 Thanks. I mean, it's an immense honor and pleasure to speak here for Ophar Gabber. I have also profited from him in many ways. I'm going to test some idea and see whether it works or not. You just say two words, and then you'll either fill in the details or tell you it wasn't mistaken. And OK, so I recently have done some work on some kind of periodic case theory. And I think I've never talked about this really. So I want to use this opportunity to speak about it here, as it's actually a couple of points related to work of Ophar. So I'll slowly make my way to periodic case theory. But first, I want to tell you how I like to think about algebraic case theory itself. So let's say R is any ring. And I mean, for me, a ring is always commutative, although strictly speaking, it's not necessary right now. Then I have approach R. It's a category of finite projective R modules. But as a category, just with automorphisms. And so this thing here will be a groupoid. A category where all maps are isomorphisms. And if you get used to thinking higher categorically, then groupoids are special kinds of spaces in the sense of Lurie. So these form an infinity category S. And roughly what this means, this translation, is that you think about this category in the following way, that it takes a disjoint union over all projective R modules up to isomorphism. And then for each such, it takes a classifying space of its automorphism group. And while there are certain very natural operations that you can do with projective R modules, for example, you can take a direct sum. Well, I mean, this certainly turns, and this is, of course, some kind of commutative operation. It doesn't matter in which order you take a direct sum. So this means that it turns projective modules up to isomorphism into a commutative monoid. But actually, and this is somewhere where algebraic topology enters, there's a way to make this statement without passing to isomorphism classes. So in fact, this thing itself, this groupoid, is a commutative monoid in spaces. And let me put this in quotation marks. So technically, this is called an infinity monoid, where the infinity there means that some are commutative up to all orders. So in this higher categorical language, there is some can have more and more commutativity constraints. And infinity means it has all of them. And so they form an infinity category, or do not they? Monoids, infinity, or technically maybe in spaces. S is the infinity category of spaces. So space for me, say, is a tan-simplical set. And so then there's this operation called group completion. So you have commutative groups. For some reason, you always say, I think commutative monoid, but you say a b in group. So there's a full subcategory of commutative monoids. And this has a left adjoint called group completion that you can also easily describe, the m-meps to m group. And the definition of the Grotten D group is that k0 of r is you take the objective modules up to isomorphism and your group complete. But what this machinery from algebraic topology, or higher category theory, gives you is a way of saying just the same words, but somehow omitting the passage to isomorphism classes in the process. So you actually have, so similarly, you somehow have infinity groups in spaces. It's full infinity subcategory of infinity monoids. So these are just all the guys such that if you look at the set of connected components that self-inherits the structure of a commutative monoid, and you ask that this is a group. And so similarly, this thing has a left adjoint, if m is the group completion of m. Unfortunately, this turns out to be a bit harder to write down explicitly. And somewhat cool, and gave some ways of doing this in terms of some plus construction. But I mean, it just abstractly, it exists. And then I think it makes it very natural to make this definition that the k0 of r just takes this category, or the groupoid, of five projective modules, which has this commutative monoid structure given by addition, and your group completes in this hierarchical way. So there's an infinity group in spaces. So in particular, you can forget all everything about the group structure now. And in particular, it's a space. So it's community, kind of. So the notion of homotopic community when associative up to IRC, this is the same as the old thing of Graham Seigel. Right, so this is basically, Seigel explained how to do this in the 70s. Yeah, but this was much before the. Right, and this is some kind of new reverting of what he did. But it's the same thing as what Seigel did. And the result is something called A infinity. Well, A infinity only makes sense for algebras. And this corresponds roughly to E1 here. So E1 would just mean associative. So I mean, a non-commutative one of it. So sometimes you have algebras. And algebras, I have an addition of the multiplication. And some of the addition implicitly, I think it's always E infinity. But then there's a multiplication which can have different levels of commutativity, which might be E1, which we said, and so on. And E1 is something that's also called A infinity then. Sorry. The case here of this is R, in particular, is homotopic groups. So here, we put this in high zero. Well, now, I mean, some of the higher homotopic groups are automatically somewhat invertible. So if you have an object in here, then it actually has some kind of inverse map. So I never understood really the plus construct or the Q construction, but I find this very intuitive. This is the right thing to do. And when I talk, tell us the number series. They don't seem to know this. When I tell to algebraic topologies, that, of course, this is a way to think about k-series. Anyway, so that's the algebraic k-series of a ring. And now, maybe I want to generalize this to schemes. And I mean, there are different ways of doing this. And I will take a route that's maybe, if you want to set up the series, not the right way to do it. Because to prove the following theorem, or I need some of the machinery that would anyway define the k-series of a general scheme. But for expository reasons, I think, OK. So there's the same of Thomas and Trouba that's the association that takes any ring R, commutative now, to its k-series, that is a risky sheaf. It's sheafed in some derived sense of spaces. But doesn't he expect that he always wants to see the finite rule dimension? Yeah, so yes. So when you talk about sheafs of spaces, you have to be careful there's a notion of a sheaf and there's a notion of a hyper-complete sheaf, which has to do with some convergence issues. And I only claim it's a sheaf. I don't claim it's a hyper-complete sheaf. And back then, Thomas and Trouba didn't really have the technology to talk about non-hyper-complete sheafs. But the theorem is true in general. It's a non-hyper-complete sheaf. It's also an itch-nevage sheaf. You use the non-connective k-series as a. No, I've. Yeah, so I'm only, for this talk, I will only consider a connective k-series because it's good enough for what I will do. Let me just add to remark that infinity groups and spaces are actually the same thing as connective spectra. And there is a kind of version to actually define some kind of non-connective spectrum whose connective cover is this thing. And then this statement would even be true for the non-connective version. But let me stick to the. OK, so the non-connective part don't matter for the sheaf, for the physical sheaf. Right. And they don't give me a rise to. Well, it's a limit. And so if you have a sheaf of non-connective things, then the connective cover is a sheaf of connective things. OK? But this makes it reasonable. So maybe I should make a warning, a very important warning, that it's not an Intel sheaf, really. A lot of the mystery of k-series is contained in the question to what extent k-series does satisfy a tall descent. So for example, all these kind of Loch-Cato conjectures are very much related to this. OK, but if you have any sheafs, then it's clear how to globalize on a scheme. So if x is a scheme, I define the k-series of x to some of these global sections on the Sariski topology of x, so an anthropological space, of some of the k-series I appreciate. Let me just call it k. And let me also define a tall k-series. So this coincides or not under some coefficient. It coincides with the actual k-series if the McDonald's and Grobo are assuming the. No, this always coincides with the k-series of perfect complexes on x. But as we said, the convergencing was for some do-ar-gamma, there was this. I mean, I might maybe want you to assume that x is qc, q-bar-gov. Do I need this? No, I don't. It's certainly I don't need any finite dimensionals in here sometimes. For the ar-gamma? Not even for the ar-gamma. So if you read Luris Hayatopo's theory, there is a way to define this even on hyper-complete sheeps, what global sections are. So it's a complete theory of sheeps of spaces or anything like this without hyper-completeness in post. Yeah, the world is all to work with Brown, Gersten. Right. But they needed some condition, which is slightly weaker than the reality, but it's still some condition for dealing with this type of. That's why part of the reason that Hayatopo's theory is such a great book. OK, I mean, you can make sense of this. I also want to define a tall case theory as the global sections on the tall side of. So this is something different. It's some of the tall sheafification of the theory. OK. The first thing that was then pretty well understood about case theory is what it looks like analytically. So after ally completion, at least as concerns the tall theory. So what do I mean by analytic here? So if I have any infinity group in spaces, I can define its analytic completion. And so there's a limit of a mod l to the n. All operations taken in the derived sense, as they must, because other operations don't make sense here. And so this itself will again be an infinity group or it in spaces, in particular itself. It's a space, and it has so much to be groups. And understanding the tall case theory, OK, now there are some convergent issues coming up, but it basically reduces to the study of the for strictly and zeal and local rings. And the structure for strictly and zeal and local rings is given by a theorem for Susslin, and I think maybe in most generality, there's something due to Gubber. So assume that r is strictly and zeal and local ring, and l is invertible on r. Then you can actually compute this thing. So then the homotopic groups of the case theory of r, analytic completed, which is a two-parallel thing and basically looks like topological case theory. The complex numbers. So what it is, it's zeal and then acetate was coming up, twisted by i over 2 if i is even and 0 else. OK? And a corollary of this. And now there are some finiteness assumptions on x in order to make some of these convergent issues go away that we just discussed. So there are certain finiteness hypotheses on x. So in general, if you have the global section of a spectrum, there's some kind of local to global spectrum sequence, or at least under convergence conditions there is, which goes from the corollary groups of the homotopy sheaves to the global thing. And the sheaves, you can understand if you understand their values at the local rings. So they are just given by these tate twists. And so there is finiteness hypotheses on x. So x is some kind of local to global spectrum sequence. I will probably get made in this next thing wrong. So it starts with the etalc homology of all the tate twists and converges to the homotopic groups of the etal and the kcp of r. Sorry, x. So you don't use the connected kql, so you have some whole genitube mix. Right. So this is if i plus j is greater or equal to 0 and something funny else that I don't want to say next thing. But you take j to be positive in there. So i and j, I guess i is greater or equal to j is greater or equal to 0 in this next thing. But I probably have the. You can have the non-connective one and get some. Well, no, no, excuse me, I'm stuck. Usually the inverted both are here. So far, I haven't converted the both element. I want to do this in a second. But because you understand the kc of stricke and z in local rings, at least most of your convergent issues, this gives you a way to compute the etal case theory. In practice, you actually want slightly finer statements where it becomes unnecessary to take the etal sheafification. And the first such theorem was proved by Thomason, so with a better statement, not involving etal sheafification. And so for this, you want to invert the both element. And there is a way to even to say this without having a both element. But I want to do it in a way where I have a both element. So for simplicity, assume that x lives over the ring z, where you join all power roots of unity. Then by the relation between the k1 of the ring and units, there is a way to define a both element, which will then live in pi 2 because in this other completion, there's some tate module coming up. So there is some both element. It's called beta l. Living in pi 2 of the k-serial of x, you can complete it. And can you revert that? And so there is something in topology called the k1 localization, which implicitly depends on your prime l. And this doesn't depend on the choice of a both element or anything. But a concrete way to compute this is to take the k-serial of r. Well, to get my both element, I already have to periodically complete the words about element. And I guess this needn't be already complete again, so I probably have to do it again. So lk1 is defined for what? So that's defined for any spectrum. And k1 means something. Yeah, so there is a thing called Morava k-series k of n. And k of 1 is basically just topological k-series. And then there's a way to localize these things. It's just that there is an intrinsic way, which doesn't depend on any of these choices of defining this thing, but which concretely just means some way to do it. So this doesn't use a multiplicative structure of r? No, it only uses the other structure. And in Thomason, I believe it's defined as using powers of the both elements, which one might believe it's different. Right, yeah. OK, and so there's the theorem of Thomason that, again, under some finiteness hypothesis on x that I don't want to spell out. And you also need the fields of the special type, because you had some way to do this. Part of the finiteness hypothesis. I mean, it needs to assume that there is some kind of finite Galois conmodical dimension things for a space field and so on. All right, so the precise hypothesis on Thomason actually had pain to write down. Let me not do it, because anyway, the second thing I want to talk about, PID case theory. So and I said I only talk about connective covers in this talk, and so the statement I want to make is only on connective covers. It's actually true that the tau case theory of r, I can complete it, is the same thing as the k1 localization of the case theory of r. So this is something for which you really need to know the tau side of x. This is something for which you only need to know the case theory of x itself. And then you just invert the both elements, and then you get this. So there is a certain map that goes from the LA state module on k1 of x to here. Just for any spectrum, there is some of the LA state module contributes to the LA computation one degree higher. And in here, in k1 of x, in particular, have all the units in x. And so in particular, the system of 1 zeta l squared lives in here, and this maps to k. And so this had an application. So there's the following theorem of Thomason, and then by Gabber, where again, in the strictly and the unlocal case, let R be strictly in the unlocal ring, invertible. I should say that this is the theorem is what was known as quotient expurity conjecture in the tau-co-omology. So then you can look at the tau-co-omology groups with support. Let's move up there with our case. Oh, sorry, yes, yes, yes. Regular. Of course I can. And also, maybe it's this one here, OK? Sorry, now I forgot to write regular. Then you can look at the conmodule with supports on maximum ideal on the tau side of zl. And this has a very simple answer. It's a dimension d. This is zl twisted by minus d, if i is 2d. And it's 0 else. So there's an application of this machinery of algebraic case theory to a very classical question in tau-co-omology. Do you use the dimension of R? Do you use the dimension here? Here is the point that we need for finite type but then it's more general. I mean, this in particular also does mixed characteristic case. No, no, only the mixed characteristic of the problem is because the equal one, they are much smaller. I mean, you can just use the professor's theorem to reduce the finite type smooth and then there it's more clear. But in one of my first papers, I use this critically. I use this theorem critically in one of my first papers. So let me just give you a very vague idea of how this is done. So you use the following thing that you can also look at the co-omology with support at the maximal ideal. So let me write x for the spec. Let me just write the spec R here. And this means Sariski. It's a case theory spectrum. This is just the case theory of little k. So k is the residue field. So this is a version of Krillin's Divisage theorem and uses it as a regular. But then this implies the same thing for after K1 localization. I mean, invertebrate element, which you have in this case because you're strictly in zero. And then the left-hand side is computed in terms of all the local co-omologies on the top side of all possible T-twists. And the right-hand side is simply Zlj. Yeah, this is what Thomas said. And so then you have a problem that there's actually some spectral sequence coming up. And the spectral sequence essentially degenerates by using evidence operations. So you can use evidence operations to essentially isolate individual T-twists. But then there is some kind of bounded torsion coming up. And then Gabber found a way to get around this. So now I would like to say something about PADK theory. And what I will say here, this relies on joint work with but and tomorrow, or second paper. And plus separate work of Clausen, Messier, and Morrow, where Messier is not the first name of Morrow. So let me say what the goal of what we're doing here is. So we want to have an analog of this relation. So we want to have a description of a TARK theory in terms of a generalization of the notion of T-twists and mixed characteristic. So R is some rather general ring. And P this time is not invertible on R. And actually, because you can always understand the ring usually in terms of the ring where P is inverted and the ring where it's PADK completed, we will always, in a second, actually assume it's PADK complete. We want to define certain etal sheaves, technically actually sheaves of complexes. Some T-twists, Cp of J, J greater or equal to 0, I mean on the etal side of R, say. Such that there is a spectral sequence, which goes from these co-amlogic groups and converges, well, again, takes a TARK theory of R and PADK complete it. And that's fine, positive degrees. And something I don't want to say anything about in other degrees. And well, the first state with other things we know and love. So Zp of 0 should, of course, just be Zp. And I mean, there's this usual issue that Zp is not really a sheave on the etal side. And there are many ways to get around this. And of course, Zp of 1 should just be the Tate module of U p infinity. Yeah, so say, use a pro etal side. You weren't complaining when I did it there. Yeah, I mean, I could. I think Iqudal theory, for example, is also general enough to handle this. I mean, I guess the way we actually set it up is used as a pro etal side. So I should make the remarks that there is previous work on this. And coming from kind of two different schools. So there is a kind of motivic school. So maybe Wojcicki, but more precisely, actually Bloch. And then maybe worked out by Geiser and Levine. So they're in a situation where R is maybe smooth over DVR. And they use Bloch side char groups or something like this. And they don't actually get etal sheaves. So they get Tsariski sheaves, or I need Snavich sheaves. And the spectral sequence converging to K-series itself, not etal k-series. And then there is a kind of different school. I mean, starting with a work of Fontaine and Messing. And then there's been the work of Schneider and Sato, where maybe R is semi-stable over DVR. So that's in particular the work of Sato. I mean, I guess I should also mention Suji. And I'm thinking of some kind of logarithmic singularities. And they use these some kind of symptomic complexes. And they certainly get etal sheaves. But I'm not sure how much is known in this case about the relation to K-series. Maybe you all have some results there. We've really got some restrictions. We're going to go to amendments and degrees. Right. So in this approach, there often is a restriction on the degrees in which you can do this. And as you get to larger and larger tates with, you usually have to invert larger and larger powers of kf2b. So what we want to do is should really work completely integrally and for all tates with. But I mean, there's been such series by Schneider and Sato in particular. They have something, I think, which works integrally and should give the correct answer. OK, so I should say what my assumptions on R are, under which we can do something like this. So I always assume that R is periodically complete. And also maybe what Barov was calling bounded, meaning that the distortion is bounded for technical reasons having to do with taking lots of purely completions. But as I said, in general, understanding this should reduce your understanding of after inverting P and on the PID completion. So we're somewhat doing the thing which is completely orthogonal to the previous work. But we need one important assumption on R ring, which is that we call quasi-satomic. And if R is in Syrian, that's just equivalent to saying it's a local complete intersection. Completions are local complete intersections. Right. I mean, there is this, I think, fixed notion of a local complete intersection. It means that the completions of the local rings are quotients of regular rings by a regular sequence. Quasi-satomic is just a condition of what's a cotangent complex. So in general, this means that if I take the cotangent complex of R over ZP and AEP complete, then it's P complete tau amplitude contained in. Well, one way to say what P complete tau amplitude is is just saying that, I mean, tau amplitude means that if your tensor was any module, then the derived tensor product will live in these degrees. And here you're just allowing yourself to tensor with P tau as a module. It's just a question on what is the most P tau. It's just a question on the most P thing. As R, are you complete periodically as an R? But you want it to be an R. It's complete periodically. When you derive complete periodically, well. That's why I make this assumption. So in particular, this includes all perfectoid rings. And all rings, which are some of formally smoothed over perfectoid rings and stuff like that. OK, so now I can give our definition, or actually a slight variant of our definition of state twists, using the prismatic side. So if you have an object AR, AI, and the prismatic side of R. And here I'm using the absolute prismatic side of R. So I'm simply allowing. So this means that this is a prism plus a map. Well, from spec A mod R to spec R, which means a map from R to M at I. Then you can define an I-guide filtration, call this n greater or equal to j of A. So this is a set of all elements X and A, set to set, like I was using this fly symbol. This lies in I to the j, which is a subset of A. This is a subset of modulo of A. So if you did this in the kind of crystalline case where I is actually generated by P, then in small degrees, this agrees with the divided power filtration. But in large degrees, it doesn't. So that's part of the reason that our theory works better in large degrees. So that's a divided power filtration, some of something that only depends on divided power structure. But this really needs the Frobenius on there. So the crystalline case is like what was explained in the previous talk that did you. Well, crystalline just means that I is generated by P. So these are some of the things that come up if you do the prismatic story in characteristic P and where it's closely related to the crystalline. Our story. And so we have a map kind of divided for Frobenius. Let's call this phi jet. Like I've used this symbol. I would like to say it goes from A to A. And this is just given by phi divided by d to the j. But I can't quite say this because I don't have, in general, a generator for my DLi. There is a way to get around this by introducing these Poisean twists. So these are trivial based principles. Phi divided by this distinguished element to the j. So if I trivialize i, then for a certain choice of the distinguished element that I used to do this, this will just be phi equal to d to the j. And put the same twist on both sides. And put the same twist on both sides. So in particular, there's also a canonical map somewhere. So there is like the inclusion still. And why you cannot eliminate the places like tensile in this ij? It's almost tensile with i to the j, except that the Poisean twist over A is slightly more complicated to write down and that they don't want to do it. But think it's just tensile by i to the j. So unfortunately, writing down the Poisean twist over A is like, I don't understand it so well, but there's a way to do it. And so the definition is that if I take my cheap Cp of j and I evaluate it on my r, then it's given by the homology on the prismatic side of r of the Poisean complex. So this is supposed to be some kind of mixed characteristic version of the art and try sequence. That first of all works a mixed characteristic, but second of all works for all t-twists. And there is some kind of, at least philosophical resemblance to these kind of symptomic complexes. R is quasi-syntomic. And so the theorem that follows from combining the works that I said is that the spectral sequence, well, I guess this is what it just called. It's a homology group of this complex. Converging to tau k through of r p, it exists. And secondly, in degree 0 and 1, we recover what we want. So you write homology, is that the prismatic homology then? Yeah, maybe I should have. Now it's just a homology group of this complex. Sorry. I mean, Zp of g is the chief of complex which associates to any r, this value, Zp of j of r. And so then school was actually adjusted to this thing again. Satisfies the tau descent. I h after j r minus h I minus j r, Zp j is not. Is it the tan cumulus vector with vanishing or just? Well, this thing, it turns out, satisfies tau descent. It's association of mapping r to this thing. And then this is, well, it's just a homology group of this complex. And r then provided which is which topology? You don't actually need to put any topology on this. You could put the servicetopology somewhere on A mod on A. So why do you write r gamma r then? Well, it's just a limit over all objects on the side of this. Sorry, I mean. This is the side of all these objects. r delta is the side of all these objects. All pairs A comma i is the map from r to A mod i. And then to any A, I can associate this complex and I can take the limit. So maybe you don't like this. You can also write the limit of all maps from r to A mod i. But when you define certain things, you assume that, ah, those things are defined in general. Or anything, the w. Right, so these things are just. Define without any assumption there, bounded and all. This is just an assumption on r, not on the. I don't think I need to put it on the present right here. Should I give sketches or should I say something about what I want to use this for? Well, let me give a very, very. By the theorem of Clousen, Messier and Moro, this can be computed in terms of something else called the topological cyclical homology of r. And so there's been long known in algebraic topology that there's a mass certain map here called the cyclotomic trace. And this turns out to be a nice morphism here. And then what we do in this BMS2 paper is analyze this topological cyclical homology. And show that it admits this description in terms of these complexes. So this is actually the homotopy fiber. So this is a paper of Nicolaus and myself. That one way to describe this is to take something called TC minus of r. Okay, completed. It's very equal. Let's say, homotopy. And then there's certain, there's a certain phi map and a certain identity map. Going to something called TP of r. And this here can be described, and this is some of the BMS2. This is described in terms of this kind of n greater or equal to j. Well, we just write n greater or equal to j of prism. But prism is one of the global sections of that left term and maybe this is twist. And this here is described in terms of these prism j's. So there's quite a bit of work going into this. And something else that Clausen, Matthew, and Moro give to you is a way to describe the difference between the k-series and the tau-k-series. So they tell you how far they actually are from each other. And they give you many situations where they're actually the same. So in some sense, the analog of the other step that after inverting the bot element, these are the same. That's also understood again by CML. We know pretty well how far they are from each other. OK, I think I started a bit late. So I think I still have maybe three, four minutes. So let me say something about a potential application of this. So it's joined with Chesnavicius in progress. And again, it's these kinds of local homology questions. And so it's about having analogs of the local homology things where now somewhere L is not any more invertible on the characteristic. So let's put ourselves, again, in the situation of a strictly local Hensirian ring. And this time, it turns out that being irregular doesn't help so much. And what turns out to be the better assumption is that it's a local complete intersection. And let's say we have any fine-effect group scheme of R. Commutative, of course. And if I look at the homology with supports on M, let's say, those that have fine-effect coefficients, I've got to use pf homology on G. So dimension is equal to G. And then this is 0 for i less than d. So let me make some remarks about this. So this is the same bound as for a curian homology. And in this situation where maybe G is Fp or something, then by R is of characteristic p, then the Fp homology is closely related to curian homology by the art in tricequence. So you would rather expect your bounds to be close to the bounds you see in curian homology. Which already means that you can't expect something like twice d to appear here. Because this is not at all what happens in the curian setting. It's easy. It's the order of G is invertible on R. I mean, there's probably some F and left shares or something in doing this. But maybe there's some work of Governturing. That's just right OK. And then a very interesting case is if G is just a mu p. So this implies the following two results. So it implies that if D is at least 3, and by U R I always denote the punctured spectrum, then any torsion line bundle or U R is trivial. And if D is at least 4, then the bra group of U R is 0. And these are actually conjectures of Governturing. And so last backboard. What can you say if G is alpha p? Can I just do this? Well, then you can probably relate it to GA, right? So that should be OK. And so this actually having this statement about the bra group here, actually, because it reproves this result about the purity of the bra group, that if you have a regular scheme and you remove something, of course, I mentioned at least 2, then the bra group doesn't change, which was recently approved by Kistutis. And he only had to prove something that I mentioned at least 4, but actually, you don't need a regular assumption. You should only need a complete intersection assumption as Governturing conjectured. And OK, so our theorem is that if R is either p tau and 3, or p R is equal to 0, so we're going to characteristic p, and G is either z mod pz or mu p, then the conjecture holds. So that's quite a weak in generality, the conjecture, but g has some theorem. I mean, that's your conjecture, or? Actually, I thought it was as obvious, but you can't, for example, be like, what do you say? After you're talking, I'm saying, in January, I tell you, please move on to conjecture of this conjecture. Yeah, for here, it's for any G. For any G, I was not sure that it should go. Yeah, so we expect that there's some kind of general version of the eudonase theory in this kind of setting, where you should be able to write any G in terms of some similar such complex, where some of the Fribinus something. And if there was such a theory of the eudonase theory in the setting, the argument should work for a general finite factor. Also, in low degrees, one can somehow use a scalar to reduce mu p. What do you say? So in small degrees, you can sometimes reduce the restrictions of scalars of mu p. So what is the definition of the y piece in this setting? Should I do the thing? Yeah? Yeah, yeah, sure. OK, so tupe, tupe, tupe. Brokeesian twist. So let me define ir to be the product of i phi of i up to phi to the r minus 1 of i. Then what will happen is that if I take the Brokeesian twist and mod it out by ir, then it is given by ir mod ir squared. And in general, the whole thing can be recovered from the inverse limit of this. But you have to be careful to define the correct transition maps here. So they have some kind of obvious transition maps, just because ir is contained in ir minus 1. But you need to divide this by p. You need to divide this by p, this transition map. And A1 is what? Is it finite? It's an invertible A module. And the general ones are tens of powers? And the general ones are just tens of powers of it, yeah? Yeah, I also have a question on the single. Maybe I don't know if you want it to be. So you mentioned this argama, the single k field, is it the k field of perfect converges that you claim, in general, argama in the sense of blue. I don't know the sense of generalization of taking hypercone module from unbounded to low converges. Not hypercone module, that's the thing. What? Well, OK, so yeah, yeah, some kind of, yeah. Mm-hmm, yeah. But now, in general, you can compute it using the hypercovering. But if you are. No, no, no, you are only lost to use nerves of chess coverings now. That's some of the difference. You're not allowed to do hypercoverings. But you can always find some kind of finite succession of nerves of chess covers, which reduce everything to the affine case, if you have at least a QCQS scheme. So the definition of the argama uses only nerves of chess? Right. I mean, first you can do this for separated schemes, and you just take any affine cover, any open affine cover. And then the chess nerve also is just affine. And so then all the terms are already defined. You can take the limit. And then for a general scheme, at least intersections are, again, quasi-completely separated. And so it's defined for them. And so then in finitely many steps you can get to any QCQS scheme. So does this argama generalize the commodity of unbounded below complexes like Spartanstein or me? No, Spartanstein, I think everything is hypercomplete. But I mean, yeah, I mean, if you restrict a hypercomplete object, it gives the same answer. It's also true. Maybe I'll pass this later. Yeah. Yeah. Cool. OK. OK.