 Thank you very much and thank you for the organizers for the invitation. It's a great pleasure to speak here in the conference in honor of Lucie Lucie. The first time I met Luc was in 2017. We happened to be visiting Chicago at the same time. He was there for a longer period. I was there just for a day or so for a seminar and I remember we well we happened to all go for lunch together and we had a very pleasant conversation and many many more of those since since I arrived since I moved from the US here here to France and Luc was especially helpful when when I just arrived and with making sure that my French is progressing well and and and that my understanding of French culture is up to date and also many many many interesting mathematical conversations and and pleasant moments over a meal or so. So thank you very much for all that. To Luc I found him extremely generous and kind and it's an honor to speak here in this conference. The subject I'd like to discuss is a Grotten Dixier conjecture which concerns torsors and their addictive groups. So let me just begin by recalling what the conjecture says. The correct conjecture originated in around at the end of 1950s. First there was an article of Serre which posed a special case. The group perhaps came from from the base field and and then Grotten Dixier and the group de Brouwer posed a slightly more general version. Later called the Lennon Oger and Gurin popularized the general form of the conjecture which is what I'm going to formulate. Still attributing this general form to Grotten Dixier the first origins are perhaps in 58 or so. It says the conjecture predicts that if we have a regular local ring regular local ring R and a reductive R-group scheme. Well the conjecture is already interesting when G is split. So for instance I don't know PGLN some SON or so on but in general a reductive group over scheme or over ring, regular local ring like so smooth affine group scheme over that base whose fibers are connected reductive groups in the usual sense over a field and that their unimportant radical is trivial. So the conjecture predicts that no non-trivial non-trivial G-torsor trivializes over the fraction field of R or in homological terms. Well really this is just a restatement, but if you if you want the map of pointed sets from the from the collection of all torsors under under under G over R to the corresponding set of torsors over the fraction field has trivial kernel. And in fact, I posteriori because one is also allowed to apply the original statement to to inner forms of G. This map is then ends up being even injective if the conjecture holds for G and all of its well twists by torsors of of G. So that's a question that I'll occupy myself with during during this talk and let me just begin with with discussing the cases. Well, the main cases in which the conjecture had been had been established. So we get we get an idea of the history of the question. So first of all, well the simplest the simplest reductive groups are commutative ones the in other words the torii and in the case when G is G is a torus the conjecture was established by Collier-Telen Hansen-Süch in in 87. They used the so-called Flask resolutions of torii to analyze to analyze torsors under under unarbitrary torus. Anyway, this case is not entirely evident, but somehow with the use of all these solutions in terms of induced torii and so-called Flask torii one one can understand one can understand the question. Well, okay, so that's that's the case when G is as simple as possible. The case when R is as simple as possible, well beyond the trivial case when R is a field, is when R is of dimension one, namely when R is a discrete valuation ring, a regular local ring of dimension one. So this was settled by Nisnevich in 84 in his Harvard PhD thesis with some help from Bruchatitz. In fact, it uses with some help from Titz who I believe was aware perhaps of the case when R is a complete discrete valuation ring and so the the argument proceeds by reducing to the case when R is complete. This step uses so-called harder approximation which Nisnevich kind of extended and adapted to this to the setting in the case when one R is complete. One uses some Bruchatitz theory to I mean to analyze torsors in that case. Alright. Now the case when R is of dimension at most one implies another case when R is rather simple or relatively simple, namely when R is Henselian local. So this case follows from a case when R is a DVR. For instance, of course, if R is complete regular local ring, it's in particular Henselian. So the complete case is known basically because when when R is Henselian, then a torsor under a smooth smooth group is is again going to be smooth and so it will be trivial as soon as it is trivial over the residue field because by Hensel's lemma a point of that torsor over the residue field will always lift to an R point granted that granted a finger smooth and so okay, so we only need to trivialize a torsor over the residue field and then we can sort of cut R up into into a chain of DVRs. I mean into we can choose a chain of primes of maximal length such that the quotients are regular and the successive quotients will be DVRs. So somehow by this by this by choosing such chain and a little argument we reduce the case on R as a DVR and then apply this Nevich's this Nevich's argument. Anyway, so somehow the upshot of this of this case free, although it's not particularly deep given given given the case too is that the conjecture is simple when R is well when R is complete or Henselian and so in particular to attack the conjecture in general we cannot reasonably hope for a strategy where we somehow reduce the Henselization or completion and then because I mean that case already that that reduction must be the main must be the main difficulty as something else that one has to come up with. All right now from from more recent more recently the conjecture has been established in the case when R contains a field. In fact this was a subject of many works which I'm not I will not be exhaustive in in mentioning there really was extended literature in this in this case with many contributions, but the final decisive decisive article settled this case completely a sequel characteristic case or by Fedorov and Panin first they were assuming that R contains an infinite field that made some geometric arguments notably ones that use Bertini lemma a slightly simpler than Panin extended the argument to the case when R contains a finite field so arbitrary equal characteristic regular local ring later Fedorov simplified that roof yet again by avoiding some how initial reduction to the case when she is simple semi-simple simply connected and just taking up a general G Right away without that initial reduction Now so these are the main known known cases that there are there are a few others for example When she is when she is of special kind I'll just say sporadic cases okay Many authors are again, I will not try To be to be exhausted for example if she is PGLN so these sporadic cases concern the cases when either R or G are Specifically the one R is of of low dimension or G is of or perhaps both are G is of Some special form for just for a simple example if she is PGLN And the conjecture is known because the torsos and the PGLN They inject into into the Browel group of R and the Browel group By a result of Grotten Dick of a regular base the Browel group injects into into a Browel group of the of the fraction field And so the case when G is PGLN is known is due to Grotten Dick In fact was one of the motivations for posing this conjecture in in in general or hoping that perhaps the statement could could be true in general and so but in in general Beyond the equal characteristic cases somehow not Not not so much had been known about this About this conjecture, especially one R is One R is ramified mixed characteristic regular regular local ring well, so Then one perhaps first needs to negotiate what classes of G1 is sort of Talking about Well, for example Well G is just an a group scheme over over R for example I can give some so if G is an abelian scheme is some kind of an orthogonal case to what we're discussing here The statement is still true the map on H1's is injective Because one can look at the dual abelian scheme basically it amounts to the fact that line bundles Extend but one uses the crucial way the dual abelian scheme and an extending torsus and properness properness of G but If G is finite just finite flat the statement is also true if More generally if a finite scheme over over a normal base has a section over a fraction field and by taking the schematic closure That section is gonna is gonna is gonna extend by normality and fineness of the of the of the torsor in in question, but Okay, I mean I I don't know but I think it's not I mean I have some ideas, but not Okay, maybe it was not Anyway, so you can look at the constant case or twist of constant groups And then I think it holds for over a field for smooth varieties. You have a twist of a console This uses some analysis of pseudo reductive and quasi reductive and using the previous arguments but in the model it's like I'm using the theory of Pseudo reductive groups and so I mean at least in the constant case one can Give it and then I think also in the twisted case if you use there So so this is probably when essentially I know in principle, I think I know how to do such things by for Just constant groups of twists of constant groups. Okay, so you're saying she is a twist of a constant Constant group if I understand correct. So this means that locally for that also on the smooth scheme over the field Locally for the topology It comes from The field of constants of the scheme Not a twist of a group so it could be a germ anyway, whatever so this kind of Situation so this means it did be some of it the behavior is very constant on the and so So even there even just to get rid of the unipotent part requires some three okay, it's not so difficult But I want us to use Some work so I think I have enough inputs to do it, but It uses just a variance of the previous ideas, but but also other things on About the two okay Okay, thanks. So so that's a more general statement that one if I understand correctly one could Pretty much expect to be true if our is of equal characteristic or perhaps a localization of a smooth variety over a field and she is twist of a group that comes from a from a base Okay, anyway, so the The the the result that I'd like to talk about this is it's about this conjecture in the case one one R is a mixed characteristic that well and un-ramified and so this the mixed characteristic cases is is is the remaining one because of this result of Fedorov and Panin and so Let me just recall for the sake of of completeness. What do I mean by un-ramified? So a regular local ring R with maximal ideal M is called well said to be un-ramified if It's residue characteristic is not in a square of its of its maximal ideal or more precisely if either If either it is of equal characteristic If either R contains a field, so it's either a q-algebra and Fp-algebra prime P or R is of mixed characteristic Are some mixed characteristics? So a fraction fields characteristic zero and the residue fields of characteristic P and this P This is prime number P is not in a square of the maximal ideal of Of R so that the the kinds of rings we're talking about just Very I mean some very basic example affine space over over Z localize that At the at the origin in characteristic P or more generally any any local ring of a smooth Of a smooth Z Of a smooth Z scheme or Or or in fact anything in equal characteristic as also all right It's just just an example of a smooth Well, I'll see mix characteristics zero P. So Z localize the P scheme Okay, so the result that I'd like to discuss in this In the stock is the conjecture of Grotten Dickens here holds in the case when R is unrammified and and g the group G is quasi-split So it has it has a borrel subgroup Okay, for example G could be split that case is already It's like new in this result and this is simpler so that one can one can think of think of that case for instance I don't know some favorite exceptional group. He's 67 or something But okay anyway, so that's that's that's the statement and a little bonus For such R for such regular local R. Namely the unrammified ones Could could be a vehicle characteristic or or possibly a mixed characteristic and deductive Reductive R group scheme H Is split if and only if it's base change to the fraction field of R is split Yeah, so in fact in general the Grotten Dickens air conjecture implies that To two reductive group schemes over over R that become isomorphic over the fraction field are isomorphic to begin with So the Grotten Dickens air conjecture implies in particular if reductive R group scheme H is is split as morphic to a split reductive group scheme Over the fraction field then it already has to be isomorphic to that over over over R itself And this implication of Grotten Dickens air conjecture to this statement about reductive groups themselves It requires the statement of of the conjecture not only for G but for also for inner forms and perhaps for also for a joint group and inner forms of those And so it's not this quasi-split assumption is a lit. I mean Anyway, we still get this conclusion about about split groups But we don't quite get that if you have two reductive groups over R which are isomorphic or a fraction field and So it does the second statement require more than the quasi-split case of the Grotten Dickens air conjecture The way I formulated here no it it does not require more Because I restricted to split to split groups and that's I mean the point is that if I restrict this H H is split if and only if the fraction field is split and it requires It falls from Grotten Dickens air conjecture and a little additional argument So you don't assume that H is causes okay H is general Yes, hey, and is it was that it is quite a split if it only if H fuck R is Aha, yes, so that I cannot quite show That's where Yes, one would perhaps wants to show that H is quasi-split if and only if over a fraction field it's quite split I cannot quite do that because if I try to do a same argument and I start needing the Grotten Dickens air conjecture for groups that need not necessarily be causes split and In equal characteristic is true because well then the Grotten Dickens air is known in fact. Yeah, so this is another conjecture of Colletelen and Pannen which says that if reductive group scheme G over a regular local ring has a parabolic of a fixed type of a fraction field and that parabolic is already I Mean not that particular but but but then it also has a parabolic of the same type over the regular local ring itself That causes splitness is just a case of Or else anyway that conjecture is kind of a bit of a story It's not a special case of Another it's another type of conjecture. Yeah, it's spiritually related, but not Okay, so for for this result In the rest of the talk I'll focus on on this first part and just Grondick's era and about these forms of reductive groups and so the proof First of all the proof Uses known cases one and two but not Three and five in other words we use the case when G is the Taurus or when G is or when R is a DVR But we're not using for instance the work of Federer von Pannen when R contains a field we recover that I mean we do prove that Although the proof is somewhat I mean it is related to to their approach as well. So We will prove that case along the way And in fact, well, I stated this in this form for for some place about in fact same statement same statement holds one R is merely semi-local and still in ramified In in a sense as local rings are Yeah, so in fact one could generalize a Grondick's air conjecture to require that the regular ring R be semi-local rather than local in some sense and a more natural starting point and This this result if one assumes G to be quasi-split is still okay in that in that case So I suppose that by purpose to R is a limit of smooth things over Yeah, okay Over ZP and so probably use it. Yes, but is it then the case of the proof works for smooth things over the VR or Right. Yes. Yes. I have such a version as well. That it still works. And so in fact, I could assume that R is R is a geometric regular over a DVR is still the statement is still okay. I'm just restricted to the subsequent case somehow without for In order not to introduce some I mean, yeah, that case is still okay Alright, so Well, let me perhaps just give a Quick corollary just to illustrate somehow the arithmetic flavor in some sense of of the result is just for if one applies this to to orthogonal groups One gets that if two is invertible in in R So that we can comfortably talk about quadratic forms then non isomorphic non degenerate quadratic forms over R Do not become isomorphic over a fraction field over yeah and ramified local R Not become isomorphic over The fraction field of R. It's a kind of kind of statements at one that one gets by specializing to the particular types of types of groups in this in this statement okay, so I Like to then proceed to to discussing the steps The steps of the proof of the main result Because these steps themselves involve Somehow self-contained statements that could could be useful else elsewhere Beyond the proof of this of this particular result. So Let me then fix fix the situation of what we have and what what we want So we have an unrammified regular local unrammified regular local ring R and The quasi-split reductive R group. Yeah, so as as in Statement of the theorem quasi-split reductive Argo G So it being quasi-split it has a Borrell subgroup defined Yeah, defined over our Borrell R subgroup B inside Inside inside of G and we have a torsor under G which happens to be generically trivial generically trivial G torsor E Yeah, and we want We want of course To show that E is trivial in other words that it has an R point Yeah, so that this E is trivial. Okay, and Well, I'd like to begin just By telling you right away how the Borrell is is used So so this is captured by by by the following claim which Which else can show a proof of which is not too long actually so okay, so The fact that we have a Borrell Will give us that there is a close subscheme of speck of R Close subscheme Y of co-dimension at least two So the complement contains all the height one points such that The restriction of E to the complement of of Y such that away from this closed Y This torsor reduces to a torsor under the unipotent radical of of of the Borrell well, okay, so this This is not Difficult at all. Let me let me sketch. Let me sketch the proof Of course, we will use the value of criterion of Properness we'll apply it to to this E modulo B B being a Borrell g mod B is proper and Any torsor E E mod B inherits that properness so you might be is a proper R scheme well, it's this Yeah, a priori an algebraic space one facts even a scheme because scheme of borrell since some in some in some in their form of of Genially the twist of of G by this torsor E. So that exists By this value criterion that exists such such Y such that E restricted to the complement of Y reduces B torsor Right because E mod B Well, E is generic to trivial So E mod B has a point over a fraction field and is being trivial that point extends to cover all the height one points of Spec of R so there is such such Y such that E mod B has a point over spec R minus Y But a point over over over there is the same thing as a reduction of a structure group of E to from G to B So E restricted that complement reduces to a to a B torsor. I'll call that B torsor E upper B and If we consider the torus Which is a quotient of a Borrell by its unimportant radical Then there is a Purity for for torsors under tori This is due to Coletel N and and and Sunsook which says that for for torsors under tori And over regular basis removing a close subscheme of co-dimension at least two does not matter that does not affect the H1 one can one can always remove a closed of convention at least two and so That that purity Implies that the quotient of of this U upper B by the unipotent radical Extends to a generically trivial T torsor over R So so this quotient is is a is a torsor under under T over Spec R minus minus Y and because because this this complement covers all the height one points purity for H1 of Tori tells us that that this torsor under torus extends to a torsor defined over all of all of R and to a generically trivial torsor for that matter and so by applying By applying Grotten-Dixier for Tori we get that That torsor that T torsor to which this uniquely extends is trivial because the generic trivial in other words this this quotient has a point has a section in particular over a complement over a complement of Y and and this this section then gives us gives us Well, this is actually what What we want because sections of this quotient are reductions of eb to torsors of the unipotent radical. So this Okay, in In short, we just we just showed show this claim that our generically trivial torsor e thanks to thanks to quasi-splitness of g over a Over a complement of a close sub-scheme of convention at least to reduces the torsor under the unipotent radical of a borrel So Okay, let's perhaps just a warm up now But let's let's proceed with with kind of over viewing the proof of the main result So the main case, I mean as I said it doesn't matter what are so equal characteristics or it's a mixed characteristic But but because sequel characters case was already known. I will Assume for the rest that are so mixed Characteristic 0p that's somehow the main case Of interest in more difficult case and so the main the main difficulty is Well, somehow slightly philosophically Speaking is that we cannot we cannot enlarge are We can somehow can only shrink it Well, what what do I mean by that? So for instance, we cannot replace arm just by its completion or so because that would Over completion the whole conjecture is known and any attempt to somehow make are larger And so the problem is that over its large ring the torsor at your studying may Become trivial. So when somehow tries to go backwards and make are simpler by by well, but by shrinking it and so Concretely, let me give an example of what to mean by this. So Popesco's result Popesco approximation is one such Structural result for for for for regular rings and that's where we use the assumption that are some ramified So it implies that our unarmed by the regular local ring is a filter direct limit of localizations Well, of yeah of smooth Z localized at PL to pass Popesco Proved that any any unarmed by the regular local ring is somehow I Mean is some geometric origin in a sense that can be obtained as a filter direct limit of of smooth algebras either over a field In equal characteristic or over Z localized at P in mixed characteristic and this results somehow allows us to to start using algebraic geometries for for this for this for this problem. So we we use We use for best co-approximation and the limit argument To reduce to the case one R is just a localization of a smooth Zp algebra. So without loss of generality R is a local ring well, a local ring of Of a smooth affine Z localized at P scheme Which I'll call X. So incidentally in this Small amount about the semi-local case Yeah in the semi-local case Can you allow residual characteristics to be different primes and in each one you will know that it isn't ramified or? Yeah, it's like that I can allow them to be of different and and in each one of them is Okay, so ours ours a local ring of a smooth Smooth affine Z localized at P scheme. I'll assume that the relative dimension of Of X is D and I'll assume that it's that it's positive because the D equals zero case It's just that basically the DVR case and that that is that case solid and also anyway so these these positive without loss of generality and our G E and And this closed Sub-scheme Y that we constructed in the claim they spread out by shrinking X We can assume that they spread out to a reductive group scheme script G a torso under its script E and a closed sub-scheme Y defined Defined over over all of X Okay, so so that's that's a setup we have we have some smooth affine Z localized at P scheme and then Reductive group scheme and a torso defined defined over it E is generically trivial. In fact Over the complement of Y it reduces the torso and the unipotent Radical and so if that complement were affine then that then that Over that complement the torso a bit trivial And we we want to show that E is E is trivial at the local ring that we're Talking about so for this is we will outline multiple steps of how to kind of Step by step simplify this situation. So first of all, we will we will simplify the geometry with with with some version of Neutral normalization. So the first step That does not use Anything about the groups. It just it just told you right geometry It's a version of notar normalization or some sort of preparation lemma Which ensures That X and Y are of particularly pleasant form. So what what does this notar normalization say what is Well, I put it in quotes. This is not really notar normalization as you see But the statement is that there are an affine open U in inside X Containing Containing spec R. So this is a smaller smaller smaller affine open neighborhood of of this local ring that we're really interested in Such that over this you will have a smooth map smooth smooth map of Relative dimension one and we'll call it we'll call that map pi So this U is a relative smooth curve over an S, which has an open of Of the affine space of Dimension of dimension one lower. It's an open containing The origin. Yeah, this S is just some affine open Okay, and crucially so so locally at enable hood off of our the the X's fiber does a relative smooth curve over over over particularly simple simple base and more over our Y Our our script Y closed subscheme Over a complement of which something good happens is such that its intersection with this with this U is finite over S Not not not not merely quasi-finite, but actually Literally finite so not not a normalization would would say that I mean ideally if there was not a normalization a mixed Characteristic it would say that well after shrinking at this at this point X Or sorry, it would say that X admits perhaps a finite map to the affine space That's what neutralization over a field says a mixed characteristic Cannot be true for instance, there are quasi-finite schemes that are not not finite and This version says that at least after knocking dimension down by one we can find a smooth Smooth relative curve such that the closed subscheme of convention at least to that we're interested is actually literally still finite and this this this So here's some ingredients Sorry Yeah, right in fact the artens good neighborhood technique is used in proving improving this so it's it's some sort of I mean It is not really not a normalization in its classical sense. It's just intuitively a statement of that sort so ingredients to this is a Albertini theorem Applied Applied to a compactification of X to a projective flat scheme X bar over over Z localized at P and in fact we use Gabbers version of Bertini theorem which valid also refined fields Well, it's also Punen's version and the in fact Gabbers version is slightly more convenient for us because it allows us to control degrees of the hypersurfaces That that occur basically the idea is we take this Compactification and we cut it iteratively by sufficiently transversal hypersurfaces and Equations of these hypersurfaces will be the images of the D minus one standard coordinates of of a one And so the fiber over zero will be the intersection of you with these hypersurfaces And so if we if we if we choose them well, then that zero fiber will be smooth somehow by Bertini and and then this will also will also hold and will spread out around around zero this is the kind of Thing that happens that happens here And of course, this is as Luke Already remarked this this is similar To Arton's method so techniques from Arton's Construction of good neighborhoods from a she for So Arton constructed such low configurations into into curves in equal characteristic over a field perhaps a large bar close field He used it to show that et alchemy ology agrees with some logic counterpart and this this result was was used I Mean his technique was used for many other purposes too and let me just mention that earlier versions of this of this type of preparation lemma versions over a field of this step one Are due to Due to Quillen who used also such kind of new turn normalizations Would especially with respect to the aspect that one wants to control a closed subscheme to study algebraic a theory and later refined by Gabber in the context of Gersten Conjecture Okay, so so that's a step one was geometric step one. Let's pass to step two which in fact this is is Is straightforward this is base change. So what what do you mean by that? So let me just summarize what what we have so far. So we have our you going to s and s is an open is Is an is an open in affine space of dimension One lower so the Z localized it P and and you well Let me just like that. Perhaps you're going to pass as this is open and you is a relative curve is a relative smooth curve of Over this s and we have we have Let me perhaps use Color chalk so so we have this why crucially we have We have why here which contains well which probably contains a point We're interested in which contains the local I mean intersects spec R which is the localization of fiber above About zero in this in this in this in this vibration into curves, okay? So what so what do we do? Well spec R is a local ring of this total space of you while we Take it one more time. That's another copy of spec R right here And so it maps to S. We just take a fiber product We get some C So what is C? Well C is a relative smooth relative curve over over over spec R Yeah, so we get C or R smooth affine our curve Basically because C had the you had these properties over s We get well it comes equipped with an R point with a section Delta which is which is an R point of of of C just because I mean there was also a copy of spec R here So surely we'll get we'll get a section from that Just by construction Z in C is a closed So it's a closed sub scheme which is finite over R. So our finite closed sub scheme Which is just a base change of Y intersect or intersect with with you that was finite over over S and Z So base change of Y intersect you we're going to be finite over R E Well We'll also have G over over C which is quasi-split By base changing the group scheme. We get a quasi-split reductive Reductive group over C With Borrell with Borrell Script B such that Delta pullback of this pair Borrell inside a reductive group scheme is Our original Borrell inside inside G Basically because our reductive group G need not be constant over S When we base change the reductive group to C we get a family over that I mean we get a reductive group scheme over C with the Borrell Which need not come from from spec R. And so but it's it's pullback along Delta is going to be the original G that we That we start that we start from and by base changing the torsor we get A torsor where this relative curve a G torsor Script G torsor Who's pullback Along Delta is is E and Such that the restriction of E to the complement of this are finally close subs scheme is E reduces To a torsor Under the unit potent radical of Off off of the Borrell Okay. Well, so From if in fact what we what managed to achieve here with this with these first two steps is that We managed to manufacture a relative curve a reductive group and a torsor over that Equipped with a section such that the original torsor we're studying actually lifts to this relative curve and so In later steps we use somehow the the flexibility of the setup to change C and Eventually reduce the case one C is the affine line and the Delta is the zero section and Z some close finite sub scheme And then you well, okay, then use techniques about studying torsos under affine line so the upshot of this is that we have the flexibility of changing of changing C and Problem becomes silt on March metric. What when did I start? Okay All right, so the next step The next step Well now the the price we paid is that our G is no longer constant We have some script G over the relative curve. We don't quite like that. So we would like to equate This this reductive group script G and the constant group just G base change to C and the idea for this I mean It's not okay. First, let me give this proposition. She's in fact a general proposition. So for for for a for a Henselian pair a But in a which is Henselian with respect to ideal I such that the quotient a mod i is normal or Or not heating on geometry in your branch should also be okay, but okay a mod i is normal then a reductive reductive groups over over Over a up to isomorphism are the same as reduct as their base changes reductive groups over a mod i up to isomorphism in other words up to isomorphism every reductive group over I mod a mod i lifts uniquely to a reductive group over over a Granted that a mod i is normal in the pair is Henselian and the case one when this is just a Henselian local ring It's an sj3 and one can well one can also obtain this more general more general version So the idea is to Henselize I mean roughly speaking the idea is to Henselize Henselize Delta Along sorry Henselize C along Delta to equate To equate G and and this constant group which is which is a base change now this because over at Henselization because Script G and and the constant group agree over of the pullback to the section Then they will agree over a Henselization and then after spreading out will have some real tool curve Where where they start to agreeing the problem is that we cannot call we cannot do that because we need this This does not retain finiteness of Z we have we need this close subscheme Z Over the component of which something good happens and we really need it to be finite over over over Z Rather than just quasi finite and if we do Henselization We will only get quasi finite keep control of the be which enters into the right. Okay. Yeah Yeah, yeah, we also there's a finer version of this proposition where in fact is to cook to a braille I mean this one is quite split if and only that one's was split. So yeah, that's a version of the bees, but I just Okay, so in fact one proofs a finer proposition where the atal neighborhood is not only at all but actually finite at all so the Proposition is as follows after shrinking after shrinking see Meaning seriously, okay, so around Around this union of of Delta and NC That exists There exists a finite at all Not merely at all, but actually finite at all C tilde over C and and the Delta tilde and our tilde and our point of C tilde and lifting lifting Delta such that The base change of this B inside inside G to C tilde is isomorphic to To just a constant family base change from from our Like so compatibility compatibility With Delta tilde pullbacks, I mean after Delta after pulling back by Delta tilde everything is identified with a constant and Identification here is compatible that's what okay, so yeah, so we can find a finite at all Cover fine at all neighborhood of this Delta Union Z where where the good thing happens the B inside G becomes becomes constant and so Ingredients ingredients for this So one uses one uses to Tauric Tauric geometry Tauric geometry to build Compactifications of torsors to build compactifications of Tarsers under Torah and then one uses Bertini theorem roughly speaking the idea is That functors which parametrize Reductive groups equipped with a borrel these are I mean relate to automorphism groups of reductive groups and By some little the massage one reduces because our cheese quasi-split when one reduces to Constantly torsors under Torah and so the same kind of proposition where one wants to equate torsors under under Torah I and to do that one first compactifies the torsor Over the torsor begins a constant also begins life or are one compactifies it Using Torah geometry. So in fact the statement is that if one has a normal normal base normal Noetherian searing for example a torus which is Which is Trivia torus which splits or find it I'll cover and a torsor under that torus Then we can find a projective compactification of the torsor such that the torsor is fiber-wise dense in that in that Compactification and to build such a competition one uses to our except division so on it's Anyway, I will not perhaps go and go go into this and proceed directly to step To step four but but let me just say the upshot So the upshot of of of this step three without loss of generality be inside This is this this borrell is just constant and and the group is also is constant so Alright, and so the rest of the argument is to Massage C into the affine line and then study the case of a affine line using affine grass manians and Some geometric property of affine grass manians and I hope I will Get anyway. So so step four is to reduce to the affine line. So the goal is to replace Replace C by the affine line. This would simplify. Okay. This would simplify the station. So So here we build a diagram We have our C and We will build a quasi finite map We build a quasi finite map to a one of our such that We have some affine open Affine open of C containing Containing Delta and and Z Containing Delta the section Delta and inside is affine open Z Z lies completely inside it The point is that this map is such that it maps the isomorphically Onto a closed subscheme of So so these are both closed Maps it isomorphically onto a closed subscheme of the affine line and moreover this right hand square is actually Cartesian in other words We realize Z as a pullback of a closed subscheme of the affine line and this quasi finite map is automatically flat because C is Well C is even regular and but C is con Macaulay and a one is regular so quasi finite it gives It's automatically flat. So this a kind of preparation llama uses well Yeah, I'm a I'm out of time. So let me Just mention so that this Yeah, so Use excision somehow to reduce To see being a one That's some subtleties in that excision, but I'll leave it there and once we have once we have the affine line We we conclude But by using our fingers minions and here I mean, okay the key key key statement Anyway over an affine line one needs to us one needs to study extensions to Projective line and the keys and this this extension somehow Given by gluing so the trivial torso are parametrized by affine I mean are related to affine grass manion and the key key geometric statement that enters is that the affine grass manion of any Of any group G one takes its derived subgroup and a simply connected cover by functionality that maps to original Have fun grass money on this relative identity component. This map is By ejective By ejective on field valued points What one shows this and I Recall that if if the characteristic of the field does not does not divide the Pi one of the Coronality of the fun of pi one of of the drive subgroup then this map f is even an isomorphism But in that characteristic there is there's no less is refinement that the map f is at least by ejective on field valued points Which helps us a lot because it tells us that points of this are invariant and the multiplication by L I mean points of this which lift to there are invariant of the multiplication by the positive positive loop group L plus of G Because that is correct. Anyway, okay, so I'm out of time and this is somehow the main geometric Point that is used in the to finish the argument, but roughly that's how Remember, I think I looked once at your And you refer to the your paper on what my qualification with some step right is it in this page in this group? Yes, so in fact So in fact in the original version of this of this of this result I was only proving the result for split groups, and I was using X bar was That compactification X bar was coin Macaulay, and the geometry was kind of complicated So I was using my qualifications and later when trying to extend to cause a split groups I realized that actually by doing arguments slightly differently one thing I should get rid of my qualifications and Geometric power becomes simpler. So anyway short answer is that yeah, I don't I don't need it in the original version I used my qualifications, and I thought that this was In turns out they're not essential one can bypass them. So I don't use my qualifications in Thank you very much unless he did the question I Stupid question in two bars. So R is the dimension at most one, but there's no And ramified in this assumption. Yeah, right mmm, and In fact the way the proof of this case to works is by first passing to completion and Then using Brouhatt's theory and nowhere there Somehow the proof there is a bit orthogonal to what we're doing here It's kind of just general facts about discrete relationings whereas what we're trying to do here is trying to use popesco to reduce to some regularings of geometric origin and then tried, you know try to do so much right geometry with it. So it It's also a main limitation of Why won't I mean wise seems difficult to to go to ramified regularings because then popesco is not available and we just don't know what to do, but I Mean, I don't know perhaps perhaps the the correct approaches to try to generalize this this to the regular local arbitrary regular localings, but it's also I Mean far from straightforward. Anyway, this current attacks on the higher-dimensional case pass from popesco. Anyway Also, I think in the sporadic case that there was this with the boonie, right? Yeah, so Ningo actually he Proved a version of this conjecture over a valuation ring. So one has A valuation ring not necessarily a theory and by resolution of single arts It's thought to be a filtered direct limit of regular local rings. So the same conjecture should be true if one has a Torso and a reductive group over valuation ring then which is generally could reveal and must be trivial and he proved that unconditionally I mean that's somehow close to the DVR case except that the valuation ring is no longer Okay