 So, let me start by, so this lecture, lecture number two. So it's titled, it's applications to a non-commutative geometry. Okay, and let me just recall you the theorem from last time. So last time, so little k was a perfect base field of characteristic p, p greater or equal than zero. And the theorem, the very last theorem was that if we start with a smooth proper skin, then we can consider conjectures over x, or they are non-commutative counterparts over the perfect complexes on x. And this are in fact equivalent for any conjecture that I described, the one of Grottyndy, of Levotsky, of Bailinson, of Bail and all these variants of the take conjecture. And the goal of today's lecture will be to explore this equivalence towards non-commutative geometry, and next time tomorrow we'll explore this equivalence towards classical geometry. Okay, let me start with the definition, the definition of what is an additive invariant. So this is simply a functor on dG categories towards an additive category, so d, it's additive, and satisfies two properties. So first of all, it sends morit equivalences, so if you have f a morit equivalent, then this invariant send it to an isomorphism. Okay, so it's defined on all dG categories, but in fact it only depends on the underlying morita class. And the second property is suppose that you have two dG categories, a and b, and you have a bi-module, so a b bi-module, then out of this data you can construct a new dG category, which I denote by t, a, b, and then b, the bi-module, which is simply you put your original category a and b, you do this d joint union, and then on this direction, or HOMS in this direction, you use the a, b bi-module, and you don't have HOMS in this direction, so you see that in this category you have an inclusion of a and b inside of this category, and you ask that these two maps, these two inclusion maps, that they give rise to an isomorphism on your, on your additive category d, so this makes sense, you are going towards an additive category. And there are a lot of examples of additive invariants, for example, algebraic k theory, cyclic homology, topological ocean homology, etc., and etc., and also all its variants, all its different variants give rise to additive invariants. And I should mention, well, this category is somehow a categorification of upper triangular matrices, so you somehow, this property is telling you that, in fact, your invariant does not see extensions, it doesn't depend on the extension that you have from a to b, it reads all these categories just as a direct sum. And now you can have, for the existence of a universal functor with this property, and that functor actually exists, and it's very concrete, you can look at the following very concrete category, so H and O little zero, so this is defined as follows. So you, on objects, you just consider all dg categories, okay? On the home spaces between two dg categories, what you do is you look at the k zero group of a certain category, wrap AB, what is this wrap AB? This wrap AB is by definition a subcategory of those bimodules in the drive category of AB bimodules that are finite on the right in the sense that if you look at the corresponding bimodule b on the right, it is compact, okay? So you have this finiteness assumption on the right, and the composition in this category, it is just a tensoring bimodule. It's induced by tensoring bimodule, okay? Something very concrete and comes equipped with a functor. So we have the following functor towards this category, let's call it U, U of universal, that does nothing on objects, the category goes to itself. And if you have a dg functor from A to B, what you can do is to look at B, where B acts on the right on itself, and xA acts on B on the left via this F, so I'll denote like this. And so this is a AB bimodule, and you look at the corresponding class in this k zero. And the theorem, so the first theorem says the following, so if you have given any additive category d, then what you can do is look at functors which are additive, so additive functors from this additive category towards d. Whenever you have one of those, you can pre-compose with U, and that gives rise to an additive invariant going from dg categories to d, and this induced functor, it's an equivalence. So in other words, this is saying that this very concrete and precise functor, it is the universal additive invariant, okay? So all invariant additive invariants will factor uniquely throughout this one, and since yesterday we talked about the Schoen character, let me just mention, so we actually use the Schoen character to define a homological equivalence, so here is just a remark saying that, well, the Schoen characters follow automatically from this theorem, because suppose that you have this, for example, this additive invariant given by periodic cyclic homology, so with values in z to graded capital, little k, so it's the same base field, then it's additive, so it will factor throughout this universal. We have a question, I can't read which one is the universal additive invariant, I think on the left-hand side. You cannot read, you don't understand which one is the universal one? Yes, the universal additive invariant. Yeah, the universal additive invariant is this functor. Possibly, possibly small font, something like that. Does it have a name? Another question. Yeah, I'm calling it U of universal additive invariant. Is this very concrete functor? Any other one will factor on this one in a unique way? And my own question, sorry, and this U also satisfies universal property with respect to localization sequences of digit categories, I guess. No, no, just no. This talk is entirely in the pure setting. I'm not considering this kind of phenomenon. Okay. So what it satisfies is this property here too, and this property here, it's another way to phrase it, is that you have a split shortest sequences become split shortest sequences. We are in the pure setting, okay? So everything splits here. Thank you. And so here you see it's additive, so it actually will factor. So the only remark is that then you can look at the induced morphism between the most simplest object that you have, which is the object base field toward your favorite digit category, and this functor gives you a morphism in this direction towards HP of the base field and HP of your A, but then you just observe that while this left hand side is the K zero of A, and this right hand side is the periodic signal homology, the positive part. And this induced map, it is the churn character. So you get for free the churn characters out of this universal property. Just a remark, because I've used these churn characters in the previous lecture. Can I interrupt with a quick question? Yes, of course. So this universal gadget is on all DG categories. All of them now. So it includes things like the DG category of coherent sheaves on some scheme, for example. So in that case, the K naught would just be the K naught of coherent sheaves. Yes, it would be the G theory, the G zero theory. Oh, okay. Great. Thank you. And in the next blackboard, I'm going to make a restriction to smooth proper things. Yes. Oh, great. Thank you. So now I'm going to restrict myself to the smooth proper DG categories that I defined yesterday in the sense of Maxime and this gives rise to this notion of non-commutative churn motives. So they are defined as follows. So I write M churn over my base fields. By definition, they are the important completion of what? And so I go inside this category here, inside this category, and I just look at those objects in here that are smooth and proper. And now I still have a filter, as Marc was saying, but now defined solely on the smooth proper DG categories towards this category. And some facts about this construction. Well, first of all, the home spaces in this category between any two objects. So in that category, it was given by this K zero of rep. But if this is smooth and proper, this becomes just the K zero of bi-modules. Second observation is that this category, it's in fact rigid tensor category. So if you take an object and its dual, it's given by the opposite DG category. And moreover, if you have any ring of coefficients, any commutative ring of coefficients, you can look at non commutative churn motives with coefficients in that ring R. And so the important, that's the facts and another important fact that will be key for us today is that if you have any non commutative churn motives with rational coefficients, then we can phrase the conjectures from last lecture to this one with C being any one of these conjectures. So in particular, in such a way that when you do the conjecture for U of A, then you get the conjecture that we defined last time for A. So in regard to the conjecture here, we actually need to work with the integral non commutative churn motive. But this is a bit technical and will not play a role in this lecture, so you can skip. So what does this tell us? Is that in fact now what is the idea coming out of this fact? Is that now if you would be able to understand the non commutative motive with rational coefficients of something, then we maybe could say something about the corresponding conjecture. And this is what I'm trying to explore today. So let's see some computations. So the first computations, so let's talk first about twisted schemes. So I have this result with Michel van den Berg saying the following. So suppose that you have a smooth, proper scheme and you have a shift of Azumai algebras, okay, over X. Then it turns out that the non commutative motive. So one thing that you can do is you can look at the non commutative motive of X with rational coefficients and also the non commutative motive of X but twisted by F by this shift of Azumai algebras. And it turns out that they are isomorphic. So the result is of course much more general. You don't need smoothness and all this and also it holds as long as we invert the rank of F. You don't need actually to go to Q coefficients. And there's a corollary, immediate corollary is that the conjectures are in fact insensitive to this twisting. So when you twist the conjecture is just the conjecture of X itself. So here I'm using the theorem from last time. So when you twist a scheme in terms of these conjectures nothing change. And for example, if we consider division algebras, so suppose that you have a division algebras, a finite dimensional division algebras, so over our base field. So you can look at its center. So its center will be a finite dimensional field extension. So this is the center. And then as a particular case, so you know that this is a central simple algebras over L. So you know that non commutative conjecture of your division algebras, it is just the classical conjecture on spec L. And so as a consequence, all these conjectures so you can get as a corollary that this conjecture for D always holds. I'm not okay. So everything works for division algebras because the conjectures are just conjectures for finite field extension. In that case, the classical ones, all of them are known to hold. So it's just an example. Another important computation is what happens with semi- autogonal decomposition. So suppose it's a fact. So it's somehow more or less a reformulation of additivity that I defined five minutes ago. So suppose that you have two DG categories inside another one. And in such a way that when you take the H0, when you pass to the H0, you get the semi- autogonal decomposition in the sense of Bondal-Orloff. So these categories generate the entire category. And there are no homes in the graded sense in one of the directions. So there are no homes in one of the directions. Can I think about these categories of non-commutative motives as the exact same things as in the paper of Gloombier-Gebner-Tubuade? Yes. I mean, there we phrase everything in the language of infinity categories. And we were using slightly different properties. We were using, as you mentioned, localization. And imposing a category that, in that case, would be triangulated, saying that these short exact sequences of Grinsfeld become triangles in this triangulated category, et cetera. But here I'm just in the pure setting. Okay. Because I want to work with smooth proper things. Why? Because these conjections were formulated for smooth proper things. I don't need to go any further. So if you want this thing embeds in what you are saying. Okay. But so a reformulation of additivity is just saying that when we have a semi orthogonal decomposition, the motive is even integral. The motive of A is the motive of B direct sum with the motive of C. And so it's also true with rational coefficients. And as a corollary, we get as a corollary that the conjectures for A, it's equivalent to the conjecture for B plus the conjecture for C. Whenever we have this situation, this is what happens with the conjectures. So let me give an example. For example, of Calabi Al DG categories. So suppose that you have an hyper surface in PN. We have a question from Mark. Is draw the full subcategory of dualizable objects in the larger category Homestar because it's difficult to read the boards. There, I imagine, no? Maybe I can just say it. So you have this category where you have the full K-naught of rep is the HOMS. And then you have this smooth proper guys inside there. Is that the whole category of just the dualizable objects in the larger category? If you look at DG categories up to Morita equivalents, the dualizable objects are the smooth proper ones. And so what I'm doing is just grabbing those and taking the full subcategory of the one with rep restricted to those objects. Great. Thank you. So it's basically the answer is yes. So let's take an hyper surface of degree less or equal than N plus 1. In that case, there is this beautiful work of Kusnetsov saying the following. So if you look at perfect complexes on this hyper surface inside of it, you can look at these line bundles going until O of N minus the degree of X. And then you can look at the orthogonal complement that appears here. So what is missing something pretty abstract. But it turns out that this category, it's in fact Calabiow of dimension of fractional dimension. So it's Calabiow of dimension N plus 1 degree of X minus 2 and then over degree of X. So another way to explain what I'm saying here is just what I'm saying is that if you look at the serf filter, so any smooth proper DG category comes equipped with the serf filter. So if you look at the serf filter and you compose it this number of times, what you get is the suspension filter composed this number of times degree of X minus 2. So as a consequence of this semi orthogonal decomposition, you get as a corollary that the non commutative conjecture for this piece here, it's in fact equivalent to the classical conjecture for your hyper surface. You see this category and perfect complexes of X are very different. This is even fractional Calabiow. It is not equivalent to any category of a scheme. Its motives you see are even different but then in terms of the conjectures nothing changes. So whenever you have a hyper surface for which you know the conjectures, you know the conjecture for this kind of non commutative gadget, purely non commutative gadget. So let's see another example of semi orthogonal compositions coming from root stocks. So root stocks. So here the setting is as follows. I'm looking at a smooth proper as usual case kin. So this can be relaxed but I want to work with smooth proper things of dimension D and I have inside of it I have a smooth effective Cartier divisor and then I have an integer which I assume to be invertible in the base field and then out of this data I can build a root stack. So I can write a stack like this which comes equipped with a map towards X and what is the idea is that well here you have your X and your Z and upstairs you have your X and your Z. So this is a delingma for stack and which is branched over X is branched on this divisor and the stabilizer here at every point is given by the end roots of unity and then outside of the divisor it is a a scheme like stack. Okay and for this kind of objects there is a theorem of Ishii and Weta that there exists fully faithful functioners going from the perfect complexes on the divisor towards perfect complexes on your stack and these for every i between 1 and the n minus 1 in such a way that we have the semi orthogonal composition of the perfect complexes on my stack given by all these pieces and then one extra piece coming from the perfect complexes on X and this function is also fully faithful. So as a corollary we get that in fact the conjecture the non-commutative conjecture for the root stack it's equivalent just to the ordinary conjecture for our underlying scheme plus the conjecture for the divisor. Okay so for example just a low-dimensional example so this conjecture actually holds so it holds when for example when the dimension is less or equal than 2 this for the Grotten-Dekend-Weyvotsky conjecture for dimensional S are equal to 1 for the Baylinson and Tate conjectures and of course it holds for any dimension for the Vail conjecture so for the non-commutative Vail conjecture holds for root stacks. Okay so this is telling you that in fact the the n roots the n is not playing a role in the conjectures. Okay it's playing a role in the dimension of the semi orthogonal decomposition but not in the conjectures. Yes? So this is the previous example in Calabi or VGL you took a hyper surface and here you took a cocktail divisor so it seems kind of both of our kind of hyper surfaces so is it related? Yes because this is cocktail divisor effectively some hyper I mean not not really not really so let me give another interesting computation coming from global orbit fold so again it's the same setting you have a smooth proper scheme of dimension D and now you have a finite group of order n and I'm assuming that n is invertible on the base field and this group is acting on x so I can this data gives rise to to this global orbit fold and then it turns out that for this kind of stacks the theorem again with Michel is that in fact a non-commutative motive of of this stack it's in fact a direct summon of here you can look something that is purely commutative you just look at the subgroups contained in G which are cyclic cyclic subgroup so the sum over all cyclic subgroups of the motive of the the part that is fixed under this action so as a as a corner so you see that here is just a scheme so as a corollary you get that well if you know these conjectures for these the cyclic groups these classical conjectures for these cyclic groups then they will imply this non-commutative version for this stack and again it turns out that well as a low dimensional example this will work exactly the same way so this this corollary will have exactly the same corollary for now the global law refold stack okay and moreover we have other uh for example another possible cases where the conjecture holds is when you are in characteristic zero and you have an abundant variety and the group that is acting on x it's acting by a group of morphisms then in that case you get that this conjecture of Grotendijk the non-commutative version holds okay because you don't leave the world of a billion varieties and for those you know that the classical conjecture of Grotendijk for a billion varieties is true okay so maybe one final example where we are able to prove the conjecture in in full in full generality is this example of a finite dimensional algebras or digi algebras so we can look at the finite dimensional even digi k algebras so you we fix a finite dimensional k algebras of finite global dimension so we a little k is a perfect field so finite dimensional gives me the properness finite global dimension it's exactly the sweetness and for example suppose that the algebra can be given by a the quiver algebra of a certain quiver module an admissible idea so if this is a finite quiver without oriented cycles then this quiver algebra is finite dimensional and if this is an admissible ideal then you have a finite global dimension okay and these are examples but a remark an interesting remark of Gabriel is that when you when the the field is algebraically closed in fact all algebras are of this form of two more equivalent so up to more equivalent is always given by a quiver with relations okay but now what can you do is okay suppose that we have one of those we can look at it's quotient by the Jacobson radical of a let's call it b so this is the largest semi-simple quotient of a so I can look at the the corresponding simple module simple b modules and and I can look at the corresponding endomorphisms over b of these simple modules and of course since they are simple modules they are a division algebras which I call the one until the end okay so they are division algebras over my base field k and for those kind of algebras we can actually compute their non-commutative motive so again it's a computation with Michel so it says the following that if you take the motive of a or the motive of its largest semi-simple quotient the non-commutative motive does not change it is even an integral statement so in particular it does not change with rational coefficients and then if you just use a art in a weatherburn theorem you know that this is more equivalent is giving you by just this division algebras here by these division algebras so you get a direct sum of motives like this and now you just need to look at the center so if you look at the center let's call it l1 until ln so they become a central simple algebras over their center and then we know that for twisted schemes that there is no difference between these two non-commutative motives so this is actually so if you want in a so only here i'm using the rational coefficients okay so if you want sorry i have a question for example by Berlinsen theorem let's take for simplicity the projective line yes and it corresponds uniquely to a quiva yes yeah the motive in your sense then how does it look like yeah the p1 it's even like as previously so the this admits a semi orthogonal decomposition or even a full exceptional collection where you have this so whenever you have a semi orthogonal decomposition this is a very particular one you know that the the non-commutative motive of p1 it's actually the non-commutative motives or the non-commutative motive of the base field plus of the base field so if you want the theorem for example recovers this computation for particular quivas for example corresponding to projective line yes yes and more generally with relations which you don't have here yes okay thank you as i said in my first lecture this is actually more retic equivalent to this algebra of matrices and you see that it's upper triangular matrices so this part vanishes somehow on the additivity it only becomes and so the difference between the non-commutative world and the commutative world is precisely here there is no tape twist the tape twist disappears becomes the tensor unit if but because if you do this with the classical motive then here you have to take this okay and here in the non-commutative world it all this one for this theorem it's hand developed yes the number i don't know i'll off the top of my head but i can give you later so as a corollary we get that so if you want in a fancy language this is just saying that finite dimensional algebras or finite global dimension the non-commutative motive is something like a an arting motive okay so the non-commutative conjecture for a it's just the classical conjecture for this field expansions and so this zero dimensional varieties you know that the conjecture is true so you get if you put all these together you know that this conjecture or any non-commutative conjecture for hay it actually holds okay so all conjectures holds for this kind of algebras so let's see for example remark about the strong tape conjecture so suppose that c is the strong tape conjecture so in particular i'm working over a finite field so let me recall you that what does it say it says that the order of the this as a val zeta function at a zero is equal to minus the dimension of your numerical k zero group so this is the conjecture and it holds why because you see here that this well you know that the k zero descends on non-commutative motives and the non-commutative motive of this is this and so it's just the k zero of fields and then the numerical equivalence relation is not playing a role so this is actually just a bunch of copies of k and copies of k and on this side we know exactly what is this what is this zeta function well it's in fact given by one over one minus so here you know that since we are over a finite field the brauer group is trivial so in fact you have an equality here and and so you know that these things are finite field extensions which are necessarily cyclic so this thing is just one minus if we call this the degree of the of the extension if we call the degree d1 until degree dn then what you have is one minus t d1 times one minus t dn and then what you need to do is to replace t by q minus s and we saw last time that you can interpret this order as precisely the algebraic multiplicity of one of this polynomial and you see that algebraic multiplicity of that is precisely n so you have n on both sides okay and maybe a remark a remark is that this is more general thanks to a recent work of Orlov you can go slightly to the dg setting so you can look at dg algebras which are smooth and that they are finite dimensional but in this strong sense that not the homologies finite dimensional but the the components themselves are finite dimensional then using recent work of Orlov it can be shown that this conjecture for this kind of algebras also works okay with a slightly different argument okay so I think you got the feeling about these computations and now I talked about these non-commutative chow motives I wanted to talk also about the numerical version of them so let me just say that we also have non-commutative numerical motives so they are defined as follows so suppose that you have an additive rigid symmetric monoidal category it's shrinking yes okay I'm going to enlarge it so you can look at this uh uh tensor ideal so look at the morphisms in this category from a to b such that for every morphism in the converse direction from b to a the trace of this composition is equal to zero so this is in fact a tensor ideal in your category and so using it you can define this category of non-commutative numerical motives as the idempotent completion idempotent completion of the category of non-commutative chow motives by this tensor ideal uh is there quasi isomorphism invariant characterization of dgs satisfying the strong finite dimension dimensionality condition uh I don't know I have to think I don't I'm not sure I fully understood the question uh you want characterization of finite dimensionality in the sense of our love imagine something like that maybe I have to yeah a priori a priori I don't have a conceptual viewpoint on these objects yes if you if you want to phrase it like that if you put them all together among all smooth property g-categories if I want to characterize them I'm not sure yeah um so just some facts about this category um so first is that when you compute the harms here between two objects uh is given by the numerical uh grotendi group that we saw uh in the last lecture in fact although the definition is different uh it's again a rigid monoidal category it's again a rigid cancer category with tools given by the opposite and then once again for any commutative ring I can look at the this category with coefficients in R okay so what do we know about this category so which is a quotient of the previous one so we know that is uh it is much simpler so it's much simpler in the following sense so we have this theorem that if you have a field of characteristic zero or more generally uh suppose that we are in a setting where the base fields and the field of coefficients they are both of positive characteristic this works also then uh it turns out that this category with coefficients in this field is a billion semi-simple okay so this naturally motivates the following question since it's a billion semi-simple then uh to understand this category it's the same question as understanding what are the simple objects of this category can we classify them and secondly uh I want I need to understand the endomorphisms of these simple objects so if I know how to answer one and two then I know completely the category and now what I would like to explain explain is a conditional a conditional answer a conditional answer to this question um to this question uh when uh the base field is actually the finite field so finite field and let me just uh recall that we denotes by wk the ring of of p-typical v-factors capital k its fraction field and by sigma the isomorphism on capital k induced by the Frobenius on little k and in order to explain this conditional answer I need to talk a bit about isocrystals so let me uh define a category which is a variant of the classical category of isocrystals so you let's take this algebra uh capital k with one variable of degree minus two and look at the raw uh an automorphism in here that uh multiplies when it changed where you send v to uh pv we multiply by p okay and then you can look at this category which I denote by isocrystals z k z plus or minus one so what is this uh so the objects the objects are pairs uh where this it's uh this is a degree wise degree wise uh finite dimensional um z graded the font is shrinking again okay module over this ring okay so it's basically just a module over this ring and I'm asking it to be finite dimensional so it's something z to graded and then this it is uh just an isomorphism between v where you twist by this automorphism towards v itself and this is uh sigma semi-linear okay and the morphisms the morphisms are uh what you expect so between two gadgets like this and w uh it's just a morphism downstairs which makes these square commutes when you put here the twist by the roll so what can we say about this kind of uh so this category is morally speaking some kind of z to graded isocrystals but then it's not actually z to graded isocrystals because the there is this p this multiplication by p it's almost that uh so remarks about this category some important remarks is that uh this category of isocrystals over z first of all is is not capital uh the k linear is just linear over the algebraic numbers sorry over qp and moreover it's equipped uh it's equipped with um with the tensor with the tensor automorphism automorphism let's call it pi of the identity factor and so if you have an object then you have an automorphism of it such that the n component is as follows so here you have your phi n of your object and then you need to multiply it by p n over two and then composite r times so r it's pr there you need to do this when n is even and you need to multiply it by n minus one over two on the phi n and composite r times when n is odd and so it comes equipped with this tensor automorphism of the identity filter and then we we have the following proposition that says that the following two conditions are equivalent so first of all if the strong plate conjecture holds for all the g categories uh for every uh smooth proper uh the g category a that's equivalent of having this uh isocrystal realization in other words uh having a fully faithful so a qp linear and fully faithful um tensor functor uh going from the numerical motives with qp coefficients uh towards this category of isocrystals and this uh functor um sends a non-commutative numerical motive to its periodic cyclical homology uh with p inverted and then with the cyclotomic Frobenius so you one way to phrase the strong plate conjecture if you put the digicator simultaneously is saying that you actually have a realization functor towards isocrystals like this and so this uh immediately tells you well there's a corollary of course of this you get then your original category the numerical motives now with qp coefficients also comes equipped under this assumption with a tensor automorphism which I still denote by pi of the identity factor and now we can explore this tensor automorphism to try to describe this category of non-commutative numerical motives similarly to the in the spirit that what's young's uh what the mill gives in the commutative world so I just need to let me just recall you the following so if you have a smooth proper digic category we look at this finite dimensional capital k vector spaces they come equipped with these automorphisms so the Frobenius zero which is the cyclotomic Frobenius composed air times and similarly this one so it's Frobenius which is this one composed air times and I recall you that the the value conjecture that we formulated last time was saying that if you have an eigenvalue of this Frobenius respectively of this one then what we we ask for is that these eigenvalues are algebraic numbers and secondly that their complex absolute value is equal to one respectively is equal to square root of k and this for every conjugate of lambda and now we can look at the strong version of this where we also impose the further property which is that there exists an n such that when you multiply by q n your eigenvalue you get this becomes an algebraic integer okay so this is a condition that it holds for example if your a is perfect on the skin you also have that and this is classical in the literature this is what people call this notation this is what people call the the veil q numbers of weight zero respectively of weight zero respectively weight one okay with all this in place now we have a conditional description of our category which is as follows so let's assume um assume that this strong version of the of the non-commutative veil conjecture and also that the strong version of the take conjecture hold for every smooth and proper dj category okay then in that case what can we say we can say that if you if you look at any object in your category with qp coefficients what you can look is look at the center of the endomorphisms of that object so clearly inside of the endomorphisms you have this element and this element it's in the center so you can look at the smallest qp algebra containing this element so this is clearly in in the center of this algebra but it turns out that it is exactly the center of this for any objects the center is precisely the smallest subalgebra qp subalgebra containing this endomorphism this automorphism okay which is this one okay okay and secondly we have a bijection we have a bijection so one side we can look at the simple objects in our category in our abelian semi-simple category and we can look at them up to isomorphism and if I have one of those one n which is simple so let me just say that of course it follows from one that if this is simple then this is a field extension okay so this will be algebraic over qp so from n you can look at this pi n this object this from if you have a simple object you can look at pi n okay and this pi n will actually land in here in the union with i from one and two from the valed q numbers of weight zero respectively or of weight i sorry okay modulo and now I need to think to consider them up to an action of the absolute galvan group of qp and what I'm saying is that you actually have a bijection like this so this gives you a bijection so this answers our question our first question which was what are the simple objects conditionally they are the valed q numbers of weight i with i being zero and one they are precisely that and then how about if I have a simple object what is its algebraic nomorphisms so it's algebra if you if you look if you consider a simple non-commutative numerical motive so what can we do so just one remark here so we have this so we have this non-archimedean local field so we have this since it's simple I have this non-archimedean local fields so I can look at this brown group and I can consider the invariant of it and this gives you an isomorphism with q mod z so we can evaluate this look at the invariant on the endomorphisms of n so endomorphisms of n it's actually a division algebra over a qp of this so and you know that the brown group classifies division algebras so if you knew this invariant you immediately know what is this completely classifies it and this is given by minus so here you have the periodic valuation of pn here you have the periodic valuation q multiplication by the degree of this field extension and you have to consider this q mod z so so that gives you a complete description of your category of non-commutative numerical motives but the conditional one where you are assuming these two things in fact okay maybe I just would like to say that the talk tomorrow will be in the same spirit so I will use that non-commutative viewpoint to attack commutative examples and to prove the conjecture in new cases and then at the very end I will talk about the Riemann hypothesis and the non-commutative version of the Riemann hypothesis okay thank you for your time okay many thanks let's thank Gonzalo any questions yeah so I have a question so it seems that Chokunet decomposition for example of classical chair motives are sent to other decompositions but you lose weights aren't you yeah for example the composition of pn will be just a sum of a unit object so do you really completely lose weights in the non-commutative motives or is there a way to recover the weight filtration things like that I mean on the category level of course on the category level you still see something there for example if you just take p1 and you take the two line bundles o and o1 so you have one x in one direction which is dimension two for example so you still have information and but as soon as you go to any invariant which is additive you lose the extension because it's like k0 k0 is the machine that doesn't see extensions treats all extensions as the trivial one so as long as as soon as you do any kind of additive procedure the type disappear okay and in somehow the non-commutative motives is the universal way to to make this additivity it's the initial way to do it like any other way will factor uniquely okay uh yes gonzala another question to you yes could you say a word on how the description differs from the classical draw motive version ah you mean this theorem possibly you mean you mean the comparison with the commutative world that's what you're asking uh remi could you please uh write uh something on that so yes i can i can absolutely say something there so gonzala the explicit description of the last theorem yes so let me i was about to say something about the comparison between what i did and what happens in the commutative world is this is of of interest to you is i'm not answering is not that no let me just say yes yes i was just saying that so you divide the world in two on one side you have smooth and proper schemes over the base field on the other world you have dg categories which are smooth and proper and you can pass from one side to the other in a contra variant way because if you have a scheme you can look at this category okay but then on this side you have a growth and dig you define this this function towards uh so this contra variant function towards numerical motives let's say that i'm working with q coefficients so inside here i have and then on the this side i can have this non commutative version of that where i go towards non commutative numerical motives with q coefficients and so the comparison between the two is as follows in here you have a tensor invertible object which is the the tight motive so tensoring with this object it's an automorphism of this category so one thing that you can do is that you can trivialize that action in the sense that you can do an orbit category so you can do this so in this category the the the tight twist becomes the tensor unit so another concrete description of this category if you want is that here the home spaces are given by correspondence of a precise codimension and here you are putting all the cold dimensions together on the home spaces but since you are working with q coefficients this is just a category where the home spaces are just the k zero homes if you want as defined for example in manin's paper and then it turns out that this embeds here by a function which is now fully fightable and this embedding is nothing but the grotten dig rim and rock theorem to guarantee that you actually have functoriality etc so this is the comparison is saying well if you trivialize the tape motive it becomes the tensor unit then after making this trivialization it embeds fully faithfully in here okay now under this size assumptions if you assume these conjectures these very strong conjectures they are going to tell you well you also know that this is a billion semi-simple this is a billion semi-simple and here I'm saying that these are the simple objects are the value q numbers of weight 0 and 1 and mill proof that these are the value q numbers of arbitrary weight so in fact under those conjectures which are pretty strong they would imply that this functor is not only fully faithful but moreover essentially subjected that it's an equivalence then the the non-commutative world it's in fact just a trivialization of the tape motive of the commutative world but this is conditional under those assumptions so morally speaking this is saying that if you assume the strong tape conjecture everything is controlled by the Frobenius and then if you force the eigenvalues of the Frobenius to be very close to what happens with the commutative world well then actually everything comes from the commutative world that's the idea behind of it and then you have a similar picture for Chao motives etc and then you have a similar picture as I mentioned yesterday for the for the mixed case where here you have Weavotsky dm maybe that's more interesting to you let me just say that you can take smooth skins and you go to dm and you do exactly the same thing here you have the tape motive and you can do the orbit category of this so you can trivialize this action of tensoring with this tape motive and that's fully faithful in this non-commutative mixed motives so more speaking the difference between is always the tape motives we lose the the weights Gonzalo another question do the conjectures imply that all non-commutative motives over finite fields possibly are actually geometric so what what geometric means you mean that it comes from here I imagine that yes so that's that's exactly what I said I said that these conjectures these two so what I don't know if you feel if it was clear what I said I'm just saying that every if we assume those two conjectures and work over a finite field and with qp coefficients then you actually have an equivalence here so saying that every gadget here it's actually coming from the commutative world okay but is very strong under those conjectures and over qp coefficients okay thank you cp decomposes as a direct sum of the drum after rationalization do you have any commutative non-commutative analog so the answer the question is so what your colleague is saying is that tp of this it's actually this so if I understand correctly your colleague is saying this yes it's true and now you want me to put here a and to have a decomposition in two pieces that's what he is suggesting well it's a question from an anonymous attendee so I think more morally speaking I think you can try to do something like that if you have extra structure for example some kind of yeah but if you have some kind of tensor and if the tensor is commutative you are morally much closer to the commutative setting so yeah I think you need the extra structure to try to to develop some kind of lambda operations and things like that to try to to do the splitings another question previously you mentioned that in some theorem rationalization is not necessary do you mean that you have more controls on things like nigh god's filtration maybe coming from the s1 actions yes I think you don't need to invert p in that case you don't think I don't think if you don't invert p you actually have a filtration yes but as soon as you invert p then you have this canonical splitting yeah so I think filtrations exist without inverting p if I remember correctly yes and the last question should this be over more general fields as well but that was the question what is the question michael can you formulate your question please so watch what fields do you suspect that all non-commutative motives come from smooth projective varieties I mean I don't know these conjectures that I'm assuming are pretty strong I don't know I mean I definitely as soon as you are outside a finite field I don't know even how to start if you are over a finite field you have this Frobenius and using this Frobenius you have some kind of classification so you can and using that that classification you can try to compare it with your commutative world but besides that I don't know yeah I don't know I expect that not to be true that there are much more things here that do not come from here yes okay thank you thank you any other questions to Bansala if not let's thank Bansala again