 So we're very happy to have this is our third short intro workshop series, this time presented by Professor Frederick de Glese. So he did his PhD in 2002, the University of Paris under Fabien Morel is currently directed at the CNRS in Colneau-Miles Superior in Lyon, who's also director of the mathematics department there. So I offer my condolences. And so he's an expert in motivic categories, there are foundations and applications. His work with Denny Charles Suzuki, triangulated categories of mixed motives, is a standard source and reference for all researchers in the field. And besides that, he has numerous publications and preprints in these foundational aspects of motivic categories. So we're very happy to have him to give this introductory series on Boyovatsky's Motivic Complexes. Thanks, thanks for that Mark Fang, thanks for the invitation. So I'll give an introduction to the notes, a survey of the notes, which are also a survey of Boyovatsky's theory. So it's a huge theory. So I guess we'll try to see the overall aspects, but of course you're welcome to ask questions for certain points if you don't understand or if you don't know some techniques. Okay, so for today, I just want to start, so I'll start with an historical introduction. So let me start with one simple formula that everyone should know from, it's called the earlier product formula. So it's like that. So if you go here, this is equal to the product of a prime, 1 minus PS. So I said historical introduction, so it's earlier product formula, it's 1734. So for me, it's one start for all this Motivic stuff and this series. So maybe now I'll draw pictures. So I like this stuff, but the software, the only problem is that I have to, I cannot put in a continuous one, so I have to start new pages each time. Okay, anyway, so let me start to draw my picture. So everything is centered around the notion of L function. So on the previous page you had the zeta function. So L functions are more or less a variant of zeta functions where you vary the one that you have seen on the left hand side. So we have two directions. So the first direction, let's say, is the function field case. And it starts by the form of the field conjectures. So L function, I don't put dates because it's, but I should, I should indicate RT and ACI, of course. So it's around the 20s. Date conjecture, I can put some dates, it's 49, okay. And so it led to two well-known developments. First one is the etala. So eladic shift theory, maybe I put shift in, she use. And so it's got a leak and it's collaborator. So I let me put SGA4 and this is 64, 63 published, maybe 66 or 8. Okay. But there was also another notion which has not appeared in, maybe in the Linus-Ajuan paper, but it's the notion of weight. So it's, you can find this in letters from Grotendik to say, okay. And so etala eladic shifts have answered the veil conjecture because of the work of the Linus. But before the last proof of the Linus, Grotendik introduced the notion of pure motives. So never published anything on this, but this is around the 69 or 70s when there were climber and several papers on this. Okay. So this is already points and, of course, weights are also a motivation for pure motives. Now let's turn to the other aspect of L functions, which is the number field case fields. So here I should put, maybe I should have put that before, the so-called class number formula, which is also a UG story called Motivation from Derrick Clay. There were several versions, but in the series time, 19th century. Okay. So there were several work on special values, but on L functions and after L class number formula. Which the thing I want to say is that it was this important work by Liechtenbaum, 73 on special values of L functions, which is, okay, and here I enter into play another actor in this thing, so it's Kefiris. So Liechtenbaum used first etalcology to formulate some conjectures and then Kefiris. So there are many people here, many works here that contributes to this, contribute to this, but I don't put everything. So of course there is Borrell and so on, but okay. And also last thing that I should add here is the theory of perversives. This is Bernstein and the line and 82 asterisks, okay. And here maybe I should put another important notion out of this structure. Okay. Now I finish my drawing because all I know I forgot something before the notion of pure motives. There was a thesis of the line. So it's a notion of mixed structure, it was before just before the pure motives. So it derives from the notion of weights, it's in characteristic zero. So all these perversives come from etalaladic sheaves, okay, and all these notions have given my last item, which is the Bernstein conjectures, Liechtenbaum also gave some conjectures like that afterwards. So this is around the 84, 86 maybe it's in high pairing. So Bernstein conjecture, the existence of a new category that we will describe actually now, I mean, what exactly, which come, which, which enlarge motives and incorporate all these aspects here, okay, maybe last, last items of what what today and what's about the, what the summer school is about is pervers keys from auto P theory and maybe for today just multi complexes, which are supposed to answer the Bainson conjecture and they are quite successful. So first, first work of the rescue was PhD physics in Princeton and it's 94. Okay. And last thing, maybe I want to advert the, it's, it's a kind of extra, I want to advert the, the talk of, of, of, of Matthew, so there is some development here. So maybe I should add, so it's, it's aside, it's today, it will not be about this. So there is a notion of pediatric health function, which are roughly health functions, but we will coefficient in ZP or rings like that. And here it apparently gives some development about a notion of p-addict motives or I don't know if it's a good name for that, but this, this will probably be so this, this is a question working. Okay. No questions so far. So we have all the, the, the aspects that I want to, to, to stay here. So the, the model for the rescue theory is the theory of et al, et al sheevs or et al torsion sheevs. Okay. And, okay. So there are, maybe I should, I should give the plan. So my first, I will first give reference for the notion of sheevs with transfers of overseas, which are, maybe I should add homotopy, which is the analog of, which is modeled on a theory of et al, et al sheevs, et al torsion sheevs. Then I will go into the case of perfect field case and main theorem, perfect field and main theorem of the theory. And lastly, we will go into the definition of multi-complexes. Okay. So my convention for today's is that so I also wanted to, to tell you about the notion of, of six transform isam. So I will work over some base scheme. So we'll fix, we'll fix S a regular notarian, maybe finite dimension or cool dimension scheme. Okay. So either you can take a smooth scheme over a field or the spectrum of a regular scheme. So otherwise it means that all the local ring are regular. So it's supposed to incorporate what you get when you have a curve over a number of field and then you can, you lift it, say over specksy. And usually you can find a model which is regular, not, not smooth for specksy, but regular. Okay. Okay. So now we, we start with a notion of sheevs with transfer. So there are three differences, three differences with respect to et al sheevs of STF4. So the first one is that sheevs admit transfers and so here I will use the notion of algebraic cycles and finite correspondences to see in a minute. So of course it's, it's this, this notion of transfers is very linked with a notion of motive. So I will not review a fear of two motives, but it's based on the notion of algebraic cycles and correspondences. Okay. The second difference is that we, we will use a big, smooth, Nisnevich site for my sheevs. So I'll give, I will give definitions. And the third difference is that it's very long. I think now is that we use the A1 homotopy relation and this allows to incorporate algebraic topology techniques into this, this fear, but you will, you will also see in a, a question. Okay. So let me start with a notion of, of a finite correspondences. So the idea goes back at least to Italian geometries. So they remarked that the notion of morphism of algebraic varieties is a bit too restrictive and so if, if, if they wanted to get some more, more flexibility and so they started to use algebraic correspondences. So that's what Bojewski has done. So all this theory of course is due to Bojewski, the theory that I represent. So let me give a definition. So now X and Y are smooth schemes and that's smooth schemes, separated or finite type, separated or finite type over my regular scheme S. Okay. And a correspondent, the finite correspondent is alpha, finite correspondent is correspondence from X to Y is an algebraic cycle, some, so this is a sum, a finite sum, one, two, and say, of a formal sum of a reducible scheme. So whereas the I inside X cross SY is closed integral as a scheme and the projection of the I to X is finite and subjective over some connected component. Yeah. So S is regular. So an X is smooth over S, so X is also regular and the regular scheme is a disjoint unit of its irreducible components, which are all connected components, okay. So it's an algebraic cycle inside X cross SY and that with a restriction that, so we also say that the support of alpha over X is finite and equidimensional. So the good point with this notion of finite correspondence is that you can, you can turn to turn them into morphisms. So this means that you can compose position, sorry, maybe I should add a notation. So we say you denote C, S, X, Y, the set of finite correspondences, which is actually an obedient group of finite correspondences from X to Y over S, okay. So now suppose we have correspondences from X to Y and beta from Y to Z. So we want to compose them, then we can, let me, I write X, Y for X cross SY, maybe I should have put so on. So if you have never seen that, this is classical in the theory of algebraic cycles. So what we do is that we consider the product of all three smoothest scheme like this and we have three projection, one from X, Y, X, C and Y, C, P, Q, R, like this, okay. So P, Q and R are smooth morphisms and we can always define the pullback of alpha, P, A star, alpha and Q, A star, beta are well-defined cycles, algebraic cycles in the product. Z for, it's easy, you just take, so if I assume that alpha is a sum of Zi, you just take this product, X, Y, Z. So the pullback is easy to define because my schemes are smooth, okay. And then it happens that the cycles, these two cycles intersects properly in X, Y, Z. So it means that using SearStore formula, but there is also you could use Samuel formula if you want, but that's not, this is correct way to do it. You can intersect them and obtain some cycle. This is another algebraic cycle in the product, okay. And now, so maybe I should picture this, we add something like this. And in fact, the intersection of, considered here is something like the fiber product. So if alpha and beta were the class of some irreducible close-up scheme, it will just be a fiber product here. So it means that what you deduce from this, let's call this gamma, you deduce from this that the support of gamma, because it's a projection here is finite and equidimensional, okay. And now also, if you take this equal t, so it means that you can, if you look at the projection r restricted to t, it induce a map from r, from t to r of t, which is well defined and which is finite, h. And then you can define the push forward of cycle, h star gamma, which is well defined also. Okay. And now, this is what we call beta composite alpha. Okay. Okay. So, sorry, I've missed some questions, but obviously, okay. So the good point is that you can, you can check associativity and also there is a neutral element. So let me give examples so I should have done that before our final correspondence. If you have a morphism of smooth scheme, x from y, you can look at its graph of f. So it's obtained. So you take, sorry, y cross s, no. So you always have a diagonal map. So because y is separated over s, this is a closed emotion. Here you can look at x cross s, y, you look at the map, f cross identity, and the graph is just pullback. Then the graph is a closed sub-steam in x cross s, y, and you can check easily that graph of this. You can associate a cycle, just take the sum of the connected components, x, and it's actually a finite correspondence. You can see because actually the projection from gamma f to x is an isomorphism. So this I will just denote it f like that. So the neutral element is just the correspondence identity of x. Okay. But I have another example of an interesting example of finite correspondence. If now you assume that f is finite and say equidimensional, so it means that every any connected component of x dominates a connected component of y, it's not much equidimensional here. Then if you look at the graph of f inside x cross s, y, you can decide, you can switch the, you can switch the factors of this scheme here. And then let me write epsilon star gamma f for a cycle in y cross sx. Then the projection from gamma f to y is just the morphism f itself. So it's also finite equidimensional. And this gives you a finite correspondence, but this time from y to x. And this I denote t of f and it's called the transpose of f. Okay. So what you can check out of all of this, but you have defined a composition product on finite correspondences with a good neutral elements and you have all this. So maybe I should give a definition here. So you have actually defined a category, a category of smooth correspondences over s whose objects are smooth s schemes and morphisms are finite s correspondences. Okay, maybe for, I should denote this gamma f. And what you can check also, so there are things to check here. What you can check also is that the graph functor defines the graph that I defined in the previous example defines a functor, which is the identity on objects. And maps morphism to this graph. Okay. So, so we have actually and not, and of course this map gamma is faithful, not full. And so we have a larger category of smooth scheme. So we have added morphisms. Okay. Thanks, thanks for sorry. Before going further, we can see that there are further operations on these categories of smooth correspondences. So let me take f from t to s any morphism of scheme of regular schemes. So I can define a pullback from some quality. So when I have a smooth scheme over s. I just take its pullback here. And when I have a finite correspondence. So I have a map so let me write x, y, it will be up obtained by pullback. So then I can define this star alpha because it will just be an exile product of cycle. Okay, so it's a base change. And you can check this is x t. Okay. But also if you have, if now you take p from t to s a smooth morphism. So when I say smooth it will always mean smooth separate of finite type. So I can define, so this is the f of a star, I can define another photo, which in some sense, forget the base. So if I have a smooth scheme, so maybe I should write this y t. I can just see why as a smooth scheme over s, just by composing with f here. So if I have some correspondence, maybe in city by prime. Now if I do that, what I have is that by cross s by prime is closed. So it's the other way. So I can associate to beta just via the direct image and you can check that this finds a correspondence form. Okay, so I have a photo which I will do not be sharp on which forget the base to define sorry maybe. To define this f of a star alpha you need to, to use the intersection theory sorry, but it works because we are dealing with finite correspondence. Okay, so we have base change for forget the base and there is another. And over, there is a monoidal structure. Same core s. So if you have on objects, if you have x, s and y or s, you just put, it's just cartesian product because it's a smooth scheme and if you have finite correspondence alpha x, x prime say and beta y to y prime. Now this time you consider the exterior product and gives you a correspondence from x, x prime. So you can check that you have in this way. So this is a monoidal structure on SM core s. So this category is good. There are some funtals and it's additive, but it's not. It's not enough to do. Well, you can do some, some homological algebra but we won't do it today. Instead, we will now go to the notion of sheaves and transfers. So this category is just additive. It's, it's not good a priority to do homological algebra. So we just, we will just enlarge it. As usual, you should be familiar with this procedure. So we consider pre sheaves on this category. So pre sheaf with transfers is just a contrarian funtals on core s. So I put up to say contrarian, which goes to the category of Abelian groups. And we add the property that it's additive, with respect to the additive structure on SM core s. Okay, so it's just the standard, the standard way to enlarge a category. And we will get a notion of natural transformation. This becomes a category and this category here has the advantage to be Abelian. And actually, even its growth and decalbillion complete and complete. Meaning it admits limits, co limits and growth and decalbillion. It implies that it admits injective resolution, for example. So it's good to do homological algebra and we still have a Yoneda embedding, which here is the following. So we get a canonical funtals, which to a smooth scheme here you associate just the pre sheaf represented by X, pre sheaf with transfers. This is denoted the str of X. Okay, but before, now the thing is that we have good example of sheaves with transfers. Example. So we have all of the representable pre sheaves also, but first one is if you take the shift GM. So restricted to smooth scheme so it's a smooth. And then associate just set the invertible global function on X, then GM is a pre sheaf with transfers. So I let that for maybe exercises or I gave references in my, in my notes. So for example, which is even more interesting so I've restricted to feel here. I'm not sure it's it's necessary but if a is a nabillion variety. Okay, then. Sheaf which I will denote by a underline which X. Okay, maps to transfer morphism of schemes. Next to a is actually a pre sheaf with transfers. Actually, it works for any commutative group scheme even group schemes actually. And it work also for semi abelian variety just for those who knows semi abelian varieties are extension of tourists by abelian varieties. Okay. So these are important. Particular cases of precedence transfers. And we have a last, last example, H star is a good. Say, for example, you take Betty Como G or et al Como G or the Ram Como G. Then, if you take S schemes. So it can be S or K can be smooth, or just regular if it's or just regular. If you use a electric et al Como G, G, L invertible on S. Okay, so we have concrete example given by this geometric example and also Como G theories. Okay, so we have concrete example given by this geometric example and also Como G theories. Okay, so what we have done here is we have enlarge the category of smooth schemes over S into this big abelian category but now we want to incorporate the other example the other ingredient which is the Niznevich topology. So we have seen that in the talk of Philip, but I will just recall. What's the Niznevich topology so Niznevich topology is topology on say SMS whose covers are given by morphism. In SMS such that for any points in X there exists a point wise in W such that P such that P of Y equal to X and the resident residue field extension X is trivial, meaning it's an isomorphism kappa of X is isomorphic to kappa of Y. So this was also called and also sorry such that P is et al. So first P is et al and so objective so this is an et al cover but we add this condition which is also called which was called completely composed by Niznevich and which means that which is which is a stronger condition and so the Niznevich topology will be different from the et al topology. Of course, a zaski cover will be a particular case of Niznevich cover, but you have you have more Niznevich cover and zaski cover. So, Niznevich is between et al and zaski. Okay, so maybe also I should say a word so about the condition of being a shift and for this I will introduce what Morale and so Niznevich distinguished square, Niznevich definition. So Niznevich distinguished square in say sm of s but it works in any for any scheme is a square over form u x w P such that j is an open immersion P is et al. And if you put x closed reduced sub scheme in x, which is a complement of you then P induces an isomorphism from P minus one. I'm not sure I will use this. Let's call this T I will use this, this notion but I, I like to use also the notion of close pair so we say XZ is a close pair. X is a smoothed scheme and this is just a close sub scheme. We say that the map P from T X to Z is an excessive morphism or close pairs. So that's it's useful to formulate some properties. Okay. So I'll let, I don't look at the question that you can interrupt me if you. We're having we're fine you just keep going. Good. Good. So the lemma due to moral and very risky is following so I will assume that you are familiar with the notion of she's so when you have covers, you have a cotton dictopogy and then you have a notion of associated she's. If you, you are not familiar, you can just use the lemma that I will state as a definition so let's take F from SMS to or actually, you can also take just sets, if you want. And F is a Niznevich if let's call this for any, for any Niznevich, I forgot this distinguished square, call it delta V image by F which is another square, either of a billion groups or sets is co Cartesian. So you can take this the second property as a definition of Niznevich. So this is also an exercise in so maybe I tried to go faster because otherwise I won't have time to finish. Okay, so that's for the recall on this news topology. Now, we want to incorporate this, this notion of topology into our, our pre shifts with transfer so we will say that if we take F a pre shift with transfer of S will say that F is a shift. I should say Niznevich but in all this talk, it will always be Niznevich approach is a Niznevich chief with transfers. If when you compose F with gamma so that's cool. Okay, so we just look only at the objects of pre shifts with transfers such that when you restrict the smooth schemes, you really have a shift. Now it's better for co mode equal reasons. So I should have said that before but why transfers, because if you take F from Y to X a finite equidimensional or subjective you can just morphism. Then you can, and you have F either pre shift or shift with transfers. F, then you can apply F the transpose of F. And this will give you a map in the wrong direction. You can do it by F low star and these are transfers. Or sometimes easy maps and so on. Besides these transfers have good properties but I won't say them. Okay, so maybe before going into next next step, I just want to add a remark about this notion of big site. So what, what, what, what can you, how can you interpret this notion of big site so of course, if X, let's take X smooth scheme over S, then you still have a small Niznevich site, like for et al topology. It's, it is made by just, you look at only the, it's a category of scheme which are V over X, which are just et al. X, okay. So, and then you endow it with the Niznevich topology. You have a notion of shift on X this. Now when you look at F, a shift with transfers, so over S it's in particular shift over SMS. So it means that you can look at the restriction of F to the category X Niz and call this FX. So, now, because you have a shift on the big site, so you automatically obtain maps. So for any morphism, Y to X in SMS, you automatically obtain maps so it goes from F up a star, F of X to F of Y, F of Y. This is a morphism, a natural transformation of sheaves on the small site, Y. Here you have, you have used this, this operation that I have not defined, which is a pullback for sheaves, but it's, it works like for size key sheaves, okay. And this map here to F is a structural map, so a shift on SMS corresponds to the data of sheaves for all SMS, which are on this small site. So this transition map to F, which are not isomorphisms. Actually, then you have, for example, you have an inclusion. You can look at sheaves on S-NIS for the Niz, and it's included in the category of sheaves on SMS plus Niz, and it goes to you associate to F just the sequence of sheaves which are obtained by pullback F star of F, smooth, okay. So all this is just for saying that sheaves on the big site, so we say big site because sheaves on this site are much bigger than sheaves on the small site. So they are given by collection of sheaves on the small site and transition map, which are not isomorphisms, okay. So this is completely linked so for those who knows to the notion of crystals, this holds. And so I'm just saying, okay, but anyway, we can, we can work with this, this notion of sheaves. So let me add another construction. So, of course, all these these notion of sheaves and so on comes from SGA4 and the very abstract topos fury. But there are, as you have said, there are several examples of advantage to work with pre-sheaves and sheaves. So we will see in a minute that sheaves with transfers is also a category of an IBM category, for example, but also you have more operation on this. So remember on this category SM4S, we add free operation, F up a star, base change, forget the base and some tensile structure maybe. Then all these operations can be extended to sheaves. So suppose, for example, let's take the first one, you have a map of regular schemes from T2S and you have a sheave with transfers over S. Then you can, you can just look at F composed with this pullback map F up a star. And then it goes to a billion. So X, as I said, we add X cross S T and then we just apply F. So this is usually called the direct image. Okay, and this you should, you shouldn't, if you have seen Zajski's sheave, this is a usual definition. And you can see immediately that if F is a niznevich sheaves, then this F lower star F is a niznevich sheaves. Just because niznevich covers are stable by pullback. Okay, and here we have another information, this, this direct image admits transfers also. So we have a full tour. I will write this in from, sorry, so F was a sheave with transfers over T. And F lower star map like that. And when you have such a full tour, you can check that formally it admits right a joint, left a joint, sorry. So you should, you should be used to these kinds of F upper star. And by the way, this F upper star is characterized by the foreign property. I happen to realize that I have forgotten example of sheaves with transfers. F upper star of C transfer S X equal to C transfer S of X cross S. So let me add an example that I should have done before of sheaves with transfers. So as I said, over S, if X over S is smooth, we add this pre-sheaf with transfer, which is just represented by X. And you can check it isn't an etal sheave. It is a niznevich sheave with transfers. Okay, when I restricted to smooth schemes. Actually, it's even an etal sheave. There are over example, all the one that I've given so GM and the line. So this is a sheave with transfers over S and this I restricted to fields where I is just an obedient varieties or even a semi-habitant variety. Okay. So actually, so as I said before, this F upper star is defined formally up here. It maps a sheaf. It's a best change for sheaves. So it takes a shift with transfers over S and it gives you a shift with transfers over T. And it's uniquely characterized by this, this property and the fact that as F upper star is a left drawing, sorry, I put it on the left. The F upper star is left to drain. So it commutes formally with code limits and then using formal stuff you can check this property. Okay, so I had my example of sheaves and now I had over example, if P from T2S is smooth, what you get is that you can, you can what you can check is that if F is a sheave with transfers over T, then when you compute P upper star of F defined in the previous blackboard, what you obtain is just a composition of F with this from top P sharp, which forget the base. Okay. And actually, so if you are followed what they do with the best change from tour. So we have this from tour P upper star which is obtained by composing with P sharp, and it admits a left adjoin P sharp such that P sharp is defined characterised by the property that maps this sheave with transfers to X, S, Y, T, S, various transfers and so on. Last thing is that so this was the second operation and then for you can define a tensor product with transfers on this category of sheaves with transfers such that the transfer X tensor is just given by X, X, Y, and this is truly this is a close symmetric model structure. So again, you have an adjoin from tour, which is the internal one. Okay, so just remark that's what we, we have defined exactly six from pause, which organising pair of adjoin from pause. Okay, so that was for the formal part of sheaf theory. So okay, I'm definitely late. So the last important property of that very risky introduces the so-called a1 invariance and for sheaves. So the definition is very easy. So you take a sheaf or it work also for pre sheaves with transfers. Just take a sheaf and you say that F is the a1 invariant. If for any smooth scheme X over S, the map induced by the projection of the affine line X to F over A1 X is an isomorphism. Okay, so very simple. So let me remark on something. So this means so this means actually that F depends only on a1 homotopic classes of final correspondence. So what do I mean by that? So if I have alpha beta final correspondences from X to Y, I can say that alpha is a1 homotopic to beta if there exists H, a final correspondence from a1 cross X to Y, such that as in topology, when you compose H with zero section of a1 of X, you obtain alpha and when you compose with a unit section, you obtain beta. Okay, so I'm cheating a little bit because this is not a transitive relation, but you can you can take the induced transitive relation. And then you get an equivalence relation and then a1 is an equivalence relation on morphisms of this category SM called S. It's compatible, you can check it's compatible with composition, this is formal, and then you get a homotopic category by SM called S whose morphisms are just the homotopic classes of correspondences from X to Y. Okay. And if F is a F is a1 invariant, it's more or less is equivalent to say that F factors through this homotopic category by SM called S. So as a funtore like this. Okay. So we will, this is how we will able to incorporate algebraic topology techniques to in this framework. So before going to the main theorem, I just give examples. So, but you can check by yourself GM is a1 invariant also underline is a1 invariant. Okay, where A is a semi-evident, not a group. Okay, so again, this is an exercise. So, maybe now I should, or the first theorem which, which allows the various key to, to make his to, to work with finite correspondences up to homotopy. So maybe I'll recall to you the relative pickup group. So, if XZ is a close pair, which means a scheme and a close up scheme. The relative pickup group of this pair is made by the couple, the pairs of L phi where L is an invertible sheath on X and phi from LZ to OZ is an isomorphism so it's a trivialization, a given trivialization of L OZ. And you look at this up to isomorphism. Okay. Okay, so the main theorem that I'll be using, which is, so maybe I should say it's true to susan invariant scheme is the following take S so I have to restrict to S and I find regular scheme. I have to take C or S curve, smooth curve, smooth, a fine curve, which admits so called good compactification such that so very exist C bar proper C is open in C bar and you also require that the complement C bar minus C, which is not by C infinity is not empty. Of course, C is a fine, of course, but it admits a fine open neighborhood in C bar. She's called a good compactification. And when you have that, you have that for all X of S move a fine, maybe I've put too much a fine scheme here, but it's not important. And then you can compute the morphism correspondences up to a motor P from X to C. It's isomorphic to a Pica group of X cross C bar relative to this closed. Okay, so the idea to prove this theorem is, if you look at the final correspondences from X to C, it's actually cartridge. It's a very divisor in X process C. It's called dimension one, because everything because C bar was normal you can see it as a divisor in X cross C bar. So this scheme is normal so it correspond to a line bundle. So, and you can see that there is a trivialization. And then you should prove that it's an isomorphism. Okay. What guess I should guess. So application of this, of this theorem. So I go to the second part. Certainly I should use my next, my next talk to finish this talk. In the second part I will restrict now to the case of this is the spectrum of K where K is a field. So I don't put perfect for now because it's not always use use but it's using the main theory. So the first PRM is the following. So because of the preceding theorem you get the following lemma, which is take any open immersion, you text in the category of systems over K, then there exists a Zarski cover. And a finite correspondence for here. So I tried with diagram commutes. Commutes. Commutes up to a motor piece. Okay, so this lemma set tells you that locally for the Zarski topology, open emotions, and it's splitting things. When you, when you have added these correspondences, and the immediate quarry is that if F is a shift with transfers over K, which is a one invariance. The map from X to FU is a split monomorphism. So this is very strange for shifts. But it already tells you that so I told you that shifts on the big side were big object. And you see that the virtue of the one invariance is to make them smaller because it adds this huge restriction here. So you can reformulate this, this, this statement like this so if X is a smooth scheme over K, and you can go from F of X to say, product, product or something of the generic point of F evaluated at function field. So, you have put F of X is just the limit of open neighborhood, F of U. Okay. And this is just a set of generic points points. And what we have obtained is that this is a monomorphism. Okay, so F of X injects in the product of F evaluated that this function fields here. So, this means so roughly space, roughly speaking, it means that fiber for dolls for the somatopoeia and I achieve with transfers. Function fields here. Okay, the function fields it means that he is function of any Z is your case move. And you define F of E by the formula. Above. So, sorry, just X. Then F is a conservative family of funtals. If you consider all functions fields. Is that clear. So because I'm lacking of time maybe I'm not so clear. Okay, so maybe now it's time to give a name definition. Okay. So, we have just seen that we have this huge restrictions open immersion must induces monomorphisms. What we can show more or less for it's not so easy is that H. H is a billion category. And actually so, of course, you can always forget that you are homotope invariant and you go into this category. So, this category is also a billion and you can show that there exists. So, this is a zero which turns you into homotope invariant shift with transfer. Okay. And as I said, maybe even more interesting thing that you have is that if you take a billion varieties of a K. There is a fully faithful embedding into this category of homotope invariant shares of transfers. So, maybe you could think with that that this category HIT of K is, is of Motivic origin because the billion varieties are the first example of motives. H zero is yes left to join to the inclusion. Okay, so. Maybe for tomorrow, for today, I will just go into the category of shifts with transverse homotope invariant shifts and so on and I reserve Motivic complex for tomorrow. Okay, so I have a second theorem for homotope shifts with transfers to state, and this uses a very, very famous construction of a minus one construction. So, it's very simple, the definition. If F is a homotope shift with transverse of K, you define F minus one of X, a kernel of a map induced from the evaluated at GM cross X. So we're going to F of X and what is this map is just look at the unit section of GM of X, and you take its inverse image. So this map S one is a monomorphism so this one is one upper star is an epimorphism and you can, you can check really that F minus one is actually homotope invariant shifts with transfers. So, I have not much time to mention that but what you can check is that F minus one is internal on computed in the category of shifts with transfers of GM. So for those room, it's, it's maybe for in fancy terms it means that it's GM look space of F. So, okay, maybe another example if you can do it by yourself, if you take GM and you apply the minus one construction. It just gives a constant shift, which is actually a constant, constant is never chief. Third example that you can also check by yourself is that if you take a nabillion a now a is a billion variety variety and a minus one is just so. Before stating the main theorem I just want to, to introduce this theorem which is not. It's a kind of devisage theorem, or it could be a lemma of Iversky so it's, it's actually a lemma for the main theorem. And it tells you, now K is a perfect field. Okay, maybe, maybe not necessary here. So, I just take Y from Z to X, close the immersion of a smooth divisor, of course, X over K is smooth. And I put you to be the complement and I look at the open immersion. And then, I always have this map that I have defined by I junction I get a map from F restricted to the Zarsky site. Remember, so this is F restricted to X, this is this new site sorry. And I can always go to jail or star F of you. So this is the map, which is a joint from the structural map of F seen as a she from the big site. We have seen by the lemma on and the fact that the chair the star is always a monomorphism or of she so we obtained that this map is a monomorphism. And actually, very risky compute the cocoon and it is why lost our F minus one, and then you restrict to see. So this is an exact sequence. Sheaves small site X. Okay. So I have no time to prove this theorem but again it really uses the existence of correspondence and the. I'm also seeing various here I've given you to be able and also you use excision to come back to a simple situation. Okay, maybe I give you a geometric interpretation of this or comical sorry interpretation in terms of comology with support. It means it implies with a notation of of preceding theorem but if you compute the co mode even isn't it co mode of X with support in Z with support. And it's actually equivalent to H zero of the F minus one. Okay, so this Davis HRM is a kind of purity here. Okay, so maybe now it's time to, to state the main theorem of a risky so I could feel that of course everything is due to risky here. So it says that now case really a perfect field. This isn't strictly necessary I think. Okay. Okay, then you can study it's cool moji and the assertion is that for all smooth scheme of okay, the co moji of F and each co moji is homotopy environment so in other words, and the co moji of X is an isomorphism. Okay, so as I said it's a main, main theorem of theory. It has also motivated several work in a one homotopy afterwards, notably by more and also when you consider different kind of transfers. I won't be able to give a proof now in except, except even because it's only five minutes I've only five minutes left. But maybe I just say that this theorem uses the davisage that I started which which allows you to to consider the case and equal on the case and equal one first then you use induction and the key step, key point is to prove that for any open immersion. J. Then our NJ. Zero. So the proof use induction and prove both property first co moji. First there is this isomorphism and co moji and you have also this vanishing. And when you have this vanishing it allows to generalize this, this purity property to higher degrees. Okay. So, as I said an important part of this theorem is the so called purity property. So now you take C. It's a x roof, but of co dimension. Take C. F as above. And in which co moji with support is isomorphic to and minus. Comoji of C but degree and minus C and with question in F minus. So if you have a proof of this query again, it's for C equal one. As I said, it's, it's a query of the proof and of the davisage theorem and then use induction on C to get the general case. Maybe I can reformulate this property as follows. So if you have a point x in x of co dimension now I take and then you can use the so called co moji local co moji F. So, this is back. Oh, X, X. So I'm just using that you can. So X localized that X is not smooth. It's not a finite type of okay but you can extend F and co moji to do this kind of schemes. So X itself is a is a close point of the scheme and as a byproduct of this query what you get that is that this is zero if N is different from C. And this is F minus. So I should just I different from. This is F minus N evaluated at the residue field so this is a function actually if I equal N. Okay, and so for those who know this, this, this property is is actually stronger than the property for the shift F seen as a shift for example XR to be co and McCollet. So, maybe I, I set another query which is almost a few rounds, it's called the Gersten resolution. So first way to say that for all X smooth. Okay, when you look at the shift X. F F at the shift F restricted to X X NIS or XR it is co and McCollet in the sense of our child residues and dualities. And then it implies that it admits a Gaussian resolution a cousin resolution so we have a cousin complex, which is C star X F, which has following form it's in degree N. So the sum of the point of co dimension and F minus N X like this, and this one sorry, and plus co dimension. Okay, so it's called the cousin complex of FX but it's also the Gersten complex of a full shift F. Frm is that the co emoji in his name each of X F is the same as the co emoji of this complex. And here you can actually put size key on his name. Maybe I use a few seconds. So just to pose an exercise so you can consider the case of GM right because in complex for GM. And maybe for Matthews talk I think so I had to to define the higher tropes so I won't do that but but at least I can I can state another theorem which is linked with all this. So, you can consider GM and the tensor product and time so let me write this. So this tensor product in H, I, R. So this tensor product is induced by the tensor product on shift with transfers. Okay, and the main theorem is that if you evaluate SNT at the function field. And it's actually isomorphic to K and the middle key theory. And what you get is that if you write the Gaston resolution again an exercise for this, this particular shift SNT, what you will get is that the co emoji in degree n is David of X with co efficient in this shift is isomorphic to. For those who know the child context in co dimension and in fact what you what we get is that's why I had new names sorry for this SNT is isomorphic to the so-called un-ramified middle or K theory. This formula is some version of blocks form. Okay, so I think I'll stop there and I'll continue either this evening or tomorrow for the category for the motive complex case. Thanks very much. Are there any questions, comments. Lively discussion the chat so I'm sure lots of people have a question. We answered all the questions in the chat. Thanks. Thanks.