 Thank you very much for the introduction, Ofer. And thanks a lot for the invitation to speak here. It's a great pleasure and in fact, a great honor for me to be speaking at this conference to Luke. So Luke's work has influenced a lot of my own work throughout my career. And in fact, my very first kind of foray into piatic harsh theory was about piatic dry dromcomology, which is essentially an object that Luke invented in his thesis. And over the years, he's sort of been extremely kind and encouraging and generous with sharing his ideas. And sometimes attributing things to me that not necessarily were due to me. And in fact, even during the pandemic, he's been, we've had sort of very nice email exchanges where he would check in and then we would discuss not only mathematics, but also he would ask about the state of the virus in Michigan and he would be slightly worried about the state here while I'm much more worried about what's going on in Paris. So it's been really nice. Thank you, Luke. Okay, so today I wanna talk about the absolute prismatic site. So what I'm discussing is joint work. It's a pair of papers joined in progress, one with Jacob Lurie and one with Peter Schulte. And it will also cover some work of Drinfeld which was done independently and it's already on the archive. And I'll be more precise when I get to the objects. Yeah, so the absolute prismatic site, what is it about? So this prismatic theory was invented a few years ago in my work with Peter. And it was sort of, it's been quite useful in understanding geometric comology theories. So in situations like you have a smooth proper variety say over OCP and you wanna compare various comology theories attached to it. And this prismatic theory gives you a tool for doing that. What I wanna discuss today is the absolute version of the theory. So this is similar to the distinction between a telecomology of algebraic varieties or an algebraic postfield versus absolute telecomology. So something that talks to Galois homology. So that's the plan. Okay, so let me begin by just recalling very quickly what the notion of a prism is. I suspect this showed up in each house talk but I couldn't attend it and the video wasn't up yesterday. So I'm not 100% sure what happened. I just need to, well, I'll give the definition anyways. So a prism is a pair, a comma i where sort of a is a commuter ring with the delta structure. And so a delta structure is essentially the same thing as a derived lift of the Frobenius. So if the ring is p-torsion tree, it's literally just the same thing as a lift of the Frobenius mod p. And if it's not p-torsion tree, then it's slightly more information. It sort of remembers why this given endomorphism is a lift of the Frobenius rather than just the fact that it is. And i is a ideal and a, which gives a Cartier divisor. So it's locally generated by a non-zero divisor. And they're supposed to interact in some way. And so the interaction is the following. Whoops. Such that there are two conditions that's satisfied. So the first one is kind of a mild completeness condition. So it says that a is, I'll put in brackets derived, p comma i complete. So p and all elements of i are topologically no potent. And this derived notion of completeness is a weakening of the usual notion of completeness. So it's just more useful in working with non-Materian objects. And the crucial condition is the prismatic condition. So it says that any local generator of the ideal satisfies the following condition. So if you hit this generator with the Frobenius that is part of the Delta structure, then you know that it looks like d to the p plus p times something because it's a lift to the Frobenius mod p, but you require that the something is a unit. And so if you like, this is saying that the first derivative of d in the p direction is a unit. And this definition, it doesn't literally pass because I've said things in terms of local generators, but maybe a more sort of global way of saying this is that p belongs to the ideal generated by i and pi of i. So this is kind of a purely intrinsic way of saying if it doesn't require choosing a generator, but in all in practice, this is sort of the useful way of thinking about it. Okay, so these are prisms. And then there are some examples that I should mention. I will not spend too much time on this. So the conditions should be a priority in terms of Delta of d being a unit. Right, the condition is that Delta of d is a unit, but that's equivalent to saying. When you write the p is in i plus pi, you have still to divide by p to get the Delta is a unit. So in general p is a zero divisor. No, it doesn't matter. The condition I wrote is equivalent to the condition that Delta of d is a unit. Even if p is zero divisor. Okay. It's a topological condition. So somewhat checking something as a unit can be checked after modding out by topologically null potent things. And that's good enough to prove this equivalent. Okay, so the examples are the following. So the most basic example of a, so elements d with this property are called distinguished elements. And so the most basic example of a distinguished element is p, the prime number p itself. So let's say a is the bit vectors of k where k is a perfect field of characteristic p or perfect ring actually. And the ideal is generated by p. And in this case, I don't need to specify the Delta structure because there's a unique Delta structure. The bit vectors of a perfect field of characteristic p has a unique lift to the Frobenius. And it's easy to see that. Okay, so if I take this definition of what a prism is then this condition is obvious that the ideal p satisfies this. But also with this definition, it's kind of clear because if you hit p with Frobenius, here you get a p, but here you'll get a p to the p. So the difference is p times a unit. And so these are crystalline prisms because they correspond in this prismatic theory to crystalline homology. And I think Arthur talked about these in his talk. At sort of the opposite extreme, there are these so-called Brøy-Kissen prisms which we'll show up in my talk later today. So these showed up in the classification of crystalline Galar representations and I will just sort of write down one example. So take a to be zp power series u. So one variable, a formal power series over zp. Phi is uniquely specified by what it does to u and I just send u to u to the p. And then the ideal is gonna be the ideal generated by Eisenstein polynomial. So an example would be something like u minus p or u to the p minus p. But of course you can do any other Eisenstein polynomial. So in these cases, a mod i is the ring of integers of a local field. And that's gonna be how these will show up for us later. I'm not gonna check this condition here, the prismatic condition. And maybe I should point out so the difference between these two examples is the following. In example one, p equals i. Whereas in example two, p comma i form a regular sequence of length two. And so I'll try to come back to this later. This somehow corresponds to a certain flatness condition on a morphism. And then maybe the third example I should mention is the example of perfect prisms. So here I will just formulate it as an equivalence. So if I work with prisms a comma i such that a mod i is perfect, a mod p is perfect. These actually turn out to be equivalent to perfectoid rings. And the functor is just a comma i goes to a mod i. So in this special case for i equals p this is the equivalence between perfect rings of characteristic p and I think what are called strict p rings. But it's a general statement about all perfectoid rings. And so the example actually that will be relevant for us is that OCP, the ring of integers of the completion of the algebraic closure of qp on this side is a perfectoid ring. And that corresponds via this equivalence to Fontaine's aint construction. Again, then there are a few other examples and one could also mention like the cute rom prism and so on, but let me not write them down. So these are what prisms are. And in earlier work, we studied this relative prismatic theory which where you fix a base prism a comma i that you work over and you have some geometric object that lives over a mod i and then you try to sort of probe it using further prisms. And what I wanna discuss today is the absolute version so I don't fix a base. So here's the definition that we wanna study. Oops, sorry about that. So the setup is gonna be as follows. So X is gonna be a periodic formal scheme. In fact, for most of my talk, it'll end up being something like Zp or the ring of integers of a local field. And then it's absolute prismatic site which I will denote by X prism is the following object. So it's the category of all pairs a comma i consisting of prisms together with the map from spoof of a mod i into X. So it's a prism over X is what I would refer to this object as. And I say spoof of a mod i this means I give it the P addict topology. In general, if you're working over a prism a comma i you always give it the P comma i addict topology. And so that determines topologies in the quotients as well. And so this is what I call the absolute prismatic site of X. And we wanna try and understand it better in this talk. So I guess so far I've only told you what the category is. I didn't tell you what the topology is. So it's topologized why I say there's a risky topology on a mod i. So for what I'm gonna talk about the topology does plays no role. So you can replace the risky with your favorite kind of topology and prisms. So like the etel topology or even the flat topology which is usually making various arguments. And there are certain obvious sheaves on this which I wanna give names to. So if I take a comma i an object of the site I can send it to the following. I can either send it to a let me write it like this. Can send it to a I can send it to i and send it to a mod i. And all of these guys give you sheaves. So this one is called oprism. The first one. Next one is called i prism. And the last one is called oprism bar which is also just oprism mod i prism. So do you do you want to use the boundedness condition? Sometimes I should probably impose boundedness here. Thank you. Otherwise you have to derive things more. Okay. So for example, right? Like with this definition if I just take x equals zp then this is like this part of the data is just extraneous. And so this is literally just the category of all prisms. Bounded prisms. Okay. And so the goal of this talk is the following. And so there are two goals. A brief question. Yep. In earlier manuscripts you say that i is in all cases known to be principal. Is that to be true in this context as well or not? So I don't think it's literally true like there are examples where it's not true but they'd never come up. Okay. That's really what I meant. Yeah. In that case, the situation with A and I is not that important. Those two sheaves. No, no, it's very important because there's a Frobenius that I guess there's a Frobenius acting on this guy. Okay. Yeah. But also there's even if there is a generator there's no canonical generator. So I don't think like you can say as like sheaves they are isomorphic. Okay. So the goal is the following. So first I want to describe quasi-coherent sheaves or really crystals, I guess. Crystals of quasi-coherent sheaves on the site X-prism. Well, actually I'll just do the ZP case today. So suppose ZP prism or prism. And so this is the part that's joined with Jacob and also independently due to Drinfeld. So if you're going to do this via a Stachy approach and the Stachy approach tends to clarify certain questions that are sort of hard to understand otherwise. So like a natural question you might ask is, what is the comological dimension of the absolute prismatic side of the scheme? For example, even for ZP itself. And this is not clear from the definitions but I will explain why it's one using the Stachy approach later in this talk. And so this part of the talk will be slightly sort of derived in the sense that there are no conditions I'm imposing on the sheaves. In fact, everything I say will work in the derived category. So you'll get sort of a reasonable understanding of the derived category of quasi-coherent sheaves or crystals of quasi-coherent sheaves. And then in the second part of the talk I want to turn on the Frobenius. So I want to talk about describing F crystals. So crystals with a Frobenius structure. But this will only be a vector bundles. And this will be in the case of stuff okay prism where K over QP is a finite extension. And so this is the part that's joined with Schultz here. We obtain a description in terms of crystalline Galar representations. And so our motivation for this was that it's not so clear what a good notion of a mod P crystalline Galar representation is. But if you have a description through the site it gives you at least a candidate definition. And so that's the plan of the talk. Are there any questions so far? So I apologize, I'm not going to be able to follow the chat in real time. So there's a question in the chat or in the Q and A, someone please ask me. Okay, so let me discuss the absolute prismatics like why are stacks, oops. And so where did the stacks come in? So let me sort of first motivate it verbally. So over the complex numbers you can ask, there's a similar picture. So given any algebraic variety over the complex numbers yet you can talk about a steromomology or you can talk about the notion of vector bundles with black connection which are sort of coefficients objects for the romcomology. And then you can ask the question whether or not there is another geometric object associated to X with the property that vector bundles with black connection on X are just vector bundles on this other geometric object. And the romcomology of X is just O-comology of this other object. It turns out the answer is yes. So Carlisle Simpson has this notion of the rom space attached to any variety and it satisfies these two properties. So it allows you to translate questions of romcomology and coefficients to just questions of vector bundles and their homology, but on some more exotic object. And that's the kind of thing. Why does the infinitesimal site itself do that? Well, the infinitesimal site is not a geometric object in the sense that it's not like a variety or a stack. I want like a variety or a stack to which I can apply methods of algebraic geometry. Not a stack? Well, I mean, it's a site. I want like something that's a functor from rings to sets or whatever. I'll explain better. Okay, so what I'm really looking for is some kind of a geometric object. So like an algebraic stack or something close enough to an algebraic stack with the property that quasi-coherent sheaves on the stack correspond to crystals of quasi-coherent sheaves on my prismatic site. And it turns out the following definition works. So Drinfeld arrived at this independently. So maybe I guess I should point. So this is joined with Luri, but also independently by Drinfeld. And our motivations were somewhat different. So I think for Jacob and I, at least for me, one of the motivations for finding the stackier approach was to just better understand questions of absolute prismatic homology, like the one I mentioned before, what is the homological dimension? And so here's what works. So we call this the Cartier-Witt stack. So it's gonna be a formal stack or, so w-cart is our notation for it. It's a formal stack. So it's a functor from P and O-potent rings to group points. If you like it's a category of fiber in group points and I'm just gonna write it down as a functor. And I'll just tell you what the definition is. So w-cart evaluated on a P-nil-potent ring R is, so I'll first sort of say how you think about it and then give you a formal definition. So it's a derived prism structure on W of R. And so what the derived part here means that if you go back to the definition of a prism over here, I had this condition that I was an ideal of A. So that's like saying it's an effective Cartier-Divisor, but in classical algebraic geometry there's this weakening of the notion of an effective Cartier-Divisor, which is a virtual Cartier-Divisor. So it's just, it's not gonna be an inclusion anymore. It's just a map from an invertible module to A. And that's the kind of thing we are demanding over here. And so one can say it extremely concretely. And so let me try to do that. So very explicitly what this means is the following. Okay. So opinion important ring is a non-unital ring or it is a unital ring? Unital. So just some power of P is zero. Ah, okay. I thought you meant the, okay. Something like this formal group. So, okay. No, no, no. Just like some power of P is zero in my ring. So it says it's gonna be like a periodic formal stack. Okay. Yeah. So I'm gonna give an explicit description of this stack as a quotient. So the numerator is gonna be the function of distinguished elements. So W card zero is the set of all vid vectors. A zero, A one, dot, dot, dot in W of R, the two conditions. So A zero is no potent and A one is a unit. So this is exactly what a distinguished element in the vid vectors looks like. So if you like, this is the set of all distinguished elements in W of R. And then W card is the quotient of W card zero by a group action, by the obvious group action, W star. So this is a explicit formula for what W card looks like. Okay. And so the claim I wanna make is that understanding the stack W card is pretty close to understanding the absolute prismatic side of ZP. And you can translate various notions back and forth. And so I'd like to explain that next. So let's try to understand the geometry of the stack slightly better. So let me first tell you what the points are. So is there a topology here? Yes, I'm implicitly actually using the flat topology, right? Thank you. Okay. And this quotient means in the sense of sheaves. Mm-hmm. Okay. Well, right. As a group, it's a quotient of an infinite dimensional affine scheme by this group action. Or infinite dimensional, a formal affine scheme by group action. Right. So let's first understand the points of the stack. And so it's a stack on P and L-potent ring. So the physical stack underlying it just lives in characteristic P. So I just need to understand its functural points on rings of characteristic P and really just the points there. So I just need to evaluate it on fields of characteristic P and I might as well take it to be perfect. And so there it turns out it has only one point. So for any perfect field K, of characteristic P, this group point W-card of K is reduced to a one element set. So that element is given by the prism structure given by a crystalline prism. Or if you like is the distinguished element P in the bed factors. And the claim is that that's the unique point and it has no automorphisms. So as a, the underlying physical point set of the stack is just one point. And so if it was an actual algebraic stack, then you would know at this point that it's the classifying stack of its automorphism group at this point. But it's not an algebraic stack. It's only a formal stack. And so it's not actually the classifying stack of any group. I mean, this phenomenon already shows up for something like A1 hat. So if you take A1 and complete it at the origin, it's a formal stack with only one point, but obviously it's not the classifying stack of a group. But it's not that far. So let me just say over here, therefore only one physical point. So I'll come back to this, how far it is from being a one point stack in a moment. But I wanna first explain the connection to prisms. So how does it connect to a prism structure? I mean, it's sort of clear that it should connect because it has to do with dried prism structures, but you can make it very explicit. So given any prism, A comma I, there's a natural map. So I'll call it row sub A comma I from spoof of A to W card. And I'll just say verbally how you construct this map. It's basically because the ring A has a delta structure. So there's a natural map from A to its width vectors. And so using that map from A to its width vectors, you can take this prism structure on A and move it over to get a prism structure on W of A. And that by definition of points of the stack gives you such a map. And so every prism has a natural map to W card. And this map, the collection of all of these maps approximates W card in a really nice way. So here's a sort of fact, whoops, which is if you try to understand quasi-coherent sheaves on W card, first of all, they map to the inverse limit of, so for each prism A comma I, I can pull back along this map, row AI and get an object in the derived category of A. The objects I get because of the way my topology setup are gonna be P comma I complete. So maybe I'll write that properly. D P comma I complete objects on A. And it turns out that this is actually an equal ones. So understanding quasi-coherent sheaves on the stack is the same as understanding crystals of quasi-coherent sheaves on the absolute prismatic side of ZP. So I mean, I would define this to be the crystals quasi-coherent of ZP prism, O prism. So I believe the limit in some kind of homotopy. Yeah, this is happening in infinity categories. Thanks, yeah. Okay, but then what is the actual definition of DQC? It doesn't have a definition direct, not with infinity category, just as a usual. So can you say, can you say what is DQC or does it require something very? Well, I mean, I'll say it in infinity categories, but I think you can just give a direct definition. In infinity categories, whenever you have this functor, it's quasi-coherent sheaves on the functor are going to be the inverse limit of quasi-coherent sheaves over all maps from spec R into the functor. So it's the inverse limit of R modules for every map from spec R into the functor. And then the assertion I'm making here is that a much smaller collection of maps suffices to compute that inverse limit. Because you want, okay. Right, like if you have a stack, you can define, you can take the definition, just take a variety, you can take the definition I give. You look at all the maps from affine schemes into your variety. You look at quasi-coherent sheaves on the affine variety and then you take that ginormous inverse limit. That's the definition. And then Zariski Descent for quasi-coherent sheaves will tell you that you can actually compute it using like a single Zariski cover rather than this kind of ginormous inverse limit. Okay, so if you just take the site of affine objects mapping to the stack with the suitable topology like Zariski et al. or something, and then you consider just sheaves over a ring side in the old sense result. And consider the unbounded, I suppose the right category of this. Is this exactly what you consider or there is some? Yes. Something like this. No, I think that's exactly what I consider. Okay, so you can just say it is, okay. Yeah. And so, right. So what this tells you is that if you want to specify, well, there's two points. So one is that I wanted to understand crystals geometrically in terms of quasi-coherent sheaves on some single object and this accomplishes that. But what it also tells you is that if you want to produce quasi-coherent sheaves over here there's a recipe for doing that, which is I need to produce for every prism A comma I an object of this dry category of A with suitable completeness. And this assignment should be compatible with base change. And so we'll see that geometry over spec ZP will give you like every time you have a geometric object with spec ZPs it's comology will give you an example of such a, such a sheep. So I'll come back to that in a moment, but I'm gonna first describe more about the structure of the stack. So there's an interesting divisor in the stack, which is one of the main tools for understanding it. So this is the hot state stack. So in relative prismatic homology passing to, if you want to understand relative prismatic homology understanding hot state homology, which is what happens when you mod out by the ideal is like one of the most important tools. And something similar happens in this kind of stacky context. So the basic point is that you can sort of taking the zero component. So I guess let me just say it this way. We have a fiber diagram, which looks like so. So here's W cart. I claim that it has a natural map to a one hat mod GM. And this is simply by taking the zero component of the bit vector in the presentation that I gave earlier. But my condition, my definition of W cart over here had this, I had this quotient description. And in the numerator, the condition on the bit vector was that the zero component is nil potent. And so it take forgetting everything else gives me a map to a one hat. And then when they're quotient out by the group actions, I get a map to a one hat mod GM. And inside here, I have this natural divisor, which is given by the origin, which is BGM. So I'll call it zero. And the fiber product is defined to be the hash state locus. So this is the definition. This is the hash state stack. And so I said earlier that W cart was not an algebraic stack because it has this formal direction coming from the, exactly this zero component being only top logically nil potent. But when you pass to BGM, you sort of, you got rid of this formal direction. And so now it has a good chance of being an algebraic stack or pediatric formal stack. So there's no I addict topology anymore. And that turns out to actually be true. In fact, we can describe this stack super explicitly. It's just the classifying stack of a certain group. And so let me record this as a llama. So there's a natural map. So the map from spouse ZP to W cart hash state, there's a natural map like this given by the bit factor V of one. So V of one is a distinguished element in the bit vectors of ZP. And so it gives me a map from spouse ZP to W cart. And V of one by definition has the property that the zero component is zero because it's the perceiving of something else. And so really it's factoring through the hot state locus because this condition over here is imposing the condition that the zero component is zero. And so it gives you a map like so. And it turns out that this map is surjective and the automorphism group is understandable. So induces an equivalence between the classifying stack of the Frobenius kernel on W star and W star hot state. W cart hot state. So the hot state locus has this very relatively simple geometric description. It's the classifying stack of this group scheme which is the kernel on Frobenius on the multiplicative bit vectors. And so I wanted to prove it but given how I'm doing on time, maybe I'm not going to prove this. It's a pleasant exercise in first, so the surjectivity part of this assertion is essentially after you unwind all the definitions that's the fact that Frobenius in the bit vectors is a surjective map of functors in the flat topology. And then understanding what the automorphism group is is also an extremely simple calculation with bit vectors. So I can give it later if someone is interested but let me avoid doing that now. Okay, so what's the upshot of all of this? The upshot of all of this is that, okay, you have this W card which is fibered over A1 hat mod GM over this kind of divisor zero. I get the hash state stack and the hash state stack is this classifying stack. The reason this is useful is that quasi-coherent sheaves on this classifying stack are extremely easy to understand. So the group scheme is slightly weird but sort of it's Cartier dual is nice. And that means that quasi-coherent sheaves on the classifying stack are also nice. And so here's kind of a concrete corollary you get out of this, which is a connection. So I'll call this Senn theory because it's a connection to the Senn operator in piatic hash theory. So we get the following explicit description of what quasi-coherent sheaves look like. So quasi-coherent sheaves on W card, hash state identifies with what we call Senn complexes which are just representations of this group scheme but we can make it super explicit. So it's the following object. It's all objects E and the P complete derived category of ZP join a formal variable. So I'll call the formal variable theta. Theta is the Senn operator and these objects have to satisfy a condition. And so the condition is that the theta action on E mod P has generalized eigenvalues in the ground field. So I'm asserting that there is a generalized eigen space decomposition and all those generalized eigenvalues is just concentrated at the point of Fp sitting inside A1 of Fp. Okay, and so this is a pretty reasonable linear algebra condition. And so therefore this category is pretty easy to understand. In particular, you can understand things like X in this category. And can I ask a question? Maybe this is not quite the right time but you missed chance talk but it seems to me these should correspond in some sense to crystals of OBAR modules in your- Yeah, so these are gonna be crystals of OBAR modules on both ZP. Right, but chance talk explained in the relative setting that a crystal OBAR modules correspond to Higgs fields more precisely quasi-nil-potent Higgs fields. The theta looks sort of like a quasi-nil-potent looks like- Yeah, so I like to think of theta as some kind of an arithmetic Higgs field. I mean, if that was non-existence base called F1 then this would be kind of the Higgs field on SPAC ZP over SPAC F1. But the quasi-nil-potent means that this looks more like a quasi-unipotent condition or a quasi-nil-potent condition because the eigenvalues are not zero but an FP. We'll consider that. Well, there could be eigenvalue zero. I mean, it's not quite- Yeah, but they have to be. Nil-potent is slightly bigger than that. For him they have to be. So I'm wondering what's- Water. Okay. No, I think this is kind of the interesting part actually that in the absolute theory you don't see as much unipotence as you see in the relative theory and I'm gonna explain why this buys you a lot. So this has some mileage. Okay. Right, so the reason I call this send theory is that absolute prismatic homology of schemes over ZP produces objects on the stack. And for those objects, this operator exactly is the send operator from Piatic-Hoch theory which gives you the Hoch state decomposition after inverting P. And so I'll try to sort of make this more precise now. Yeah, maybe one thing I should say before I actually say that is just so that if you concentrated this part, this means that any object in this category has a Z mod P grading. I can grade it by the generalized eigenvalues mod P. And so every object in here having a Z mod P grading is actually very easy to understand in terms of this picture because this group scheme has a mu P inside it. W star of F has the kernel of probanus on GM inside it. And so any representation of W star of F in particular gives you a representation of mu P and that corresponds to a Z mod P grading which in this picture is the grading by generalized eigenvalues. Okay, so let me now talk about the absolute prismatic homology and how it gives you, why you can use this description to study it. Okay, so let's say X over ZP is a smooth formal scheme, smooth, Piatic formal scheme of relative dimension D. And so attached to this object, I'm gonna define a positive coherence sheet from the Cartier-Witt stack. So I'm gonna do it as follows. So I'll call it H prism of X. It's the following collection of things. So for every prism, A comma I, I'm gonna define a module. And the module is just a relative prismatic homology of X over that prism. So I take X, I extend scalars from ZP to A mod I, and then I look at this relative prismatic homology over A. And so the collection of all the relative prismatic homology groups attached to X over varying prisms is base change compatible by the hostage state comparison. And so this defines an object of this category that I had before at the inverse limit over all prisms of suitably completed derived category of A. And that I told you earlier was the Cartier-Witt stack. And so this is the prismatic, if you like, this is the push forward of the prismatic structure sheet from the prismatic side of stack attached to X to the prismatic stack attached to ZP. I'm not talking about the relative prismatic stack here. So let me just say it this way. And so this satisfies a bunch of nice properties. So this is the definition. It's not proper. It's not proper. But it is quite a compartment for the separated. Uh-huh. Oh yeah, sorry. Thank you. You want to be the smooth there, okay. Thank you. Okay, so we have the following, this has the following features. So first is that there's an absolute comparison, meaning if I take global sections of the sheet, I just get the absolute prismatic homology of X. So it's factoring the process of taking absolute prismatic homology into two steps. I mean, so if you like, it's just the Loray spectral sequence for the map from X to stack ZP. But this is useful for us because we understand something about the stack, right? So we understand that the hostage state locus, oh yeah, I forgot to say this, right? One thing that follows from this, maybe I said it verbally, but I didn't write from SEM theory is that a W card has homological dimension one, quasi coherent homological dimension one. And so sorry, I should have said this earlier. So the hostage state locus because it's described by this linear algebra conditions, it's easy to see that it has homological dimension one. And a W card is some kind of a defamation of W card hostage state by one parameter. And so this condition of having homological dimension one survives the defamation. And so you can make some limit argument to go from knowing something about the homological dimension of the hostage state locus, to knowing something about the homological dimension of the whole stack. And so going back to this absolute prismatic story, this is very useful because we know that this stack has homological dimension one. And this object is defined using relative prismatic theory. So we understand it pretty well. And so an upshot of this is that the absolute prismatic homology of access dimension at homological dimension at most t plus one. And so this kind of makes precise this intuition that if something had relative dimension over D, it sort of has absolute dimension D plus one over some non-existent base. So yeah. Two x is an affine, those are two D plus one. Oh yeah, thank you, thank you, thank you. X is affine here, yeah. Right, otherwise the coherent homological dimension of x would also have to be accounted for. The other thing I wanted to record is the Durham comparison. And this is just straightforward from the relative theory, but it's necessary to say for what I want to say next. And so I have this quasi-coherent sheaf on this Cauchy-Ewitz stack. I told you that the Cauchy-Ewitz stack has one physical point of characteristic P. And so you can ask what you get when you pull back the sheaf to that physical point. And you get exactly the Durham homology of the special fiber. So pull back along the unique physical point. Actually, this isomorphism absolute comparison, does it hold only in the Japan case or more generally? It's only maybe the colloquial dimension. Yes, yes, the affineness is only relevant for this part. But the first part, yes, okay. Thank you, yeah. The first part holds there generally, I think. Right, so if I pull back along the physical point, what do I get? So xfp upper star of this prismatic complex of x is just the Durham homology of the special fiber. And so here for the experts, I wanna point out that if this is why I chose to work over zp rather than a general unramified base, there would be a Frobenius, there's a Frobenius twist involved in the Durham comparison, which I can ignore if I work over zp. So that's why I did that. Okay, so if you have this sheaf, it's value of the unique physical point is given by Durham homology. But we all can also do something else. We can pull it back to the hodged state locus, which is something we understand now. So we have the following hodged state send comparison. So it says the following. So the object that you get by, I take h prism of x, I restrict it to the hodged state locus. So this is now a quasi-coherent sheaf on the hodged state locus. But this I described earlier in terms of the send complexes. And so you can ask what is the send complex that you get? And it turns out you can describe it pretty nicely. So first of all, ignoring the send theory, in the relative prismatic case, there is this kind of natural filtration on relative hodged state homology called the hodged state, the conjugate filtration. And so you still have that over here. So the object has a natural increasing filtration with good eye given by differential forms, shifted by minus I. So it's an integral lift of the conjugate filtration under arm homology and characteristic P. That's just formally coming out of the relative period. But now you can ask, how does this interact with the send structure? So how does this interact with the send operator? And so it turns out it interacts rather nicely. So moreover, oops, the send operator, theta. I'm sorry, do you have smoothness assumptions here for the assertion C? Yeah, and all of this I'm assuming access smooth over ZP, right? If you don't assume smoothness, you can say something similar with the quotient. And can I ask briefly? Yeah. Oh, okay. Can I ask briefly for B here? No, for A I think for the homological dimension, do you have a sort of whole shift spectral sequence which relates the absurd homology to some group action on the relative homology? It's not quite a group action. So it's like, it's related to the fact that- Oh, something of this kind, yeah. Yeah, so this is not quite a group quotient. And so it's not gonna be a group action, but there is a description of this in terms of group quotients. It just, it's slightly more involved than what I wanna say right now. I can answer this later if you want. So there is like a natural group quotient that maps to W-card, and then you have to modify it along a certain locus. And so that gives you like a spectral sequence to compute what happens over the group quotient and then also then a fiber diagram to compute what happens when you modify. Okay, so thank you very much. I'm sorry for the interpretation. No, no, it's fine, I love the questions. Right, but the point I was making is that the Senn operator theta acts, oops, it acts on gir i by minus i. And this is also again, something that should be familiar from piatic-Hoch theory. Like if everything was torsion-tree, then this is what happens in the Hoch state decomposition. The Senn operator is the operator that picks out the Hoch state decomposition precisely because it's acting on i-forms but weight minus i. And so when you invert P, things split apart completely. But now the nice thing is that everything is happening integrally. Right, I mean, this was an integral statement. And this was also an integral statement. So I can combine B and C. And what happens if you combine B and C is that you get a decomposition for Durand-chromology or rather you get an operator on Durand-chromology. And so here's the corollary of all of this. So this is noticed by Drinfeld. And it says that in the above setup, there is a natural operator theta on the Durand-chromology of the special fiber that preserves the conjugate filtration and acts by minus i on Geri. This is simply because the conjugate filtration is the mod P reduction of the Hoch state filtration. So I have this natural filtration integrally that's preserved by theta. And so I can reduce the picture mod P and still get the conjugate filtration being preserved by theta. And on the other hand, I identified the total complex with the Durand-chromology of the special fiber. And so you get this kind of nice operator that acts on everything. And so in particular, if the dimension is small, so the relative dimension is less than or equal to P minus one, all the eigenvalues that show up are distinct. They don't interact with each other because the integers from zero through P minus one are distinct mod P. We get the delinu z decomposition. But in delinu z you need only to make small p squared. Yeah, I'll come back to this in a moment. Thank you. And also I think you get a grading in any case. Right, you always get a grading. Yes, okay. I was gonna make exactly those two comments. For the section of dimension, you get the z mod P grading. So that's the problem. And also as over said, maybe later you would discuss improvements. Right, so all I wanna say is exactly the things you guys said. So first of all, this tells you that you get a z mod P grading because I can just look at the distinct eigenvalues mod P and that gives me a z mod P grading without any assumptions in the relative dimension. And then the second thing was that as over said in delinu z, you only assume there's a lift mod P squared rather than all the way to ZP. And so you can refine everything I've said to work in that setting. I only talked about the Cartier-Wittstack for ZP, but there's a relative version of the Cartier-Wittstack that makes sense. In particular, there's a Cartier-Wittstack for Z mod P squared. And in the Cartier-Wittstack for Z mod P squared, you still see this group scheme, W star of F showing up as a stabilizer. So you can say all the same words. And so yeah, one thing that I should point out is that something that's still quite mysterious is that how does this operator theta act? So we know that it induces this FP grading, Z mod P grading, but on each individual graded piece, you still have a residual nilpotent operator because I'm not saying that theta is acting semi-simply. And so it would be nice to have examples where this operator is actually non-zero because it's giving you some canonical extra structure on the Dirac homology of varieties and characteristic P with the lift to W2. But we don't have an example yet that the operator is non-zero, except that, so you can prove that it's not always, like it cannot be functorially zero. So that's the best I can do so far. Yeah. Okay, so I think this is all I wanted to say about the absolute prismatic story. I mean, there's a lot more to say. I could have talked about the Nygard filtration and so on, but I think for the purposes of this talk is all I want to say because I want to talk about F crystals. So maybe I can ask, are there any questions about this, what I've said so far? Sorry, I have a question. Is there a version of Cartier stack for schemes over ZP or for more schemes over ZP? Yeah, yeah, there is. And the definition is, I mean, somehow once you understand the Cartier stack for ZP, it's not that hard to do the general case. You just copy the definition that shows up in the prison structure. So let me not give it, but yeah, I think it's the obvious thing you might imagine. Okay, so in that case, so you can also define the cogitated locus, then that would be an analog of the theta, which would interpret the arithmetic same operator and also the geometric operator somehow. Yeah, right? Yeah, and you can describe the situation pretty nicely. So if you have a smooth scheme over ZP, you get this hot state stack for the smooth scheme, it's fibred over the hot state stack for ZP, and then the fiber looks like a jerb for the tangent bundle or rather a jerb for the PD completion of the tangent bundle at the origin. And then the jerb is trivialized exactly when you have some lifting data. So probably over ZP, it's always trivialized, but in general, it's okay. You need some lifting data and that gives you the connection to Higgs field as well. Yeah, thanks. Thank you. Okay, so let me talk about prismatic F crystals in my remaining time. So this part is joined in progress with Peter. And so this was an attempt, so pretty much soon after the prismatic side was invented, we had this question of how does this prismatic story relate to the usual notion of crystalline Galar representations? I mean, somehow the intuition is that the prismatic picture gives you a good integral coefficient theory. And so you want to, on the other hand, crystalline Galar representations are sort of Galar representations with good integral behavior. And so you expect that there's some interaction between them. And this is what we wanted to make precise. And so here's what we got. So let me first give some general setup. So fix a periodic formal scheme max. Here, I'm not gonna assume smooth over ZP because in fact, the most interesting example is gonna be rings of images of ramified extensions, potentially ramified extensions. Here's a definition of a prismatic crystal. It's more or less what you might expect. So an F crystal on X prism or prism is the following pair of things. So you have a vector bundle on the prismatic side. So just a crystal of vector bundles and then you have some F structure. So the condition you have is that there's an isomorphism between the Frobenius pullback of the vector bundle and itself, but only after you invert the ideal of the prism. So it's this kind of a stuka type condition. And I mean, the reason this is a reasonable notion is that if you're in a geometric setting, so if you have some Y over X and you look at the relative prismatic homology, then this is the structure that you get there. So if you're trying to abstract from the Gauss-Mannin case, this is kind of the natural notion you would come up with. And so we wanted to understand these objects better. And so here are some sort of two examples to keep in mind. So the first one is what I said. So actually, let me do the Tate twist first and then I'll come back to the Gauss-Mannin example. So the Tate twist is, so this is something rather nice. So it turns out that you can make sense of an infinite tensor product on the prismatic side. So O prism, Brouck isn't or Tate twisted by one. I'm gonna define to be the following infinite tensor product. So it's I prism tensor phi pullback of I prism tensor phi squared pullback of I prism tensor and so on. So this naturally is a line bundle on the prismatic side of X. And so you have to do something to actually make sense of this. The implicit assertion I'm using over here is that the tail end over here, so each high enough for Baynes pullback of I is trivial modular big power of I. So the tensor product and canonically trivial impact. So the tensor product converges to something reasonable. And so this is some object. And the reason to make this a definition is that you want the structure, right? You want the Frobenius pullback of the vector bundle to be related to the vector bundle up to inverting the ideal. If you just took the ideal itself, then you would be trying to relate this guy and this guy and there's basically no hope of doing that outside the crystalline case. But now you can do it. So you immediately sort of concede that at least if you believe that infinite tensor products behave like normal tensor products, then what you see is that phi pullback of oprism of one is I prism of one tensor, well, it's I prism tensor oprism of one. And so what this tells you is that this bundle and this bundle are isomorphic to each other after you invert the ideal of the prism. So you get this F crystal structure. And then there's also a Gauss-Mannian example which this will be a special case of. So for y to x. There is seem to be an inverse when you do the formal saying this infinite tensor product you, it seems. Oh, sorry, yeah, yeah, thank you. Yeah, thanks. Actually, let me not write the Gauss-Mannian example because I'm really struggling with time here. Let me just, I said it in words and the connection with the Tate twist is that if you do H2 of the relative prismatic homology of P1, you'll get oprism of minus one as one might expect. Okay, so these are F crystals. I was trying to make the point that these are kind of reasonable coefficient systems on X. And so here's one way they're reasonable. So there is a construction which takes such an object and produces a local system on the generic fiber. So there's a natural function which goes from F crystals on X prism to, well, what I can do is I can consider F crystals on X prism again, but with a different structure sheet. So instead of using oprism, I can look at oprism, I can work the ideal of the prism and then I can be adequately complete. So this is just the base change function. I'm certainly allowed to talk about this, but it turns out that this guy over here is equivalent to local systems on the generic fiber, ZB local systems. So this is a theorem. And I should say it's this work of Jiu Wu, which proves this theorem for X equals okay, but the theorem is true generally. And so you get this functor, which I'll call TX, which is taking an F crystal on X and producing a ZP local system on the generic fiber. So if X was the spectrum of the ring of integers of a local field, this is producing a Galar representation of the Galar group of the local field. And then I can at least formulate the theorem that we prove. So say X is okay, spiff okay per K over QP finite. Then the statement is that TX gives an equivalence between F crystals on X prism and representations, ZP representations of GK. So the local systems on generic fiber, which are crystalline in the sense of fontan. And so crystalline ZP representations can always be sort of spread out to F crystals. And it's important to work with ZP over here in the sense that it won't be true if you sort of work mod P or anything like this. Okay, and so this at least suggests that this category on the left is sort of a reasonable integral theory of coefficient objects. It recovers the nice ones that we know about in the case of the ring of integers of a local field. Obviously I don't have time to give a proof of this theorem which I wanted to, but let me just say what goes into it. So the two ingredients. So one is essentially it's the work of Kissen and then also which builds on ideas of Berger and results of Kidlaya. And so the way to be proved this theorem is by taking an object on this side and building an F crystal. The way we build an F crystal is via descent. So if you go up to the ring of integers of OCP, there you have the corresponding prism which is on 10s a nth. And there you can build a module using ideas of these guys. So this is to build F crystal over O.C, sorry. And then we also need descent data. So we want to construct the F crystal by descent. We have some huge cover of my faithfully by cover of my final object and I've built some object there. So I need isomorphisms over the overlap. And so the descent data comes from the balance and fiber sequence. So this is work of Antio, Matthew, Mauro, and Nikolas. And what it allows you to do is, so this is over some really exotic looking objects as the prism of O.C, tensor O.C over O.K. So we have this kind of really horrible looking ring and their theorem allows you to control some algebra over this ring. And then we use that to build the descent data that we need. Okay, I think this is all I can say. So I will stop here. Thank you. Are there any questions? Maybe other people in the computers are questions. Yeah, so, I seem to recall Drinfill was hoping, Drinfill was suggesting a way, I think he had to change the stacks a little bit of getting coefficient systems on the stack to correspond to F gauges in the sense of Fontaine Janssen. Do you see that in this theory? Yeah, so I should say that Drinfill has these three stacks, sigma, sigma prime and sigma double prime. And so what I talked about is sigma. Now, it's absolutely true that sigma is not quite good enough. So if you want a good coefficient theory that works not just with integral coefficients, like Zp, but also accounts for mod p coefficients, you wanna work with those finer objects. And so there, I mean, we have a way of understanding his stacks, the fancier versions of the stacks as well, but it's more complicated. But I absolutely agree that that's the right thing to look at for the good integral theory. Okay, I was hoping it could be simplified, okay. Well, if you get something slightly better, so we can describe quasi-coherent sheaves on a stack without talking about this notion of admissible modules, which I find quite mysterious in his work, but yeah. Thank you. There is a question in the, I don't know, I don't know if it's interesting. Just to clarify, acts by minus i and the i mean with eigenvalues minus i, not with minus i identically. I don't know, so what? Just to clarify, acts by minus i and gar i means with eigenvalues minus i, not by minus i identically. No, no, no. So it means minus i identically. I have a complex and theta equals minus i as an endomorphism of this complex. Okay. And there was also a question in the chat about under the equivalence of the last theorem, is there a nice way to read off the hot state weight of the crystalline representation from the F crystal? The hot state weight just comes from the SENT operator, which is part of the definition of the, just a crystal. So actually implicitly, what this is saying is that if you want the hot state weight on this side, you can forget about the F structure. All you need is just the underlying crystal. Yeah. So I think it's related to, it was questioned, it's a, we did the SENT operator. So can you say a few words about diffraction? Ah, thank you. So, right. Luke knows this. So Jacob and I have been calling this object H prism of X restricted to the hot state locus, the diffracted hot complex. So diffracted, because it's kind of a slightly twisted version of Hatch homology. And it turns out to be an extremely kind of useful invariant to understand the structure of prismatic homology. I give you one example, which is through Trinkel's theorem. But also if you want to understand things like the associated grader of the Nygaard filtration, the diffracted Hatch complex is the way to go. So we have a fiber sequence, for example, that understands GER I of the Nygaard filtration on absolute prismatic homology in terms of the action of data on the diffracted Hatch complex and its conjugate filtration. And so this, for example, helps you understand. So there's a connection between this stuff I've talked about today and THH topological Hatch homology. And the statement that the fiber sequence I just made helps you understand the associated grader of the so-called motivic filtration on THH. What is the final statement for some particular keys? Well, the final statement is, let me just, I can just write over here. So GER I Nygaard of the prismatic homology of X, it sits in a triangle where in the middle you have a fill I conjugate of this diffracted Hatch complex. We can't read what you are writing. You can't read what I'm... Where are you writing? It's the red. It's the red thing. Oh, it's the red. Okay. Is he okay? Okay. Sorry. Now I see. Yeah. So on this diffracted Hatch complex, it's a fiber sequence. It's a fiber sequence. So that helps you understand this guy in terms of the conjugate filtration on diffracted Hatch homology and the action of theta plus I. So for example, this recovers the box stats calculation of the topological Hatch homology of ZP, but also some other examples. That's great. Thank you very much. So no play-point available yet. No what? Sorry. No play-point, no manuscript available yet on these things. There's one on my Dropbox folder that I'm happy to send to you if you want, but it's not ready for public consumption yet. Okay. Thank you. So I guess the other questions I see are probably the answer, but I am not. How many don't close the. Like, is that at the right for when you sleep on W card? Oh yeah, there is. I should have said that. Right. So W card is naturally a Delta stack. And then there is a question. I assume that refers to this theorem over here and I'm not sure because I don't know what the analog of the right side is for a general smooth, I mean, okay, I know crystalline Gallar presentations, but I don't know what the relative, what a good relative notion of a crystalline Gallar presentation is. So I don't want to comment. So anyway, so, so let us thank the speaker again.