 Okay, it's, I think, time to listen to the second part of Frédéric Deglises' lectures on Motivation Homotopy Theory and categories of motifs. Please go ahead. Thanks so much. So we will resume from the lecture we have left. But maybe I'll start by giving more focused or more precise description of the end of my diagram when I started my introduction. So we were at the Baylinson Conjectures. So I did them to 1984. It's published in 1986 actually. And it's not only about existence of material complexes but also of mixed motifs and so on. So I should add, so we have seen now that almost at the same time, Bloch introduced these higher trial groups that we have seen in Mathieu's talk. Okay, so Bloch in 1986. Okay, so this was very, very connected. And in the 90s, maybe I'm not so precise, so there were several consequences of this higher trial groups, one by Anna Moura and one by Marc. So a good definition of motifs over fields with some of the expected properties, which derived from higher trial groups. Let's recall we had this background, pure motifs by Grotendic above. So I just put G because, okay. And there was also at that time, these very good ski motifs that we have started to see to which are I think I have. I tried to show you because of this move, this category of smooth correspondences, which is close and in spirit with with pure motifs and which also were trying to answer this question. And finally, there was a connection that was, there was a comparison between these motifs that was proved by, so some of them were proved by Marc, but I think Ivor gave the most complete answers. Okay, but I want to say that today I'll try to go to the homotopy setting and what was exciting about the, this idea of over the ski was that he most maybe since the beginning include this theory into the homotopy call from war. Now we call, we know this as Morale by Witzki A1 homotopy category. And with this new idea, a new wall, new word of possibility appear because all the techniques, all the techniques from algebraic topology and phenomena from algebraic topology were transportable to A1 homotopy and that's what we have seen in Kirsten. So for today I'll try to present mostly A1 homotopy theory, but I'll try also to give you a presentation of motivate complexes of Witzki in the same time. Okay. So now we will try to go into definition. And especially for multi complexes and for A1 homotopy. So I'll give definition in the language of infinity categories because it allows me to give you precise definition without giving too many technical details. So infinity categories. So I will use two infinity category. The first one is category of spaces pointed spaces. I'm not against models so you can, you can see them as a simple sure sets. Okay, and the other one is the infinity derived category of a billion groups. So I write curly D to indicate that I'm working with infinity categories but it's sometimes not so important. Okay. And the advantage of the infinity categorical language is that you can present definitions in with very shortly and very concretely. Okay, so let me draw a line. I will define both unstable A1 homotopy category. I have pointed, I have taken this pointed. And maybe I should have said that this is now any notary and scheme. Any scheme so there are some finiteness condition but not much. Okay. And in the same time I will define the effective or just motivate complexes and you will see that the definitions are very close. So I will use the D M FET of S so maybe now S is regular if you. But it's it's actually not necessary in the theory if you are more clever with finite respondents so here we you are just looking to infinity funtals. Say it like that from the category of smooth schemes over S to the category of pointed spaces. Okay. And you will look only at some checks which satisfies two properties so the first one is excision. And it's like that for any excessive morphisms so let's F from YT to C. Excessive so if you remember it means F from Y to X is et al and the inverse image from T is the inverse image of Z and it's an isomorphism. Then the map so I write it like this C is an isomorphism so it's a kind of supported sections I will give a definition afterwards and the other property is the A1 autopy property, which says that for all X in smooth schemes, the map induced by the projection. When I say ISO I should say week equivalent sorry. But well in infinity categories you say you say ISO for week equivalences. Okay and now here I look at almost the same maybe I change the letters but now I take this category of finite correspondences. And I go to the hub, you could add if you want that these functions are additive in the infinity categorical sense it means that it commutes with product but it's not necessary. And you have the same excision. Same same action so you just replace X by K. A1 homotopy. Okay, so very fast definition which tells you exactly that the basic action that you want are this excision, A1 homotopy property, maybe just a remark so first of all XZ is the homotopy fiber of the natural map from X to X. Similarly for when you have a complex you take you can take the con just the homotopy fiber. This is why I take pointed spaces, maybe a remark is that a more more convenient things to you can replace one by asking that for any distinguish distinguished square, delta, then say X of delta, homotopy current diagram induced by delta after applying X is homotopy Cartesian. So I like this excision formulation because it makes it makes it very close to what we know in topology, but if you use this definition you can, you can forget about both base points. Okay. So, the example that you can get our following first. You can take care of pointed simplicial scheme. Set, sorry, simplicial set. And you can look at the constant, constant, which say to X maps. In some, let me put some racket. Just K itself. Okay. Okay, you would like to to, if you X is a smooth scheme. You would like that the full tour. You would like to have a unit I'm betting. Okay, you would like that the just you take the shift of set represented by X and you would like that it's an object like that, but, or even if you take this in the infinity category whole sense it will not be the case because this excision and one homotopy property will not be true. It's the same here so you could take so to anticipate you can as an example you couldn't you can take GM and it will work but you won't have if you want to. There is a problem if you want to take what I call a shift of transfers represented by X. Okay, this will not work as it is because you need an intermediate theory. So, to be able to work with this you need to slightly more information. Okay. I have a problem with zoom. Okay. Okay, now I can add a new page. Actually, what I what you need to do in order to have these these object X, what you will want to do for a smooth scheme over S as an object of a category is that you want to use the, the infinity category call localization theory for saying it shortly you want to describe H, you want to describe your category so H a one as localization of a category of simtical pressure so let me say in the infinity categorical sense for SMS to pointed space by So, let me say in this name is topology a one, a one, a one equivalence is with means which means here the map T transfer S a one X. Okay. So here same thing instead that you, you will be she is with transfers from or she is on SM called S, going to be, which is the same thing as the giraffe category, in fact, with this category. Okay. And what I describe in the, in the, so in the, in the notion in the theory of localization. Because this category are present table. And here you just invert sets of maps so here this is connected with the infinity to post theory I won't enter into the details but you invert certain the same is the which I procuring for simplification. And here you just in this map, but when you do that you can, you can always describe the object as a certain local objects inside this category here, and this is what I've described here. But when you, you, you just decide to invert maps in this category now you can just, you can just consider your any example so if now X is any, any smooth scheme. You can look at its pre chief, and then localize it with respect to this, this, this map here on something here. Okay, so just to say that. There is this localization theory which is hidden here, but now I want to give a concrete definition which is what do we, we use in practice which is the original definition of, of more and very risky. Which using it's in some sense it's hide it hide some some localization here. So you can define the one homotopy category as localization. And at times you don't take pre sheaves, but you take sheaves for venisonous topology. SNS of simple to group so as a model you can just take funtals from SNS. Now these are, I'm mixing things right I'm now. You're looking at models but you can just take simple sheaves like this, these are called spaces pointed. Okay. With respect and now because you are you have taken these, these, these categories of sheaves, you have to localize only with respect to a one respect to a one. So which means you just invert. The map. Sorry. A one. A one text where you are the base point. Sorry. Here also I should have put this. Transfers is for the right side. Okay. You take the category of shift with transfers that I have introduced and localized with respect to these maps. This is my team effective. Yes, we will take all coefficients. Okay. As I said before, there is always a notion of a one local object and a one localization. So here, the a one local object maybe the complex of sheaves with transfers. I can write this such fat for all x over s smooth. The co-mology of with question from Disney's co-mology of tech or with question and car is on the top invariance. This is an isomorphism here I have a similar description for a one local object I have to take mapping spaces but okay. In any case in this abstract infinity category called word. Language or you can also use model categories varies a kind of a there is a functon abstract which a one localized. Object and which gives you object of this form here. But the problem is to in the problem in all these theories to describe this a one localization from. So now I want to explain a bit more this in the case of sheaves with transfers and now we restrict to the case where K is a perfect field. So you can notice that I have slowly passed from the infinity categorical language to that of a model and model category but it's it's I mean I think we should be smooth about this passing from one language to another. In any case, if you consider now a complex of sheaves with transfers so let's say you can I recall that you can always define its co-mology. Okay, so we have this map here. This one you can define its co-mology sheaves H I K. As you just take canal. As usual in plus one. This is computed in the in the category of sheaves with transfers so I have briefly said that it's an Abelian category so you can do that. Actually, it's it's okay. It's almost like you take the co-mology stupidly as pre sheaves and then you take the associated associated chief and the theorem of it tells you that it will admit transfer so this leaves in the category of sheaves with transfers. And then you can translate the main theorem of this key as follows. We have a K as above is a one local. If and only if for all integer is co-mology pre sheaves are a one invariant in the belong in this category but. H I TR of K. And so actually this is a this is a formal consequence of the fact that if you take an object, which is a one environment face a sheaves with some shares which is a one environment then it's co-mology is a one environment so it's just it just fall off from the main theorem of a risky and you definitely need that case perfect here. Okay, so. Actually, maybe. So maybe I just to be a bit more clear very risky define the category of multi complexes now maybe as a triangulated category of a K has a full subcategory. The, the category derives sheaves of the derive categories sheaves with transfer made by whose a one local. Okay, but this is just what you will get from the localization theory, the important part is that you can describe this a one local object. Okay. The bonus is that you, you have now following construction, which is the system complex construction which is very important and which will give you, which will give you a way to compute in this strange category. So, we have seen that the delta delta and the simple sure scheme. Okay. Okay. In the talk of Matthew. Okay, and now you can just if you have a shift with transfers, a complex of ships. You can define the system complex of K. Just by associating to X. Okay, sorry. You take the total complex for direct sum of the double complex that you will get. Like K to delta dot cross K X. Okay, so here you have a simple sure scheme, you apply contra round from thought so you will get a co simple sure objects or co simple complex to turn it into be a by complex and you take the total, total space. Okay. So the corollary of the previous theorem is that for any K as above the complex system S of K is a one local and the canonical map so you have a canonical map like that system S goes to K is a one week equivalent so an isomorphism in this category. Okay. Okay. So, now examples. So, first one is X over K is smooth. You can define the various key motifs. M of X, it will be the complex. And then you can also sling of this shift. The transfer of K. So the shift with transfer represented by X over K. So, over a definition you can define the test twist Z one. So, let me take the reduced motive of P one, pointed by infinity so to insist that it's really close to what we have seen for us P one, P one seen as a sphere but you should this time minus so various key motifs and of X are do you have to thought about them as as the single homology of a space X. So, when you compute the homology of P one, it will be in two degrees say for singular homology, degree zero and two, and the twist here correspond to what appear in homological degree to okay. Maybe isomorphic to reduce motive of GM pointed by one shifted minus one because these two spheres are equivalent from the A1 homotopical point. Okay. And then twist the N as just so I have not, I have not given detail here but you can define a tensor product on this particular of complexes. And so product is such that M X tensor one. I think someone has his mic open. Okay, so. These are the definition of the twist. The thing is that with what we have said, you can compute some part of this these objects so first of all, the motive of GM. And as this complex will be isomorphic to a sum of a constant shift plus a shift GM that we have seen. So in particular it's in degree zero or seen as an object in the derived categories of transfers. And that follows from Susan various his computation so this implies by definition that the one is just the shift GM placed in comodical degree one. Okay, so it belongs in zero. With respect to the structure here. So over you can show that see and so it's reduced motive of GM pointed by one smash and so let's say it's direct factor of a motif of GM. Sorry, there is a minus here. Times and shift minus and so by definition of this in complex it's in comodical degree so maybe minus infinity and as a complex of Jesus transfers. Okay, so we have seen this. So we can reformulate the beings and so that's conjecture that we have seen tells you that for and strictly positive CN should be in one and for this is inside the category of the direct category of shoes with transfer. So it's slightly more precise when Matthew here Matthew said zero and but it's conjecture that actually it's in one and we have not much that's big just because I'm based on the Z one. Okay. So the word this this conjecture is equivalent to say so if you want to try to prove that for a function field of a variety. But you obtain so I just I'm just taking the definition of finite correspondences from the over E from delta. To GM. To N is. Sorry, has no. Homology. In degree. Equivalent to that and as as been as has already indicated it's sufficient to do this with rational coefficients. So, so now it's you see this is this final correspondences are this is very explicit almost combinatorial but still nobody has succeeded to do that for in case any cool one you can you can, you can work with it because these these cycles will reduce two devices and then you can use the trick of basically to reduce to pick up and that works but otherwise, there's no nothing to do at the moment. I leave it to you now. Okay, so maybe I go now with motive emoji just to connect to you. So we have these complexes recall the end. We have axes of shifts with transfers over S over case right. And then we define h I, comma, and in the one homotopical notation motif, motif emoji degree. Sorry, sorry. Invert. And I. We have a smooth scheme here. As a co emoji for the ski on this nevich it's the same of x with correction in these complexes so it's not exactly the same definition of a lot of blocks I trouble, but it's also morphism in dm effective of k x, see, sorry, see I shift and and is the degree and I twist. Okay, so we have from this point of view so there is a comparison of this definition due to very risky with a higher troubles for in this generality so we have also all the computation that Matthew has given but I should say a last last computation, which is you to see numbers here we have seen that. So, I said that ZN is in degree minus infinity and and we can still compute its 10th homology. And this is the so-called un-ramified min or k theory. And this comes from what I said in the previous talk so it's also if you want, I denoted that SNT but it's GM tensor in the category of shoes we have homotopy environment shoes and times. Okay. These shoes that appear live in this category HIT, you know, this is why it's it's motivaking nature and it's very important. Okay, no more questions. So now I'm going back to A1 homotopy. I want to indicate the development on the six-hunter formalism so we first see a glimpse of the six operation on H on the unstable homotopy category H star A1 so pointed. And now we go back to some scheme as any material and finite dimensional. So I want to insist on the fact that the first description I gave in terms of infinity category allows you to construct the so-called basic six operation maybe. Let's let's start with, if you recall, we have a morphism F from T to S. We have let's let's go back to an infinity from Paul like this S star. Then, so this is a space in the infinity categorical sense. Okay. Then I can just look at something like X composed with F up a star. So if I have a smooth scheme over S so I define this base chain for for correspondences but it of course it works for smooth schemes. Just recall. It's just a base change and then I compose with my phone call here. Okay. So this is a well defined and you can check that it's it satisfies the the excision if X satisfies excision. So let's say shortly it lives in this category so it's local for this name which I'm a one. Then you can check that this is also it satisfies also excision plus a one on the P on the notes. Okay, so it directly gives you an object in each hour is subject we noted F lowest of X. So that easily we have obtained a full tour. From to S. Okay, this is a direct image from it's like for she is defining a flow store is is is easy. But now because I'm working with presentable infinity category I have an adjoint from top of your M which will say that very stuff. The right infinity from pause, which is formally. Okay, so this gives me two operation if I repeat this, we from morphism from T to S, which is smooth. And I will get two from Paul so I get the best changes if I repeat this operation so you have to prove this. And here I get this operation that I've been noted by P sharp with in some sense forget the base. And the two over operations are a bit hidden so we are the fact that H star a one of T is a monoidal infinity category so which means that there is a smash product and internal. So we all these operations so, as I said the right adjoints are easy to define even for the home. But the left adjoints are characterized by properties so many minutes. The left adjoint sorry are characterized by properties like. So if you take a point the pointed superficial schemes and you apply this to this subject if it's smooth. Okay, so say that this exists by the localization property. Then you just have you just obtain case. And so on for the object we have. Okay. And this characterizes the basic six operations. Okay, we have also some formal property that I let to you there are the so smooth base change formula and smooth projection formula for lack of time I want. So these are formal everything of this is for one thing which is not for more we try with that now is for moral. Localization theory, which says the following take any closed in motion. Emotion. You open complement. Let she J be it's the open emotion. Then for any point in space. The following sequence chair up a star chair. Close to x close to our star star star x is a homo to P. Frederick, can I ask a question at this point. Yeah, this does this rely essentially on the nasnevich localization does it is actually not. It really relies on it really rely on each and a one homo to. Yeah, yeah, but I mean it wouldn't it wouldn't work for the risky. That's right. Absolutely. That's right. It works for the et al, but not for us. So that's right. Absolutely. So it really using this section a one. And this is one of a reason why the nasnevich topology was a good choice. It also works for the et al topology but the et al is somewhat too strong for some other perspective like quality forms, because quality forms are trivial and So maybe I can so this proof is really the central point here. Maybe I can write this more more concretely it it's equivalent to prove that for any x of s smooth. Then I just write this sequence for x, maybe. No, I think it's okay. Then I can write x mod x, x minus x z so may I just see. Let me just finish. So the map that you will get from x mod x minus c to x c plus is weaker one. So nice morphism. H star. Yes. Yeah, yeah sorry to interrupt I think you mean homotopic co fiber sequence. So where is homotopic co Cartesian. Yes. Okay, sorry. So co fiber. Yeah, sorry so this is because I'm so used to stable. Okay. So this is how is this one. Yeah yeah sorry co fiber sequence of course I want this to be. I want this to be right exact absolutely and this this this fits well with this property. Okay so geometrically you are just proving this kind of thing here. So this this theorem is, is very related to the one of the main problem of higher trouble, which is proving the localization, exact, exact sequence for higher trouble. So it's not known. And it's not known for the category DM effective of S, except if the close immersion I admits a smooth retraction or over over possibility you can, if you restrict to regular schemes over of characteristics or it works also. But the general case is unknown at the moment. Okay so it's really the central point of. So, now to go to go to the six month or formalism there's no there's no six month or formalism unstable. You'll have a trace of it but not the full one and so now we I want to go to the stable. category. So I will give a quick definition. Using still is still using this language of infinity categories. So recall that we have several spheres. So now we work in this pointed a one. Okay, so we are P one pointed by infinity. And as Kirsten recall this is isomorphic to S one smash GM by one so these are the two sphere that have also appeared to define the dead twist. We have also over models of Seattle sphere but I won't I won't use them. So the trick is that we in some sense we want to invert both both sphere as fun and GM so we will take a spectrum respect to P one. So for this we introduced the omega P one new space space X which is just internal on P one X for X appointed space. Each time I use I write internal home they should be derived if you work within with model categories and so on but I'm a bit sloppy here. And now you can define if you follow the definition of all battle you can take the following definition for the stable heaven on the topic category as H. Now I don't I don't put a name one anymore of S. The problem one more to pick out of category of P one spectrum is the call limits. Following diagram of infinity categories so you take each one S. Apply this omega P one. And you apply it indefinitely. In this direction. Okay, so in some sense the object of SH of S are here you make you make objects objects become infinitely maybe I can say infinitely P one divisible in some sense. Okay. So this is a beautiful definition calling it into the category of present table may monoidal present table infinity category. So, but I'll give a concrete. I'm really confused here. You should call limit. If you go to the right and it stops you're not doing anything. If you mean limit. Is it clean? Maybe I don't know I mean, or do you have the arrows coin in the infinite and the other direction I'm not sure I'm not an infinity guy. No, you mean limit. I'm sorry. Again. Thanks. But let me give a concrete. Concrete meaning of this concrete construction concrete model for this so if you compute this is the object in this. The limit. Then what you get is sequences of objects in SH of S, which are called here you want spectra. So as in the classical case you will get sequences of spaces. We have an isomorphism from x and omega p one. So we have one evidence of spaces. Okay. So this really correspond to what we should call omega p one. Omega. Yeah, omega spectra. Mega p one spectra. Okay. Usually, this map here corresponds to p one smash x and goes to x and plus one. Okay. The classical model you just, you just asked for the distance of this suspension map. So you should, if you know classical spectra you should be back in classical. Okay, so I have given the concrete model now let's let's state the universal formula, which is absolutely not trivial. Universal property, sorry. SH of S. Maybe it is a presentable maybe I should not put this monoid or infinity category universal with a full power sigma infinity from h a one h star a one. Two s h of s such that the infinite suspension of p one is invertible. So, or so when I consider the monoid or searcher on spectra I just use tensor product because a spectra closer to homology and so that's our invertible. But the usual convention is to take smash problem. Okay, so just a remark so, as I said we have inverted two sphere when we have inverted only p one so it's, it's, it's, you can see it formally here, but you can also look at s one spectra. And then you will factor this map here. So, as one spectra, I will denote it as such F. Okay, so you invert only you apply the construction of the construction above but to s one which is the simple show spheres. Okay. And then it factors through this category. It's useful. But but now it's also useful for. Define this category here. So it was only for regular schemes but you can launch it. So by using this gamma, this fun total adding transfers you can define something like this. So it's, it's, you know, right this gamma star composed with a billionization so to simply show up chief it will associate first complex with a billion groups, and then it had transfers. Okay, and now I want to insist that if we do the same construction of pure one spectrum but for this category, we obtained the non effective category of motives, and it fits into this diagram here. So it's just to. Okay, now I'm back and I'm on good food to state the main theorem. So, are you going to escape. So now, I have all the definition to study this theorem so it was stated by the rescue. I gave sketchy notes and it was completely proved by I have been his PhD thesis so it says basically the following for any f separated a finite type. This is a system of schemes. There exists a pair of pair of a joint from poor. So the exception of fun total such that which are characterized by the property so a freak a freak is compatible with composition so and so I don't give a presage account here it's a bit formal it's a so so called physical condition but also, if F is proper. F lower strict relates to the basic fun total so it's F lowest on the fun previously. And if F is smooth, we have F low, sorry. The trick again is isomorphic to F lower sharp. But here you have to add something. This is a space tensor product where T of F is the tangent bundle of TF is a stable term space so it's infinite suspension of space T of F mod by F, the very open complement. Isn't that credit isn't that backwards. It's the F lower sharp as you have to put a palm space the minus TF there. Correct way like that. No, the upper shriek is the upper star composed of the Tom space. Yeah, on the left. Yeah, pretty sure. Okay, so, so I have to add that. TF is tensor invertible in the stable category. That's why you have a stable category T and exactly that's a virtue of passing to you invert only P1 but but by by this procedure you invert all the time space and minus TF is the tensor inverse. Conventions. Okay, the good thing is that these properties determine the F low three. So there are some infinity categorical issue but I want to enter into this. These details here. Oh, I have plenty of times. I've gone faster than I expected. Okay, so, and I have to take another blackboard sorry. And you have also the so called best chain formula so maybe if you haven't seen that I will set it because I have more time. So if you have a pullback square in the category of scheme so let's do it proper. So if you have a division TQ, then you have the upper star F low shriek. And you assume that F is fine separate the finite type. The upper star F low shriek is isomorphic so these isomorphism are canonical as much as they can. It's a isomorphic to T low shriek. And likewise so these these are all left adjoints so now if you have an isomorphism of left adjoints will have isomorphism of right adjoints so you have a dual isomorphism. So this is a very important form of a projection projection formula, which so you take F, F separate the finite type as before, and this time, it says that isomorphic to F low shriek. So if anything is that you see that you have, you have formulated in terms of this category of coefficients. These properties that are known for commodity theory so this, this, this, this, this property is known for transfers and for example chart groups. And this one also it's also called the projection formula for the chart or in commode. So you have translated all these properties of commodity theory in terms of the objects of a category of coefficients. So in brief. So I mentioned that this key had extended the real more mostly complexes but now the category SH with schemes very answers to answer a kind of a generalized in some formalism in the sense that now this SH this category SH so it's triangulated so I've not said that but when you invert P one, you also invert S one and you obtain a stable infinity category which means that the associated homotopy category is triangulated. The SH, the category SH of S is a triangulated category, and it's equipped with the six funtals formalism of growth of a formalism of growth and that is the formalism that you, you, you get for shoes. Classical shoes on the analytical and equal varieties for example but also electric shoes. For this category I said you have good realization funtals in for example in the electric, in the electric category. You know, you know, you know much. Also, you have also this, this very enthusiastic program to compute the homotopy shifts inside the stable homotopy category that Kirsten has as mentioned in her book. So, to finish the talk I want to go now to the, so the, the, the integral category SH category is very mysterious at the moment or it's mysterious but it's also very exciting here you have many phenomena, and so on. But with rational coefficients now we know much more about this stable homotopy category and it will allow me to connect this with material complexity so last part I will talk about rational part. Okay. Let's take S scheme as before and then I can look at this triangulated category and I can always take its rational part so there are several formal formal ways to do that, either you can just you take the almost so you take the homotopy category, and you, you, you can solve all the om with q and it gives you a category and you can check it's still triangulated, or you can do in a more topological way and you invert more spectra as, as, or as Mark has said so you invert maps in stable homotopy and you can use localization here. Anyway you get a good triangulated category. And the first thing is that it can be decomposed into two parts. So, there is plus part and it's a product with a so-called minus part. The plus part is so that so you so we recall now that Kristen has described the stable homotopy group source here, and it's given by, if you consider the GM grading by, by this Milnovit K theory of fields. And in this case, in these groups you have, you have several elements, notably you have the algebraic opf map. So the plus part is characterized by the fact that the opf map, the algebraic opf map acts by zero. It's also characterized by the fact that the map epsilon, which should be also in a presentation of Milnovit is minus one. The minus part is, the minus part is characterized by the fact that it is invertible. The opf map is invertible and epsilon acts by plus one on the whole category. The good thing is that this, this category be composed like that for any schemes actually. And here is isomorphic to the M effective of S q, not effective, sorry, the stable category. So at least for S regular or, but actually with a good definition of this category of multiple complexes of rational coefficients, you can extend this and you can, you have it for all schemes. So if you really want to use shift with transfer as you can do a little bit more you can use any normal or even geometrically unibrow scheme. And the important thing here is that this part also is a completely determined so maybe I should write S zero for you have a scheme so it's over Z, you can always take the part which are points which are of characteristic zero. So this part is something like the modules over V chief. So this is this represents over S zero and ramified sheath. So this minus part here is completely connected with a quality force. Okay. So for non regular you have to be slightly more careful. So now concretely what does it mean on the home so let me so we you can compute the co homotopy groups by an eye of S say again S regular to simplify. So these are just the map so in the stable homotopy category between says your spectrum and here you take so as and minus. So I said I didn't smash GM to the eye. Okay, and you compute this in SH of sq. Then it will be isomorphic to two part for the plus part, what you have is a credit part for the say gamma filtration. Let me check. I have care to I minus N of S so it's K theory algebra K theory of S regular scheme in degree two I minus 10 and you take the graded path for the gamma filtration you can take also the graded path for topological filtration it will give the same answer. So gamma filtration is referred to the lambda structure that you have on the hierarchy theory, which is formally more comfortable to work with. So this is why we take it. It's also the Adam's credit parts. The interesting thing is that you have a complement which correspond to the minus power. And here this is H and minus I miss nevich of the scheme as you're all with correction in this and run me for each. Okay. So, okay, maybe if if I'm, if you want more more concrete description you take to be a field. And then the comot of the field. With q coefficients will be so same things. Let me write what we come with H and I will take a few. And here, you will have the value of the question of q if and equal I and zero. So we, I'm saying that just to, to, to make my answer to kiss Kirsten more concrete so you seem Bainson Suley conjecture tells you that there's no combo g if n is negative here. But there will always be co homo topi if n is negative because you can take i positive and there will always be this big part here. It's zero or so. So this group is also zero if he is a field of characteristic. This is because so this part of the answer why this this this things work well is that the featuring is all too torsion if I feel he here is of characteristic. So when answering with you we are killing a lot of information but still it remains this quality information, for example, if we work with schemes of car testing zero. So, all this picture has been obtained by the long series of effort and many mathematicians but maybe if you want. So, the last the last point was the was always was proved by in a recent paper by Jean Fazel. Jean. So if you want to see to see this but the first steps were taken by by mark. And there was also an input by Garcon Sean also input by by more and so. Okay, but still it's a complete computation for rational coefficients. If there's no question so I just want to finish with some a word about the six or formalism so it's a kind of accomplishment accomplishment of a six or formalism. It's the gotten the idea that you obtain very generally. So the six or formalism was invented by growth and in particular to be to be able to be able to get this this kinds of charity statements as one knows so you can see that in the ICM conference in 58 I think when he was starting about thinking about the surgery. So it's a kind of surgery but in a very, very general high generate. So here you take F from T to S separate the final type such that S is regular. And the assertion is that if you let me take X rather than T because it will be pretty. So you can define the X to be the F upper streak of the constant object so let me write this one as this leaves in SH. So I take rational coefficients. Okay, and say what this is isomorphic to the top space of cotangent complex of F if X is a regular. But what I'm saying will work for X single. Then what you have if you if you if you take e to be a compact object of SH or S, which means that e commutes with direct sums, infinite direct sums so this is in the language of this formalism is called constructible in reference to what we have for electric shoes. Then you can define the dual, the weak dual of E as being internal arm of E with quotient DX. And the statement of Quotanic Verde duality says that DX is a dualizing complex so it means that the map E is isomorphic to its B dual. E is a nice also stable wiki one equivalence SH or SQ. Okay, so you get out of this world machinery a kind of very strong duality statement that you can after that apply to co-mology and get more concrete form. So this maybe I should say that this Quotanic Verde duality is true here in this generality for rational coefficients but it was also proved by Joseph Iubini's PhD when S is with integral coefficients when S is of character 60. There are several generalization. Okay, so I think I think I took much faster than last time so maybe we can stop here and give some time for questions. Yes, yes, that's perfect. So thanks. Thanks a lot. Let's give Frederick some applause for his beautiful lecture. So we have ample time for questions are there some more than already have been posted in the chat. What does constructible mean other than compact. So, usually what you take as a definition is you take SH of index X index C is subcategory generated by generated by all C minus infinity X plus with some twist and finite coordinates. Okay, so this stable homotopic categories stable by infinite sounds. And you of course you won't have duality results for for these infinite sounds. So you want something which which is smaller and the good generalities to take only the object which comes from X over S smooth. Okay, so you take all the objects which are finite obtained by by taking extension and finite sounds and suspension of these objects here, maybe I should add some high. Okay, and and from it's very meaningful from the motive point of view because it corresponds to so what basically called geometrical motives. It's so when you try to make this definition for after some realization in a category of coefficients, what you will get is the coefficient of geometric origin. Not exactly what people call, for example, in Eladik co-co-moji or in net actually, it's not exactly the analog of this is not exactly the category of bonded complexes with constructible co-co-co-moji, it's really objects of our coefficients of geometric origin. And it's a quite complicated notion, mysterious. But for formative it's work finds and it's equivalent to compact and I should say also that this this compact equal contractible it reminds should remind you about the fact that perfect complexes in the category of which modules are exactly the compact object is this beautiful theorem of Thomas. So, yeah. It's a natural notion of finance to be brief. Are there any further questions. Can I ask a very general one. Yesterday we were working with just set valued pre-sheaves like initially right and this kind of give us category that we could, you know, do some kind of homotopy with algebraic geometry. So I guess for me and I mentioned this yesterday I'm not a homotopy theorist so in infinity categories to me when they come up or always a little bit like was a little bit spooky. I feel like bias here like what by like taking, you know, she's right in spaces, like how does that improve our theory and like why would I think to do that or go that route, I guess. So, the answer is that first of all you get the homotopy category. So, so unstably you get a category which is closer to classical topology I mean, and you have access to invariance such as by one which are very important topological invariance. So the second answer which is my which might be more comodical is that you get more comod if you if you start with a simple show sets, and you get these kinds of extraordinary comod if such as a cobaltism or K theory. And if you're a for example cannot be constructed as a category of complexes of shapes with transfers. It's an extraordinary comod theory. And in topology to get these these comod you really need to make a consider spectra s one spectra respect to simply shows it. So you get more, more theory. And this simplifies this simplifies a lot and there's no difference now, but it's a kind of long story that with integral coefficients it's really different. Okay, that makes sense. Thank you. Any further questions. I guess I also had one that was kind of related to the general scheme of constructions here. I noticed I haven't really said the word Mackie functor because we're not really working with that category of correspondences but a lot of the theory seems set up in somewhat of a similar way and the six functor formalism seems to be giving us that sort of. I forgot the name of the condition on it but it seems to be giving us the sort of like Mackie functor condition with the case that we're working with here and so the thing I was kind of curious about was like, how much of that intuition from like the theory ports over like, you know, is there a good theory of like norms of like an infinity ring spectra structured over this category of correspondences. There are several answers to this so. The first point of view is that if you use transfers and things like that and she's, and she's with transfers or not to be she was transferred by very risky. They relate to a notion which is not so well known of cycle modules, which is completely, which is a generalization of Mackie functor value. So if you are having transfers restriction on course fiction you had another direction which is residues you had residues operators like by that. So, and what you can prove is that she was transfers are completely related to with the cycle of module so in some sense they are some generalized Mackie functor. So the stable homotopy category is actually closer to the equivalent stable homotopy. The stable homotopy category of schemes is actually closer to the stable homotopy category of equivalent spaces, because, for example, when you consider SH over a field such as air or even a number field, then the category is really with a category of, of, with the equivalent category of space with an action of a Galois group of the base fields. And finally, finally for no maps you want to come to look at this at a beautiful paper by Marco you are under. And Paul and Tom Bachman, where they define the study multiplicative transfer it's not exactly transfers for Mackie functor but just just to advertise. So okay, that's, is that answering the question. I was also curious for computing stuff with motives like is there are with with the motives that we just defined like is there a good notion of slice filtration and slice spectral sequence like are those computations that go through here as well. Well, so Oliver knows, knows that better and Mark also knows that better than me but the slice filtration is first important in the stable homotopy category because it gives a filtration which allows first to relate SH with DM. For example, the zero slide of the sphere spectrum overfield is, is known to be isomorphic to the island of McLean, what you expect to which represents much more. But it still exists on on the M. I don't know exactly what the results are. What you have what one can prove with a slice filtration on the, but maybe Oliver or something to say about that. There are so many full on the stable homotopy cut, and it has plenty of applications. Okay. Oh, there's some more questions. That's a slightly vague question but this this DM, the category of month is it close to the category of motifs that growth and it had in mind so does it come with a collection of. So, I've hidden all this picture is beautiful much a picture. I've not said that but on or I've used that on DM of care you have a T structure, the homotopy T structure we just come from the direct category of series transfers and you, you can use this to prove a lot of things, but it's expected that you have another category, which is definitely not the homotopy T structure which is the motivic T structure whose whose heart is the, the, the, the, the, the category of mixed motifs you have been in category of mixed motifs which is, which contains the category of pure motifs as, as the full subcategory and pure motifs are supposed to be the simple object in this opinion category. Okay, so this picture is beautiful you have even more things to say about the decomposition this category and so on. It's, there are many, many papers on this, maybe the book of Ivandre, and also proceedings by and so many, many, many things. But so about, I should say that the Baylinson conjecture is known so as I stated for, for some fields so it's known for just, just the one I stated about the complex. It's not for number fields and finite fields and function fields of curves of the finite fields. And Mark Levin has proved that you can define this motif structure if you restrict two motifs over these kinds of fields, which which comes only from say finite extension, or some twist so the art intake motifs. So we have a trace of this, this motility structure, if you restrict to these motifs we can also restrict to the so called one motive so you can, you can look at the literature so it's not hopeless to say something on this but definitely the obstruction, the number one obstruction is this vanishing conjecture. Actually Mark proved that existence of the motility structure on these art intake motifs is equivalent to the Baylinson-Soulet conjecture. So that settles the matter. Okay, as we see there's a lot left to do. Many conjectures are still there to resolve. So there are no further questions. I would say we, we thank Frederick again for his beautiful lectures and don't forget in about 15 minutes. There's a problem and discussion session where experts will be present, and where you can ask for the questions and in the meantime you can stroll around in so coco. But yeah, let's thank Frederick again.