 Thank you very much. Let me begin by thanking the organizers for the invitation to speak and to be part of this conference in honor of Lucie Luzi. Like so many people, Luc's work lies at the foundation of much of my own work, including the work I'm talking about today. And his generosity with ideas and encouragement has been tremendously important for me. I'm really delighted and honored to take part in this event. Let me also apologize in advance. I'm still struggling with the slides, so I'm going to do my best to give the closest thing to a Blackboard talk that I can and handwrite the talk. Please let me know if my handwriting is hard to read, and I'll try to do better. Okay, so I'm talking about representability results for Flat Comology. This is joint work with Dan Bragg, who's here at Berkeley. Let me begin with some motivation and the main results. Is the writing showing up okay? Okay. So let's consider a proper smooth morphism. Let's say geometrically connected fibers. So what Dan and I wanted to start it out trying to understand was various properties of the divide, push forward, even, well, we started with mu P. So here I'm talking about FPPF Comology. So I'm using the FPPF topology. And in particular, we were interested in representability results for these sheaves. So these are sheaves on the big flat side, and you can ask whether they're schemes or algebraic spaces, and then do things from there. So, well, let me do some easy cases. So if I0, because of my assumptions on the morphism, R0, F lower star, mu P is just mu P on the base. So that's good. If I1, already there, it gets somewhat interesting. You use the Kuhmer sequence, 0 here. So this is multiplication by P. And you see that R1, F lower star, mu P is the kernel of multiplication by P on the relative Picard scheme, which again is a nice, well, at least it's a group scheme. And so it's representable also in that case. So sort of the first question we thought about was, what about? So the Picard scheme is a group algebraic space in general. Yeah. So I guess when I say representable, I will usually think about algebraic space. Yeah, thank you. So next case R2, F lower star mu P. So already this case seems to have sort of in the current literature, not complete results. So for surfaces over a field, this was studied by Arden and Arden and Milne in particular for K3 surfaces and more recently in a relative situation by Bragg and Lieblich. So, but already for i equal to 2, there's work to be done. And so the goals of this project, well, there's a variety of goals. So one is we want to understand representability results for all integers, i bigger than or equal to zero. Two, we want to consider more general group schemes. So consider arbitrary, finite, flat, abelian group schemes. And three, we want to relax the assumptions on F. And maybe four, which I won't talk much about today, but is a significant part of the project is to study Cartier-Judoné theory for the comology groups and really understand what's going on there. And maybe I should have said this at the beginning throughout. Let me put this over here. If I case I forget, I'll work over field K, which will be perfect of characteristic P positive. Okay. Right. Okay, so let's see. Okay, so let me get just to a simple example. Aside from UP, well, the next easiest group, well, you can study alpha P. So what can we say about alpha P? So let's say F from X to S is proper and flat. So for alpha P, the story is actually much easier than mu P because you have this sequence with the additive group, Y goes to Y to the P. Let me call label this so I can refer to it later. I'll call that one. And so you get here the comology of alpha P sits in a distinguished triangle. And so, well, that gives good, fairly good control over the comology of alpha P. Let me just mention that this is given by coherent comology. So I'll say coherent complex. So it's comology of OX. And then you have this map here, which is not an OS linear map. But so if you look at this, you see that if you want to think about what kind of group schemes or group spaces might you get if you take comology of a finite flat group scheme? Well, of course, you have to include finite flat group schemes, but you also have to include coherent sheets because they arise naturally in this example. So let me state the first theorem. So again, so F, so we're over this perfect field K. So F from X to S will be a projective morphism. And these are finite type K schemes. And let me further assume that S is reduced. And let me let GX be a finite flat abelian group scheme over X. And the theorem is that there exists a dense open subset U and S, such that, well, okay, so we're good. We're ending up in some category of abelian sheaves on the FPPF side of U. I'm going to consider the restriction. And so here's R, I, F lower star GX. And I want to say that it lies in some more restrictive sub category inside here. So let me put that there. And so, well, I have to include finite flat abelian group schemes. And I have to include the abelian sheaves. So I'll say so I'll write it this way, abelian sheaves, FPPF locally given by vector bundles. And so what do these brackets mean here? I will try not to write it. What I mean is you take those objects, so finite finite flat abelian group schemes over U, and sheaves, which are FPPF locally given by vector bundles, and take the smallest abelian subcategory of the category of abelian sheaves on U. And the theorem is that the comology, you can find a dense open such that the comology lies in this subcategory. So concerning this, maybe I misunderstood, so abelian sheaves, FPPF locally given by vector bundles. So I believe you mean abelian sheaves on the side of all the schemes of finite type with the FPPF topology. But then if it is locally, then usually the same theory will give you vector, but I mean, it seems that what is the difference between this and abelian sheaves associated with vector bundles? So that's a question. Yes. So I don't know the answer, and that's why I'm being cautious and saying it this way, but I'm not specifying the OX module structure. Oh, there's no OX, excuse me. Right. So that's why I mentioned, yeah, so let me just elaborate a little on that point. So that's why I mentioned in this example, sort of what we're doing, even in the comology of alpha P, you have this term is given by coherent sheaves, but somehow we are forgetting the OX module structure because we're passing to just, you know, the maps between them are not OX linear. So I'm just thinking about underlying abelian sheaves. So I don't know. Okay, shall I go on? Yeah. Okay. All right. So here's a corollary, corollary one. So this implies in particular that you can find an open U as above so that this is an algebraic space. And so in particular, if you take S in the case when you start over field, if S is just a field, then RIF lower star GX is a finite type group over, okay, algebra groups, group scheme. Okay. All right. So further remarks. So one, I think in general, is that right? Say again, F is not even flat in this assumption. Is that right? Let's let's see. I think I want, well, I'm assuming S is reduced. And so and I'm allowing myself to shrink. So I could assume it to be flat to start. Yeah. Yeah, thanks. So in general, shrinking on U seems to be necessary. Shrinking on S, S appears necessary. Let's see. Ah, I see there's a question. Oh, I see beyond. Yeah, let's see. I think I can do that. Please let me know if I put a link in the chat if you want to go back and forth among the slides, please let me know if that, oops, I think I send it to the panelists. Sorry. Okay, thanks. All right. So shrinking seems necessary. And two, I expect, but so that projective should be replaced by proper. But we do use projectivity in our arguments. So for now, it's that's, that's a genuine projectivity assumption. We do have some results, which don't have that. Let me say theorem two, which is, again, let me assume some strange things are happening. Okay, so F from X to S is proper and smooth. So that's proper and smooth. And here we could even assume algebraic spaces. So I'm not, I seem to have some, I'm not sure if people are joining, it's causing some confusion, but okay. Finite flat group scheme. I apologize. It seems that when people are joining the jam board, it's causing some issues of height less than or equal to one over X. So if we have a height one group scheme, and let's, so fix n, fix n and assume. So, sorry, we have a question. Yes. So you also have, maybe it's automatic and I'm not so sure, based on compatibility. Yes. So, right. So, I mean, so I guess I'll say stronger is sort of variant statements later, but let me go back here. So what, I mean, I guess what I'm, so if I think about a vector bundle, what kind, you know, that has my definition of the abelian sheaf associated to vector bundle has sort of base change built in me. So I mean, it's the big sheaf defined by the vector bundle. And similarly for the group scheme. So I'm not sure. I guess it's, I'm not sure how to formulate base change other than to say that it lies in this category, these sheaves that have these properties, but perhaps I misunderstand the question. But you could just ask about the change for homology itself like our I have lower star gx and you have the map that's the formation of that. So I'm always on the big side. Right. So so the piece is automatic. So maybe again about this vector bundle issue. So if the sheet is actually given by vector bundles, and it is also a usual technique is you can have place for the flat quasi finite, but since you're allowed to shrink, you can assume finite plot. And then you can use, let us say you prime over you and then you can do you prime over you prime over you. So you will find that your kind of mysterious sheet is inside the direct image from your prime of such which is really a vector bundle. That is the kernel of a double arrow. So finally, you describe it in terms of vector bundles and kernel of maps. So you don't need this given locally. It's the same as a billion sheaves given by vector bundle. Then you perform operations like kernels. Yes. Okay. Okay. Okay. Thank you very much. Yes. Okay. Thank great. Thank you. Also, this probably one is the algebraic space of finite type. Yes, a finite type. Yeah. Are you asking about the finite type part? Yeah. Yeah. Yeah. Yeah. Great. Yeah. Yeah. Okay. So I'm fixing an n and let's assume that the lower comologies are representable and flat over s for i less than n. Then r i f lower star g r n, sorry, is an algebraic space. So in this case, we can get resolved over the whole s, but it requires additional assumptions. So this height one condition and that the lower comology groups are flat, which is something you might expect from the theory of coherent sheaves. Okay. All right. So let me make a few remarks. So one, of course, if g is a tau, tau or prime to p, so remarks on theorem one, if the group scheme is a tau prime to p, this is of course, s g a four and a half there. And if you take g x to be z mod p, well, here, you can consider probanius one minus f g a. And again, let me label that sequence two, because I'll use it again later. And so in this case, again, you can get the result from coherent comology. And again, and actually Arden and Milne use this to sort of get a similar sequence if the Cartier dual of g has height less than or equal to one. So in these cases, one can do things with different techniques than the ones I'll talk about in a moment. And I also just want to mention that we have drawn significant technical tools and inspiration from the work of Cessna Vissius and Scholze on purity for flat comology, which is just a paper filled with beautiful ideas. And so I just want to mention that, that that plays the their workplace a significant role. And in particular, there's the systematic use of what they call animated rings. So it seems necessary in our context at least to work in a sort of derived setting, which I'll explain. Okay, and so the key ingredients that I'll talk about today. So it's sort of longer project. But let me, the two things I want to emphasize is one, the calculation of comology of finite flat abelian group schemes of height one. And two, and I'll have to be a little sketch on this, but I'll explain. Well, some version of compactly supported comology in the flat context. So in the context of finite flat group schemes. So why is that necessary? Well, the reason is that we want to do devisage, devisage to, well, particular group schemes to special cases, special GX, let me put it that way. And so you have to deal with things like a degeneration. So you can have a scheme, or which you have a group scheme that did say degenerate from an alpha, from a Z mod p to an alpha p, or a mu p to an alpha p. So how does one do devisage? Well, if you have a group, if GX is one of these sort of degenerations, and the way we approach it is to try to study a sort of a compactly supported theory, which lets us sort of shrink and go smaller and bigger to do our devisage to a nice group scheme. So I hope this compactly supported theory may be of independent interest. So I'll say a bit about that. Okay. All right. So let me talk about the first part. Comology of finite flat group schemes. All my group schemes will be a billion of height less than or equal to one. And let me start with the smooth case. So let's say X over K is smooth. And if you look in, well, SGA3, there's a reference, there's an equivalence of categories, finite flat, a billion group schemes, G over X of height less than or equal to one. That category is equivalent to pairs, v, rho. So since I'm in an abelian case, there's no the algebra, the bracket. So v is a locally free OX module, and rho is a semi-linear map, fx upper star v to v. Okay. So we can actually, so more is known. So if we have a G, so let's, so I guess by work of Hubler and also work of Arden and Milne. In this case, you can calculate the comology using differentials. So let's say G corresponds to a v, v, rho. And let me let epsilon be the projection to the etalotopos. Then they, in these papers, they show that if I take the comology and one has to put a shift there, this is given by the complex. So you take the push forward of the closed forms, z1 X over K, and then you have this nonlinear map. One is you take rho tensor one, and the other is one tensor, the Cartree operator, which goes to v tensor omega one X over K. So that's a tensor there. Sorry. Okay. So let me just explain. So this is an X here. So I can include the closed forms into omega one. And so that's what this one in this coordinate is referring to. Then I have my map rho, which is a semi-linear map, or I can take it with this way. And then I have the Cartree operator there. And so I can take the difference of those, and that gives a calculation of, so there's a question that's had why assume height less than equal to one. Well, that's the only situation where this, well, I'm not sure I have an answer to that question, but I think you, I mean, you could try to modify this, but that's, I think to have this statement, this equivalence over here, I need to have the height less than or equal to one. Okay. So I have this way of calculating the homology in the smooth case. And so let me just note that you can take this right side and do things more generally. So let me do a construction, which is let's start with just a perfect complex on X, and I can take a map fx upper star V to V. So that's, so I'm just going to let my V be a perfect complex. Then I can consider the functor so I can take schemes over X. And well, let me be a little bit imprecise here. I wanted to take values in complexes of a billion groups, which is just send Y to the homology Y. And then let me take the tensor f Y lower star. I'll take the derived functor of closed forms. And then I can take again, do this row minus C procedure and end up in V and now differentials become the cotender complex. And so what is this? And sort of fancier terminology, this is the left con extension of, well, let me give it a number. If I call this here three, so I specified this functor on smooth schemes, or in particular, like in the theory of the cotender complex polynomial rings, say over X, and you can do what's called left con extension, do some simple resolution, and you get this functor here. So you can derive the right side. Okay. And here's a theorem. The theorem, let me say it in words. So, and then now, so let's say G is finite flat billion height less than or equal to one on X. And let's say it corresponds to V row. Then the homology, and let's say we have some Y to X, not necessarily smooth. Then the homology of YG shifted by one is isomorphic to Y. And then this, let me just write it again, f lower star, LZ1Y row minus C, the tensor LY. So in this height less than or equal to one case, you can really compute the homology using the cotender complex. What is this saying? It's really the statement that this side is again obtained by doing this con extension business, or more concretely taking some partial resolutions as in the theory of the cotender complex. Let me also remark. So when you take some partial resolution of the ring, the point of the group scheme is defined only on your initial ring. So you cover it by something nicer. The group doesn't extend, so it's not clear how to do this procedure. So that's why I'm starting. So X, I'm starting with X smooth, and then I'm looking at Y over X. So I should really work relative. So let's say X was affine, so I have a smooth algebra, and then I can take polynomials. I can resolve Y by smooth algebra. Polynomial rings over the coordinate ring of X. Because I need the group scheme, as you say. So remarks for G equal to mu P. This is a result I learned from BOT. I believe it's due to BOT and Luri, and I understand that it had also been observed independently by Scholze. So let's see. I don't think I have time to talk about it today, but it is interesting. One thing that is important for us, maybe I'll just make a few remarks, is once you describe the homology of the group scheme by this two-term complex, it is very natural to start thinking about the additional structure you get from this representation of the homology. That's one of the key ingredients that ties into our sort of study of jewellery theory for the homology groups, which is sort of to use. This gives you additional structure, or lets you see additional structure on the homology. Maybe I'll leave it at that. Okay, so let me say a word about a compactly supported homology. So again, we're viewing this homology as necessarily involving a mixture of coherent sheaves and group schemes. So let me begin by mentioning very briefly the theory for coherent sheaves, which I believe is first deline in the appendix to residues and duality, and then Hartzhorn has another article about this. So what's the setup? Let's start with x finite type and proper over our field over k, and we have some open u and some other z, the complement, and we start with a coherent sheave here, coherent on u. So then I guess what we should do is you choose an extension f to x of this starting sheave and then define the compactly supported homology u, f, u to be, well, the cocoon of the map from the homology of f to the homology of the formal completion of x along z. Okay, and then in these works, various nice properties are discussed. Okay, so what about, so now we're interested in sort of doing a similar thing, except now we have this g, let's say is a finite flat, a billion group scheme over x. And so, well, we have to, this is where I don't want to go into too much detail, but this is where we have to use a bit of derived geometry. What do I want to do? So first, let me define, let me write hg. So we're working on a big site, so we'll go from schemes over x to some category of complexes of abelian groups. I'll send a t over x goes to the homology of t is f p p f homology with coefficients in g. And then I can also think about the nth infinitesimal neighborhood of z in x. So I have I start with my z over here. I can look at each of the neighborhoods. And let me write i n lower star g, oops, where is that? h sub g n, this will be the functor from schemes over x up. So I want to send this to, I'll write it this way. So basically, I want to take the derived tensor product of the rings here with coefficients in this group g n, the restriction of g to z n. So there's, of course, an issue about how to make sense of this, and I found the exposition in this paper of Cessna Vissius and Schultz to be really clear. So if anyone sort of wants more details, I mean, there are other references, of course, too, but I think that that was a really nice place to read. And so we can think about this and then we can take, let me write h g hat. Well, now I have to, as in Deline, go to some pro categories. So I'll just write whole limb over n of these i n lower star h g n. So we take, this place is going to play the role of this formal completion here. And so we take the limit of these functors. This lives in some pro category of certain sheaves. I'll write it this way just to have it written, but I don't want to emphasize it too much is this she sort of category of sheaves on some class of animated rings in the language of that says a and i in the language of Cessna Vissius Schultz. Okay. And then what we can then do is think about and I'll write this reflecting the starting x and z. So let me write it this way. The compactly supported homology of G will write, consider as the cocoon of h g of x to h g hat of x. And so, but there's also a relative version, which is if you have a morphism, let's say proper, we can talk about RF lower streak x z g. So defined analogously. So I think, you know, so one can make the definition. But sort of one of the key results is that so let me say it sort of loosely. We show that this has nice properties, or at least as nice properties as one might hope for. We show that if you do this, divide factor co homology, it lies in a certain pro category, generated by, I'll say, perfect complexes. And finite, flat, billion group schemes, at least after shrinking in the setup I had before. So if we have a proper morphism, or projected morphism with S reduced, then you can shrink on S to make this object lie in some category that you build out of perfect complexes and finite, flat, billion group scheme. So it's a little bit awkward to say more precisely because one has to deal with the pro category and these kind of higher categorical, this sort of infinity category thing. So I don't think I want to say it much more precisely than that. But in particular, what we show is that it has reasonable properties and it depends essentially only on the restriction of G to you. So that's kind of the key property. So if you go back to the origin of the work of Deline and Hartzhorn, the first thing to ask is, well, you choose this extension here. And of course, you want it to not depend on the choice of the extension, right. And so that gives you some freedom. And so in our situation, the analogous statement is that if you have a sort of G and a G prime, which isomorphic over U by some map over X, then in fact, this sort of compactly supported cosmology is the same. And so it's sort of an analogous statement to this extension property. Okay, so that's what I want to say about that. Let me just conclude with sort of, is it in fact sort of you or that's not cool. You said something seems to be weaker than that. It's a little bit weaker. Because you see, I mean, one problem we deal with is that we have to sort of work around is over here. You can always choose this extension just by general properties of coherent sheaves. But in our situation, if I just give you a finite flat group scheme over U, it's not so clear how to extend it always to any X, you can do some alteration or something like that to extend. But so we have to do something, we do something a little bit weaker, namely, here I really start with the G over X. And then but and that's enough for sort of the applications we have in mind. But yeah, so there's more work to develop this theory. I mean, but in our setup, we start with the G over X. And then, yeah, I guess that's all I have to say about that. I mean, that's also why I sort of make it part of the notation here. In the end, it would be good to develop that theory better, I think. But yeah, thank you. Okay, so let me, I think I have five minutes. Okay, so let me just say sort of again, actually related to that question, which is some of the aspects are a little still a bit clumsy. And so I mean, what what we don't have. And so I'll say towards some kind of DB CTF. Right. So I mean, what what we're sort of moving around is that, well, we have to shrink because we only can say good properties over some open and so on. And so I think that to me is reminiscent of sort of the problems when you talk about Lee sheaves and so on. So so what can we say, sort of at a derived category level. So let me say, so if I have a scheme, let's say a finite type over K. What sort of the the categories we can think about. Well, again, inside here, there's I can write, let me write this DFX. Let me again, use sort of brackets. So what I mean is you have to take the triangulate thick, triangulated subcategory generated by certain objects. And so the objects I want to consider here are I lower star in the notation before you start with G on some closed subscheme. So we want to throw in by, you know, push forwards in this derived sense of finite flight group schemes on closed subscheme. And also sheaves defined by perfect complexes. And let me write here D and am concerned that maybe I'm saying something redundant, but I guess that's would be good. So here, again, I'll worry about this local, just I want to put locally, perfect. So maybe this is, in other words, I just, I want to only assume complexes of abelian sheaves which are locally given by perfect complexes. Maybe this is redundant. So but anyway, so let me still do that to be careful. And then what's the theorem that we actually prove? So let's start if I have f from x to s as in theorem one. So projective as reduced. Then for f in this D f of x, there exists dance open u and s such that if I take this our floor star, f stricter to you, this lies in D lf of u. So it's not quite a stability property of the sort of like the BCTF, but it's the best we can do at this time. Okay, thank you very much. Thank you very much. Are there any questions? So I have several questions, but I don't know where to begin. Maybe the mu p is related to omega one log. So height at most long, so if you take me p and then of course it's related to W n omega one log. And so there is this whole theory of logarithmic there are big sheaves and so it should enter there some place. So this construction you mentioned is equivalent to a category should have some variance of higher level, I think. And so in the, of course, if you can extend, like it is such a good to say today, then you can also replace the one beat by the right one beat and you can construct and perform some construction. But I don't know if you have looked into that. So that's my first question. The second thing is an observation that I think what you are looking for at the end in section four, already you can find some sort of a first idea of this question in this expose by Grotendieck on christening corology, knows by Kultz and Tuxila. So he looks for such a category of the, including the vector boundaries and flat goals. And maybe also including also a billion schemes maybe because you see you're the bt's which are kernel. So they should come also. So what kind of complexes do you get? And I think the motivation for Grotendieck was to find suitable category stable under some suitable duality. So what kind of duality is here and can you get in the question to the long term project? Yes. Yeah. So I, well, first of all, thank you for that. I guess regarding the first question, if I understand correctly, and this is something Bargav has also mentioned to me, I think if we want to talk about new P to the end or something like that, that then we should be looking at the wrong bit. And I think there should be a generalization that we haven't worked it out. So it's definitely Or maybe F gauges, F gauges, right? In the sense of Fontaine Johnson, maybe. Yes. Yes. I mean, I, yes. So you mean so to go from what goes here? Yes, this gets really sort of due to an a theory proper, I think. But I would expect, I mean, I'm hopeful that that should be possible, but we just haven't done it. But that's right. And then here now, let's see, I'm not. I'd have to think more about whether the logarithmic sheaves enter in. I mean, here, I guess for Mu P, I guess the only logarithm I see is to, the way you get this isomorphism is by taking D log from the Kuhmer sequence. And then that sort of connects the two. But okay, I don't have anything intelligent to say about that. So maybe I'll, I'll think more about it. Thank you. For the second question. So I haven't looked at that, but I guess I, so we don't have a, right now we don't know about duality. I'm not sure I see where the abelian schemes enter in. Because the finite code schemes are sometimes a resolution by abelian scheme. So it's sort of abelian schemes. And for example, the older stuff, and we have to do, going missing, then this was important to look first. So you consider abelian scheme and then finite. Right. So yeah, so that's a little bit of a different direction, actually. So one thing that confused us for a very long time is we were trying to do exactly working with resolutions like this to get, to understand the homology. And it seemed difficult for us in general, if you know, if you have some arbitrary finite flat abelian group scheme over some scheme X. So maybe we just missed something, but we kind of got stuck trying to work with resolutions in that setting. So I mean, I think maybe another way to say what our concern was, like if you think about GM, and you think about the homology of GM, like R2F lower star, and you think about the formal Brouwer group, I think it's possible to make examples where the order of an element grows as you go up meal potent thickenings. And so that is something we want to avoid, because that's going to imply non-representability if you sort of keep. So for us, it's quite important to fix. In the study of mu P say to not let the do mu P to the end and sort of let N grow. I'm not sure of that. So I'll have to, I haven't looked at, I'll look at the reference, but yeah, there's much work in that area. I don't have much to add. Thank you. Relic questions? So you mentioned that for duals of high ground groups, there is a similar description of homology. So is it, I don't remember, so this should generalize the case of Z mod P. So this works for duals of flat high ground groups on X. Is it, what is the remark on duals that you made? Oh, okay. So let's see if I can remember. So I mean, so you start with, let's see. So basically you start with, you want to take harm, let's see. So you have, you take GM, you take D log, which goes to F lower star Z1 X. And then you take 1 minus C to omega 1 X. And then here you have GM, you have multiplication by P. So I think, let's see, you do this and you then take harm from the Cartier dual of your group scheme into this, mapping into this sequence. And then, let's see, I hope I'm, this is in this paper of art and Milne, for example, in Hoobler. And so if you take this, if you do this, then here you get G and G. And then you get, then you analyze this and see that you really get the Li-algebra tensor, this F lower star Z1 and V tensor omega 1 here. As you get a similar complex that allows you to get the comb. Yes. And so then you can, and then they check that that gives you a sort of a sequence with the G now and G and something like, let's see, I hope I'm not, I may be messing this up. So it's in this, there, so I probably didn't say it correctly, but it's something like that, that you use the sequence for GM and then you harm into it with the Cartier dual to understand the G. Okay. So this in this other paper is probably in the six, seven days though? Yes, yes, yes. Okay. I think it's to add in the flat homology of curves, I believe. Any final questions? Could you go back to theorem one just for a moment? Right. So this theorem is described in terms of RIF lower star, which is good for old fashioned people like me, maybe not so good for a young hawk shot like Luke Edizee. And I'm wondering if maybe there's a way to reformulate it using the language of n stacks that you described in the talk you gave here at Berkeley a while ago. And I just sort I think I understood you're talking for a little while. Would that be a suitable language here for this? Well, I think that's so first of all, well, thanks for asking that. I think so I think the reformulation of this here is what I was doing at the end. That's right. Let me just pull it up here. Right. So it's a stability property. Now this theorem two that I mentioned is I think the one you're referring to. So there's a different argument in this case where so I mean the proof of theorem two here is a little bit different. I mean what one can show what we show is that this if you just take the derived homology r flower star g x some kind of complex that that actually gives some kind of no sort of algebraic n stack or geometric n stack in the language of Simpson. And so sort of the whole complex like if you have a you know the commodity of a perfect complex is better behaved than the sort of individual homology. So that's the reflection here that it's some kind of a nice higher stack or that a nice complex. And then you have sort of special things you have to deal with to pass to individual homology groups. Yeah, yeah, that's that's the idea there. So there's a stronger statement than theorem two. Thank you. Okay, thank you.