 Thank you very much for the introduction and also for the perfect organization. So we are online, so I'm far off of you, so I hope that everything will go smoothly. So let me tell you that before starting, I put some links on the chat about my notes. What I present here is, I will say, a short version of a longer text with references and so on, which is the first thing. So if you want still to come back to this Beamer file, it is public also and you like exercise, there is one. I will put the links after in a proper way on my home page. For example, I don't know, maybe it will, there will be on the site of the summer school. Okay, so repost the link. Philippe, maybe a little bit of organization. I'll try and feel most of the comments from the chats and if there's something that I think you want, I'll interrupt you. Thank you. So please interrupt me and of course, I cannot be concentrated on everything, so I cannot look at the chat. Okay, so let me start with the screen. Okay, so this is not the real place, so it's a good try. So before to talk about the third one's correspondence, I would like to do a brief introduction about the topic. So just to test my material. So what we do is, as you know, this is algebraic geometry. So you see my writing and you understand what the reason why I make notes. And what we do today and the other days is to take topics coming from differential geometry. And so one topic is known for a long time. This is vector bundle. So in a differential thing, so we can and we see that today how to make them in a nice way in algebraic geometry. So this is before I will say the second world war. But what is more interesting and today this is a notion of friendship principal g bundle or principal homogenous spaces. And so this thing was put in the way in algebraic geometry by Jean-Pierre Serre and Alexandre Crotendic. So there is a date which is 1958, the official birth. But actually if you go back before, you will see there is an unpublished memory by Crotendic. So it is on the note. When he was postdoc in I call it the Kansas paper in Kansas University. And which is kind of predatory. So it's a Crotendic way to see about principal g bundles in differential geometry and analytic geometry. And many things actually are already in that paper. So the material I'm talking about today is in many places. So there are several references. So for example, you have the book by Max Douce, which is called quadratic and emission forms over rings. You have the famous book by Demasur and Gabrielle, which is in French. So we have a short version in English. So this is not recent, but this book is still the best one, I think. So group algebraic. And you have also, of course, other references. So the reason why I'm talking about those references is because they are exactly in the sitting I working today. So today, I work over rings, or if you prefer, over affine schemes. So I will not do the theory of general skin to simplify. So for people aware of the general theory, it's quite easy to generalize. So this is because most of the definition of the work is of local nature. And for people knowing only about rings, and even about rings, which are algebra for some fields, they cannot turn the leg. So let me start with the most, with the beginning. So the beginning is with vector bundles. This is called the Soins-Saire correspondence. So what is this correspondence? It's the correspondence between projective finite modules of finite rounds or projective modules of finite rounds and vector bundles. And this correspondence was known in the differential geometry setting on paracompact topogical spaces. So there is a beautiful paper by Robert Swan. So this is this reference 37. And I will recall it explicitly because this is really the first stone of the construction in the setting of affine scheme. And what I follow, it's more or less the book of Gersvedon, which is an excellent book, by the way. Okay. So what are vector group schemes? So before talking about the vector groups, so we are given a ring, so ring of algebraic geometry, so an emicomutative unit. And we take an air module. And with this air module, we can attach this symmetric algebra. Okay. And we can take the spectrum of this emicomutative algebra. Okay. So this is an affine scheme over air. And in terms of functor, so what I call an air functor, this is a, if I take air algebra S, I can consider the home of M to S, which is the same thing there. Okay. So this is just the fact that we have an air functor. This is the universal property of the symmetric algebra. Of course, if you take M to be air, okay, what you find is just the affine line. Okay. So this is very classical. So this is this construction. And so it's called the vector group scheme attached to M. I realize that I slice it too much, but it's too late to change. And the important thing, if you are in the, you work in the content next setting, this is that it commutes, that construction commutes with arbitrary is the change of race. Okay. So there is, and then what the nice thing is that we can really like that, see modules like schemes and conversely. So in a clean way, this is to say that the functor which associate to a module M, the vector group M is an anti-equivalence of categories between the category of air module and that of vector group schemes. Okay. So be careful. This is anti-equivalence. It permits the sense of the arrows. And the inverse vector is clear. It's just, you are, it's just to take the section. Okay. So you can, from VM, if you evaluate to the air point, what you get is precisely M. Okay. So, okay. So I continue on vector group scheme. I don't know why. Okay. And so a very special case, but which is the main case of interest for us is when M is locally free or final run. So in that case, we can consider the dual and the dual is also locally free or final run. And then the air algebra, the symmetric algebra is a finite presentation. Okay. So the reference here are in the, this is the content of your donate. And then what happened is that the functor which is more natural to think about in as per glance, namely to take the M tensor RS is representable by the affine air scheme V of the M dual. Okay. And another notation for this guy is WM. Okay. So of course you can ask about representation of WM about this functor here for an arbitrary M. But it turns out that the finite locally free assumption is a necessary condition for representability. And this thing, I trusted it in a lecture by Romany. I did not trace it anywhere else. So if somebody knows another reference, I will be interested in that. Okay. So this is not in the basic reference for sure. Okay. So vector bundle. So this is the definition of which is in the girls they don't, which is perfectly good. So a vector bundle of our air, air can be zero. Be careful of our speckers. The affine scheme speckers is an affine air scan X subject. You can find a partition of the unity. So one is F1 plus Fn and isomorphism of the localization. So when you take your local X, localize X to RFI, what you get is just V over RFI power air. So namely the affine space of the motion air. Okay. And we want one more condition. This is not enough. Okay. We want that the cars maps. Okay. So what happened when you take phi i minus i, phi minus 1, phi g. And we want this map to be linear automorphism. Okay. Else, if you remove this second condition, what you get, this is so-called, so in geometry, they call it the theory of affine varieties. Okay. In remaining geometry, for example. Okay. So do not forget the second condition. So I will be more precise later about what I call a linear automorphism. But this is an automorphism which is induced by a linear map between these two motors. Okay. So this is, sorry, I go to two. And so what is the Schwan-Seers correspondence? So this is the above functor. So I mean the functor I had before. When you restrict it to the locally free air module, so it induces an equivalence of categories between the locally free air modules of air and the vector bundles. But you have to be careful in a category. The important things are morphism. Okay. So this is Schwan-Seers correspondence. We really look about the group weight. So namely, we want all automorphism to be isomorphism. Okay. So this is quite very easy from the other theorem to deduce that. Okay. It's just to check that essentially that everything is well defined. Okay. So it's done more properly in the book. Okay. So let me give examples of vector bundles. So maybe the most important vector bundle, this is the tangent space. So what is the tangent space? So we are given a smooth map of affine schemes. Okay. So I'm not going to define what is a smooth map, but okay. If you don't know, this is something which is, for example, which is flat and such that all fiber are smooth in the sense of algebraic variety over a field. Okay. And what is the tangent bundle? This is the preceding construction attached to the coton-joint things. Okay. So this is V of omega 1 S of L. So the reason why we need smooth, that is we want this thing to be a vector bundle. Else, he has no reason to be a vector bundle. And if this is, because this is a vector bundle over spec S, over X. Okay. And then a very important case in the history, this is the case of the tangent bundle of the real sphere. So what is the real sphere? This is when you take the ring over the field of real number. So here I need my pen because something is missing. So we have to put minus one. Okay. So this is an example. So the tangent bundle is an example of vector bundle of the motion two, because this is the motion two, but it is not trivial. But the fact is not trivial. It's actually hard to establish. Okay. So you can prove it by using some differential geometry because all this notion of the counterpart in the differential geometry setting. And what happens? This is the so-called Erie-Bolteogram. So maybe you have done that. So which is a stronger statement. Okay. We still use that there is no way to on the sphere. So I'm not good to this. There is no way to put an entry vector field, which does not vanish anyway. Okay. So it has, if you have a vector field, so then you attach in a continuous way a tangent vector to each point. Okay. It has to vanish somewhere. Okay. And a consequence of that statement is deep, actually. One is that there is no, you cannot equip the real sphere with the league group structure. Okay. So we have in a forest one, you have forest three. Yes. This is my own question. Sorry. So is there any understanding of for which fields K is the sphere, same construction, also give you the tangent bundle with the non-trivial tangent bundle? Okay. So that's the first question. And I'm not sure of myself. So I will answer it for tomorrow. Okay. So for which field it works? Because I, no, no, no, because, because I, I caught this number 38, I can tell you what it did. This is a Robert Swann's paper. I caught it, but I did not read it. Okay. So, so, so in this paper, this is, this is, there is a algebraic proof of this result. Okay. Without using any differential geometry or, or, or measure theory, whatever. Okay. And I don't know for which field this is true, but I cannot swear that for tomorrow. Okay. Okay. So that's a good, no, but that's a good question. Okay. But maybe someone in the audience, no, I don't know. No. Okay. Okay. So I come to, I come to linear group. So to precisely the idea of linearity. So we, we take no or M, a locally free model of finite rank, and we can tensorize with this dual, with this dual. And what you get, as you know, this is the, the endomorphism ring of M. And, and this is, this endomorphism is, is all, is also locally free model of finite rank. So this over S has, has to be, has nothing to do here. Okay. So we can consider the vector, air group scheme attached with this under viewing. And actually, but this is, this is, it has an extra structure of, because no, it's an air functor in associative and unital algebras. And on the other hand, we can consider the automorphism group of M functor. So, namely, it's for H ring S over air, we, we, we look at this automorphism. So, and under the condition which is above, okay, it's, it's representable by an open or sub scheme of W of this one. Okay. Which is a, which is nice object. And it is denoted by GLM or GL one of M is the point of the authors. Okay. So here we have to pay attention a bit with left and right. Okay. So the problem of this topic, because we, we have this kind, this equivalence of categories. So we bear in mind that this group scheme GLM on WM as, as comes on, as comes the right here and as comes on here on the, no, sorry, last comes the left on WM and on the right on the end. And by the way, so there, there is a result button needs to, so, which is in the note if air in notary, okay. So M locally free of finite round is a, is equivalent to say that this GLM is representable. Okay. So once again, if you want to, to deal with linear groups, we need to put strong assumption on, on, on the model you work with. Okay. So we are, we have linear groups and, and of course, the, the, so this is the same story. But you cannot, the most famous one, okay, is when we take the free guy, okay, the air power and this is, we did not did by GLM. Sorry. Okay. No. Okay. So need to, I wrote it on the next page. Okay. And there is another related construction, slightly more general. So if you are given be locally free OS algebra of finite ground, okay, you can look at the functor of invertebrate elements. Okay. It's a, and similarly, this is representable by Nathaniel groups. Okay. So, so for example, this thing is useful if you take a metal algebra something like that. Okay. Okay. So, so this is a linear group we will face today and tomorrow. Okay. So, so now I come to, to, to, to the, the, the co-cycle description of vector bundles and which is totally similar with what you find in any differential geometry book. Okay. So you are given a locally free or modular front air and since it's locally free, you, you can find a partition of unity and isomorphism between the, the, the free guy, okay. And the, the, the given module. So my conventions is to, to take trivialization maps from the, the, the split objects or the most common object to the non-standard one. Okay. So you, of course, you can, you can reverse that convention and some, some people do. Okay. So don't be afraid if you open a book that the convention are not what, what, what I do here or the convention you can find in other books. So that's okay. And then what, what you do, you, once you have a trivialization, you, you look at the transition function. Okay. So you look at fi minus one Fiji. So it will go from this one. Okay. Because the RFE FG, this is the, in term of, of, of algebraic geometry. This is the intersection of RFI and RFG. Okay. And so, so this is, this is linear. And so, and then that's defined on an element of the linear group with these things. And what I mean by linear. Okay. So we have this compatibility. Okay. So as this is by acting on the left and, and what, what, well, another way to say the same thing is that we see this module as colon vectors. Okay. Okay. And it turns out that this function, G, E, G, sorry. Yeah. Okay. Yes. It is a one core cycles. So what, what happened is that if you, if you look, if you take three, okay, so E, G, I, K, you have a comparability, okay, which, which is done exactly to have the, that way. So G, E, G, G, G, K, this is G, E, K over the intersection of the triple intersection. And this is very easy to show. Okay. So it's what we call a core cycle. And, and, and how we prove it. And we just write the proof is we just write the one in one line proof is you decompose it. Okay. And then you have to be careful with the left and the right. But if you, if you are careful, you, you, you get the, the, the right formula. And when, when we do, when you take privatization, we have to see what happens if I change the privatization. But for a vector of bundles, I have no much choice. Okay. So the, the only other privatization I can take out of this shape. So Phi prime I has to be Phi I composite with some GI. So yes, this is not G, this is GLR. Sorry. And what happens is when you, what, if you change like that, you, you get a new on the core cycle, okay, which has this shape. Okay. And we say that these two core cycles are common logos. Okay. So very important things we, which already occur at this level. This is when you have two core cycles, which are common logos, there is no way to, to choose a better, I mean, to choose a canonical or even a nice GI. Okay. So the fact that you have some freedom here when, when dealing with, with, with common logos core cycle. This is the reason what we, which makes a one reason making the theory subtle. Okay. So, okay. So, so this is the definition. So and, and I have problem with my mouse. Okay. And then if, if we look at this affine cover, so you were, okay, we can, we, I mean, we can abstract the scenes by, by denoting that one of your GLR, the same of one core cycles. Okay. And if we mod out by this, when, and we, we did not buy H1. Okay. When we mod out this set of one core cycles, the most of the comological relation, commercial relation. So, so this thing is called a, a church, non abbey, a first example of not church non abbey and homology, homology set. So be careful. This is just a set, a point on set because you have just, you have the, the, the, the lemon one, which is a class of the trivial element, which is trivial core cycle, but, but there is no way to, to multiply elements. Okay. And so, so this is this set of, of comology, of check comology. So it's written again. And then if we summarize the whole construction, so if you have a vector bundle, VM or M, that's the same thing, we, we, we have a class gamma M in this commercial set. And, and what happened is that by, by the risk relaying, if we have a core cycle GEG, we can attach to it a vector model VG over F, equipped with trivialization like that, just, we satisfy these things. Okay. So we can go backwards. The reason is just algebraic geometry, by definition, allows the risky gluing. Okay. And, so we have two maps and, and if you, if you, if you look at these two maps, what they do, what you get at the end of the day is that the point and set classifies the isomorphism classes of vector bundle of air over spec air, which are trivialized by this cover. Okay. So this is this correspondence between co-citals and vector bundles. And I prefer always to work at finite level with a given cover, but of course, there is one thing we can do always is, is to pass to the limit. We can pass the limit of this, to pass to the limit of this construction by taking all covers. Okay. Because of you, if you have two, two, two covers, right, you, you, you, you can intersect all pieces and make a, a, a nice refinement of the two covers you have. And then you have transition maps and everything goes nicely. And then we can, we can define. So the church, the risky homology of air GLR, okay, by this limit, and by passing to the limit, a formal thing that this inductive limit classifies the isomorphism classes of vector bundle of air over spec. Okay. So, so what we have done, so we have a geometric object. So, so vector bundles and, and if we, if we just look at isomorphism classes, okay, not the category of the other classes, you, you can describe that just by, by some coordinate things, which is H1. And I, I will say this is quite efficient. I will say for tomorrow for curves. Okay. So I'm interested in functoriality. So I have functoriality for wings, okay, and for covers. But I have also functorialities by playing on the groups. And those are very important. So what I mean is that if you have a, if you have a map of algebraic groups, group scheme from GLR to GLS, okay, then when you look at these co-cycle things, it means that we can attach to a vector bundle VG where G is some co-cycle of wrong air, we can attach to him to it, the vector bundle V of FG. You just take the image by F of, of your co-cycle. Okay. So what I'm doing here, it's, it's a bit deduce. Okay. We could try to think in a, in a tricycle way. Okay. So that's possible. But I prefer to stay elemental. Okay. And no, but, but you, you can believe me that this construction, so there is natural functor from X to, so F lower star from X, from vector bundle of wrong air to the two vector, not from, but two vector bundle of wrong S. Okay. So let me give an example of, so, and, and, and actually this, this thing is also true for, for many other groups. Okay. So when we, when you have a const, a nice constructions, often there is some underlying morphism of group scheme or morphism of groups. Okay. So example, example of that. So, so the first one, I mean, the first one is the total inventory. So this is a direct sum thing. Okay. So you are, you decompose air as a sum of two integers and you, you, you have, you take the block map from JLA 1 times JLA 2 to JLA. Okay. But then actually, so what happened, what you get is just the direct sum of vector bundle. Okay. So this is not very surprising. Okay. And of course you can generalize that with, by decomposing more and, and, and, and interesting case is when you take one air times and then you have GM power air. So in, in, in, in term of, of, of the theory of group schemes, this is a so-called maximal torus. Okay. And, and actually, so what happened is that if you, if you have a vector bundle, so since, so namely a co-cycle in JLA, which can be arranged in such a way it comes from this thing. This is this notion of decomposable vector bundle. So, which are direct sum of rank one vector bundles. And, and as you know, and I think we will do that. We'll see that with my tumor was a decomposable vector bundle are very important. Okay. And, and, and in term of group scheme is it is, it is really about this torus inside JLA. So a more complicated construction. This is a standard product. So here we decompose air in the product. Okay. And when you do that, so this is a connector products of, of, of matrix. And this is the, the vector, what you get is a vector product for vector bundle. Okay. And, and the last example, which is the, the most important for us today, this is a determinant. Okay. So you have the map from GLA to GM, which is the determinant. So it makes sense for this reason to take about the determinant of a vector bundle. Of course, you can construct it by cars locally. But, but, it in my eyes, this is really the determinant of a vector bundle. This is an avatar. And it's a consequence of the determinant map from GLN to GM. Okay. So almost what everything today is, is, is quite general. So I want to, to say, I mean, specific things. And specific things can be set for, for, for, so, for easy, easy rings. Okay. So after a field, the, the, the easiest case of, to deal with offerings, of course, this is DVR or the, the, the king ring. Okay. Well, so the VR is not very interesting for what we do here, because on a DVR, and actually over every local ring, a finite locally free model is free. Okay. So the first interesting example, I rise with the ducking rings. So I think you know what is a duking ring. So this is an Italian domain, such as the localization of at each maximum ideal is a discrete variation ring. Okay. So for example, ring of entegers, rings of function of affine smooth curves are the duking rings. And so a theorem which you can find in a, in any book of algebra is the following. So that we, we know that a locally free air model of, of, of positive rank over a duking ring is the sum of a free object plus an invertible air model. So I mean an air model, a projective model of Frank one. And actually this I is unique up to isomorphism. Okay. So there is a, we have a very good classification. And for example, for Z, since I, since Z is principal, that a consequence of this theorem is that a projective module of, of final one over Z are all free. Okay. But my, my, my interest today is, is to see this theorem in a, in geometric way, using this, this, this theory of, of, of course, cycle and so on. Okay. So theory of bondage. And so, so what, and actually this I, I did not say that, but this I is the determinant. Okay. And, and then if you see this is the determinant, it tells you that that, that it's isomorphism is unique. Okay. And sorry. And I'm interested in a, I'm interested in the, in the corollary. And so corollary of that. Okay. So the statement, a corollary of the statement is that if you have a locally free air model with determinant is trivial, then it is trivial. Okay. And, and this statement, I will, I will try to show you by, by, by my, in my own way. Okay. Not using. Okay. Okay. So let me, let me, let me sketch a proof of that. Okay. So how, how we can prove that. And so this is not mine. Okay. So this is a harder viewpoint on the, on the topic. Sorry. I will do. Yes, this is better. Okay. So, so, so we are given a vector bundle vm. Later, we will assume that m is of determinant one. So it travels over an open affine subset. Okay. But we just pick one f spec ff and we look at the, at the complement. Okay. So the complement, I denoted by sigma. This is spec r minus spec rf. And so, so what I get are maximal ideas, PG. Okay. And there are finitely many of them. And, and what I can do, I can complete. So there are DVR. So I can complete. And, and I can consider the fraction field of the completion, which is nothing but the transfer product of the, of the fraction field, big capital K, by, by the, the completion of the, of the field. Okay. So, so, so what, so I, I already stated it. This is Nakaya Malema. So over, over, over local ring, every projective module of finite, finite quantity is free. Okay. So we can pick a trivialization. So here, because full, I take the presentation at the, at the level of, of the completion. Of course, it, it exists already at the level without completion. But I want, I want, I want really to specify that I with the completion. And then I take a specialization, I take this fee f, a trivialization over this rf. And, and as before, okay, I look on the intersection. Okay. So intersection being, being the transfer, I mean, some of the interactions being the, I look at what happened on, on, on this completion. Okay. And on this completion, what I get, I get some element GI with entry in the, in the, the completed field. And as before, I have to take in account choices because I have here, I have some choice at the level of phi, okay, which, which is reflected here. Okay. And I have also a choice at the level of, of here and with this choice. Okay. But if I take into account the choice, what I get is a well-defined element. Okay. In this double coset. Okay. So you take the product of the, of the completions of the glr of the completion, model, model the, the, the, the glr of the integer and you model by the glr of the rf. Okay. So this is, this kind of set you can see, for example, when, when dealing with Shimura varieties. Okay. And, and the point is that this map is injecting. Okay. So, so this is the variant that the co-cycle determines the, the, the, the vector abundant. Okay. And so I have no time to prove it. So it's done in the node. Okay. It's not hard. Okay. But it's, this is true. And then I assume knows that the determinant of VM is trivial. And then to show here, what, what happened is that I have functionalities. So if my determinant in is a, is a bundle is trivial, it means that the class of my GI in this sigma belongs to the kernel of this determinant. Okay. Ternel is a sense of potency. It's got to one. But I had, I had, I had some choice. Okay. So up to a bit of trivialization, we, I cannot assume that my GI. Okay. So that my co-cycle in that sense belongs to SLM. Yes. But SLM is a group, which is, which is nicer than GLM for at least one thing. Okay. Is that is generated when I take it over a field is generated by elementary matrices. Okay. So that that I'm sure you know that. And then I can use an argument of stronger approximation. So this RF here is done in this product. Okay. So weak approximation is just a capital K is done. But here we have something stronger. We have this RF to be done here. Okay. But in the other one. Okay. So and then by decomposing elements here in a product of elementary matrices, I see that this group SLR RF is done to this product. Okay. But on the other one, when I look at SLM, this is, this is an open and actually is closed in the same time, but it's open in SLR here for, for, for the, for the topology of, of this complete field. Okay. So things about SLM of ZP, this is open in SLM of QP. Okay. Yes. But if you have, if you have, you know, dance plus open, it makes, this is everything. So, so, so what happened is that if you look at this C sigma SLR, this is one. Okay. And then my GI has to be trivial and then it is trivial in GLA. Okay. So it's a, it's, if I summarize is by a kind of stronger, stronger, stronger approximation argument. Okay. Involving these completions, I can show a part of my statement. And actually with some more effort, you can prove the statement with GLA. Okay. With the same method, we need to require only some more work. Okay. And we will see this, this method again tomorrow because this is very efficient. Okay. So the, also again, I took it in another, maybe it was not as before. And, and so, okay. Philippe, are you still there? Did we lose you? We've got a problem. The zoom is still working. So I'll quickly send Philippe a message to let him know that we're no longer seeing him. Anybody know any good stories? No, well, sometimes the magic works. Sometimes it doesn't. Send him an email. There's not much else you can do, I think. Anybody have any questions so far to discuss? Yeah, can I ask a question? Sure. So in differential geometry, to define the check cohomology, we usually don't need to go to the direct limit that we can choose a fine color, a good color. In algebraic geometry, do we have the similar thing? In general, no. In general, no. Yeah. I mean, you can have schemes where each affine has, so you'd like to know that your affine pieces, say those are the small pieces, right? They'd satisfy this condition if the isomorphism classes of vector bundles or even, let's say, line bundles would be trivial on each of the overlaps, on each of the affine pieces, right? That's what you need for the covering. But that's often not the case. You have schemes with very large sets of isomorphism classes of even line bundles. Like if you take even an elliptic curve over the complex numbers, you can't find one open cover that will trivialize every line bundle just because there are uncountably many, right? The open covers, you can only remove finitely many points from the elliptic curve. That's a risky open cover, but there are uncountably many different line bundles. If you're talking about cohomology of coherent sheaves, that's a different story because those are trivialized on affine open subsets. Then if you have a separated scheme, then you can always achieve that by just taking any affine cover. But for these other types of cohomology, then unless you have some finite generation properties which you often don't have, so that's a good question. Thank you. Other questions or comments while we kill time here? Is there some way to fix this problem you just mentioned by moving to a different topology? Could you repeat that? I couldn't. This problem that you just mentioned by moving to a different topology, since it's a risky topology, you can work on. Yeah, but then the co-cycle may not compute the isomorphism classes. Well, yeah, I mean, it depends on the topology. It turns out for vector bundles, you can actually build this theorem, Hilbert theorem 90. So as you can also use the etal topology, but you still have the same problem. You won't be able to trivialize every vector bundle on a fixed finite cover. I have a quick question. How much simpler is it to compute the determinant like that Professor Ziele is talking about? How much does it simplify? Oh, it's a lot simpler. I mean, I think so because the group you're dealing with is just the multiplicative group of units. It's a commutative group. So I mean, it's easier to deal with multiplication of numbers than it is to deal with multiplication of matrices. And so I think it's easier. I mean, for example, the set of isomorphism classes of line bundles, I think somebody mentioned this in the chat forms a group. And so you can try and understand the set by its group structure. And I mean, this is classical theorems about smooth projective varieties over an algebraically closed field that the group structure is given by a torus, you know, an abelian group variety times some finite, you know, times some finitely generated abelian group. So you can get a pretty good idea of this structure. But if you take the set of isomorphism classes of vector bundles, that can be a very strange object. There's you have this whole, which something I'm not so familiar about, but modularized spaces of vector bundles, it can be quite mysterious, especially some modularized spaces of vector bundles become even non separated as schemes. So you can, they're not going to be group varieties and they can be quite mysterious in terms of their structure. Or let's say more mysterious than the case of line bundles. But there are certainly other people in the audience who know more about modular vector bundles. Up, do we get it? Yes, I'm back. I'm back. Okay. I'm sorry. It's a problem of connection here where I am. I have to restart the system, but okay. That's all stories while you're away. Okay, certainly. Okay. Okay, let me, let me share the screen once again. Okay. So let, yes, no, I'm okay. So, okay. So the good thing, the interruption was when I was changing of chapter. Okay. So this is the second chapter. So why is the risky top position final enough? So the one argument I will give two arguments. One is with quadratic bundles. Okay. So, so quadratic bundles in algebraic geometry, we see them in terms of quadratic forms. So what is a quadratic form for, for us? Okay. So this is a model M with a map, which is quadratic and Q from M. So the first condition is quadratic and associated. We wanted that this form BQXY, okay, which is defined as it written, we wanted to be symmetric and billionaire. Okay. And as it is written, it's not clear, but this is tab bar, this definition is tab bar by, by, by best chance. Okay. So, and the, the form Q is said regular if this, this BQ induce an isomorphism between M and is dual. Okay. And a fundamental example, this is hyperbolic form attached to a locally free, a model of finite form. Okay. So here, this is just a partial viewpoint of the topic, okay, which, which essentially is the viewpoint of form of even ranks. Okay. But that's enough for, for, for, for consideration of today. And, and then we consider a regular quadratic form MQ where M is locally free or from air. And it's, it's tempting to make analogy with the vector bond dogs. Okay. So you can see a quadratic bond forms as a, with some unregistered of your vector bond. And then we can consider the, the orthogonal group scheme, scheme OQM. This is a closer group scheme, closer to the group scheme of GLM. And for an open cover U of air as before, and we can define similarly, co cycles and also church H1. Okay. And actually, what I did for this orthogonal group scheme, I can do it for any group scheme. Okay. And we do that. And, and what we get is the following that this set classifies the isomorphism classes of regular quadratic forms, which are locally isomorphic over E2 QA. So this is nice. Okay. Because that gives you something, but it's not exactly what we want. So what is not what we want because regular quadratic form over air of the ocean air have no reason to be locally isomorphic to MQ or to any of the ones. Okay. So you can think, for example, just as the case of a field, okay, the field of real numbers where the regular quality form are just classified by signature. Okay. And over a field, of course, the risk is, okay, speaking, there is nothing interesting. Okay. So there is there are one single cover. And then the set is this set H1 that is key is only of piece of what we would like to obtain. And what we have to obtain is a set of H1, which really classify isomorphism classes of regular quadratic forms. Okay. So this is one thing. Okay. So for quadratic, for the quadratic form things is this H1 is not very sweet. And also the other important thing. So this is yes, more crotendic, I will say this your point. This is what happened with this change of groups operation. Okay. So when we have a map of GH of group schemes, we would like to understand to have some control and this is vague about on the map H1 from from G to H1 from H. So to say things about image about about the kernel about fibers. Okay. And when you start to investigate this problem, you face difficulties. So let me take an important example, which is a CUMER map. So what is a CUMER map? You take GM, this is GL1, and you take the map, which goes from T to T power D for an integer D. So H1 of GM, as we have seen, this is, we can call it also the PICAR group that classifies invertible modules. So locally free modules of one. And this map, if you think from PICAR to PICAR, so this map FD lower star and lower star, this is the map which takes an invertible module to M times the sensor product data. And so what can we say about this? So about the image, I don't know, maybe Frédéric or somebody will talk about that. So it will involve higher etalcomology. So right now, I'm just saying things about the kernel. Okay. And so the kernel, it has a concrete description. So we take an element M, so namely an invertible module, with the trivialization of its D power. Okay. And then the natural things to do is to introduce a commutative group ADR. So this is my notation. This is not of isomorphism classes of couple M theta, where M is an invertible model. And theta is a trivialization of its D power. Okay. So, of course, you can multiply elements like that. And this is a group because you can take M, M dual and so on. Okay. And then when you do that, so from this ADR to pick area, you have a forgetful map, just by forgetting theta, you just get, you just keep the invertible module. And then what I claim is that we have an exact sequence like that. Okay. So ADR maps onto this kernel. So, of course, it was tailor made to do it. But the interesting thing is about this kernel. Okay. And what is this kernel? This is actually the, it's covered by a R star mode, R star power D. And the explicit map is, you take R. So this is a trivial R module, but the trivialization is not trivial. This is the map with taken element X. No, I'm sorry. No, no, my map is not, this is not the right map. Sorry. I have to take, I have to take, sorry, this is not A, this is theta A. And here, I think I take, it's not XD, it's just AX. Yes. Sorry. I have to change that to minus. And then you can check that. And for those aware of etalcology, this thing can be explained by etalcology. Okay. But not at the Zarisky level. Okay. So this ADR has no, I do not see a way to interpret it in a, in a comogical way with what we have done. Okay. Now, and then, well, this is an example, but of course there are others. So, Kotendixer ID is to become the notion of cover in algebraic geometry. Okay. So they did it originally with etalcovers, but it turns out that the flat covers setting is simpler in a first approach. Okay. For what we do. Okay. So, and actually, this is what is done in the Mazur Gabriel on Ignous. And so what is a flat cover? In French, this is called FPPF, Fidel Marplat de présentation finie. So it's a finite collection of airwings, satisfying two properties. So first, we ask SI to be flat of finite presentation. Okay. And secondly, we want that speaker is the union of the, of the image of the spec SI, of the image of the spec SI. Okay. We want it to be subjective as topological space. So the second notion is natural. The first one may be a bit less for you. Okay. But so an important thing is that if you, if you have a such a cover like that, you, you can take the product. Okay. And the product will satisfy the things too. So, so the condition one and two can be expressed by saying that as is fairly, fairly flat algebra or finite presentation. So in practice, if we want, and it is simpler to deal like that, okay, we can always deal with a unique ring. Okay. So we have a finite collection, but actually, one ring is enough to make the whole theory. So what you should first notice is that this theory generalized the other one, what we have done before. Okay. So if you take a partition of unity, okay, but then this rings, this collection of ring is a flat cover of, of air and so is the product. Okay. And so, so, so here is called a risky cover. And what I'm saying is that the risky cover is a flat cover too. Okay. Well, so what is shesh non-navian technology. So as I said, I deal only with a single fast fully flat algebra s. And then I have maps. Okay. So I advise them in terms of rings, of course, you can dualize and buy in terms of spectrum. So we have, we have projections and projection to two and to three. Okay. And also other projection from two to three. Okay. So, and there are formulas relating all those maps, but I do not advise them, because I'm not given, I'm not giving any proof here. So I start, I'm given an air group scheme, for example, GLN or Togal group, whatever. Okay. And what is the core cycle from G and S This is an element of G of this tensor product S, tensor R by S, which satisfies this rule, which is, so this rule is with, with the tree. Okay. Here is the two. And we ask Q1, 2G, Q2, 3G to be like that. And, and we denote by Z1. So this is the set of one core cycles. Let me say, and maybe this is later written, I forgot, but in the case of, of the risk he covers, this is exactly the same definition that what we have done before. Okay. So this, this is a generalization. And with, and two such core cycles are set common rules. If there is some H in JS, such that G can be written like that. G is P1 star H minus one G prime P2 star H. Okay. And similarly as before, we, we can denote by H1 as that SRG, the product set of one core cycle up to common G equivalent. Okay. So just a definition. And so this is what I said before. So, and here there is a subtle point that the fact we can pass on the limit on all flat covers. Okay. So here, this is important to use this final presentation six. Okay. So there are more details of the note about that. Okay. Because of course, if you have two flat covers or two, two rings I had before, you can take the top product, but you have to be a bit careful with the set theory of, of doing limits. Okay. But, but this is done properly in the, in the Gabrielle. Okay. So, and, and so the good, the, the good news is that all my construction is functorial in air and also in the group scheme. Okay. As before I can allow morphism of group scheme. And no, I come to torsos. So a right G torso X is an air scheme equipped with right action. So be careful with the right action. That's the promotion which satisfies the foreign properties. So for the action map, which is take XG to X, XG is an isomorphism. And secondly, there exists a flat cover air prime of air such as XR prime is not empty. Okay. So if you, so I will, I will comment about, about these two sections. So, so the first section is always present. The second option, there are variants of, of it, which are active and depending of, of, of your reference. So the first condition reflects, reflects the simple, the simply transitivity of the action. So what means simply transitivity in that context is means that GT for T, an air scheme, an air ring acts simply transitively on XT. Okay. So of course, if, if XT in empty, this is simply transitive. The action is simply transit. Okay. So, so, and the second condition is a local priority condition. Okay. So an example, okay. So, so for example, if I take the, the, the empty scheme, the empty scheme is going to be, to satisfy the first property. Okay. So, so if you want to check that something is a torso, you have to check both. Okay. And an example of torso, the most common torso is called split torso. You take X to be G and G acting by right translation. Okay. So, so this is the, the, the simplest example of torso, but it's really important. Okay. So I recollect the, the axiom of, of, of torso and an important thing is that we, we, we know when a torso is, is a, is isomorphic to the split torso, it's when it has a point, because if you have a point of, if X is not empty, a point X define a morphism G to X, which takes G to XG. So just orbit. And, and this thing has an, it must be an isomorphism by the simple transitive property. And, and then in that case, we say that X is trivial. So, namely it's isomorphic to, to, to, to the split torso. And, and one of the way to, to rephrase the condition to taking in account what I've said is that tooth says that a torso X is locally trivial for the flat topology. So maybe it's not trivial locally for Zariski. Okay. So it's too, too much to ask. But if, if I allow a flat base change, it becomes trivial. Okay. So, well, and, and no, I can be more precise about morphism. So it's just a G we have our map. And also again, by the simple transmissivity property, any such a map has to be an isomorphism. So, so what I mean is that the, the category of G torso is, is a, is a group point. Okay. So, so this is a very nice category. Good. And, and so what is the relation with co-cycle? So is that the relation co-cycle comes from the fact, the following fact that the air functor of the morphism of the trivial G torso G is represented by G by, I think by left translation. Okay. Because once you, once you know where, where one goes, okay, then is defined. So, so, so this is very rigid. Okay. So, so any morphism of this trivial torso, it just define with the, with the value of, of, of, of your transformation at one. And then that's the reason why the, the only morphism are the left translation. Okay. So, no, I, I, I take a formal definition. This H1 is a, so here I know IP, FPPF. That, that the set is a morphism class of G torso for the flat topology. Okay. And I can look at a smaller subset. Okay. For each S, I can look at things at, at, at classes which are split. So, which are split by, or trivialized by S. Okay. Well, and then as in the vector model case, we shall construct a class map. Okay. Coming from the torsos. Okay. To the co-cycle. And to the, to the set of course. So it's a, it's a, I'm already late. So I will go fast on that. Okay. But this is the same, always the same ideas. Okay. You take a trivialization, you compare them. Okay. And, and the, the, the comparison give you a co-cycle and this co-cycle depend on the choice, but the choice are up to commercial co-cycle. And then it works well. Okay. Sorry to, to, to, to, to, to go too fast, but you, at this point. Okay. So, and, and once again, what, what you get, this is formal, when you, this is kind of, of, of computation, what you get are co-cycle. But, so, so this, this, sorry, so this class map here is well defined. The problem is to, to, to this map. And here comes the technicalities. Okay. So, so it's to involve the technicality, which is called the, the, the, the, the first fully fledic center frame of content. Okay. So what, so here I need this technical factor, which is this entire lip. So I hope that you have already seen that in your, in your medical life is that that's a beautiful and amazing result. Okay. So. So this result holds without any condition of finite represented as a result why I changed the letter. So I take t to be a first loop, first loop extension of the green air and I look at transfer product of t d times with itself. And the first tone of this theory, this is so-called the Amitsur complex, okay, so which is a map, so I will write the differential, so we have a complex from m to m tensor t, m tensor like that, and the differentials are defined, okay, so this is the complex, but if t is faithful, is flat, the first result is that this complex is exact, okay. And so this is important for several reasons, but one is that for any affine-air scheme x, okay, x-air, I can identify x-air with an element of x t such that p1 upper star x is p1 upper star star in here, okay. So this thing tells you that you are not losing any information to go from air to t, if you control the what happened here, okay. So this is the consequence, so this is clear because if x is affine, so x is spec r, okay, I'm looking a map from air to t, and if a map, sorry, if I take a map from a to t, but is a map such that this to define a map like that is to define a map here which goes to zero, okay, so this is a conflict, and actually, but this is true for any scheme, not necessarily affine, okay. Okay, so what is Fesulilla descent? So here, one t is missing on the, no, sorry, on the left, okay, so given a t-module, we consider the t-tosser t-module p1 star n, so this is np2, so here this is p2, and then we are interested in in diagram like that, so an isomorphism phi, so descent data, this is an isomorphism like that, okay, such that it's, we have a compatibility, okay, with phi3, phi2, phi1, where phi3, phi2, phi1 are deduced by phi, by this formula, okay. Of course, it has to be read and you can learn that in one minute, okay, but and the interesting thing is that, so here I have still problem with my, sorry, so there is a clear notion of morphism for t-modules equipped with the descent data, okay, and if M is an R-module, so the identity of M gives rise to a, to connect an isomorphism between this p1, M-tosser t, and with p2, okay, and this thing we can check, it's a bit, of course deduced, but we can check this is a descent data, and the theorem is the following, this, the cotendic phase result descent theorem, that this functions from M, which are associated to M with this canonical descent data, is an equivalence of categories between category of R-module and that of t-module with the descent data, okay, and then suddenly like for, for you, you are not losing any information, but instead to consider M, you consider this gadget, and the way to go back is we have N-phile, so N is a t-module with the descent data, and, and you look, and you have a precise formula in terms of tonsils, which define R-module, so this thing is R-module, and also when you tonsilize this R-module to t, you get the original N, so this is the first statement, and the, the second part of the statement is that you, you can, it's, it's so nice, okay, so you, you, it's, you, you can also play with R-algebra here, okay, so, so the above from Tor, you, and there's an equivalence of categories between category of R-algebras and t-algebras with the descent data, okay, so this is, this gadget, and this theorem, this, this is, this permits to do the same gluing things that you do usually for the risk topology for these flat things, okay, and here this is about affine, and if it's not affine, there are extra difficulties, and extras, certain, certain things I don't want to talk right now, okay, so I, so this is, this theorem is one good reason to concentrate on the affine setting, so let me go back to the vector bundle, so, so remember one series correspondence, okay, between projective modules and vector bundles, actually we can add one more categories, okay, so if you, a consequence of a fessuelive descent theorem, and also the fact that the property to be locally free is so-called local for etal topology, so I mean a module is finite locally free of frank air, if only if it is, after some flat bashing, fessuelive flat bashing, so a consequence of, of, of this result, is a, is a following theorem, okay, so let them be a locally free model of frank air, then the air front top, which will go from S to Ezone to SRM, also, also RS, is representable by a Giller torso, okay, and so this isome construction permits to associate to any locally free model of frank air, a Giller torso, and then when you write the things in, in a categorical words, what you get is that this function M to XM induce an equivalent of categories between the group point of locally free arm modules of frank air and the category of Giller torso, okay, so, so this one there is even rich correspondence is even richer, okay, yes, but you could say there is nothing new, because what, what is true is that Giller torso and locally free arm modules are, are free after some localization, this is true, but actually just to say this sometimes give, give us very important corollaries, okay, so, so the, the, the, the important corollaries, this is, this in the 90 theorem, okay, which reflects that Giller torso are the same with flat topology or with the risky topology, and, and then the precise statement is that H1 the risky air Giller is H1 FPPF air Giller, okay, and so in particular, if you take a, if you take a local ring, sorry, if you take a local ring, H1 FPPF air Giller or local ring or even somitocale, this is not, okay, so this is not the more general statement, the more general statement is GL with a, with a separable algebra, okay, so for example, for today, this is already very nice to have that, okay, of course, I did not prove the, the, the, but this is, this is a straightforward consequence of it, okay, so, I realize that I'm out of time, but let me just, Mark, I have five minutes because of the interruption, certainly, okay, well anyway, if, if I do not finish all my things tomorrow, this is not a problem, the, the most important thing are the foundation and mathematics as you know, not the, not the roof, okay, so, so, so I'm come back to my initial perspective and which was to compare the H1 here with Torsor and H1 with Cossack, well, so the first statement which is totally right, that this map is injective, okay, so, I will not prove it, but this is a consequence of a Facebook data center issue, for example, it, and the, the hard part, no, okay, I proved it, but I have no time to prove it, okay, so we can see the notes, the proof, the, the hard part is, is the surjectivity, okay, so, namely, if, if, if G, sorry, if G is alfine, so here, the reason why I put alfine, because I need the Facebook data center, the class map is an isomorphism, so it means that the, we have exactly what we want, namely Torsor's and Cossack's up to equivalence are really the same things up to isomorphism, okay, and, and then let me, let me explain why, why we can go from the right to the left, so, namely, having a, a Cossack's, we can, we can construct a, a Torsor's, okay, so, so this is, okay, so I sketch a proof of the theorem, so, so we are given a Cossack's, okay, and so G is alfine, and then, since G and alfine, I have the left translation by G, and when, of course, when I look at the level of function, it takes a start up, okay, so, okay, so I get a map like that, and then I want to use the fast forward flat descent terrain, and I want to, to define a, an identification function 5G, and 5G has to go from S tensor SG to SG tensor R over S, and there, and from this LG star, so there is a natural way to, to construct some 5G like that is, is to take here alpha and beta, okay, and which are natural function here, so natural isomorphism, and to, to, to, to verify 5G like that, okay, so here, you have to believe me, that is 5G defines a descent data, okay, so this is the next step, so the concicle condition, because we have a position on this G, when you write it at the level of this 5G, this is exactly the condition to be a descent data for this S algebra. Yes, but the descent theorem defines an R algebra RX, okay, and then by functionality, you can check that this X comes with the left action of, of G, and, and, and when you extend to, to S back, you see this is, this is a torso, but then it is a torso, it has to be a torso, because, and to be an isomorphism, this is something which is local after, after fast forward flat bed change, okay, and actually this, this, this algebra defined by, by this, by this descent method, okay, is, is a torso, and if you specify what I have done here, just to the Zariski 6, you will see what we are doing here will be just Zariski glowing, okay, and, and then this construction I have done is called twisting, okay, so the, and this is a special case, and so I have twisted a torso, but more generally I can, I can twist an affine scheme equipped with a left action of G, okay, so then as before I can use G or G lower star, this G define, define, define a descent data, okay, and, and, and then we can twist this Y by this cross cycle, and what you get is this YG, and, and the nice thing of course this is, it is affine over here, okay, and, and also, sorry, this is functorial, okay, so in a special case is when you take G acting on itself by an isomorphism, then the GG is called the twisted group scheme, okay, so if, of course if G is commutative you, you get nothing new, but if G is no commutative, you, you, you, you, you get a new group scheme, well actually I use cross cycle for simplicity, but the above construction do not depend on any choice, okay, of cross cycle trivialization, so actually you can vice the thing in an intrinsic way by twisting by a torso, okay, and then the notation is with, like that, Y twisted by E and G twisted by E, okay, and, and, well, so I will use tomorrow a bit these things, okay, and let Jimmy just say that in practice the, if we ask all the scheme to be affine, this is too strong, okay, so for example we like projective spaces and things like that, which are not affine, so, so if you want to work with non affine scheme, so you need more technicalities, well, time to time descent is not possible, but for example for, for J scheme, a keeper with, with an ample invertible generalized wonder, which is a case of, for example, if you take GLN and projective species, then it can, it can, it can be done, okay, so, so I use, I use that the affine setting for simplicity, but many things can be pushed further, okay, so tomorrow I will, I will end with the generalities and go, and go to the curves, okay, which are the, the main topic, and, and I hope that I will manage to, to give some, some, some statement with, with, I mean, we, we, we, we, which are really relevant for, for homotopy theory and things like that, okay, thank you very much. Oh, thanks very much, really, lovely talk. Are there any questions or comments, questions? Maybe, maybe I can ask a question. So, you're talking about the risk topology is not, not enough, so is a dollar topologist enough or is there, I mean, you use a FPPF, so is there counter examples used that the eight dollar topology is not enough? Okay, so ethyl topology is good, for example, if you take the cum, the cummer cover, okay, so this T, T power D, so if D is invertible in air, this is nice because it is going to be an ethyl cover, okay, but if D is not invertible, this is only a flat cover, okay, so, so what I'm saying is that if you want to extract D roots of your rings, okay, and you, and when D is not invertible, and it happens, then you cannot limit yourself to, to, to, to, to the, to the flat city, to the ethyl city, for torsos, so two more, two more, but there is one case I did not have the time to say, I will go on that tomorrow quickly, this is, if we deal with a smooth group scheme, ethyl covers are enough, okay. Okay, thanks. Yeah, so it's where D always unremifed kind of, so I want to know this case, so, okay. But, but, but let me just say that the, the, the flat theory, at first glance, this is simpler, ethyl, ethyl is a, is a more sophisticated object, and, and one, one, one thing you have with ethyl, okay, and is that you have a notion of local rings, maybe you know that, I mean, this Enselian construction, okay, so Schroer has a kind of construction like that in the flat setting, but this is very abstract, okay, this is not a ring we can really think about, okay, so ethyl has many advantages, but it, it, it requires more, more, more work, because, because smoothness is a more elaborated concept, that flatness, I mean, that, that's just what I'm saying. Okay, thanks. Okay, so I think we could save further questions for the discussion session, so I'll have a discussion session after the next lecture, and I encourage you to go to that and pose your questions there.