because the way motivic tame vanishing cycles are defined it's not an object in this category in SHS but it's a diagram in this category so we have to make sense of this statement in a more precise way, but I think it's correct. Okay, so now I want to define, so this was just a summary of Marco's talk and then I want to move to trace formulas for dj categories and then apply this to Bloch's conjecture. So first step is something that was already known but we do it in a slightly more general situation so again A is our DVR, Vase and Celian DVR and we fix B which is an infinity algebra over A, okay? And we always work relative to A so I would just write hql without referring that everything is over A. And not that if you take the ellatic realization of B, so B is an infinity algebra so you can view it as a dj category and if you take the ellatic realization this is an infinity algebra in ellatic SHS and now you take dj categories which is linear over B so it's a module in dj categories over A over B which is an infinity algebra, okay? So think about this as being B linear dj category. Then if since t is over B then the ellatic realization is a module over hbql which is the infinity algebra in ellatic SHS and we can consider this composition so you go from dj categories over B, you apply the realization, you arrive in ellatic SHS and you take the right global section, you arrive in ql complexes of ql vector spaces and then you apply Eilenberg-McLean construction, you get the spectrum and this composition is like symmetric monoidal. So then by the universal property of algebraic theory in non-commutative motives so algebraic theory is going from dj categories over B to spectra there is a unique natural transformation up to a unique in the infinity category sense so there is its unique up to a contractible space of choices and this is what is called the Charon character goes from algebraic theory to this ellatic realization but viewed as a spectrum and it's a laximetrical monoidal transformation of laximetric monoidal functors both k and realization of h are laximetric monoidal functors and this transformation is laxmonoidal. So did you have a characterizing operative fit when you say it's unique? Yes. It's unique because basically in non-commutative motives the mapping spaces are given by algebraic theory, this algebraic theory. Mapping spaces between two dj categories given by algebraic theory of t tensor t op, something like that. So this is the universal property I meant. There is no trivial choice in viorex. No, it's a universal property so whenever you have something which is a laximetric monoidal functor then you have a functor from k to this, that's it. So k theory is representable in non-commutative motives and the mapping spaces are given by k theory which is essentially the same state. Okay, so a consequence which is not super interesting is that by definition essentially you have a built-in growth of the remark rock theorem here just by functoriality. This is just a curiosity, it's not super useful. And, but the interesting thing is that we can use this channel character to define, to prove a trace formula for dj categories. So, again, the setting is as before, we fix b which is in the infinity algebra over a then I will explain why we want this extra generality in a while. So again, dj categories over b, this is a symmetric monoidal category because, or a infinity monoidal category because b is infinity. And then you say that the dj category over b which is b linear is smooth and proper if it is dualizable in this infinity monoidal category. So it has a dual in the sense of, say, symmetric monoidal categories. Okay, but remember that Marko explained that the ladic realization functor hql is only lax monoidal. So we don't, so we need to give the following definition that we say that as smooth and proper dj category is over b is tensor admissible if the Kulnet map is an equivalent. So this is the map given by lax monoidality and we ask that, we don't say that h is symmetric monoidal. We just say that for this t and t op, this thing works. This is very special. It's much less strong that saying that h is symmetric monoidal. Symmetric monoidal. Okay, so the Kulnet holds for t op and t. This is what it means. And not now that if you have this property for t, so admissibility, then you take h, you realize this dj category, then this becomes a dualizable object in hbql modules. And the dual is just the realization of the dual of t, which is t op, because t is smooth and proper. Okay, now the trace formula, okay, let, so fix this t, so t is smooth and proper, so dualizable over b, and it's admissible for this tensor product, so for this realization. And you fix an endomorphism over these dj categories over b. So giving this endomorphism is the same thing as giving a graph, which is just a perfect complex of t op tensor t modules, but the product is made over b. And so we can consider a composition given by the graph, followed by evaluation. And remember that there is an evaluation because I'm supposing that t is smooth and proper, so it's dualizable. So there is a co-evaluation, evaluation, et cetera. So I compose the graph with evaluation. This is the same, so it's a mat from b to b of b module, so it's a perfect b modules which I denote by hh, this is essentially the oxyl homology object associated to f. And so I can take the class of this perfect complex in k0 of b, so algebraic k theory of b. But I can also, and I call this the hh theoretic trace of f. I can also do, I can also apply h, sorry, h, the realization hql to this, and by admissibility, we can play the same game with h of fql, because now h, the realization of t is dualizable, so I can also do the same thing. And I obtain a map from the realization of b to the realization of b, which is just when I take the spectrum associated, it's just an element in pi zero of the associated spectrum of hb. And I denote this element by the trace over b of the realization of f. And I call this the erladic trace. So we have an hh theoretic trace of f, an erladic trace, and the non-commutative erladic trace formula tells us that these two traces are related by the term character. The theorem is this one. If you have a digit category over b, which is both improper and admissible, then you take the term character level zero of the class of the hh theoretic trace of f. This is the trace of the realization in the pi zero of the spectrum hbql. So there is an easy case when you have canonical map from z to k zero b and ql to pi zero. When these maps are isomorphized, then churn is just a natural inclusion, and it formalize just an equality of erladic numbers. And an intuitive way to see this formula is that the left hand side, so chau zero of hh, is the virtual number of fixed points of f. So because they are intersection of number of the graph of gamma f with the diagonal. So in some sense it's counting the fixed points in a virtual sense. So let me give you the corollary here. Now suppose that you work over a finite field, so now your infinity algebra b is equal to a, so it's commutative, and it's fq. And t is the perfect digit category of proper smooth deline map for stack over a, so over fq. Then you have this growth and decay left-shed formula that counts the points of x over fq, where on the right hand side is like the growth and decay formula, but you replace the talc homology with the orbital homology, which is actually the erladic homology of the inertia stack of x. Everything over fq bar, of course. So this is a slight generalization of, I think this was already known, of growth and decay formula. Now it comes to the problem. So far I was talking about an infinity algebra, so b was, so the base was this DVR, then I was fixing some, you didn't understand why, because I didn't tell you, there was an infinity algebra over a, but the problem is that, so remember that the trace formula holds when the dg category is smooth and proper over b. We cannot use this infinity, we cannot only work with infinity algebras, we need to work with b, which is an e2 algebra to make these works, but I will explain carefully this point now. You can, the first thing I want to tell you is that this formula, the trace formula, also works when b is just an e2 algebra, over a. And the problem is, there are some problems, that it's not an easy consequence of the previous theorem, because in this case, dg category is over b, is no more a monoidal category, because b is only e2, is not e3. So it's not infinity in particular. So to define dualizability, what it means to be smooth and proper, which was, by definition, being dualizable, it's a bit tricky, but you can do it. And then the trace formula, you obtained a trace formula in the e2 case, so which basically say more or less the same thing, so b is now an e2 algebra over a, and then in the next slides, I will tell you why we need an e2 algebra, we need this hypothesis. T is a dg category over b, which is smooth and proper in an appropriate sense, and it's admissible. Then for any endomorphism of T over b, you have this formula, it's the same formula as before, but this is now, this is now inequality in the pi zero of the spectrum, but the spectrum now in the pi zero of the spectrum, it's easier to write it this way. So this is the etalc homology, h zero of h, h of the realization over ql beta, so hh is the axial homology. So remember hbql, now it's a priori in e2 algebra over ql beta, and you take hh, viewing this as an associative algebra, it's just hh, it's not hh e2, it's hh e1, so it's usually the axial homology. So there is this light change because of trace, defining trace in this case where dj category of b is no more monoidal, it involves axial homology. Okay, this is the formula we have and why do we care about allowing b to be just e2 algebra? Well, the reason is very easy if you think about Markov's talk. So in the approach to Bloch's Conductual Conjecture, we want to take this dj category t to be singularities of x with respect to f, okay? And this category, I mean, we know it's too periodic, okay? So it has no chance to be proper over a base which is not itself too periodic, it's completely impossible. So in particular, it will not be proper over the base ring a, which is a usual commutative ring. So we need to modify, so we need to find an algebra, an a algebra, something like that, on which this singularity category is proper. And, yeah, exactly this, okay? And luckily, there exists a natural algebra, I will describe it in a moment, b, such that t is this singularity category is a bdj category, which is proper over b and also smooth and admissible, but that's another story, okay? The difficulty here is proper, okay? That's why we need this fancy e2 stuff. So I want to describe the application to blocks conductor conjecture, so let me state the conjecture. So we start with an insilient trait with perfect residue field k. You pick a scheme, which is over s proper and flat with smooth genetic fiber, and then you assume also that x is regular. You fix l, which is different from the residue characteristic, and then you have this formula here, which is also appeared in Saito's talk. So the left-hand side, so what is this equality? So the left-hand side is an intersection theoretic term, and the right-hand side is analytic shift theoretic terms. So kajoveli is the eladic color characteristic, swank over x k bar, bk is the fraction field, is the swankonductor of the Galovar representation given by eladic homology, and the left-hand side is the degree in the Chao zero of k, which is just z of blocks localized self-intersection of the diagonal. So the negative of the right-hand side, so the different of value characteristic minus one conductor is called the arting conductor of this p, so of x over s. So this is a, I found it as a fantastic conjecture because it's really a wild generalization of Gauss Bonnet. So as you see, I mean, it's supposed to describe the change of value characteristics, change of topology with special and generic fibers using some representation theoretic ingredients, which is one conductor taking into account the wild inertia, and most importantly the intersection theoretic number, which is the self-intersection of the diagonal. So some known cases are the relative dimension zero, it's just the conductor discriminant formula in algebraic number theory. Relative dimension one was the proved by block himself when he proposed the conjecture in 87, and then the most general result I know is arbitrary relative dimension. The conjecture is true by Cato and Saito 2004, supposing that reduced special fiber is a divisor with normal crossing. I think this is very much related to the fact that maybe Saito can comment on this, that they use log geometry too. To prove the conjecture. So these hypothesis I think is related to that. Anyway, so let me also remark that this block conductor conjecture implies that the linear conjecture for isolated singularities is proved by Olga Gauss. So the theorem that we have, it's that block conductor conjecture is true when the inertia acts with unipotent monodromy. No hypothesis on the reduced geometric fiber. So in some sense, so first of all let me tell you why it's a theorem star. It's a theorem star because we are double checking the details these days. So that's why I want to be prudent on that. Anyway, under unipotent monodromy is more general than Cato Saito because we have no hypothesis on the reduced special fiber but of course we have this hypothesis of unipotent monodromy. So in the, when the action of the monodromy is unipotent, this one conductor is zero, so the formula is this one. So self-intersection of the diagonal is equal to the difference of heladic or the Euler characteristics. So let me give a sketch of the proof. So again the setup is the one of the theorem. So p is the map, little k is the fraction field, bk is the, sorry, little k is the residue field, k is the fraction field. And the first thing we do is that we want to use non-commutative geometry so we do the following. So we compose, as Marco explained also in the other theorem, we compose the map p with a uniformizer and we get so a Landau-Ginsburg model given by x over s and the composition, which is a map to a1 over s and we take singularity of this Landau-Ginsburg model. This is our non-commutative space. Ok, so we want, basically what we want to do now, we want to prove that the Euler characteristic of this, of the realization of this is exactly the difference of the two Euler characteristics. Generic fiber, special fiber mass. Ok, this is the aim of the proof. So, but still we need to find an algebra. So this singularity category is not proper, is never proper over a. Over a is back, s is back of a, no way. So we need to find an intermediate algebra on which this t is proper. So what if you look at the geometry, actually it's, there is a, at least there is a t is linear over some algebra, ok, b, which is this algebra here. I think the, basically, ok, let me, basically what I do, I have this close point, spec ok into s, I take the nerve of this map and the nerve of this map is a group point by definition, I take the algebra of this group point. It's pretty clear that, I mean intuitively that this guy acts on t, right? And this algebra is what I call b plus and it's described in that way. I mean, when I say tensor product and the morphism, everything is derived, so I don't write it. What is the lean conjecture? The lean conjecture tells you that, so if you look at the definition of b plus, b plus is the endomorphism of k tensor a over k over kk. This is by definition the oxydomology of k over a and tells you that if you have a, if you have an e to a, an e n algebra, and if you take h h, so homology, oxycomology, then this is e n plus 1. In this case you take something which is e1, you get an e2 algebra. This is why another way to see it. But there is, in this case it's completely, it's clear, right? There is, it's endomorphism, so there is a composition, but it's also a group point algebra, so you can multiply the, right? And these two operations are compatible, so this is why it's an e2 algebra. But of course, I mean, if you want to prove it that it's an e2 algebra, it's better to use this. The lean conjecture, it's not a conjecture, it's a theorem, that's why I'm using it. It's called the lean conjecture, but it's a theorem. And it's pretty clear that by the geometric definition that b plus x on t. Actually, more is true, because you can describe this b plus as e1 algebra, so it's associative algebra, it's just k joined variable in degree 2. But this b plus is not infinity. It's not infinity, it's easy to see that there is a, if you, for example, if you go from from an algebra to Poisson algebra, you see that the bracket is is non-zero, so it's impossible that it's even an equivalent, so b plus and k u are equivalent as associative algebra, but not as e2 algebras. So it's really something. Of course, if you are in inequicaracteristic, this is true. They are equivalent as infinity algebra, but it's important to take into account the mixed characteristic case. What do you mean? When you are overfilled, in the geometric case, this is true for infinity, there is no problem. But in the mixed characteristic case, it's completely false. So they are equivalent as e1 algebras, but b plus is not, it's only e2. It's not infinity, so it's not commutative in particular. And there is an invariant that detects that it's easy. Anyway, this b plus is e2 and as an associative algebra, it's easy, it's a polynomial algebra in a generator of degree 2. Then I invert this generator as an e2 algebra and I get something which is in e2 algebra and which as an associative algebra is, obviously, this K u, u minus 1. But remember that singularity category, as Marko explained, it's two-periodic, so it's an alge, it's a singularity category, it's a dc category, u minus 1 linear and so it's easy to see that this t is actually b linear, so it's a module over b, which is an e2 algebra in dg categories over a. So it's a b linear dg category and where b is this thing, which is the periodized version. I also invert u. So I have really the analog, so this is really two-periodic. T is a two-periodic category because it's b linear. Now we... This is completely useless, sorry. Now observe that this is a dg category over b and this is, as I told you already, it's a refined version of two-periodicity and the other result you have that there is a natural equivalence of dg categories over a, which is the singularity category of pi, underline pi is the composition of p with the uniformizer, opposite tensor b, singularities of pi is the same thing as singularity of absolute singularity category of the tensor of the fiber product. Ok, this is something you have to prove, it's not obvious. And in these equivalence you see on the left-hand side you have the... you can view singularities, singularity category as a b-module over singularity times singularity and this goes to the diagonal on the right-hand side. From this result, so for the second point here, you can deduce that this T, which is now we know that it's b linear, it's also smooth and proper over b. And the hard point, I mean the point where you have to use the result above is for properness. Ok, now this singularity category is linear over e to algebra b, which is the one described and it's smooth and proper over b. Now you work a bit and this is not easy to prove that it's also admissible, so that basically the realization, eladic realization factor on this T op and T it behaves like a symmetric monoidal factor. Ok, now you have, at this point you have all the proper, so T is smooth and proper and it's admissible over b. So we can use the trace formula for these categories. And we use it in the easiest case when the endomorphism F is just the identity of T. So you put this trace formula for the identity together with the result that Marco was explaining this morning, so the relation between vanishing cycles and singularity categories, you get the Chao zero of oxydomology of this T over b is this difference of alegrity. So it's the right-hand side of Bloch's conductive projection that one conductor vanishes under hypothesis. And basically this formula follows from some easy manipulation plus the fact that it's known that Euler characteristic of the vanishing cycles is this difference of Euler characteristics by this specialization sequence. So it's an elaboration of this thing. So the hard work is to prove that you can apply the trace formula to this singularity category over b. So this is a conductor formula so it tells you that this difference of Euler characteristic the arty conductor in this case minus the arty conductor is something, is some chi or Chao zero of something. So this is already a conductor formula. But to get to Bloch's formula you need to prove that this is Bloch's intersection number. So we prove this and then we conclude. So I'm almost finished. I wanted to make some comments on the proof in this unipotent monodermic case. So as I said, the admissibility is the hardest part. Its faults, admissibility is completely faults if the action is not unipotent. So because if the action is unipotent you can commute taking h invariant under inertia with tensor products, otherwise it doesn't hold. So you need to do whether this means that you need to do something to adapt the proof if you want to prove the general case. When the action is not necessary unipotent. And then we have to say between the churn zero of chi of h of t over b with Bloch's number is was already known in by twisted the ram complexes by ideas of conservation and then more recently and more generally by Pregel. And of course you need, we have a strategy for the general case, which means basically the mix characteristic case. So what about the case, the general case? So when you the monodromi action is arbitrary. So of course the first thing so the first thing in your service that this proof doesn't work because you have no more admissibility. Are you okay? You can naive way I mean use grothendic unipotent monodromi theorem to say that you can base change through a ramified covering such that the inertia of the base change is it has an action of unipotent in unipotent action. The point is that the base change is no more regular so you don't even expect the Bloch's formula holds in this case. So you don't know what is a conductor formula in this case in general. So our idea is that we can use in some sense local computation. So we choose the h-hyper cover of the base change and we show that in some sense it's truncated. So it's not going to be truncated as an hyper cover but for our purpose because we want to evaluate the homology this becomes truncated. So then you have a finite sum and you can use the trace formula for the singular pieces and you can try to do this. Or you can just do the base change so suppose g is the group the quotient of the Galois group of the separable closure of bk of the fraction field that defines the base change the ramified covering s prime to s. So call it g then I can consider of course xs prime over s prime and I can also consider the quotient stack xs prime over g which map to the quotient stack of s prime over g. And you can try to prove a trace formula for this sticky version. In this case these are stacks so the trace formula is not already proved. It's not there, we have to prove it. So one of these two techniques or the composition is to solve the problem in general but of course this is very much in progress. And I wanted to finish with mentioning few applications I mean other applications that we have in mind is getting wilder and wilder as soon as I go down so the first thing is that what is the block conductor formula when x is not regular so what is the formula so maybe it's not true as it stated and with this approach we can say something because we know that in the non-regular case we can use a quotient matrix fertilization so we can at least guess what the formula is and maybe try to prove it. Then you can ask what is the conductor formula when x is a formal scheme or a stack. We know that vanishing cycles only depend on the formal completion so this is a meaningful question and we expect to need some rigid geometry to say to guess what is the conductor formula in this case so even more wildly when suppose now the base is not a DVR of dimension so not a DVR but it's evaluation ring of dimension bigger than one so we are working on higher local fields for example the bk becomes a higher local field so of course here the problem is that these rings are non-notherian so you need to to have a theory of commutative so more moral bojvockim motifs over some base s which is no more noetherian so this this has been studied but there are no definite complete results so you can then of course you can globalize this you can think of a work of a basin which is no more evaluation ring just a scheme of some dimension anyway can ask what vanishing cycles maybe we can make sense then but what is blocks conductor formula what can you expect for this formula in this case so here the idea is that we can go from order characteristic to complexes so basically the equivalence blocks conductor formula is an equivalence of inequality in numbers so we can go from numbers to complexes which is not so difficult to imagine how to do this but then you have to work with basically we want to work with families where locally at the point you know what is the formula which is given by blocks formula so we need to shiftify in some sense this singularity category of x over f and we expect to that Baylien Sonadelsk could be useful there are other conjectures like Nikke's conjecture that Nikke's conjecture is a conjecture concerning essentially the rational volume of a proper variety over a field and this rational volume essentially counts rational points and this is expressed as a trace formula in the conjecture so we think we can we can replace the trace formula that it has with the proof that the trace formula that it has is the same as a non-commutative trace formula and then this is very quite a long shot I mean we think that maybe there is a way to formulate not to prove I'm not claiming anything about proving this but to formulate the weight monodernic conjecture in a non-commutative setting so say that this non-commutative motive of seeing is pure over b in some sense but of course we need to make very precise but it means pure over this e to algebra so thank you very much did you have an example of a variety where you know that the homology is unipotent without knowing in advance that it's a mistake no, I think there is a conjecture, right? let's say that no no, I don't know I think people don't expect that there is a homological criterion ok, ok, no I don't so the question your algebra b was only e to sorry, I thought you would need e infinity to be able to to use it somewhere so why is that I mean I explained for the infinity to state that the trace is easier it's like for symmetric monoidal category but this is e to so the problem is that when it's e to so the evaluation it's ok goes from 1 to some tensor product the co-evaluation goes to h h so actually the homology and then when you want to compose you have a problem because they are not the same object so when you compose you can compose for dg categories infinity to categories which are of the form like you have fixed symmetric monoidal category take algebras, the infinity to category of algebras over there so which means objects are algebras bimodules and morphism bimodules then you have a special duality that makes it possible to define a trace to compose the graph with the evaluation it's a bit tricky but it doesn't work in general for infinity to categories it's not a question of adjoint you need some hypothesis I was not so clear on that point so you can do the same but it's more complicated for an e to algebra and the point is that we only have an e to algebra to work with so we have to prove a trace for that Other questions? Gabriel again