Prismatic Dieudonné theory
Bras, Arthur-César Le
I would like to explain a classification result for p-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Johannes Anschütz.
 So, welcome to the Paris Peking Tokyo seminar. So it's my great pleasure to introduce the speaker, the first speaker on this Zoom session today. So, the speaker is Arthur Seijou-Rubula, also he will tell us the prismatic doness theory. So please start. Okay, well, hello everyone. And well, thank you very much to the organizers for giving me the opportunity to speak and for setting up everything in this very particular context. So my talk of today is about prismatic doness theory. And so everything I will discuss is a joint work with UNS Anschutz. And for all this talk, I will fix the prime number P once and for all. So the goal of our work was to prove some classification results for P-isible groups of various kinds of rings. And the main tool we used to do this is the recent theory of prisms on prismatic homology, which has been developed by Bat and Scholtz. So the plan of my talk will be the following. I will first spend some time, maybe 20 minutes, half an hour to discuss a little bit prisms on prismatic homology. So recall the basic definitions and constructions. And then I will start speaking about the joint work with Uranus. So first of all, I need to tell you about which kind of rings we want to classify P-isible groups. And these rings are called quasi-syntomic rings. So this is what I will do next. I will explain the definition of these rings. Then I need to tell you by which kind of objects we classify P-isible groups over such rings. And we call them fitter prismatic geodontic crystals. So I will explain the definition and some basic properties of these objects. And then finally, I will explain our main results and say a few words about the proofs and some corollaries of these results. Okay. So let me start with prisms on prismatic homology. And I should make it clear. Everything in this part is true to but and also. So the theory of prismatic homology relies on two important basic definitions, the notion of a delta ring and the notion of a prism. So let me start with delta rings. So I will assume always that my rings leave over Z localized away from P. And then the delta ring will be a commutative ring A together with a map of sets, just of sets, delta, which goes from A to A, which has the following properties. So first of all, it maps 0 and 1 to 0. And then you want to prescribe how delta behaves with respect to multiplication and addition. And this is given by these two formulas, which may look a bit strange the first time you see them. So delta of a product if x, y for all x, y in A is x to the P delta y plus y to the P delta x plus P times the product of delta x by delta y. And then you have a similar formula for the addition. So delta of a sum x plus y is just delta of x plus delta of y. And then you need to correct it by this term here. And so observe that if you use a binomial formula to expand this x plus y to the P, then all the terms except x, P and y to the P, which are cancelled here, are actually divisible by P. So this expression makes sense in any ring. And you don't need to assume that the ring is P times 3 or anything like this. OK. So this is what the delta ring is. So the ring A together with a map of sets satisfying these three properties. So if one has a delta ring, A delta, so you can do the following operation. So you can define a map phi, which goes from A to A, and which sends x to the P plus P times delta x. And then you check an exercise that the identity is satisfied by delta, which I had before over there. Actually, exactly what you need to check that this map phi, or just a map of sets from A to A, is actually a ring morphism. And moreover, just by definition, it's a lift of four binos. So if you kill P, then these terms disappear. And it just becomes x. It just becomes a four binos. X goes to x to the P. And conversely, assume you start with a ring, commutative ring A, together with ring morphism phi, which lifts four binos modulo P. Then if you assume that the ring you have is P torsion free, you can actually just divide phi x minus x to the P by P and define this way, a delta structure on your ring. So in other words, in first approximation, you can think of delta rings as rings together with a lift, with a ring morphism, which lifts four binos. But the two notions are not exactly the same thing when you have P torsion in your ring. OK. Another point of view, which I just want to mention about delta structures, is the following. So if you have a ring A, then giving yourself a delta structure on A is in fact the same thing as specifying a ring morphism from A to the ring of length to V vector over A, which will be a section of the natural projection on the first component. And the recipe for this is assume you start with a delta structure, delta on your ring A. Then you just look at the map from A to W2 of A, which sends x to x, delta x. And once again, you can check that the axioms about delta, which tells you that you have a delta ring, is exactly what you need to verify that this is a ring morphism. OK. And this point of view is useful because it allows you to prove that this category of delta rings, contrary to the category of rings with a lift of four binos, has all limits and collines, which you can just compute on underlying rings. And so in particular, this forgetful factor from delta rings to rings, we have both the left and the right adjoint. And the right adjoint is given by the V vector's function. OK. So that's the first remark. And then another remark, which is more an exercise that you can do. I said that there are examples of delta rings with p-torsion, but it can never happen that you have a delta ring in which p to the n is 0 for some A. OK. This is something you can check just using the definition of the delta. OK. Very good. So we will see some examples later. Oh, wait a second. It's a trivial remark that 0 is a delta ring, so p to the n. Yeah, OK. I assume that 0 is different from 1. But OK, yeah. If you want. No, but if you want all limits and collines, then maybe. OK. Yeah. Then I should have said that there is no non-zero delta ring in which p to the n can be 0. Thank you. OK. So as I said, we'll see examples soon, but no, I'll come to the next definition, namely, the definition of a prism. So what is a prism? It's just a pair, a, i. Well, a is a delta ring. So usually in the notation, I will just forget the delta. OK. I'll just say a is a delta ring without mentioning delta. It's implicit. So you have a delta ring, a, and then you have some ideal i inside a. And again, this pair has to satisfy some properties. So first of all, you require that I define the Carti divisor on spec a. So it's just like locally principle generated by a non-zero divisor. Then you also require that I is p iidically complete. So for technical reasons, you should mean this in the derived sense. Also, soon we will make some assumption on the ring, which I'm sure that in practice, like the derived on the classical completions agree. So first approximation, you can just think that this is classically p iidically complete. And then the last condition, the most important one. So I has to be a pro-zariski locally generated by a distinguished element. And what is a distinguished element by definition is just an element of D, which has the properties of delta of D is a unit. Okay. And once again, in practice, I will always be principal. So you should just remember that a prism is a pair a, i, where a is data ring and let's say i is principal generated by a non-zero divisor, which is distinguished in the sense that it's made by delta is a unit. And moreover, the ring should be p iidically complete. Okay. The important condition is the last one. And so now I can give two examples of such prisms. So first of all, okay, if you have a p complete on p torsion free delta ring, a, then the pair formed by a and the ideal generated by p is a prism. Okay, so I mean, remember, so you need to check three conditions. So first of all, I had assumed that p, there is no p torsion, so please non-zero divisor. And by assumption, I also require that my ring is p complete. In this case, I is just p. And finally, the only thing you have to check is actually that delta of p is always a unit. Okay, and this, I mean, this is also what you need to do this little exercise I gave before that it cannot happen that p to the n is zero. I mean, the way to prove this is just to check that delta p has to be unity in any data. Okay, so that's one first class of example. And now here is another interesting example of prisms. So, let me give one more definition first. So we will say that the prism is perfect. If it's for Benus phi. So remember, whenever you have a prism, sorry, a delta ring, you have this delta, then you can define a phi which leaves for Benus. There is a formula that phi to the x is x to the p plus p times delta x. So what you require is that this ring morphism phi is actually an isomorphism. And then the claim is that actually the category of perfect prisms is the same as the category of integral perfect with rings. And how do you see that? Well, I mean, you can define a function in both directions which are quasi inverse of each other. You how so in one direction assume you start with an integral perfect to a dream. Then you can do this classical font and construction. You can consider a inf of r. So what you do is first you tilt your perfect ring. So you get perfect ring of characteristic p r flat. And then you take its string of bit vectors. This is what is called a inf of r. And a inf of r comes with a natural map sitar, which goes from from it towards our. I mean, part of the definition of at least consequence of the definition of an integral perfect to a dream is that this will be principle generated by a non zero divisor. And you check that in fact, this generator which is usually denoted by Xi in p. So the first theory is in fact distinguished in the previous sense. So, namely, so I should have said first that this ring is p torsion free. Our flat being perfect. And so it has, it comes with a natural for being used. And you just take the delta structure attached to this to this for being used to. And then my claim was that the generator of this idea is actually distinguished. Well, the way I mean you can check this by proving more generally that in such a delta ring and that delta ring of this form bit vectors of some perfect ring. An element is distinguished it on only if it is primitive of degree of degree one, which means that when you write the expansions of some of the time times powers of p, then the coefficient of p has to be a unit. And this you can check for. Okay, so this is one one factor. And then if you want to go in the other direction. It's even more simple you just mod out. You have a prison AI, which is assumed to be perfect. And then you just look at a mod. How do you recover the data. How do we get delta. Yeah. Well, okay, I said that if the ring is p torsion free, which is a case for this ring a info far, then having a data structure is the same as having a problem you see. And you have a problem you send the ring a bit like. So you just take the data structure attached to this to this for being. Okay, and so this, this proposition is the I mean this example is the reason why I guess button shows up. Like, describe prisms as some kind of deep perfection of the category of perfect to do it because you see that inside the category of all prisms I have this. And subcategory from my perfect prisms and this are exactly the same thing as perfect. And I is like choosing I my ideal is be like is the same as choosing some until my of the tilt of my perfectly. Okay. Okay, so once you have done this, namely introduce data rings and prisms, you can define the prismatic side. And there are several versions of it. So, for us, the one which is really relevant is the absolute version. So, let's start with our question. Hmm. Could you unmute the person. Yeah, I did not hear anything. Okay, she could you unmute the way the person who asked the question. Yeah, so I'm trying to do that but it doesn't work. Maybe. Yeah, there's a question on the microphone. There's a problem on the microphone. Okay, so maybe he can ask his question by chat. Okay. Okay, so I define those are the absolute prismatic site. So let me fix a ring R, which is assumed to be purely completely. And then the absolute prismatic site of the ring R, which will be denoted by our prison. Okay, so this symbol is supposed to be present, even if it appears as a delta here. So as a category, it's just the opposite of the category of all bounded prisms BJ together with a ring map from R to be more J. Okay, so here there is one adjective which I did not define yet, bounded. So it's just again, a technical condition, but which you can forget in first approximation. It's telling you that you have your prison BJ. And what you require is that be more J here as a bounded P infinity torsion. And this just means that if there exists some integer n big enough, so that any element which is killed by your power of P in this ring P mod J is actually already killed by P to the end. And this is a condition you put for technical reasons, which have to do with with derived versus classical completions. Okay, but I mean basically an object of the site is a prism with a map from R to its reduction be module J. And then you put a topology on this category so you define covers to be morphism of prisms. PJ goes to be prime shape prime. So morphism of prism is an obvious notion. So it's just a morphism of data rings compatible with a data structure, which sends J into J prime. And you say it would be a cover if when you just look at the underlying ring map from B to B prime. And it's just a P J completely facefully flat. So this means that if I take the derived answer product of B prime with B module P J over B. I mean, first of all, it has to be concentrated in the zero. And it is then facefully flat over B module P J in the usual sense. So slightly weaker notion that the notion of facefully flat ring morphism. Someone would just require a condition module of PJ. For the reason that again everything is assumed to be complete. And you want to notion which is table under completion. So instead of looking at facefully flat morphism you just look at PJ completely facefully flat. So there's a question. Okay, please. Okay, so first for the prison several technical so in the morphism of prison you require the J goes to J prime but I suppose it's for it should follow the J generate J prime is it correct. Yeah, this is true. Yeah. And also you need to have a finite disjoint unions of I mean when you have the topology also wants a risky covering of this and like an opens. Here you just have covering by one thing of course you need to add like the disjoint union of and load or the couple. Well, if he's a product of situations then it's taking all of them will be a covering by and things. Okay, so yeah I agree. Okay, so it should just be generated by this. This covers. Okay. Okay, no, no other question. Okay. So this is the definition of the prismatic side. And then you have two natural pre shifts on this site. So one is denoted is denoted by oprysum. And the other one is denoted by oprysum bar. This is a functor which define on the prismatic side which send a prism BJ on the prismatic side to be. So this is for for oprysum and oprysum bar will send BJ to be more J. And something bad should check is that these two functors are actually shifts on this prismatic side. And they both have a name so this chief or prism will be just called the prismatic structure shift. And the other one oprysum bar is called the reduced prismatic structure shift. And then I mean, okay, something you can deduce from this is that you can also you could also consider the functor I prism which sends just PJ to J itself. And then you this is also a shift on this side. Okay, so you have these two shifts oprym these three shifts on this prismatic side. And this is the ones we will use later on. But before doing that I want to mention that okay, I define the absolute version of the prismatic side, you could also do the following so let me fix to start with a bounded prism a comma I. And then I require that my ring R is leaving over AI so R is a P complete a mod i algebra. And then you can define the variant of the absolute prismatic cycle where everything leaves over a. So it will be denoted by R over a prism. And it will be the category of all prism BJ, which live over AI so together with a map of prisms from AI to BJ. As before, you want that be more J receives a ring morphism from R, which now is required to be a morphism of a mod i algebra. So everything leaves over my bounded prism AI which I fixed at the beginning and the topology is as before. And this is the version that batches will use and what they do I mean one of the main objectives of their paper is to compare what you get using this notion with other classical more classical p. Addic homology series. And so to do that, you, you, I mean, once you have defined the site you can define prismatic homology. So, I keep the same notation as before so AI is my fixed prism and R is living over a mod i. And then the prismatic homology of R over a, which will be denoted by prism of R over a is simply the homology of my shift or prism on this relative prismatic site. R over a prism. And well, actually, to be more precise you only make this definition when R is assumed to be a formally smooth over a mod i. You could do it always, but it's not well behaved. And the way batch also define prismatic homology for general a mod i algebra is using left hand extension from the smooth case. In the same way as you, you would define the cotangent complex from the shift of degree one differentials in general. But at least when, when everything is smooth, then the definition is just the community of the shift on this side. And as I said, what batch also do is they compare this new community theory with other periodic homology series. And I just want to mention two comparison results I prove there are many of them. Namely the hot state and the crystalline comparison series. So for this, because I recall notation AI is fixed prism, which is assumed to be bounded. And R is formally smooth over a mod i. Okay, so the first result is the hot state comparison. So it tells you that. Let me start with the right hand side. So here you have my prismatic homology complex, which I defined before prism arm of a. And you take its direct answer product with a mod i over a. So it would be the same as considering the community on the prismatic side of my reduced prismatic super shift or prism bar. And I consider a community of this in some degree. And the claim is that this is kind of canonical isomorphic as an arm module with the module of degree I differential forms on R over a mod. And here I mean it's implicitly assumed to be purely completely up to some small twist, which is denoted by the symbol is an I. So boy kissing kind of twist. So I recall the notation below. If you have some a mod i module M, you will be not by M twisted by I the tons of product of M with I mod I square I times over a mod. So this is a rather surprising result when you sing a little bit about it because I mean you have made this definition of prismatic community just using delta rings and prisms. And you see that naturally when you compute it, I mean when you compute the reduced prismatic homology the community groups of this complex. So you see differential forms showing up. Okay, and as a remark, I said before that the other question. Yes. So, you didn't define what is be completely smooth I imagine that completions of smooth things or direct limits, fitted limits of those are completely smooth but what is the exact definition. So I would say that I have a map a to be, I would say it's be completely smooth if I take the derived answer product of be with a mod P over a and I require this to be sitting in the zero and being smooth over a mod P in the classical finite type sense. Yes. Okay. Other question. Okay. Great. Okay, that's a remark that I said before, if you want to define prismatic homology in general, you don't do it just by computing homology of the structure shift on the prismatic side. You do this process of life can extension. But once you have done this, I don't want to explain it in detail, but you can check that this hot state comparison result will actually generalize as follows. Namely, if you have some a mod i algebra are P complete but not necessarily smooth. Then it's reduced prismatic homology so this complex over there. The base change to a mod i of prismatic homology. It actually comes equipped with a natural filtration which is increasing. And as the properties of the great pieces of this filtration, which is called the conjugate filtration are given just by which powers of the cotangent complex of our over a mod i and suitably shifted on break is interested and periodically completed. So, I mean, this is something you can directly deduce from the definition from the previous hot state comparison, plus the definition of both sides in general. I wanted to point this out because one, I mean, we will see the cotangent complex appearing later one later again. Basically the moral what you can remember from the statement is that hot state comparison gives you a way to like have some control on prismatic homology or at least its reduction module i in terms of the cotangent complex. So if you have some information on the cotangent complex you can usually deduce some interesting properties of prismatic homology. Okay, and then the next statement is the so called crystalline comparison. So this is the case where you assume that in your fixed prism AI is generated by P. Okay, then in particular because our leaves over a mod i it means that P zero in your ring. Then you could ask the question how does this prismatic homology relate to another interesting crystal, come on, this year in in characteristic P, namely crystalline commodity. And the answer is that, actually, they are almost the same. So if you compute crystalline commodity of our A. Then, well, it is the same as prismatic homology except that here on the right hand side, you have to twist by four minutes so you take pullback along the four minutes of a. Okay, so in particular if you know what prismatic homology looks like you recover crystalline commodity. You can not necessarily go the other way because a is not assumed to be perfect so five is not necessarily an isomorphism. But at least if you know prismatic only you recover crystalline commodity. And this is compatible with the strict with the Frobeno structure on both sides. This is also quite surprising because this way you get a definition of crystalline commodity without choosing divided powers and anything like that. And the key technical statement to check this is funny exercise that if you have a repeat ocean free data ring. If you have some element in this delta ring, so that it's first divided power is in your ring, then actually all the other divided powers are also in the ring. This is something you can check using the existence of the delta structure. This is one of the key inputs to in the in the proof of this crystalline comparison. Okay. Oh, there's a question. Just a clarification in the theorem you wrote the upper stuff, the kind of pullback. But since you're always working the derived the complete things is it's either some completion there or is that the bright. Maybe I'm confused but wait. I'm not sure now. I don't think you need to complete. I think what you say is that the delta. Yeah. No, go ahead. No, I as far as I understood you characterize the your you characterize your your things like guarz vitz forms is only after modding by I and also maybe you to be complete so it seems that everything is something derived complete relative to your finals and so no, and when you take fear, drying anche start by destroying this, I'm not Yeah. Okay. Okay. Okay. Good, so, okay, that's all I wanted to say about prismatic homology in general. Now I turn to prismatic doodonissiri itself. So as I said, I need to explain over which rings we want to classify feasible groups and by which kind of objects we want to classify. So I start with the rings. So there will be again definitions. So the rings we will consider are called quad asymptomic. So ring R is said to be quad asymptomic if it satisfies the following conditions. So first of all, it's P complete. So this we always assume everywhere and with bounded P infinity torsion. So I recall that this just means that there exists some integer n so that everything killed by power of P is already killed by P to z. Okay, and then the really important condition in the definition is that you want the cotangent complex of R over ZP to have P complete tor amplitude in degree minus one zero. So this means that you take this cotangent complex and you take its derived answer product with n for any R mod P module n. And then you want that these objects leaves in degrees minus one zero and the complex of R mod P modules. Okay, so this is the absolute notion somehow and then you can also define what the quad asymptomic morphism is. So it will be a morphism of P complete with bounded P infinity torsion rings. R goes to R prime which, okay, first you want that R prime is P completely flat over R. So I already explained what that means. And then you want that the relative cotangent complex of R prime over R as P complete tor amplitude in minus one zero. Okay, and you can also define what the quad asymptomic cover is. This would be useful later. It's the same definition, but instead of requiring that the map is P completely flat, you want it to be P completely faithfully flat. Okay, but so, yeah, the important condition is really the condition on the cotangent complex. And this definition is due to, I mean, it appeared in the paper of Batmouro and Schultzern topological orgy-domology. And the idea is that it should extend in the world of periodically complete rings. So there's some trouble. We lost that speaker. Ah, yes, coming. Hello. Ah. Okay, does it work? Sorry, I think we... Yeah, no, we convert our connection. Well... Okay. Is it okay? Yes, sorry. So is it good now? Yeah, it's good now. Okay, sorry, I think the connection, okay. That was, I don't know. Okay, so I was just saying that this definition is an extension of the classical notion of symptomic, well, of LCI ring and symptomic morphism. But you don't make any Nussarian or finite type assumption in this definition. So, yeah, before giving examples, one more piece of notation. So I will denote the category of all quasi-syntomic rings by QC. And then you can look at the opposite category and consider the topology which is defined using the quasi-syntomic covers in the above sense. So as I said, like maps which are quasi-syntomic and which are P completely phase-free. No, sorry, P completely phase-free. Okay, and then as a notation, if R is an object of this site, I would just denote by R with small letters QC, the sub-site which is formed by all rings which are quasi-syntomic of R. And again, undone with this quasi-syntomic topology. Okay, so I give examples of such rings now. The first example is just to justify the claim before that this generalizes the classical notion of LCI ring. So the claim is that any P complete and SIN ring which is locally complete intersection is quasi-syntomic. And well, this is checked using, I mean, actually Avramov gave a characterization of such rings LCI in terms of the cotangent complex. But here you just need the easy direction of Avramov C of M to prove that any such ring is quasi-syntomic. Okay, so that's one first class of example, but you also have like huge rings in this category of quasi-syntomic rings. So, namely I claim that any integral perfected ring is quasi-syntomic. And the reason for this is, okay, well, remember you have to check, okay, first of all, it's purely completely by definition. Then you can also check that for perfected ring, if there could be some P torsion, but like the P torsion is just the same as a P torsion. So the first two conditions are checked. And then you need to check this condition in the definition about the cotangent complex. And for this, well, you observe that this map from ZP, I mean, the canonical map from ZP to R, it actually factors through, you can factor it through the CTA map. So I should have written that this second map here on the right is Fontaine Zeta map. And whenever you have such a composite, you get a triangle for the cotangent complex. And now you observe that, well, what is a inf of R once you mod R of P? It's just R flat, which is a perfect ring of characteristic P. And whenever you have a perfect ring of characteristic P, its cotangent complex over FP is just zero. Basically here is just that. If you take any X, like it's always of the form Y to the P for some Y, because the ring is perfect. And then DX is just like P, Y, P minus one, DY. And if P is zero, this is just zero. So this way you can check that the cotangent complex of something perfect over FP is zero. But this just tells you that mod P, I mean the P completions of my cotangent complex of R over ZP and the cotangent complex of R over a inf of R agree. Right, because in this triangle, the other term will vanish after P completion. But so then once you know this, you can just, you are reduced to describe this cotangent complex of R over a inf of R. But then this map is a subjective and the kernel, theta is by properties of perfected rings, is principal and generated by a non-zero divisor. So this means that then the cotangent complex is just the same as R, but shifted leaving in common, common logical degree minus one. Okay, so this way you check. That in fact, in this case, the cotangent complex even has term amplitude in degrees minus one, minus one. It's even better. And then from these two class of examples, you can construct other examples if you want. So you can take a smooth algebra of our perfected ring and take its P completion. Or you can take a perfected, sorry, yeah, take an integral perfected ring and just mod out by a finite regular sequence. So if it's again, if it is again, bounded P infinity torsion, then this would give you another example of quasi-syntomical rings. And here I list some examples. So you can take the theta algebra in one variable over OCP. You can take OCP mod P or you can take this characteristic P perfect ring, FP T one over P infinity and mod out T minus one. So these are all examples of quasi-syntomic rings. Okay, so I think that's basically all I wanted to say about quasi-syntomic rings, but I just wanted to try to convince you that many interesting examples of rings are actually quasi-syntomic. Okay, and one good point of this about Mohr-Scholz's definition is that it's purely a definition in terms of the cotangent complex. And we have seen before, this was a hot state comparison theorem that if you have some control on the cotangent complex, because of this hot state comparison, you can usually deduce things about prismatic correlation. Okay, now I turn to third part. So filtered prismatic do-donate crystals. This will be the objects we will use to describe our P-zibber groups. And as you can guess, the definition will use the prismatic side. So to state it, I first need one observation. Let's take R to be quasi-syntomic. And then the claim is that, okay, you have a natural morphism of topos, which goes from the category of sheeps on the prismatic side, the absolute one, to the category of sheeps on the small quasi-syntomic side of my ring R. And if you wonder how this, what does this morphism of topos come from? Well, it's defined as a composition. So first of all, you observe that if you take a prism AI, so I mean, if you take some object of this prismatic side, you have a prism AI, then you can look at a mod i. So it will be P-complete. And the claim is that, this defines a, a co-continuous function from the prismatic side of R towards the big quasi-syntomic side of our ring R. And then you just restrict. Yeah, but I'm, only difficulty is checking this co-continuity of this, of this function. But if you are familiar with crystalline common logic, it's like, very similar to what you do when you, you go from the crystalline side to the etal or the risky side. Except that here we work with something a bit more general. We work with the quasi-syntomic topology. Okay, and now what I will do is I will take my prismatic structure shift or prism on this ideal prismatic shift. I prism, I just push everything using this morphism V. So V was my notation for this morphism of topos. I just push everything down to the co-syntomic side. And then I claim that, okay, actually you have a natural surjection from oprys to O. And I will give a name to the kernel of this morphism of shifts. So O here is just the structure shift on the quasi-syntomic side. This kernel is denoted like this. So it's what is called the first piece of the Niagara filtration on this prismatic shift oprys. Well, the reason for this notation is that, I mean, there is this notion of Niagara filtration of a prism, which is defined for any, you could define for any positive integer I. Here I only need the first piece of the Niagara filtration. I would just take as a definition that it is the kernel of this surjection, which I did not explain. And then a property of this is that, well, because any data in other Frobenius, so this shift oprys will come with a Frobenius morphism, Phi. One can check that this kernel has the properties at Phi of the kernel. So Phi of the first piece of the Niagara filtration for oprys actually lies in high-price times oprys. So morally on this first piece of the Niagara filtration, the Frobenius is divisible by, well, let's say the locally I, the ideal is generated by non-zero distinguished element. The idea is that on this first piece of the Niagara filtration, Phi is divisible by this distinguished element. So there is question from over here. Yeah, there's a question. I cannot hear. Can you put the microphone on? You should activate his. I guess you can unmute him. Oh, yeah, now his microphone must be on. Internet, I don't. Oh yeah, now you can speak. So why do you write oprys times oprys? I think oprys is an ideal in oprys. Oh, yeah, sorry, sorry. Yeah, it's a typo. Yes, thank you, sorry. Yeah, it's just because later it will appear. Yeah, sorry. Okay. Okay. Okay, so then the definition of a filtered prismatic dunic crystal is, so let me again fix a quadsentamic ring R. And then a filtered prismatic dunic crystal is by definition of collection, a triple M, fill M and five M. So M is finite located free oprys module. First of all, then fill M inside it will be some oprys sub module. And finally, five of M is morphism, ring morphism from M to M, which is assumed to be a file in R. Okay, where five is a Frobenius of oprys. Okay, and then you ask three conditions on this triple. So first of all, you want that this Frobenius five M sends a fill M to I press times M. Okay, but then there is an obvious oprys sub module of M which also have this property that five M of it is leaves inside I press times M. Namely, consider the first piece of the Nagat filtration of oprys times M. Then because of the previous slide, we know that this is contained in I press. And so this sub module, first piece of the Nagat filtration times M also has a property that five of it is contained in I press times M. And then the second axiom you have is you want that this sub module is in fact contained in fill M. And once you have done this, you want to fill M and once you have done this, then you know that M module of fill M will be a module over oprys module of the first piece of the Nagat filtration. But remember that oprys module of the first piece of the Nagat filtration was by definition is just the structure shift of the quiescentomic site. Oh, so M mod fill M will be an O module and you ask that it is finite locally free. And then the third condition is that the image of the filtration by the Frobenius is big enough in the sense that it will generate I press times M as an oprys module. So if you are familiar with crystal and deudonis theory, again, it's very reminiscent of the usual notion of deudonis or filter deudonis crystal. And the third condition is just in this setting which has been a formulation of the condition that the filter crystal is admissible in the sense of quotient. So that's somewhat just the obvious extrapolation of this classical notion to the setting of the prismatic site. Okay, and then notation is if R is my quiescentomic ring, I will denote by Df of R the category of all filtered prismatic deudonis crystals over R. And the morphism are the obvious ones. I mean, they should be oprys linear and they should be compatible with Frobenius and with the filtration. Okay, and so now I can come to the statements of the two main results we proved. So I will fix again once and for all now quiescentomic ring R. And let me take G to be a feasible group over R. Then I can define M prism of G to be X1 of G by oprys. And here when I write this curly X1 is supposed to be like the local X groups in the category of abelian sheeps on the quiescentomic site. So you check that your feasible group G, where the reduction to the case of finite locality free group schemes defines a sheep, an abelian sheep on the quiescentomic site. And oprys is also an abelian sheep on this site. So you can take X1 in this category. And then you also define fill M prism of G to be the same thing except as you replace oprys by the first piece of the Naga filtration. And then the first main result is that, so if G is as before, then this triple M prism G, fill M prism G and for benus of M prism of G, which is just by definition the for benus coming from the for benus on the prismatic on the prismatic sheep. Then the claim is that this is actually an objective of this category DF of R. So this is a filtered prismatic dodomic crystal over R. And in what follows, I will denote it by M prism G underline. Okay, and as a remark, if P is zero in my ring R, so in the characteristic P situation, you can check using the graph, you can check using the crystalline comparison theorem, which was discussed in the first part, that in fact, this is just the same thing as the usual function you will find, people have studied in crystalline dodomy theory, which you find, for example, in the book of Bertolo, Brin and Messing. So this is nothing new in characteristic P. Okay, and then our second main result is that, well, this filtered prismatic dodomy factor underline M prism, which associates to G, M prism of G underline. It actually realizes an anti-equivalence between this category of BT of R of visible groups over R, and the category DF of R, which I defined before. And as a bonus, I mean, as a byproduct of the proof, you actually obtain that the prismatic dodomy factor, G goes to M prism of G, is already fully phase-free. But if you really want to get an equivalence of categories, so if you want to be able to describe the essential image, you need to add the filtration in the picture. Okay, so now I make a few remarks about the, these two results and about the way we prove these two results. So first remark is that, as I said before, in characteristic P, you just recover the usual factor from dodomy theory. And so in particular from the serum two, this classification result, well, you can deduce that the crystalline dodomy factor is an equivalence for all quasi-syntomic rings in characteristic P. And well, this was actually already known, of course, in some cases. So if you look at LC irons, which are also excellent, then fully phase-freeness was proved in the end of the 90s by To-Yong and Messing. And more recently, Eichelot has proved that this factor is actually an equivalence when the rings is, okay, Nocerian LCI and more of a F finite. So this means that the Frobenius is finite. And this particular implies XM. So in this case, the result was already known. Okay, then I wanted to explain now that this category DF of R may seem a bit abstract, but there is an interesting class of quasi-syntomic rings for which you can make it more explicit. Namely, you will say that the ring is quasi-regular semi-perfectory. If it is quasi-syntomic, of course, first, and then you also want that there exists that a perfected ring which maps subjectively onto R. It has to be big enough. Examples of such in the list of examples I gave before, where you can look at any perfected ring is obviously, because we know it's quasi-syntomic and the second condition is trivial. Any perfected ring will be quasi-regular semi-perfectory, but also like a quotient of such a ring by a finite regular sequence, which has bounded P and P torsion. So these are examples of such rings. Well, for such rings, it's like for perfected rings, and the example we saw, in fact, the Quotantin complex after P completion is just sitting in the minus one. And for this reason, you can check, using Hatch-Tech comparison, that for such a ring, the prismatic side has a nice feature that it admits a final object in this case, which you can just describe as, if you want the absolute prismatic homology, comes with a natural idea. And then also you can check in this case that if you take the first piece of the Naga-Tutration on this prism to be just the inverse image of I by Frobenius, then the quotient is isomorphic to I itself. So two examples of computation of this initial object, or some final object. The first case is if I is perfect to E, then this final object is just given by the ring A-info of R together with the kernel of the theta map. So, okay. In this case, the prism is perfect. So Frobenius is an isomorphism. So there is always some choice that you can work with the map theta, or with its pre-composition with Frobenius minus one, and then consider theta tilde. It's just a matter of conjunction. Okay, so that's the first example. And then another example, if you take a ring which is called a regular somaiparfectoid and with P zero in this ring, then this initial prism is just the same as a crease of R together with the idea generated by P. So you can compute it in several situations. And then we make a definition. It's very similar to the one we had before. A field of prismatic design in module over R will be a collection M, field M, and phi M. But no, I just have, instead of having finite locality free modules over the prismatic shift, I just have a finite locality free module over this ring, prism R, a certain sub-module of it, field M, and a phi in R map, phi M. And you just ask the exact same actions as before, but no, you only work with modules over this ring, prism R. So I want to repeat them because I don't have much time left. But okay, so you can make this definition and then the claim is that if my ring is quasi regular somaiparfectoid, I can in fact just evaluate all the objects I have on this, so I should have written final on this final object. And the claim is that this gives an equivalence between the category of filtered prismatic due to the crystals over R and the category of filtered prismatic due to the modules over R. So in other words, in this case, you can make the category more explicit. You can just work with modules instead of working with this object, this category DF of R. And moreover, if the ring is perfectoid, then you can do even better. Then you can even forget the filtration. So just look at the forgetful function for this category of filtered prismatic due to the modules over R to simply what you can call a prismatic due to the module, but in this case, it already has a name. People call them minuscule break is in FARG modules. And then the claim is, in this situation, this forgetful function is in fact an equivalent. So in other words, as a special case of the CRM2, you see that you recover the fact that for perfectoid ring, peaceful groups over such a ring are classified by minuscule break is in FARG modules. But this is a bit cheating because in fact, we need as an input for the proof, for the proof of CRM2, we need a special case of this. Namely, we need that peaceful group over a variation ring, which is perfectoid and has algebraically closed fraction field is the same thing as a minuscule break is in FARG module. And actually, it's not difficult to do. You can deduce the case of all perfectoid rings from this special case using VT-SAN target. But so we need this as an input. Okay, so can I just take five minutes to finish? Or should I stop now? Okay, go ahead. So I try to finish quickly. So I also want to mention that, as I said, in general, you really need the filtration if you want to describe the essential image of this prismatic do-don't-functor. But I just said before that in this remark that for perfectoid ring, the filtration is actually unique. So it's not needed to state the classification theorem in this case. This is not true in general, but this also works for P-complete regular rings. And as an example of this, I just take the ring of integers in some discreetly valued extension K of QP with perfect residue field. So for example, take a ring like ZP. And then also this case was, of course, already known before. So then you can prove that P-complete groups over this such a ring are classified by so-called minuscule brachycin modules. And this has been done by Bray and Kissing, at least for all P, and then extended by Kim, Lau and Yu to all P. But we can also recover this, namely by first proving that for such a ring, you can forget the filtration. And then checking that you can just evaluate and, well, there is a natural prism attached to such a ring once you choose a uniform measure. It's not a final object, but still you can check by some reduction to the perfect situation that in this situation, evaluation on this object is again unequivalent. And so, and then we check that the factor we have is actually the same as the one which has been studied by Bray and Kissing, all these people. But now the good point is that some of you directly learned in the correct category. So the proof works uniformly for all P. You don't need to make a special argument when P is too. Okay, and then just two words about the proof. So for theorem one, it's not surprising. So you just follow the strategy of Bertel and Bray and Messing in their book. So you have this definition of this triple. You want to check that it is a filtered prismatic dosonary crystal. And the idea is, well, you have to understand how what is X1 looks like. And in fact, for any group object in a topos, you have some device to make computation about this X groups, at least in low degrees. So this is explained in the book of, Bertel and Bray and Messing. And you are reduced to compute some prismatic commodity groups. So first step, you use a theorem of Reno to reduce to the case of Psemer groups which comes from some Abelian scheme. And then, via this Bertel and Bray and Messing partial resolution of any group object in a topos, what you just need to do is understand precisely prismatic commodity of Abelian schemes. And a key tool for this is provided by the Hatch state comparison theorem for prismatic commodity. So, but the ideas are really similar to the one of Bertel and Bray and Messing. And then, theorem two, the proof is more difficult, but I just want to point that the key idea is to use quasi-syntomic descent. So here, I introduced just before the notion of quasi-regular, so my point is that the very nice feature of this site, this quasi-syntomic site, which was observed by Bayt-Maurot and Schultz-Hou, is that this quasi-regular semi-perfected ring actually forms the basis of this quasi-syntomic topology. Or by any quasi-syntomic ring, by extracting enough piece-roots, you can always make it quasi-regular semi-perfected. And this means that once you have defined this factor, so once you have proofs here on one, then you can, to prove that it's unequivalent, you can somehow reduce to the quasi-regular semi-perfected situation. And then everything is more concrete, as I said. Instead of having crystals, we just have modules over this ring prism arc. Okay, and then the hard work starts, but I wanted to say this because it shows that even if you, if you want to, if you are a non-proliferation like getting classification results over like Nucleus and rings, then the proof is such that actually you really need to work with this big category QC, because the argument is by descent. This is very big quasi-regular semi-perfected ring. Okay. Okay, and then I will skip this point. I will skip this point. I just want to finish by saying that there are some natural questions which are left open. First of all, it would be interesting to see what happens for more general baserings. We only prove results about quasi-syntomic rings, but for example, in the work of Tzink, you can find some results for a very general, periodically complicated ring. So, for example, in the work of Tzink, you can find some results for a very general, periodically complete rings. But I have no idea how to do it for more general rings. And then another thing we don't have is something like deformation theory in this setting. So for this prismatic duodonné functor, so for the usual crystalline duodonné functor, you have this gotennik-messing deformation theory, which is very powerful. But here, we don't have any analog of this. And then finally, a final remark is more anecdotal, but it's about the case of perfect rings in characteristic P. I just want to point that in this case, well, because if you live over like an object of the prismatic side of such a ring R, it will be P torsion-free in particular. So you can always map this prismatic structure shift on this side to the same thing where you invert P and take Q to be the quotient of this injectable. And then you check that, well, in this situation, the prismatic duodonné module of any P-zero group G is in fact the same thing as home from G to the push forward to the quadrathymtomic side of this shift on the prismatic side. I mean, this is just obtained by looking at the associated long exact sequence. And then a natural question is whether the, I mean, front end has, was the first to give a general definition of the duodonné module over such a perfect ring of characteristic P. And the definition looks a bit similar, except that instead of this mysterious push forward here, you have this shift of VIT covectors. And so it would be also interesting, and we did not do it to know how to relate this object here to Fontaine's original definition of the duodonné functor with VIT covectors, but without choosing the general crystalline comparison C-ray, which of course implies this comparison, but just directly using the two definition. Okay. Thank you very much. So thank you very much for the interesting lecture. So are there any questions? Yes, Luke. Could you unmute him? Yeah, I'm trying, but it doesn't work. I don't see one. Luke, could you unmute yourself? Luke, can you put the microphone on? Now you can hear me. Yes. Okay. So I think you skipped one of the last slides where you were writing something about classification of finite, locative free community group schemes. You discussed PDVS groups. I presume that you, but presumably you can also classify truncated BTs and maybe more general. So that is the, Oh, I can't hear you well. Okay. So it was just that I wanted to see this slide again. Yeah. But then is it in terms of a module or whether a complex maybe? Okay. So we only do it for over perfected rings. So I think that's the question. I think that's the question. I mean, the idea is the same as the one you find in the paper of kissing. Well, actually you use the fact that you can just like identify the category of final locality free group schemes with the category of like two times complex. Of peace by groups. With an isogenic between them. And the only difficulty in general for general quasi-syntomic rings is I'm not sure by which kind of objects you would classify fine and local if we group schemes. So for perfected rings, as I said, for peace by groups, you can forget the filtration. You just have this minuscule boy kissing fog modules. So it's not difficult to guess what, which can, by which kind of objects you will classify fine and local if we group schemes. So it would just be a module of your ring prism, R, which is just a inf of R in this case of projected dimension less than one killed by your power of P with a four been used on a Vershebu. But in the general situation where you should also take the filtration into account, I'm not sure how to describe the very local if we group schemes. So that's why we only did it for perfected rings. Actually, this was already known in all cases except maybe when peace to my work of law. I had another question you mentioned work by think. So is there a relation between this place and the theory? Yes, this is another slide. I had to skip here. So let me take our game to be one first one reduces to the case of quasi regular semi perfected rings by quite a systemic descent. And then, well, you have this natural map from prism R to R was kernel is the first piece of the nugget filtration on this prison. It's a map of Delta rings. So it's a map of rings. But I said that the forgetful functor from delta rings to rings as a right adjourn, which is the big vectors fun. Yes. So this mapping juices a map of delta ring from prison R to W of R. And using this observation, you can actually check that you have a functor from our category DF of our filter prismatic modules. To the category of displays of our in the sense of. It's not an equivalent. I mean, the classification of sink is both more general and more restrictive. It's more general in the sense that I think he only assumes a ring to be periodically complete. But then he has to restrict to formal feasible groups because when he is to like some difficulties. So this is due to the fact that this functor we get from DF of R to this place of R is not an equivalent. But it is when you restrict to any important objects. The terminology of. Thank you. Thank you. Okay. Thank you. Then offer. Okay. So I remember the concern in this. Welcome to the net. There were some after seeing the way in particular some papers of law where he. He looked at the for example, complete. Mix characteristic. The local rings with. Is it perfect or not perfect. Then he gets some using saying he get, I mean, he gets. He does things which was using windows and friends. So they get a very simple linear algebra. Sinks which classify PD visible groups. Over mix characteristic regular rings, but sometimes he needs to. He needs to get a. To assume it is a. The result is different and the rest of it is not perfect. But any case he has some. Do you get a relation between. What you do and use to, but maybe there are several such references. I don't remember exactly, but I remember it's quite simple. Like you only need to give. Some. Some final free modules and some maps. Composition equal to the equation or something like this. Is it possible to relate it to your. Yes. That's what I mentioned here. So. As I said, in. For such. A P complete regular rings. This the filtration is also not relevant. So you can describe everything. I think, as I said, using currency realms, you can. Get a natural prism attached to the situation. So for example. In the simple case where I just okay. Yes. You just take this break is in prison. So you just take like. Is it your field double brackets. A. U for some formal variable U. And he is some ascent-style polynomial, which is like determined by after you choose a uniform either you. You have a natural map from this track S. Towards. Your ring R and either generator of this. Of the kernel. And then yes, then, then you get a quite simple classification in this case. in this case as modules, find locally free modules over such a ring together with the Frobenius which has the properties that it's, like after linearization it's co-carnalized. After you invert E it becomes, the linearization becomes an isomorphism, the linearization of Frobenius. So, but for checking that the functor is really the same as the one which is used in law. At least we checked it in this particular case, but this is the same as the functor that people usually consider. But in the marginal situation, I'm not sure we checked it. So there was also a technical question about whether the resdiel field is, when the resdiel field is not perfect, I think he needed to work like sync with only formal pt-visual group, but maybe I don't, do you get, do you get easy to, that is classified, well I don't have a, now I'm sorry. When difficulty that low on contours is that, I mean he uses crystalline Jordanian theory, yes, so somehow you have your ring R, some of us you reduce to, to arm of p where you can use crystalline Jordanian theory, but then you need to go back from arm of p to R. So you need to use gotonic messing theory. But the issue is that this, like this divided power structure on the idea and generated by p is not important when p is two. Gotonic messing theory only works well when the divided power structure is important, right? So I think for this reason, you have to do some, usually you have to do some extra work when p is two. But here somehow I think this issue does not appear because in some sense you directly, with this prismatic Jordanian function, you directly land in the correct category. Like again, back to this example where R is okay, in the original work of Boy and Orkissin. Actually you, what you first do is you produce a factor from the category of feasible groups over this ring R towards the category of filtered modules over another ring that's frack S, which is like the p completion of the pd envelope of E inside this ring frack S. And this is usually just called curly S. And then you have to do some semi linear algebra to check that this is indeed the same as the category of minuscule Boy and Orkissin modules. But here we directly get a factor from feasible groups over R towards minuscule Boy and Orkissin modules. For some of we avoid this difficulty and that's why we don't have any assumption on p. I have another small question. Maybe what is, so you said that in the definition of a prism you have a Cartier divisor. You can, are there examples where it is not globally principle? I know I don't know any example writes, writes not principle actually. I think all the examples I'll know the ideals are principle. I see basically one question. Can you hear me, Ophar? Yes. Okay, so the question was, does there exist any coincidence between the filtration of prismatic Jodonaic crystal and the filtration on bun G on the farfontaine curve since Fark proposed that using chromatic filtration one can correspond perfect with BT1 with bun G. One can what? Correspond perfect with BT1 with bun G. I don't know what that means. I'm just reading what it is. BT1 is like Berserk's detail of level one. I guess so, but I guess what is perfect with BT1 is, BT1 over perfect with bun G. I don't know what it means. But maybe BT1 over perfect, okay, I don't know. But I don't know any relation with bun G. I mean, of course, Burkis and Fark modules are related to modifications of vector bundles on the Fark-frontaine curve. But bun G is like the bun G bundle, so. Yes, so I don't see. For a group, they do it with a. I guess, well. Yeah, so front-end. The G should be. G should be reductive group or something. So I don't see any relation. Okay, well, if I, so it's not, I'm not, I will need more to see it in more precise form. So, yeah, so as far as I understand, okay. Yeah, okay, so by any case, you prove that the same construction of display is canonically correspond to what you do, using these functors to display, you get exactly things construction, which is defined for more general rings. Right, yes. Okay, all right, so. Yeah, so then this partly answer, because the work of Lowentz-Holm is kind of trying to use this display to do some complicated tricks to which I don't, okay, so it's, and of course you get also the crystalline deodorant theory from your. Yes, as a. You need to use some morphism between relating the crystalline, okay, but you wrote it, I don't remember if you put it in the slides. I just put a remark, I mean, yeah, basically you can check that the usual, I mean, you have this prismatic side, you have the crystalline side, let's say you just push everything to this color-syntomic side and make the comparison there. And then the claim is that the two categories you get are actually the same, so. For the ring. So the ring is killed by P. Exactly, yes. So you get a, okay, I have to, this was a. I mean, basically this follows from this crystalline comparison theorem. Okay. Okay, thank you, so I will have to think. Okay, not there. The slides will be online at some point. Okay, thank you. You're welcome. Okay, goodbye. Bye. Thank you very much, so goodbye. Thank you.
 So, welcome to the Paris Peking Tokyo seminar.  So it's my great pleasure to introduce the speaker, the first speaker on this Zoom session  today.  So, the speaker is Arthur Seijou-Rubula, also he will tell us the prismatic doness  theory.  So please start.  Okay, well, hello everyone.  And well, thank you very much to the organizers for giving me the opportunity to speak and  for setting up everything in this very particular context.  So my talk of today is about prismatic doness theory.  And so everything I will discuss is a joint work with UNS Anschutz.  And for all this talk, I will fix the prime number P once and for all.  So the goal of our work was to prove some classification results for P-isible groups  of various kinds of rings.  And the main tool we used to do this is the recent theory of prisms on prismatic homology,  which has been developed by Bat and Scholtz.  So the plan of my talk will be the following.  I will first spend some time, maybe 20 minutes, half an hour to discuss a little bit prisms  on prismatic homology.  So recall the basic definitions and constructions.  And then I will start speaking about the joint work with Uranus.  So first of all, I need to tell you about which kind of rings we want to classify P-isible  groups.  And these rings are called quasi-syntomic rings.  So this is what I will do next.  I will explain the definition of these rings.  Then I need to tell you by which kind of objects we classify P-isible groups over such rings.  And we call them fitter prismatic geodontic crystals.  So I will explain the definition and some basic properties of these objects.  And then finally, I will explain our main results and say a few words about the proofs  and some corollaries of these results.  Okay.  So let me start with prisms on prismatic homology.  And I should make it clear.  Everything in this part is true to but and also.  So the theory of prismatic homology relies on two important basic definitions, the notion  of a delta ring and the notion of a prism.  So let me start with delta rings.  So I will assume always that my rings leave over Z localized away from P. And then the  delta ring will be a commutative ring A together with a map of sets, just of sets, delta, which  goes from A to A, which has the following properties.  So first of all, it maps 0 and 1 to 0.  And then you want to prescribe how delta behaves with respect to multiplication and addition.  And this is given by these two formulas, which may look a bit strange the first time you  see them.  So delta of a product if x, y for all x, y in A is x to the P delta y plus y to the  P delta x plus P times the product of delta x by delta y.  And then you have a similar formula for the addition.  So delta of a sum x plus y is just delta of x plus delta of y.  And then you need to correct it by this term here.  And so observe that if you use a binomial formula to expand this x plus y to the P, then all  the terms except x, P and y to the P, which are cancelled here, are actually divisible  by P. So this expression makes sense in any ring.  And you don't need to assume that the ring is P times 3 or anything like this.  OK.  So this is what the delta ring is.  So the ring A together with a map of sets satisfying these three properties.  So if one has a delta ring, A delta, so you can do the following operation.  So you can define a map phi, which goes from A to A, and which sends x to the P plus P  times delta x.  And then you check an exercise that the identity is satisfied by delta, which I had before  over there.  Actually, exactly what you need to check that this map phi, or just a map of sets from A  to A, is actually a ring morphism.  And moreover, just by definition, it's a lift of four binos.  So if you kill P, then these terms disappear.  And it just becomes x.  It just becomes a four binos.  X goes to x to the P. And conversely, assume you start with a ring, commutative ring A,  together with ring morphism phi, which lifts four binos modulo P. Then if you assume that  the ring you have is P torsion free, you can actually just divide phi x minus x to the  P by P and define this way, a delta structure on your ring.  So in other words, in first approximation, you can think of delta rings as rings together  with a lift, with a ring morphism, which lifts four binos.  But the two notions are not exactly the same thing when you have P torsion in your ring.  OK.  Another point of view, which I just want to mention about delta structures, is the following.  So if you have a ring A, then giving yourself a delta structure on A is in fact the same  thing as specifying a ring morphism from A to the ring of length to V vector over A,  which will be a section of the natural projection on the first component.  And the recipe for this is assume you start with a delta structure, delta on your ring  A. Then you just look at the map from A to W2 of A, which sends x to x, delta x.  And once again, you can check that the axioms about delta, which tells you that you have  a delta ring, is exactly what you need to verify that this is a ring morphism.  OK.  And this point of view is useful because it allows you to prove that this category of  delta rings, contrary to the category of rings with a lift of four binos, has all limits  and collines, which you can just compute on underlying rings.  And so in particular, this forgetful factor from delta rings to rings, we have both the  left and the right adjoint.  And the right adjoint is given by the V vector's function.  OK.  So that's the first remark.  And then another remark, which is more an exercise that you can do.  I said that there are examples of delta rings with p-torsion, but it can never happen that  you have a delta ring in which p to the n is 0 for some A.  OK.  This is something you can check just using the definition of the delta.  OK.  Very good.  So we will see some examples later.  Oh, wait a second.  It's a trivial remark that 0 is a delta ring, so p to the n.  Yeah, OK.  I assume that 0 is different from 1.  But OK, yeah.  If you want.  No, but if you want all limits and collines, then maybe.  OK.  Yeah.  Then I should have said that there is no non-zero delta ring in which p to the n can be 0.  Thank you.  OK.  So as I said, we'll see examples soon, but no, I'll come to the next definition, namely,  the definition of a prism.  So what is a prism?  It's just a pair, a, i.  Well, a is a delta ring.  So usually in the notation, I will just forget the delta.  OK.  I'll just say a is a delta ring without mentioning delta.  It's implicit.  So you have a delta ring, a, and then you have some ideal i inside a.  And again, this pair has to satisfy some properties.  So first of all, you require that I define the Carti divisor on spec a.  So it's just like locally principle generated by a non-zero divisor.  Then you also require that I is p iidically complete.  So for technical reasons, you should mean this in the derived sense.  Also, soon we will make some assumption on the ring, which I'm sure that in practice, like the derived on the classical completions agree.  So first approximation, you can just think that this is classically p iidically complete.  And then the last condition, the most important one.  So I has to be a pro-zariski locally generated by a distinguished element.  And what is a distinguished element by definition is just an element of D, which has the properties of delta of D is a unit.  Okay. And once again, in practice, I will always be principal.  So you should just remember that a prism is a pair a, i, where a is data ring and let's say i is principal generated by a non-zero divisor, which is distinguished in the sense that it's made by delta is a unit.  And moreover, the ring should be p iidically complete.  Okay.  The important condition is the last one.  And so now I can give two examples of such prisms.  So first of all, okay, if you have a p complete on p torsion free delta ring, a, then the pair formed by a and the ideal generated by p is a prism.  Okay, so I mean, remember, so you need to check three conditions.  So first of all, I had assumed that p, there is no p torsion, so please non-zero divisor.  And by assumption, I also require that my ring is p complete. In this case, I is just p.  And finally, the only thing you have to check is actually that delta of p is always a unit.  Okay, and this, I mean, this is also what you need to do this little exercise I gave before that it cannot happen that p to the n is zero.  I mean, the way to prove this is just to check that delta p has to be unity in any data.  Okay, so that's one first class of example.  And now here is another interesting example of prisms.  So, let me give one more definition first. So we will say that the prism is perfect.  If it's for Benus phi. So remember, whenever you have a prism, sorry, a delta ring, you have this delta, then you can define a phi which leaves for Benus.  There is a formula that phi to the x is x to the p plus p times delta x.  So what you require is that this ring morphism phi is actually an isomorphism.  And then the claim is that actually the category of perfect prisms is the same as the category of integral perfect with rings.  And how do you see that? Well, I mean, you can define a function in both directions which are quasi inverse of each other.  You how so in one direction assume you start with an integral perfect to a dream.  Then you can do this classical font and construction.  You can consider a inf of r. So what you do is first you tilt your perfect ring.  So you get perfect ring of characteristic p r flat.  And then you take its string of bit vectors. This is what is called a inf of r.  And a inf of r comes with a natural map sitar, which goes from from it towards our.  I mean, part of the definition of at least consequence of the definition of an integral perfect to a dream is that this will be principle generated by a non zero divisor.  And you check that in fact, this generator which is usually denoted by Xi in p.  So the first theory is in fact distinguished in the previous sense. So, namely, so I should have said first that this ring is p torsion free.  Our flat being perfect. And so it has, it comes with a natural for being used.  And you just take the delta structure attached to this to this for being used to.  And then my claim was that the generator of this idea is actually distinguished.  Well, the way I mean you can check this by proving more generally that in such a delta ring and that delta ring of this form bit vectors of some perfect ring.  An element is distinguished it on only if it is primitive of degree of degree one, which means that when you write the expansions of some of the time times powers of p, then the coefficient of p has to be a unit.  And this you can check for.  Okay, so this is one one factor. And then if you want to go in the other direction.  It's even more simple you just mod out. You have a prison AI, which is assumed to be perfect.  And then you just look at a mod.  How do you recover the data.  How do we get delta.  Yeah.  Well, okay, I said that if the ring is p torsion free, which is a case for this ring a info far, then having a data structure is the same as having a problem you see.  And you have a problem you send the ring a bit like.  So you just take the data structure attached to this to this for being.  Okay, and so this, this proposition is the I mean this example is the reason why I guess button shows up.  Like, describe prisms as some kind of deep perfection of the category of perfect to do it because you see that inside the category of all prisms I have this.  And subcategory from my perfect prisms and this are exactly the same thing as perfect. And I is like choosing I my ideal is be like is the same as choosing some until my of the tilt of my perfectly.  Okay.  Okay, so once you have done this, namely introduce data rings and prisms, you can define the prismatic side.  And there are several versions of it. So, for us, the one which is really relevant is the absolute version. So, let's start with our question.  Hmm.  Could you unmute the person.  Yeah, I did not hear anything.  Okay, she could you unmute the way the person who asked the question. Yeah, so I'm trying to do that but it doesn't work.  Maybe.  Yeah, there's a question on the microphone. There's a problem on the microphone.  Okay, so maybe he can ask his question by chat. Okay.  Okay, so I define those are the absolute prismatic site.  So let me fix a ring R, which is assumed to be purely completely.  And then the absolute prismatic site of the ring R, which will be denoted by our prison.  Okay, so this symbol is supposed to be present, even if it appears as a delta here.  So as a category, it's just the opposite of the category of all bounded prisms BJ together with a ring map from R to be more J.  Okay, so here there is one adjective which I did not define yet, bounded.  So it's just again, a technical condition, but which you can forget in first approximation. It's telling you that you have your prison BJ.  And what you require is that be more J here as a bounded P infinity torsion.  And this just means that if there exists some integer n big enough, so that any element which is killed by your power of P in this ring P mod J is actually already killed by P to the end.  And this is a condition you put for technical reasons, which have to do with with derived versus classical completions.  Okay, but I mean basically an object of the site is a prism with a map from R to its reduction be module J.  And then you put a topology on this category so you define covers to be morphism of prisms.  PJ goes to be prime shape prime. So morphism of prism is an obvious notion. So it's just a morphism of data rings compatible with a data structure, which sends J into J prime.  And you say it would be a cover if when you just look at the underlying ring map from B to B prime.  And it's just a P J completely facefully flat. So this means that if I take the derived answer product of B prime with B module P J over B.  I mean, first of all, it has to be concentrated in the zero. And it is then facefully flat over B module P J in the usual sense.  So slightly weaker notion that the notion of facefully flat ring morphism.  Someone would just require a condition module of PJ.  For the reason that again everything is assumed to be complete. And you want to notion which is table under completion.  So instead of looking at facefully flat morphism you just look at PJ completely facefully flat.  So there's a question. Okay, please. Okay, so first for the prison several technical so in the morphism of prison you require the J goes to J prime but I suppose it's for it should follow the J generate J prime is it correct.  Yeah, this is true. Yeah.  And also you need to have a finite disjoint unions of I mean when you have the topology also wants a risky covering of this and like an opens.  Here you just have covering by one thing of course you need to add like the disjoint union of and load or the couple. Well, if he's a product of situations then it's taking all of them will be a covering by and things.  Okay, so yeah I agree. Okay, so it should just be generated by this.  This covers. Okay.  Okay, no, no other question.  Okay. So this is the definition of the prismatic side.  And then you have two natural pre shifts on this site. So one is denoted is denoted by oprysum. And the other one is denoted by oprysum bar.  This is a functor which define on the prismatic side which send a prism BJ on the prismatic side to be. So this is for for oprysum and oprysum bar will send BJ to be more J.  And something bad should check is that these two functors are actually shifts on this prismatic side.  And they both have a name so this chief or prism will be just called the prismatic structure shift.  And the other one oprysum bar is called the reduced prismatic structure shift.  And then I mean, okay, something you can deduce from this is that you can also you could also consider the functor I prism which sends just PJ to J itself.  And then you this is also a shift on this side.  Okay, so you have these two shifts oprym these three shifts on this prismatic side.  And this is the ones we will use later on.  But before doing that I want to mention that okay, I define the absolute version of the prismatic side, you could also do the following so let me fix to start with a bounded prism a comma I.  And then I require that my ring R is leaving over AI so R is a P complete a mod i algebra.  And then you can define the variant of the absolute prismatic cycle where everything leaves over a.  So it will be denoted by R over a prism.  And it will be the category of all prism BJ, which live over AI so together with a map of prisms from AI to BJ.  As before, you want that be more J receives a ring morphism from R, which now is required to be a morphism of a mod i algebra.  So everything leaves over my bounded prism AI which I fixed at the beginning and the topology is as before.  And this is the version that batches will use and what they do I mean one of the main objectives of their paper is to compare what you get using this notion with other classical more classical p.  Addic homology series.  And so to do that, you, you, I mean, once you have defined the site you can define prismatic homology.  So, I keep the same notation as before so AI is my fixed prism and R is living over a mod i.  And then the prismatic homology of R over a, which will be denoted by prism of R over a is simply the homology of my shift or prism on this relative prismatic site.  R over a prism.  And well, actually, to be more precise you only make this definition when R is assumed to be a formally smooth over a mod i.  You could do it always, but it's not well behaved.  And the way batch also define prismatic homology for general a mod i algebra is using left hand extension from the smooth case.  In the same way as you, you would define the cotangent complex from the shift of degree one differentials in general.  But at least when, when everything is smooth, then the definition is just the community of the shift on this side.  And as I said, what batch also do is they compare this new community theory with other periodic homology series.  And I just want to mention two comparison results I prove there are many of them.  Namely the hot state and the crystalline comparison series.  So for this, because I recall notation AI is fixed prism, which is assumed to be bounded.  And R is formally smooth over a mod i.  Okay, so the first result is the hot state comparison. So it tells you that.  Let me start with the right hand side. So here you have my prismatic homology complex, which I defined before prism arm of a.  And you take its direct answer product with a mod i over a.  So it would be the same as considering the community on the prismatic side of my reduced prismatic super shift or prism bar.  And I consider a community of this in some degree.  And the claim is that this is kind of canonical isomorphic as an arm module with the module of degree I differential forms on R over a mod.  And here I mean it's implicitly assumed to be purely completely up to some small twist, which is denoted by the symbol is an I.  So boy kissing kind of twist. So I recall the notation below.  If you have some a mod i module M, you will be not by M twisted by I the tons of product of M with I mod I square I times over a mod.  So this is a rather surprising result when you sing a little bit about it because I mean you have made this definition of prismatic community just using delta rings and prisms.  And you see that naturally when you compute it, I mean when you compute the reduced prismatic homology the community groups of this complex.  So you see differential forms showing up.  Okay, and as a remark, I said before that the other question.  Yes.  So, you didn't define what is be completely smooth I imagine that completions of smooth things or direct limits, fitted limits of those are completely smooth but what is the exact definition.  So I would say that I have a map a to be, I would say it's be completely smooth if I take the derived answer product of be with a mod P over a and I require this to be sitting in the zero and being smooth over a mod P in the classical finite type sense.  Yes.  Okay.  Other question.  Okay.  Great.  Okay, that's a remark that I said before, if you want to define prismatic homology in general, you don't do it just by computing homology of the structure shift on the prismatic side.  You do this process of life can extension.  But once you have done this, I don't want to explain it in detail, but you can check that this hot state comparison result will actually generalize as follows.  Namely, if you have some a mod i algebra are P complete but not necessarily smooth. Then it's reduced prismatic homology so this complex over there.  The base change to a mod i of prismatic homology.  It actually comes equipped with a natural filtration which is increasing.  And as the properties of the great pieces of this filtration, which is called the conjugate filtration are given just by which powers of the cotangent complex of our over a mod i and suitably shifted on break is interested and periodically completed.  So, I mean, this is something you can directly deduce from the definition from the previous hot state comparison, plus the definition of both sides in general.  I wanted to point this out because one, I mean, we will see the cotangent complex appearing later one later again.  Basically the moral what you can remember from the statement is that hot state comparison gives you a way to like have some control on prismatic homology or at least its reduction module i in terms of the cotangent complex.  So if you have some information on the cotangent complex you can usually deduce some interesting properties of prismatic homology.  Okay, and then the next statement is the so called crystalline comparison.  So this is the case where you assume that in your fixed prism AI is generated by P.  Okay, then in particular because our leaves over a mod i it means that P zero in your ring.  Then you could ask the question how does this prismatic homology relate to another interesting crystal, come on, this year in in characteristic P, namely crystalline commodity.  And the answer is that, actually, they are almost the same. So if you compute crystalline commodity of our A.  Then, well, it is the same as prismatic homology except that here on the right hand side, you have to twist by four minutes so you take pullback along the four minutes of a.  Okay, so in particular if you know what prismatic homology looks like you recover crystalline commodity.  You can not necessarily go the other way because a is not assumed to be perfect so five is not necessarily an isomorphism.  But at least if you know prismatic only you recover crystalline commodity.  And this is compatible with the strict with the Frobeno structure on both sides.  This is also quite surprising because this way you get a definition of crystalline commodity without choosing divided powers and anything like that.  And the key technical statement to check this is funny exercise that if you have a repeat ocean free data ring.  If you have some element in this delta ring, so that it's first divided power is in your ring, then actually all the other divided powers are also in the ring.  This is something you can check using the existence of the delta structure.  This is one of the key inputs to in the in the proof of this crystalline comparison.  Okay.  Oh, there's a question.  Just a clarification in the theorem you wrote the upper stuff, the kind of pullback.  But since you're always working the derived the complete things is it's either some completion there or is that the bright.  Maybe I'm confused but  wait.  I'm not sure now.  I don't think you need to complete.  I think what you say is that the delta.  Yeah.  No, go ahead.  No, I as far as I understood you characterize the your  you characterize your your things like  guarz vitz forms is only after modding by I and also maybe you to be complete so it seems that everything is something derived complete relative to your finals and so no, and when you take fear, drying anche start by destroying this, I'm not  Yeah.  Okay.  Okay.  Okay.  Good, so, okay, that's all I wanted to say  about prismatic homology in general.  Now I turn to prismatic doodonissiri itself.  So as I said, I need to explain over which rings  we want to classify feasible groups  and by which kind of objects we want to classify.  So I start with the rings.  So there will be again definitions.  So the rings we will consider are called quad asymptomic.  So ring R is said to be quad asymptomic  if it satisfies the following conditions.  So first of all, it's P complete.  So this we always assume everywhere  and with bounded P infinity torsion.  So I recall that this just means  that there exists some integer n  so that everything killed by power of P  is already killed by P to z.  Okay, and then the really important condition  in the definition is that you want the cotangent complex  of R over ZP to have P complete tor amplitude  in degree minus one zero.  So this means that you take this cotangent complex  and you take its derived answer product with n  for any R mod P module n.  And then you want that these objects leaves  in degrees minus one zero  and the complex of R mod P modules.  Okay, so this is the absolute notion somehow  and then you can also define  what the quad asymptomic morphism is.  So it will be a morphism of P complete  with bounded P infinity torsion rings.  R goes to R prime  which, okay, first you want that R prime  is P completely flat over R.  So I already explained what that means.  And then you want that the relative cotangent complex  of R prime over R as P complete tor amplitude  in minus one zero.  Okay, and you can also define  what the quad asymptomic cover is.  This would be useful later.  It's the same definition,  but instead of requiring that the map is P completely flat,  you want it to be P completely faithfully flat.  Okay, but so, yeah, the important condition  is really the condition on the cotangent complex.  And this definition is due to,  I mean, it appeared in the paper of Batmouro  and Schultzern topological orgy-domology.  And the idea is that it should extend  in the world of periodically complete rings.  So there's some trouble.  We lost that speaker.  Ah, yes, coming.  Hello.  Ah.  Okay, does it work?  Sorry, I think we...  Yeah, no, we convert our connection.  Well...  Okay.  Is it okay?  Yes, sorry.  So is it good now?  Yeah, it's good now.  Okay, sorry, I think the connection, okay.  That was, I don't know.  Okay, so I was just saying that this definition  is an extension of the classical notion of symptomic,  well, of LCI ring and symptomic morphism.  But you don't make any Nussarian  or finite type assumption in this definition.  So, yeah, before giving examples,  one more piece of notation.  So I will denote the category  of all quasi-syntomic rings by QC.  And then you can look at the opposite category  and consider the topology which is defined  using the quasi-syntomic covers in the above sense.  So as I said, like maps which are quasi-syntomic  and which are P completely phase-free.  No, sorry, P completely phase-free.  Okay, and then as a notation,  if R is an object of this site,  I would just denote by R with small letters QC,  the sub-site which is formed by all rings  which are quasi-syntomic of R.  And again, undone with this quasi-syntomic topology.  Okay, so I give examples of such rings now.  The first example is just to justify the claim before  that this generalizes the classical notion of LCI ring.  So the claim is that any P complete  and SIN ring which is locally complete intersection  is quasi-syntomic.  And well, this is checked using,  I mean, actually Avramov gave a characterization  of such rings LCI in terms of the cotangent complex.  But here you just need the easy direction of Avramov C of M  to prove that any such ring is quasi-syntomic.  Okay, so that's one first class of example,  but you also have like huge rings in this category  of quasi-syntomic rings.  So, namely I claim that any integral perfected ring  is quasi-syntomic.  And the reason for this is, okay, well,  remember you have to check, okay,  first of all, it's purely completely by definition.  Then you can also check that for perfected ring,  if there could be some P torsion,  but like the P torsion is just the same as a P torsion.  So the first two conditions are checked.  And then you need to check this condition  in the definition about the cotangent complex.  And for this, well, you observe that this map from ZP,  I mean, the canonical map from ZP to R,  it actually factors through,  you can factor it through the CTA map.  So I should have written that this second map  here on the right is Fontaine Zeta map.  And whenever you have such a composite,  you get a triangle for the cotangent complex.  And now you observe that, well,  what is a inf of R once you mod R of P?  It's just R flat, which is a perfect ring of characteristic P.  And whenever you have a perfect ring of characteristic P,  its cotangent complex over FP is just zero.  Basically here is just that.  If you take any X, like it's always of the form Y to the P  for some Y, because the ring is perfect.  And then DX is just like P, Y, P minus one, DY.  And if P is zero, this is just zero.  So this way you can check that the cotangent complex  of something perfect over FP is zero.  But this just tells you that mod P,  I mean the P completions of my cotangent complex of R over ZP  and the cotangent complex of R over a inf of R agree.  Right, because in this triangle,  the other term will vanish after P completion.  But so then once you know this,  you can just, you are reduced to describe this cotangent  complex of R over a inf of R.  But then this map is a subjective and the kernel,  theta is by properties of perfected rings,  is principal and generated by a non-zero divisor.  So this means that then the cotangent complex  is just the same as R,  but shifted leaving in common,  common logical degree minus one.  Okay, so this way you check.  That in fact, in this case,  the cotangent complex even has term  amplitude in degrees minus one, minus one.  It's even better.  And then from these two class of examples,  you can construct other examples if you want.  So you can take a smooth algebra  of our perfected ring and take its P completion.  Or you can take a perfected, sorry,  yeah, take an integral perfected ring  and just mod out by a finite regular sequence.  So if it's again, if it is again,  bounded P infinity torsion,  then this would give you another example  of quasi-syntomical rings.  And here I list some examples.  So you can take the theta algebra  in one variable over OCP.  You can take OCP mod P  or you can take this characteristic P perfect ring,  FP T one over P infinity and mod out T minus one.  So these are all examples of quasi-syntomic rings.  Okay, so I think that's basically all I wanted to say  about quasi-syntomic rings,  but I just wanted to try to convince you  that many interesting examples of rings  are actually quasi-syntomic.  Okay, and one good point of this  about Mohr-Scholz's definition is that it's purely  a definition in terms of the cotangent complex.  And we have seen before,  this was a hot state comparison theorem  that if you have some control on the cotangent complex,  because of this hot state comparison,  you can usually deduce things about prismatic correlation.  Okay, now I turn to third part.  So filtered prismatic do-donate crystals.  This will be the objects we will use  to describe our P-zibber groups.  And as you can guess,  the definition will use the prismatic side.  So to state it, I first need one observation.  Let's take R to be quasi-syntomic.  And then the claim is that,  okay, you have a natural morphism of topos,  which goes from the category of sheeps  on the prismatic side, the absolute one,  to the category of sheeps  on the small quasi-syntomic side of my ring R.  And if you wonder how this,  what does this morphism of topos come from?  Well, it's defined as a composition.  So first of all, you observe that  if you take a prism AI,  so I mean, if you take some object  of this prismatic side,  you have a prism AI,  then you can look at a mod i.  So it will be P-complete.  And the claim is that,  this defines a,  a co-continuous function from the  prismatic side of R  towards the big quasi-syntomic side of our ring R.  And then you just restrict.  Yeah, but I'm,  only difficulty is checking this co-continuity  of this, of this function.  But if you are familiar with crystalline common logic,  it's like,  very similar to what you do when you,  you go from the crystalline side  to the etal or the risky side.  Except that here we work with something a bit more general.  We work with the quasi-syntomic topology.  Okay, and now what I will do is I will take my prismatic  structure shift or prism on this ideal prismatic shift.  I prism, I just push everything using this  morphism V.  So V was my notation for this morphism of topos.  I just push everything down to the co-syntomic side.  And then I claim that, okay,  actually you have a natural surjection from  oprys to O.  And I will give a name to the kernel of this  morphism of shifts.  So O here is just the structure shift  on the quasi-syntomic side.  This kernel is denoted like this.  So it's what is called the first piece of the  Niagara filtration on this prismatic shift oprys.  Well, the reason for this notation is that,  I mean, there is this notion of Niagara filtration  of a prism, which is defined for any,  you could define for any positive integer I.  Here I only need the first piece of the Niagara filtration.  I would just take as a definition that it is  the kernel of this surjection, which I did not explain.  And then a property of this is that, well,  because any data in other Frobenius,  so this shift oprys will come with a Frobenius morphism,  Phi.  One can check that this kernel has the properties  at Phi of the kernel.  So Phi of the first piece of the Niagara filtration  for oprys actually lies in high-price times oprys.  So morally on this first piece of the Niagara filtration,  the Frobenius is divisible by, well,  let's say the locally I, the ideal is generated by  non-zero distinguished element.  The idea is that on this first piece of the Niagara  filtration, Phi is divisible by this distinguished element.  So there is question from over here.  Yeah, there's a question.  I cannot hear.  Can you put the microphone on?  You should activate his.  I guess you can unmute him.  Oh, yeah, now his microphone must be on.  Internet, I don't.  Oh yeah, now you can speak.  So why do you write oprys times oprys?  I think oprys is an ideal in oprys.  Oh, yeah, sorry, sorry.  Yeah, it's a typo.  Yes, thank you, sorry.  Yeah, it's just because later it will appear.  Yeah, sorry.  Okay.  Okay.  Okay, so then the definition of a filtered prismatic  dunic crystal is,  so let me again fix a quadsentamic ring R.  And then a filtered prismatic dunic crystal is by definition  of collection, a triple M, fill M and five M.  So M is finite located free oprys module.  First of all, then fill M inside it will be  some oprys sub module.  And finally, five of M is morphism, ring morphism  from M to M, which is assumed to be a file in R.  Okay, where five is a Frobenius of oprys.  Okay, and then you ask three conditions on this triple.  So first of all, you want that this Frobenius five M  sends a fill M to I press times M.  Okay, but then there is an obvious oprys sub module of M  which also have this property that five M of it  is leaves inside I press times M.  Namely, consider the first piece of the Nagat  filtration of oprys times M.  Then because of the previous slide, we know that  this is contained in I press.  And so this sub module, first piece of the Nagat  filtration times M also has a property that five of it  is contained in I press times M.  And then the second axiom you have is you want  that this sub module is in fact contained in fill M.  And once you have done this, you want to fill M  and once you have done this, then you know that M  module of fill M will be a module over oprys module  of the first piece of the Nagat filtration.  But remember that oprys module of the first piece  of the Nagat filtration was by definition  is just the structure shift of the quiescentomic site.  Oh, so M mod fill M will be an O module  and you ask that it is finite locally free.  And then the third condition is that the image of  the filtration by the Frobenius is big enough  in the sense that it will generate I press times M  as an oprys module.  So if you are familiar with crystal and deudonis theory,  again, it's very reminiscent of the usual notion  of deudonis or filter deudonis crystal.  And the third condition is just in this setting  which has been a formulation of the condition  that the filter crystal is admissible  in the sense of quotient.  So that's somewhat just the obvious extrapolation  of this classical notion to the setting of the prismatic site.  Okay, and then notation is if R is my quiescentomic ring,  I will denote by Df of R the category of all filtered  prismatic deudonis crystals over R.  And the morphism are the obvious ones.  I mean, they should be oprys linear  and they should be compatible with Frobenius  and with the filtration.  Okay, and so now I can come to the statements  of the two main results we proved.  So I will fix again once and for all now  quiescentomic ring R.  And let me take G to be a feasible group over R.  Then I can define M prism of G to be X1 of G by oprys.  And here when I write this curly X1 is supposed to be  like the local X groups in the category of abelian sheeps  on the quiescentomic site.  So you check that your feasible group G,  where the reduction to the case of finite locality free  group schemes defines a sheep, an abelian sheep  on the quiescentomic site.  And oprys is also an abelian sheep on this site.  So you can take X1 in this category.  And then you also define fill M prism of G to be the same  thing except as you replace oprys by the first piece  of the Naga filtration.  And then the first main result is that,  so if G is as before, then this triple M prism G,  fill M prism G and for benus of M prism of G,  which is just by definition the for benus coming  from the for benus on the prismatic on the prismatic sheep.  Then the claim is that this is actually an objective  of this category DF of R.  So this is a filtered prismatic dodomic crystal over R.  And in what follows, I will denote it by M prism G underline.  Okay, and as a remark, if P is zero in my ring R,  so in the characteristic P situation,  you can check using the graph,  you can check using the crystalline comparison theorem,  which was discussed in the first part,  that in fact, this is just the same thing  as the usual function you will find,  people have studied in crystalline dodomy theory,  which you find, for example,  in the book of Bertolo, Brin and Messing.  So this is nothing new in characteristic P.  Okay, and then our second main result is that,  well, this filtered prismatic dodomy factor  underline M prism, which associates to G,  M prism of G underline.  It actually realizes an anti-equivalence  between this category of BT of R of visible groups over R,  and the category DF of R, which I defined before.  And as a bonus, I mean, as a byproduct of the proof,  you actually obtain that the prismatic dodomy factor,  G goes to M prism of G, is already fully phase-free.  But if you really want to get an equivalence of categories,  so if you want to be able to describe the essential image,  you need to add the filtration in the picture.  Okay, so now I make a few remarks about the,  these two results and about the way we prove these two results.  So first remark is that, as I said before,  in characteristic P, you just recover the usual factor  from dodomy theory.  And so in particular from the serum two,  this classification result,  well, you can deduce that the crystalline dodomy factor  is an equivalence for all quasi-syntomic rings  in characteristic P.  And well, this was actually already known,  of course, in some cases.  So if you look at LC irons, which are also excellent,  then fully phase-freeness was proved in the end of the 90s  by To-Yong and Messing.  And more recently, Eichelot has proved that  this factor is actually an equivalence  when the rings is, okay, Nocerian LCI  and more of a F finite.  So this means that the Frobenius is finite.  And this particular implies XM.  So in this case, the result was already known.  Okay, then I wanted to explain now  that this category DF of R may seem a bit abstract,  but there is an interesting class of quasi-syntomic rings  for which you can make it more explicit.  Namely, you will say that the ring  is quasi-regular semi-perfectory.  If it is quasi-syntomic, of course, first,  and then you also want that there exists  that a perfected ring which maps subjectively onto R.  It has to be big enough.  Examples of such in the list of examples I gave before,  where you can look at any perfected ring is obviously,  because we know it's quasi-syntomic  and the second condition is trivial.  Any perfected ring will be quasi-regular  semi-perfectory, but also like a quotient of such a ring  by a finite regular sequence,  which has bounded P and P torsion.  So these are examples of such rings.  Well, for such rings, it's like for perfected rings,  and the example we saw, in fact,  the Quotantin complex after P completion  is just sitting in the minus one.  And for this reason, you can check,  using Hatch-Tech comparison,  that for such a ring, the prismatic side has a nice feature  that it admits a final object in this case,  which you can just describe as,  if you want the absolute prismatic homology,  comes with a natural idea.  And then also you can check in this case  that if you take the first piece of the Naga-Tutration  on this prism to be just the inverse image of I  by Frobenius,  then the quotient is isomorphic to I itself.  So two examples of computation of this initial object,  or some final object.  The first case is if I is perfect to E,  then this final object is just given by the ring A-info  of R together with the kernel of the theta map.  So, okay.  In this case, the prism is perfect.  So Frobenius is an isomorphism.  So there is always some choice  that you can work with the map theta,  or with its pre-composition with Frobenius minus one,  and then consider theta tilde.  It's just a matter of conjunction.  Okay, so that's the first example.  And then another example,  if you take a ring which is called a regular  somaiparfectoid and with P zero in this ring,  then this initial prism is just the same as  a crease of R together with the idea generated by P.  So you can compute it in several situations.  And then we make a definition.  It's very similar to the one we had before.  A field of prismatic design in module over R  will be a collection M, field M, and phi M.  But no, I just have, instead of having  finite locality free modules over the prismatic shift,  I just have a finite locality free module  over this ring, prism R,  a certain sub-module of it, field M,  and a phi in R map, phi M.  And you just ask the exact same actions as before,  but no, you only work with modules  over this ring, prism R.  So I want to repeat them  because I don't have much time left.  But okay, so you can make this definition  and then the claim is that if my ring  is quasi regular somaiparfectoid,  I can in fact just evaluate all the objects I have on this,  so I should have written final on this final object.  And the claim is that this gives an equivalence  between the category of filtered prismatic  due to the crystals over R  and the category of filtered prismatic  due to the modules over R.  So in other words, in this case,  you can make the category more explicit.  You can just work with modules instead of working  with this object, this category DF of R.  And moreover, if the ring is perfectoid,  then you can do even better.  Then you can even forget the filtration.  So just look at the forgetful function  for this category of filtered prismatic  due to the modules over R  to simply what you can call a prismatic due to the module,  but in this case, it already has a name.  People call them minuscule break is in FARG modules.  And then the claim is, in this situation,  this forgetful function is in fact an equivalent.  So in other words, as a special case of the CRM2,  you see that you recover the fact  that for perfectoid ring,  peaceful groups over such a ring  are classified by minuscule break is in FARG modules.  But this is a bit cheating because in fact,  we need as an input for the proof,  for the proof of CRM2, we need a special case of this.  Namely, we need that  peaceful group over a variation ring,  which is perfectoid and has algebraically closed  fraction field is the same thing  as a minuscule break is in FARG module.  And actually, it's not difficult to do.  You can deduce the case of all perfectoid rings  from this special case using VT-SAN target.  But so we need this as an input.  Okay, so can I just take five minutes to finish?  Or should I stop now?  Okay, go ahead.  So I try to finish quickly.  So I also want to mention that, as I said,  in general, you really need the filtration  if you want to describe the essential image  of this prismatic do-don't-functor.  But I just said before that in this remark  that for perfectoid ring,  the filtration is actually unique.  So it's not needed to state the classification theorem  in this case.  This is not true in general,  but this also works for P-complete regular rings.  And as an example of this,  I just take the ring of integers in some  discreetly valued extension K of QP  with perfect residue field.  So for example, take a ring like ZP.  And then also this case was, of course,  already known before.  So then you can prove that P-complete groups  over this such a ring are classified  by so-called minuscule brachycin modules.  And this has been done by Bray and Kissing,  at least for all P,  and then extended by Kim, Lau and Yu to all P.  But we can also recover this,  namely by first proving that for such a ring,  you can forget the filtration.  And then checking that you can just evaluate  and, well, there is a natural prism attached  to such a ring once you choose a uniform measure.  It's not a final object,  but still you can check by some reduction  to the perfect situation that in this situation,  evaluation on this object is again unequivalent.  And so, and then we check that the factor we have  is actually the same as the one which has been studied  by Bray and Kissing, all these people.  But now the good point is that some of you  directly learned in the correct category.  So the proof works uniformly for all P.  You don't need to make a special argument when P is too.  Okay, and then just two words about the proof.  So for theorem one, it's not surprising.  So you just follow the strategy of Bertel and Bray  and Messing in their book.  So you have this definition of this triple.  You want to check that it is a filtered  prismatic dosonary crystal.  And the idea is, well, you have to understand  how what is X1 looks like.  And in fact, for any group object in a topos,  you have some device to make computation  about this X groups, at least in low degrees.  So this is explained in the book of,  Bertel and Bray and Messing.  And you are reduced to compute some prismatic  commodity groups.  So first step, you use a theorem of Reno  to reduce to the case of Psemer groups  which comes from some Abelian scheme.  And then, via this Bertel and Bray and Messing  partial resolution of any group object in a topos,  what you just need to do is understand  precisely prismatic commodity of Abelian schemes.  And a key tool for this is provided by the Hatch  state comparison theorem for prismatic commodity.  So, but the ideas are really similar to the one  of Bertel and Bray and Messing.  And then, theorem two, the proof is more difficult,  but I just want to point that the key idea  is to use quasi-syntomic descent.  So here, I introduced just before the notion  of quasi-regular, so my point is that  the very nice feature of this site,  this quasi-syntomic site, which was observed  by Bayt-Maurot and Schultz-Hou, is that  this quasi-regular semi-perfected ring  actually forms the basis of this quasi-syntomic topology.  Or by any quasi-syntomic ring, by extracting  enough piece-roots, you can always make it  quasi-regular semi-perfected.  And this means that once you have defined this  factor, so once you have proofs here on one,  then you can, to prove that it's unequivalent,  you can somehow reduce to the quasi-regular semi-perfected  situation.  And then everything is more concrete, as I said.  Instead of having crystals, we just have modules  over this ring prism arc.  Okay, and then the hard work starts, but I wanted  to say this because it shows that even if you,  if you want to, if you are a non-proliferation  like getting classification results over  like Nucleus and rings, then the proof is such  that actually you really need to work with this  big category QC, because the argument is  by descent.  This is very big quasi-regular semi-perfected ring.  Okay.  Okay, and then I will skip this point.  I will skip this point.  I just want to finish by saying that there are some  natural questions which are left open.  First of all, it would be interesting to see what happens  for more general baserings.  We only prove results about quasi-syntomic rings,  but for example, in the work of Tzink, you can find  some results for a very general,  periodically complicated ring.  So, for example, in the work of Tzink,  you can find some results for a very general,  periodically complete rings.  But I have no idea how to do it for more  general rings.  And then another thing we don't have is something  like deformation theory in this setting.  So for this prismatic duodonné functor,  so for the usual crystalline duodonné functor,  you have this gotennik-messing deformation theory,  which is very powerful.  But here,  we don't have any analog of this.  And then finally, a final remark is more anecdotal,  but it's about the case of perfect rings  in characteristic P.  I just want to point that in this case,  well, because if you live over like an object  of the prismatic side of such a ring R,  it will be P torsion-free in particular.  So you can always map this prismatic structure  shift on this side to the same thing where you invert P  and take Q to be the quotient of this injectable.  And then you check that, well, in this situation,  the prismatic duodonné module of any P-zero group G  is in fact the same thing as home from G  to the push forward to the quadrathymtomic side  of this shift on the prismatic side.  I mean, this is just obtained by looking at the associated  long exact sequence.  And then a natural question is whether the,  I mean, front end has, was the first to give  a general definition of the duodonné module  over such a perfect ring of characteristic P.  And the definition looks a bit similar,  except that instead of this mysterious push forward here,  you have this shift of VIT covectors.  And so it would be also interesting,  and we did not do it to know how to relate this object here  to Fontaine's original definition of the duodonné  functor with VIT covectors,  but without choosing the general crystalline comparison  C-ray, which of course implies this comparison,  but just directly using the two definition.  Okay.  Thank you very much.  So thank you very much for the interesting lecture.  So are there any questions?  Yes, Luke.  Could you unmute him?  Yeah, I'm trying, but it doesn't work.  I don't see one.  Luke, could you unmute yourself?  Luke, can you put the microphone on?  Now you can hear me.  Yes.  Okay.  So I think you skipped one of the last slides where you were  writing something about classification of finite,  locative free community group schemes.  You discussed PDVS groups.  I presume that you, but presumably you can also classify  truncated BTs and maybe more general.  So that is the,  Oh, I can't hear you well.  Okay.  So it was just that I wanted to see this slide again.  Yeah.  But then is it in terms of a module or whether a complex maybe?  Okay.  So we only do it for over perfected rings.  So I think that's the question.  I think that's the question.  I mean, the idea is the same as the one you find in the paper  of kissing.  Well, actually you use the fact that you can just like identify  the category of final locality free group schemes with the  category of like two times complex.  Of peace by groups.  With an isogenic between them.  And the only difficulty in general for general quasi-syntomic  rings is I'm not sure by which kind of objects you would classify  fine and local if we group schemes.  So for perfected rings, as I said, for peace by groups,  you can forget the filtration.  You just have this minuscule boy kissing fog modules.  So it's not difficult to guess what, which can,  by which kind of objects you will classify fine and local if we  group schemes.  So it would just be a module of your ring prism,  R, which is just a inf of R in this case of projected dimension  less than one killed by your power of P with a four been used  on a Vershebu.  But in the general situation where you should also take the  filtration into account, I'm not sure how to describe the  very local if we group schemes.  So that's why we only did it for perfected rings.  Actually, this was already known in all cases except maybe when  peace to my work of law.  I had another question you mentioned work by think.  So is there a relation between this place and the theory?  Yes, this is another slide.  I had to skip here.  So let me take our game to be one first one reduces to the case  of quasi regular semi perfected rings by quite a  systemic descent.  And then, well, you have this natural map from prism R to R  was kernel is the first piece of the nugget filtration on this  prison.  It's a map of Delta rings.  So it's a map of rings.  But I said that the forgetful functor from delta rings to rings  as a right adjourn, which is the big vectors fun.  Yes.  So this mapping juices a map of delta ring from prison R to W  of R.  And using this observation, you can actually check that you have  a functor from our category DF of our filter prismatic  modules.  To the category of displays of our in the sense of.  It's not an equivalent.  I mean, the classification of sink is both more general and  more restrictive.  It's more general in the sense that I think he only assumes a  ring to be periodically complete.  But then he has to restrict to formal feasible groups because  when he is to like some difficulties.  So this is due to the fact that this functor we get from DF  of R to this place of R is not an equivalent.  But it is when you restrict to any important objects.  The terminology of.  Thank you.  Thank you.  Okay.  Thank you.  Then offer.  Okay.  So I remember the concern in this.  Welcome to the net.  There were some after seeing the way in particular some papers  of law where he.  He looked at the for example, complete.  Mix characteristic.  The local rings with.  Is it perfect or not perfect.  Then he gets some using saying he get, I mean, he gets.  He does things which was using windows and friends.  So they get a very simple linear algebra.  Sinks which classify PD visible groups.  Over mix characteristic regular rings, but sometimes he needs to.  He needs to get a.  To assume it is a.  The result is different and the rest of it is not perfect.  But any case he has some.  Do you get a relation between.  What you do and use to, but maybe there are several such references.  I don't remember exactly, but I remember it's quite simple.  Like you only need to give.  Some.  Some final free modules and some maps.  Composition equal to the equation or something like this.  Is it possible to relate it to your.  Yes.  That's what I mentioned here.  So.  As I said, in.  For such.  A P complete regular rings.  This the filtration is also not relevant.  So you can describe everything.  I think, as I said, using currency realms, you can.  Get a natural prism attached to the situation.  So for example.  In the simple case where I just okay.  Yes.  You just take this break is in prison.  So you just take like.  Is it your field double brackets.  A.  U for some formal variable U.  And he is some ascent-style polynomial,  which is like determined by after you choose a uniform either you.  You have a natural map from this track S.  Towards.  Your ring R and either generator of this.  Of the kernel.  And then yes, then, then you get a quite simple classification in this case.  in this case as modules, find locally free modules  over such a ring together with the Frobenius  which has the properties that it's,  like after linearization it's co-carnalized.  After you invert E it becomes,  the linearization becomes an isomorphism,  the linearization of Frobenius.  So, but for checking that the functor is really the same  as the one which is used in law.  At least we checked it in this particular case,  but this is the same as the functor  that people usually consider.  But in the marginal situation, I'm not sure we checked it.  So there was also a technical question about  whether the resdiel field is,  when the resdiel field is not perfect,  I think he needed to work like sync  with only formal pt-visual group,  but maybe I don't,  do you get,  do you get  easy to,  that is classified,  well I don't have a,  now I'm sorry.  When difficulty that low on contours is that,  I mean he uses crystalline Jordanian theory,  yes, so somehow you have your ring R,  some of us you reduce to,  to arm of p where you can use crystalline Jordanian theory,  but then you need to go back from arm of p to R.  So you need to use gotonic messing theory.  But the issue is that this,  like this divided power structure on the idea  and generated by p is not important when p is two.  Gotonic messing theory only works well  when the divided power structure is important, right?  So I think for this reason,  you have to do some,  usually you have to do some extra work when p is two.  But here somehow I think this issue does not appear  because in some sense you directly,  with this prismatic Jordanian function,  you directly land in the correct category.  Like again, back to this example where R is okay,  in the original work of Boy and Orkissin.  Actually you, what you first do is you produce a factor  from the category of feasible groups  over this ring R towards the category of filtered modules  over another ring that's frack S,  which is like the p completion of the pd envelope of E  inside this ring frack S.  And this is usually just called curly S.  And then you have to do some semi linear algebra  to check that this is indeed the same  as the category of minuscule Boy and Orkissin modules.  But here we directly get a factor  from feasible groups over R  towards minuscule Boy and Orkissin modules.  For some of we avoid this difficulty  and that's why we don't have any assumption on p.  I have another small question.  Maybe what is, so you said that in the definition  of a prism you have a Cartier divisor.  You can, are there examples where it is not globally  principle?  I know I don't know any example writes,  writes not principle actually.  I think all the examples I'll know the ideals are principle.  I see basically one question.  Can you hear me, Ophar?  Yes.  Okay, so the question was, does there exist any coincidence  between the filtration of prismatic Jodonaic crystal  and the filtration on bun G on the farfontaine curve  since Fark proposed that using chromatic filtration  one can correspond perfect with BT1 with bun G.  One can what?  Correspond perfect with BT1 with bun G.  I don't know what that means.  I'm just reading what it is.  BT1 is like Berserk's detail of level one.  I guess so, but I guess what is perfect with BT1 is,  BT1 over perfect with bun G.  I don't know what it means.  But maybe BT1 over perfect, okay, I don't know.  But I don't know any relation with bun G.  I mean, of course, Burkis and Fark modules are related  to modifications of vector bundles on the Fark-frontaine curve.  But bun G is like the bun G bundle, so.  Yes, so I don't see.  For a group, they do it with a.  I guess, well.  Yeah, so front-end.  The G should be.  G should be reductive group or something.  So I don't see any relation.  Okay, well, if I, so it's not, I'm not,  I will need more to see it in more precise form.  So, yeah, so as far as I understand, okay.  Yeah, okay, so by any case, you prove that the same construction  of display is canonically correspond to what you do,  using these functors to display,  you get exactly things construction,  which is defined for more general rings.  Right, yes.  Okay, all right, so.  Yeah, so then this partly answer,  because the work of Lowentz-Holm is kind of trying  to use this display to do some complicated tricks  to which I don't, okay, so it's,  and of course you get also the crystalline deodorant theory  from your.  Yes, as a.  You need to use some morphism  between relating the crystalline, okay, but you wrote it,  I don't remember if you put it in the slides.  I just put a remark, I mean, yeah,  basically you can check that the usual,  I mean, you have this prismatic side,  you have the crystalline side,  let's say you just push everything  to this color-syntomic side  and make the comparison there.  And then the claim is that the two categories you get  are actually the same, so.  For the ring.  So the ring is killed by P.  Exactly, yes.  So you get a, okay, I have to, this was a.  I mean, basically this follows  from this crystalline comparison theorem.  Okay.  Okay, thank you, so I will have to think.  Okay, not there.  The slides will be online at some point.  Okay, thank you.  You're welcome.  Okay, goodbye.  Bye.  Thank you very much, so goodbye.  Thank you.
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So, welcome to the Paris Peking Tokyo seminar. So it's my great pleasure to introduce the speaker, the first speaker on this Zoom session today. So, the speaker is Arthur Seijou-Rubula, also he will tell us the prismatic doness theory. So please start. Okay, well, hello everyone. And well, thank you very much to the organizers for giving me the opportunity to speak and for setting up everything in this very particular context. So my talk of today is about prismatic doness theory. And so everything I will discuss is a joint work with UNS Anschutz. And for all this talk, I will fix the prime number P once and for all. So the goal of our work was to prove some classification results for P-isible groups of various kinds of rings. And the main tool we used to do this is the recent theory of prisms on prismatic homology, which has been developed by Bat and Scholtz. So the plan of my talk will be the following. I will first spend some time, maybe 20 minutes, half an hour to discuss a little bit prisms on prismatic homology. So recall the basic definitions and constructions. And then I will start speaking about the joint work with Uranus. So first of all, I need to tell you about which kind of rings we want to classify P-isible groups. And these rings are called quasi-syntomic rings. So this is what I will do next. I will explain the definition of these rings. Then I need to tell you by which kind of objects we classify P-isible groups over such rings. And we call them fitter prismatic geodontic crystals. So I will explain the definition and some basic properties of these objects. And then finally, I will explain our main results and say a few words about the proofs and some corollaries of these results. Okay. So let me start with prisms on prismatic homology. And I should make it clear. Everything in this part is true to but and also. So the theory of prismatic homology relies on two important basic definitions, the notion of a delta ring and the notion of a prism. So let me start with delta rings. So I will assume always that my rings leave over Z localized away from P. And then the delta ring will be a commutative ring A together with a map of sets, just of sets, delta, which goes from A to A, which has the following properties. So first of all, it maps 0 and 1 to 0. And then you want to prescribe how delta behaves with respect to multiplication and addition. And this is given by these two formulas, which may look a bit strange the first time you see them. So delta of a product if x, y for all x, y in A is x to the P delta y plus y to the P delta x plus P times the product of delta x by delta y. And then you have a similar formula for the addition. So delta of a sum x plus y is just delta of x plus delta of y. And then you need to correct it by this term here. And so observe that if you use a binomial formula to expand this x plus y to the P, then all the terms except x, P and y to the P, which are cancelled here, are actually divisible by P. So this expression makes sense in any ring. And you don't need to assume that the ring is P times 3 or anything like this. OK. So this is what the delta ring is. So the ring A together with a map of sets satisfying these three properties. So if one has a delta ring, A delta, so you can do the following operation. So you can define a map phi, which goes from A to A, and which sends x to the P plus P times delta x. And then you check an exercise that the identity is satisfied by delta, which I had before over there. Actually, exactly what you need to check that this map phi, or just a map of sets from A to A, is actually a ring morphism. And moreover, just by definition, it's a lift of four binos. So if you kill P, then these terms disappear. And it just becomes x. It just becomes a four binos. X goes to x to the P. And conversely, assume you start with a ring, commutative ring A, together with ring morphism phi, which lifts four binos modulo P. Then if you assume that the ring you have is P torsion free, you can actually just divide phi x minus x to the P by P and define this way, a delta structure on your ring. So in other words, in first approximation, you can think of delta rings as rings together with a lift, with a ring morphism, which lifts four binos. But the two notions are not exactly the same thing when you have P torsion in your ring. OK. Another point of view, which I just want to mention about delta structures, is the following. So if you have a ring A, then giving yourself a delta structure on A is in fact the same thing as specifying a ring morphism from A to the ring of length to V vector over A, which will be a section of the natural projection on the first component. And the recipe for this is assume you start with a delta structure, delta on your ring A. Then you just look at the map from A to W2 of A, which sends x to x, delta x. And once again, you can check that the axioms about delta, which tells you that you have a delta ring, is exactly what you need to verify that this is a ring morphism. OK. And this point of view is useful because it allows you to prove that this category of delta rings, contrary to the category of rings with a lift of four binos, has all limits and collines, which you can just compute on underlying rings. And so in particular, this forgetful factor from delta rings to rings, we have both the left and the right adjoint. And the right adjoint is given by the V vector's function. OK. So that's the first remark. And then another remark, which is more an exercise that you can do. I said that there are examples of delta rings with p-torsion, but it can never happen that you have a delta ring in which p to the n is 0 for some A. OK. This is something you can check just using the definition of the delta. OK. Very good. So we will see some examples later. Oh, wait a second. It's a trivial remark that 0 is a delta ring, so p to the n. Yeah, OK. I assume that 0 is different from 1. But OK, yeah. If you want. No, but if you want all limits and collines, then maybe. OK. Yeah. Then I should have said that there is no non-zero delta ring in which p to the n can be 0. Thank you. OK. So as I said, we'll see examples soon, but no, I'll come to the next definition, namely, the definition of a prism. So what is a prism? It's just a pair, a, i. Well, a is a delta ring. So usually in the notation, I will just forget the delta. OK. I'll just say a is a delta ring without mentioning delta. It's implicit. So you have a delta ring, a, and then you have some ideal i inside a. And again, this pair has to satisfy some properties. So first of all, you require that I define the Carti divisor on spec a. So it's just like locally principle generated by a non-zero divisor. Then you also require that I is p iidically complete. So for technical reasons, you should mean this in the derived sense. Also, soon we will make some assumption on the ring, which I'm sure that in practice, like the derived on the classical completions agree. So first approximation, you can just think that this is classically p iidically complete. And then the last condition, the most important one. So I has to be a pro-zariski locally generated by a distinguished element. And what is a distinguished element by definition is just an element of D, which has the properties of delta of D is a unit. Okay. And once again, in practice, I will always be principal. So you should just remember that a prism is a pair a, i, where a is data ring and let's say i is principal generated by a non-zero divisor, which is distinguished in the sense that it's made by delta is a unit. And moreover, the ring should be p iidically complete. Okay. The important condition is the last one. And so now I can give two examples of such prisms. So first of all, okay, if you have a p complete on p torsion free delta ring, a, then the pair formed by a and the ideal generated by p is a prism. Okay, so I mean, remember, so you need to check three conditions. So first of all, I had assumed that p, there is no p torsion, so please non-zero divisor. And by assumption, I also require that my ring is p complete. In this case, I is just p. And finally, the only thing you have to check is actually that delta of p is always a unit. Okay, and this, I mean, this is also what you need to do this little exercise I gave before that it cannot happen that p to the n is zero. I mean, the way to prove this is just to check that delta p has to be unity in any data. Okay, so that's one first class of example. And now here is another interesting example of prisms. So, let me give one more definition first. So we will say that the prism is perfect. If it's for Benus phi. So remember, whenever you have a prism, sorry, a delta ring, you have this delta, then you can define a phi which leaves for Benus. There is a formula that phi to the x is x to the p plus p times delta x. So what you require is that this ring morphism phi is actually an isomorphism. And then the claim is that actually the category of perfect prisms is the same as the category of integral perfect with rings. And how do you see that? Well, I mean, you can define a function in both directions which are quasi inverse of each other. You how so in one direction assume you start with an integral perfect to a dream. Then you can do this classical font and construction. You can consider a inf of r. So what you do is first you tilt your perfect ring. So you get perfect ring of characteristic p r flat. And then you take its string of bit vectors. This is what is called a inf of r. And a inf of r comes with a natural map sitar, which goes from from it towards our. I mean, part of the definition of at least consequence of the definition of an integral perfect to a dream is that this will be principle generated by a non zero divisor. And you check that in fact, this generator which is usually denoted by Xi in p. So the first theory is in fact distinguished in the previous sense. So, namely, so I should have said first that this ring is p torsion free. Our flat being perfect. And so it has, it comes with a natural for being used. And you just take the delta structure attached to this to this for being used to. And then my claim was that the generator of this idea is actually distinguished. Well, the way I mean you can check this by proving more generally that in such a delta ring and that delta ring of this form bit vectors of some perfect ring. An element is distinguished it on only if it is primitive of degree of degree one, which means that when you write the expansions of some of the time times powers of p, then the coefficient of p has to be a unit. And this you can check for. Okay, so this is one one factor. And then if you want to go in the other direction. It's even more simple you just mod out. You have a prison AI, which is assumed to be perfect. And then you just look at a mod. How do you recover the data. How do we get delta. Yeah. Well, okay, I said that if the ring is p torsion free, which is a case for this ring a info far, then having a data structure is the same as having a problem you see. And you have a problem you send the ring a bit like. So you just take the data structure attached to this to this for being. Okay, and so this, this proposition is the I mean this example is the reason why I guess button shows up. Like, describe prisms as some kind of deep perfection of the category of perfect to do it because you see that inside the category of all prisms I have this. And subcategory from my perfect prisms and this are exactly the same thing as perfect. And I is like choosing I my ideal is be like is the same as choosing some until my of the tilt of my perfectly. Okay. Okay, so once you have done this, namely introduce data rings and prisms, you can define the prismatic side. And there are several versions of it. So, for us, the one which is really relevant is the absolute version. So, let's start with our question. Hmm. Could you unmute the person. Yeah, I did not hear anything. Okay, she could you unmute the way the person who asked the question. Yeah, so I'm trying to do that but it doesn't work. Maybe. Yeah, there's a question on the microphone. There's a problem on the microphone. Okay, so maybe he can ask his question by chat. Okay. Okay, so I define those are the absolute prismatic site. So let me fix a ring R, which is assumed to be purely completely. And then the absolute prismatic site of the ring R, which will be denoted by our prison. Okay, so this symbol is supposed to be present, even if it appears as a delta here. So as a category, it's just the opposite of the category of all bounded prisms BJ together with a ring map from R to be more J. Okay, so here there is one adjective which I did not define yet, bounded. So it's just again, a technical condition, but which you can forget in first approximation. It's telling you that you have your prison BJ. And what you require is that be more J here as a bounded P infinity torsion. And this just means that if there exists some integer n big enough, so that any element which is killed by your power of P in this ring P mod J is actually already killed by P to the end. And this is a condition you put for technical reasons, which have to do with with derived versus classical completions. Okay, but I mean basically an object of the site is a prism with a map from R to its reduction be module J. And then you put a topology on this category so you define covers to be morphism of prisms. PJ goes to be prime shape prime. So morphism of prism is an obvious notion. So it's just a morphism of data rings compatible with a data structure, which sends J into J prime. And you say it would be a cover if when you just look at the underlying ring map from B to B prime. And it's just a P J completely facefully flat. So this means that if I take the derived answer product of B prime with B module P J over B. I mean, first of all, it has to be concentrated in the zero. And it is then facefully flat over B module P J in the usual sense. So slightly weaker notion that the notion of facefully flat ring morphism. Someone would just require a condition module of PJ. For the reason that again everything is assumed to be complete. And you want to notion which is table under completion. So instead of looking at facefully flat morphism you just look at PJ completely facefully flat. So there's a question. Okay, please. Okay, so first for the prison several technical so in the morphism of prison you require the J goes to J prime but I suppose it's for it should follow the J generate J prime is it correct. Yeah, this is true. Yeah. And also you need to have a finite disjoint unions of I mean when you have the topology also wants a risky covering of this and like an opens. Here you just have covering by one thing of course you need to add like the disjoint union of and load or the couple. Well, if he's a product of situations then it's taking all of them will be a covering by and things. Okay, so yeah I agree. Okay, so it should just be generated by this. This covers. Okay. Okay, no, no other question. Okay. So this is the definition of the prismatic side. And then you have two natural pre shifts on this site. So one is denoted is denoted by oprysum. And the other one is denoted by oprysum bar. This is a functor which define on the prismatic side which send a prism BJ on the prismatic side to be. So this is for for oprysum and oprysum bar will send BJ to be more J. And something bad should check is that these two functors are actually shifts on this prismatic side. And they both have a name so this chief or prism will be just called the prismatic structure shift. And the other one oprysum bar is called the reduced prismatic structure shift. And then I mean, okay, something you can deduce from this is that you can also you could also consider the functor I prism which sends just PJ to J itself. And then you this is also a shift on this side. Okay, so you have these two shifts oprym these three shifts on this prismatic side. And this is the ones we will use later on. But before doing that I want to mention that okay, I define the absolute version of the prismatic side, you could also do the following so let me fix to start with a bounded prism a comma I. And then I require that my ring R is leaving over AI so R is a P complete a mod i algebra. And then you can define the variant of the absolute prismatic cycle where everything leaves over a. So it will be denoted by R over a prism. And it will be the category of all prism BJ, which live over AI so together with a map of prisms from AI to BJ. As before, you want that be more J receives a ring morphism from R, which now is required to be a morphism of a mod i algebra. So everything leaves over my bounded prism AI which I fixed at the beginning and the topology is as before. And this is the version that batches will use and what they do I mean one of the main objectives of their paper is to compare what you get using this notion with other classical more classical p. Addic homology series. And so to do that, you, you, I mean, once you have defined the site you can define prismatic homology. So, I keep the same notation as before so AI is my fixed prism and R is living over a mod i. And then the prismatic homology of R over a, which will be denoted by prism of R over a is simply the homology of my shift or prism on this relative prismatic site. R over a prism. And well, actually, to be more precise you only make this definition when R is assumed to be a formally smooth over a mod i. You could do it always, but it's not well behaved. And the way batch also define prismatic homology for general a mod i algebra is using left hand extension from the smooth case. In the same way as you, you would define the cotangent complex from the shift of degree one differentials in general. But at least when, when everything is smooth, then the definition is just the community of the shift on this side. And as I said, what batch also do is they compare this new community theory with other periodic homology series. And I just want to mention two comparison results I prove there are many of them. Namely the hot state and the crystalline comparison series. So for this, because I recall notation AI is fixed prism, which is assumed to be bounded. And R is formally smooth over a mod i. Okay, so the first result is the hot state comparison. So it tells you that. Let me start with the right hand side. So here you have my prismatic homology complex, which I defined before prism arm of a. And you take its direct answer product with a mod i over a. So it would be the same as considering the community on the prismatic side of my reduced prismatic super shift or prism bar. And I consider a community of this in some degree. And the claim is that this is kind of canonical isomorphic as an arm module with the module of degree I differential forms on R over a mod. And here I mean it's implicitly assumed to be purely completely up to some small twist, which is denoted by the symbol is an I. So boy kissing kind of twist. So I recall the notation below. If you have some a mod i module M, you will be not by M twisted by I the tons of product of M with I mod I square I times over a mod. So this is a rather surprising result when you sing a little bit about it because I mean you have made this definition of prismatic community just using delta rings and prisms. And you see that naturally when you compute it, I mean when you compute the reduced prismatic homology the community groups of this complex. So you see differential forms showing up. Okay, and as a remark, I said before that the other question. Yes. So, you didn't define what is be completely smooth I imagine that completions of smooth things or direct limits, fitted limits of those are completely smooth but what is the exact definition. So I would say that I have a map a to be, I would say it's be completely smooth if I take the derived answer product of be with a mod P over a and I require this to be sitting in the zero and being smooth over a mod P in the classical finite type sense. Yes. Okay. Other question. Okay. Great. Okay, that's a remark that I said before, if you want to define prismatic homology in general, you don't do it just by computing homology of the structure shift on the prismatic side. You do this process of life can extension. But once you have done this, I don't want to explain it in detail, but you can check that this hot state comparison result will actually generalize as follows. Namely, if you have some a mod i algebra are P complete but not necessarily smooth. Then it's reduced prismatic homology so this complex over there. The base change to a mod i of prismatic homology. It actually comes equipped with a natural filtration which is increasing. And as the properties of the great pieces of this filtration, which is called the conjugate filtration are given just by which powers of the cotangent complex of our over a mod i and suitably shifted on break is interested and periodically completed. So, I mean, this is something you can directly deduce from the definition from the previous hot state comparison, plus the definition of both sides in general. I wanted to point this out because one, I mean, we will see the cotangent complex appearing later one later again. Basically the moral what you can remember from the statement is that hot state comparison gives you a way to like have some control on prismatic homology or at least its reduction module i in terms of the cotangent complex. So if you have some information on the cotangent complex you can usually deduce some interesting properties of prismatic homology. Okay, and then the next statement is the so called crystalline comparison. So this is the case where you assume that in your fixed prism AI is generated by P. Okay, then in particular because our leaves over a mod i it means that P zero in your ring. Then you could ask the question how does this prismatic homology relate to another interesting crystal, come on, this year in in characteristic P, namely crystalline commodity. And the answer is that, actually, they are almost the same. So if you compute crystalline commodity of our A. Then, well, it is the same as prismatic homology except that here on the right hand side, you have to twist by four minutes so you take pullback along the four minutes of a. Okay, so in particular if you know what prismatic homology looks like you recover crystalline commodity. You can not necessarily go the other way because a is not assumed to be perfect so five is not necessarily an isomorphism. But at least if you know prismatic only you recover crystalline commodity. And this is compatible with the strict with the Frobeno structure on both sides. This is also quite surprising because this way you get a definition of crystalline commodity without choosing divided powers and anything like that. And the key technical statement to check this is funny exercise that if you have a repeat ocean free data ring. If you have some element in this delta ring, so that it's first divided power is in your ring, then actually all the other divided powers are also in the ring. This is something you can check using the existence of the delta structure. This is one of the key inputs to in the in the proof of this crystalline comparison. Okay. Oh, there's a question. Just a clarification in the theorem you wrote the upper stuff, the kind of pullback. But since you're always working the derived the complete things is it's either some completion there or is that the bright. Maybe I'm confused but wait. I'm not sure now. I don't think you need to complete. I think what you say is that the delta. Yeah. No, go ahead. No, I as far as I understood you characterize the your you characterize your your things like guarz vitz forms is only after modding by I and also maybe you to be complete so it seems that everything is something derived complete relative to your finals and so no, and when you take fear, drying anche start by destroying this, I'm not Yeah. Okay. Okay. Okay. Good, so, okay, that's all I wanted to say about prismatic homology in general. Now I turn to prismatic doodonissiri itself. So as I said, I need to explain over which rings we want to classify feasible groups and by which kind of objects we want to classify. So I start with the rings. So there will be again definitions. So the rings we will consider are called quad asymptomic. So ring R is said to be quad asymptomic if it satisfies the following conditions. So first of all, it's P complete. So this we always assume everywhere and with bounded P infinity torsion. So I recall that this just means that there exists some integer n so that everything killed by power of P is already killed by P to z. Okay, and then the really important condition in the definition is that you want the cotangent complex of R over ZP to have P complete tor amplitude in degree minus one zero. So this means that you take this cotangent complex and you take its derived answer product with n for any R mod P module n. And then you want that these objects leaves in degrees minus one zero and the complex of R mod P modules. Okay, so this is the absolute notion somehow and then you can also define what the quad asymptomic morphism is. So it will be a morphism of P complete with bounded P infinity torsion rings. R goes to R prime which, okay, first you want that R prime is P completely flat over R. So I already explained what that means. And then you want that the relative cotangent complex of R prime over R as P complete tor amplitude in minus one zero. Okay, and you can also define what the quad asymptomic cover is. This would be useful later. It's the same definition, but instead of requiring that the map is P completely flat, you want it to be P completely faithfully flat. Okay, but so, yeah, the important condition is really the condition on the cotangent complex. And this definition is due to, I mean, it appeared in the paper of Batmouro and Schultzern topological orgy-domology. And the idea is that it should extend in the world of periodically complete rings. So there's some trouble. We lost that speaker. Ah, yes, coming. Hello. Ah. Okay, does it work? Sorry, I think we... Yeah, no, we convert our connection. Well... Okay. Is it okay? Yes, sorry. So is it good now? Yeah, it's good now. Okay, sorry, I think the connection, okay. That was, I don't know. Okay, so I was just saying that this definition is an extension of the classical notion of symptomic, well, of LCI ring and symptomic morphism. But you don't make any Nussarian or finite type assumption in this definition. So, yeah, before giving examples, one more piece of notation. So I will denote the category of all quasi-syntomic rings by QC. And then you can look at the opposite category and consider the topology which is defined using the quasi-syntomic covers in the above sense. So as I said, like maps which are quasi-syntomic and which are P completely phase-free. No, sorry, P completely phase-free. Okay, and then as a notation, if R is an object of this site, I would just denote by R with small letters QC, the sub-site which is formed by all rings which are quasi-syntomic of R. And again, undone with this quasi-syntomic topology. Okay, so I give examples of such rings now. The first example is just to justify the claim before that this generalizes the classical notion of LCI ring. So the claim is that any P complete and SIN ring which is locally complete intersection is quasi-syntomic. And well, this is checked using, I mean, actually Avramov gave a characterization of such rings LCI in terms of the cotangent complex. But here you just need the easy direction of Avramov C of M to prove that any such ring is quasi-syntomic. Okay, so that's one first class of example, but you also have like huge rings in this category of quasi-syntomic rings. So, namely I claim that any integral perfected ring is quasi-syntomic. And the reason for this is, okay, well, remember you have to check, okay, first of all, it's purely completely by definition. Then you can also check that for perfected ring, if there could be some P torsion, but like the P torsion is just the same as a P torsion. So the first two conditions are checked. And then you need to check this condition in the definition about the cotangent complex. And for this, well, you observe that this map from ZP, I mean, the canonical map from ZP to R, it actually factors through, you can factor it through the CTA map. So I should have written that this second map here on the right is Fontaine Zeta map. And whenever you have such a composite, you get a triangle for the cotangent complex. And now you observe that, well, what is a inf of R once you mod R of P? It's just R flat, which is a perfect ring of characteristic P. And whenever you have a perfect ring of characteristic P, its cotangent complex over FP is just zero. Basically here is just that. If you take any X, like it's always of the form Y to the P for some Y, because the ring is perfect. And then DX is just like P, Y, P minus one, DY. And if P is zero, this is just zero. So this way you can check that the cotangent complex of something perfect over FP is zero. But this just tells you that mod P, I mean the P completions of my cotangent complex of R over ZP and the cotangent complex of R over a inf of R agree. Right, because in this triangle, the other term will vanish after P completion. But so then once you know this, you can just, you are reduced to describe this cotangent complex of R over a inf of R. But then this map is a subjective and the kernel, theta is by properties of perfected rings, is principal and generated by a non-zero divisor. So this means that then the cotangent complex is just the same as R, but shifted leaving in common, common logical degree minus one. Okay, so this way you check. That in fact, in this case, the cotangent complex even has term amplitude in degrees minus one, minus one. It's even better. And then from these two class of examples, you can construct other examples if you want. So you can take a smooth algebra of our perfected ring and take its P completion. Or you can take a perfected, sorry, yeah, take an integral perfected ring and just mod out by a finite regular sequence. So if it's again, if it is again, bounded P infinity torsion, then this would give you another example of quasi-syntomical rings. And here I list some examples. So you can take the theta algebra in one variable over OCP. You can take OCP mod P or you can take this characteristic P perfect ring, FP T one over P infinity and mod out T minus one. So these are all examples of quasi-syntomic rings. Okay, so I think that's basically all I wanted to say about quasi-syntomic rings, but I just wanted to try to convince you that many interesting examples of rings are actually quasi-syntomic. Okay, and one good point of this about Mohr-Scholz's definition is that it's purely a definition in terms of the cotangent complex. And we have seen before, this was a hot state comparison theorem that if you have some control on the cotangent complex, because of this hot state comparison, you can usually deduce things about prismatic correlation. Okay, now I turn to third part. So filtered prismatic do-donate crystals. This will be the objects we will use to describe our P-zibber groups. And as you can guess, the definition will use the prismatic side. So to state it, I first need one observation. Let's take R to be quasi-syntomic. And then the claim is that, okay, you have a natural morphism of topos, which goes from the category of sheeps on the prismatic side, the absolute one, to the category of sheeps on the small quasi-syntomic side of my ring R. And if you wonder how this, what does this morphism of topos come from? Well, it's defined as a composition. So first of all, you observe that if you take a prism AI, so I mean, if you take some object of this prismatic side, you have a prism AI, then you can look at a mod i. So it will be P-complete. And the claim is that, this defines a, a co-continuous function from the prismatic side of R towards the big quasi-syntomic side of our ring R. And then you just restrict. Yeah, but I'm, only difficulty is checking this co-continuity of this, of this function. But if you are familiar with crystalline common logic, it's like, very similar to what you do when you, you go from the crystalline side to the etal or the risky side. Except that here we work with something a bit more general. We work with the quasi-syntomic topology. Okay, and now what I will do is I will take my prismatic structure shift or prism on this ideal prismatic shift. I prism, I just push everything using this morphism V. So V was my notation for this morphism of topos. I just push everything down to the co-syntomic side. And then I claim that, okay, actually you have a natural surjection from oprys to O. And I will give a name to the kernel of this morphism of shifts. So O here is just the structure shift on the quasi-syntomic side. This kernel is denoted like this. So it's what is called the first piece of the Niagara filtration on this prismatic shift oprys. Well, the reason for this notation is that, I mean, there is this notion of Niagara filtration of a prism, which is defined for any, you could define for any positive integer I. Here I only need the first piece of the Niagara filtration. I would just take as a definition that it is the kernel of this surjection, which I did not explain. And then a property of this is that, well, because any data in other Frobenius, so this shift oprys will come with a Frobenius morphism, Phi. One can check that this kernel has the properties at Phi of the kernel. So Phi of the first piece of the Niagara filtration for oprys actually lies in high-price times oprys. So morally on this first piece of the Niagara filtration, the Frobenius is divisible by, well, let's say the locally I, the ideal is generated by non-zero distinguished element. The idea is that on this first piece of the Niagara filtration, Phi is divisible by this distinguished element. So there is question from over here. Yeah, there's a question. I cannot hear. Can you put the microphone on? You should activate his. I guess you can unmute him. Oh, yeah, now his microphone must be on. Internet, I don't. Oh yeah, now you can speak. So why do you write oprys times oprys? I think oprys is an ideal in oprys. Oh, yeah, sorry, sorry. Yeah, it's a typo. Yes, thank you, sorry. Yeah, it's just because later it will appear. Yeah, sorry. Okay. Okay. Okay, so then the definition of a filtered prismatic dunic crystal is, so let me again fix a quadsentamic ring R. And then a filtered prismatic dunic crystal is by definition of collection, a triple M, fill M and five M. So M is finite located free oprys module. First of all, then fill M inside it will be some oprys sub module. And finally, five of M is morphism, ring morphism from M to M, which is assumed to be a file in R. Okay, where five is a Frobenius of oprys. Okay, and then you ask three conditions on this triple. So first of all, you want that this Frobenius five M sends a fill M to I press times M. Okay, but then there is an obvious oprys sub module of M which also have this property that five M of it is leaves inside I press times M. Namely, consider the first piece of the Nagat filtration of oprys times M. Then because of the previous slide, we know that this is contained in I press. And so this sub module, first piece of the Nagat filtration times M also has a property that five of it is contained in I press times M. And then the second axiom you have is you want that this sub module is in fact contained in fill M. And once you have done this, you want to fill M and once you have done this, then you know that M module of fill M will be a module over oprys module of the first piece of the Nagat filtration. But remember that oprys module of the first piece of the Nagat filtration was by definition is just the structure shift of the quiescentomic site. Oh, so M mod fill M will be an O module and you ask that it is finite locally free. And then the third condition is that the image of the filtration by the Frobenius is big enough in the sense that it will generate I press times M as an oprys module. So if you are familiar with crystal and deudonis theory, again, it's very reminiscent of the usual notion of deudonis or filter deudonis crystal. And the third condition is just in this setting which has been a formulation of the condition that the filter crystal is admissible in the sense of quotient. So that's somewhat just the obvious extrapolation of this classical notion to the setting of the prismatic site. Okay, and then notation is if R is my quiescentomic ring, I will denote by Df of R the category of all filtered prismatic deudonis crystals over R. And the morphism are the obvious ones. I mean, they should be oprys linear and they should be compatible with Frobenius and with the filtration. Okay, and so now I can come to the statements of the two main results we proved. So I will fix again once and for all now quiescentomic ring R. And let me take G to be a feasible group over R. Then I can define M prism of G to be X1 of G by oprys. And here when I write this curly X1 is supposed to be like the local X groups in the category of abelian sheeps on the quiescentomic site. So you check that your feasible group G, where the reduction to the case of finite locality free group schemes defines a sheep, an abelian sheep on the quiescentomic site. And oprys is also an abelian sheep on this site. So you can take X1 in this category. And then you also define fill M prism of G to be the same thing except as you replace oprys by the first piece of the Naga filtration. And then the first main result is that, so if G is as before, then this triple M prism G, fill M prism G and for benus of M prism of G, which is just by definition the for benus coming from the for benus on the prismatic on the prismatic sheep. Then the claim is that this is actually an objective of this category DF of R. So this is a filtered prismatic dodomic crystal over R. And in what follows, I will denote it by M prism G underline. Okay, and as a remark, if P is zero in my ring R, so in the characteristic P situation, you can check using the graph, you can check using the crystalline comparison theorem, which was discussed in the first part, that in fact, this is just the same thing as the usual function you will find, people have studied in crystalline dodomy theory, which you find, for example, in the book of Bertolo, Brin and Messing. So this is nothing new in characteristic P. Okay, and then our second main result is that, well, this filtered prismatic dodomy factor underline M prism, which associates to G, M prism of G underline. It actually realizes an anti-equivalence between this category of BT of R of visible groups over R, and the category DF of R, which I defined before. And as a bonus, I mean, as a byproduct of the proof, you actually obtain that the prismatic dodomy factor, G goes to M prism of G, is already fully phase-free. But if you really want to get an equivalence of categories, so if you want to be able to describe the essential image, you need to add the filtration in the picture. Okay, so now I make a few remarks about the, these two results and about the way we prove these two results. So first remark is that, as I said before, in characteristic P, you just recover the usual factor from dodomy theory. And so in particular from the serum two, this classification result, well, you can deduce that the crystalline dodomy factor is an equivalence for all quasi-syntomic rings in characteristic P. And well, this was actually already known, of course, in some cases. So if you look at LC irons, which are also excellent, then fully phase-freeness was proved in the end of the 90s by To-Yong and Messing. And more recently, Eichelot has proved that this factor is actually an equivalence when the rings is, okay, Nocerian LCI and more of a F finite. So this means that the Frobenius is finite. And this particular implies XM. So in this case, the result was already known. Okay, then I wanted to explain now that this category DF of R may seem a bit abstract, but there is an interesting class of quasi-syntomic rings for which you can make it more explicit. Namely, you will say that the ring is quasi-regular semi-perfectory. If it is quasi-syntomic, of course, first, and then you also want that there exists that a perfected ring which maps subjectively onto R. It has to be big enough. Examples of such in the list of examples I gave before, where you can look at any perfected ring is obviously, because we know it's quasi-syntomic and the second condition is trivial. Any perfected ring will be quasi-regular semi-perfectory, but also like a quotient of such a ring by a finite regular sequence, which has bounded P and P torsion. So these are examples of such rings. Well, for such rings, it's like for perfected rings, and the example we saw, in fact, the Quotantin complex after P completion is just sitting in the minus one. And for this reason, you can check, using Hatch-Tech comparison, that for such a ring, the prismatic side has a nice feature that it admits a final object in this case, which you can just describe as, if you want the absolute prismatic homology, comes with a natural idea. And then also you can check in this case that if you take the first piece of the Naga-Tutration on this prism to be just the inverse image of I by Frobenius, then the quotient is isomorphic to I itself. So two examples of computation of this initial object, or some final object. The first case is if I is perfect to E, then this final object is just given by the ring A-info of R together with the kernel of the theta map. So, okay. In this case, the prism is perfect. So Frobenius is an isomorphism. So there is always some choice that you can work with the map theta, or with its pre-composition with Frobenius minus one, and then consider theta tilde. It's just a matter of conjunction. Okay, so that's the first example. And then another example, if you take a ring which is called a regular somaiparfectoid and with P zero in this ring, then this initial prism is just the same as a crease of R together with the idea generated by P. So you can compute it in several situations. And then we make a definition. It's very similar to the one we had before. A field of prismatic design in module over R will be a collection M, field M, and phi M. But no, I just have, instead of having finite locality free modules over the prismatic shift, I just have a finite locality free module over this ring, prism R, a certain sub-module of it, field M, and a phi in R map, phi M. And you just ask the exact same actions as before, but no, you only work with modules over this ring, prism R. So I want to repeat them because I don't have much time left. But okay, so you can make this definition and then the claim is that if my ring is quasi regular somaiparfectoid, I can in fact just evaluate all the objects I have on this, so I should have written final on this final object. And the claim is that this gives an equivalence between the category of filtered prismatic due to the crystals over R and the category of filtered prismatic due to the modules over R. So in other words, in this case, you can make the category more explicit. You can just work with modules instead of working with this object, this category DF of R. And moreover, if the ring is perfectoid, then you can do even better. Then you can even forget the filtration. So just look at the forgetful function for this category of filtered prismatic due to the modules over R to simply what you can call a prismatic due to the module, but in this case, it already has a name. People call them minuscule break is in FARG modules. And then the claim is, in this situation, this forgetful function is in fact an equivalent. So in other words, as a special case of the CRM2, you see that you recover the fact that for perfectoid ring, peaceful groups over such a ring are classified by minuscule break is in FARG modules. But this is a bit cheating because in fact, we need as an input for the proof, for the proof of CRM2, we need a special case of this. Namely, we need that peaceful group over a variation ring, which is perfectoid and has algebraically closed fraction field is the same thing as a minuscule break is in FARG module. And actually, it's not difficult to do. You can deduce the case of all perfectoid rings from this special case using VT-SAN target. But so we need this as an input. Okay, so can I just take five minutes to finish? Or should I stop now? Okay, go ahead. So I try to finish quickly. So I also want to mention that, as I said, in general, you really need the filtration if you want to describe the essential image of this prismatic do-don't-functor. But I just said before that in this remark that for perfectoid ring, the filtration is actually unique. So it's not needed to state the classification theorem in this case. This is not true in general, but this also works for P-complete regular rings. And as an example of this, I just take the ring of integers in some discreetly valued extension K of QP with perfect residue field. So for example, take a ring like ZP. And then also this case was, of course, already known before. So then you can prove that P-complete groups over this such a ring are classified by so-called minuscule brachycin modules. And this has been done by Bray and Kissing, at least for all P, and then extended by Kim, Lau and Yu to all P. But we can also recover this, namely by first proving that for such a ring, you can forget the filtration. And then checking that you can just evaluate and, well, there is a natural prism attached to such a ring once you choose a uniform measure. It's not a final object, but still you can check by some reduction to the perfect situation that in this situation, evaluation on this object is again unequivalent. And so, and then we check that the factor we have is actually the same as the one which has been studied by Bray and Kissing, all these people. But now the good point is that some of you directly learned in the correct category. So the proof works uniformly for all P. You don't need to make a special argument when P is too. Okay, and then just two words about the proof. So for theorem one, it's not surprising. So you just follow the strategy of Bertel and Bray and Messing in their book. So you have this definition of this triple. You want to check that it is a filtered prismatic dosonary crystal. And the idea is, well, you have to understand how what is X1 looks like. And in fact, for any group object in a topos, you have some device to make computation about this X groups, at least in low degrees. So this is explained in the book of, Bertel and Bray and Messing. And you are reduced to compute some prismatic commodity groups. So first step, you use a theorem of Reno to reduce to the case of Psemer groups which comes from some Abelian scheme. And then, via this Bertel and Bray and Messing partial resolution of any group object in a topos, what you just need to do is understand precisely prismatic commodity of Abelian schemes. And a key tool for this is provided by the Hatch state comparison theorem for prismatic commodity. So, but the ideas are really similar to the one of Bertel and Bray and Messing. And then, theorem two, the proof is more difficult, but I just want to point that the key idea is to use quasi-syntomic descent. So here, I introduced just before the notion of quasi-regular, so my point is that the very nice feature of this site, this quasi-syntomic site, which was observed by Bayt-Maurot and Schultz-Hou, is that this quasi-regular semi-perfected ring actually forms the basis of this quasi-syntomic topology. Or by any quasi-syntomic ring, by extracting enough piece-roots, you can always make it quasi-regular semi-perfected. And this means that once you have defined this factor, so once you have proofs here on one, then you can, to prove that it's unequivalent, you can somehow reduce to the quasi-regular semi-perfected situation. And then everything is more concrete, as I said. Instead of having crystals, we just have modules over this ring prism arc. Okay, and then the hard work starts, but I wanted to say this because it shows that even if you, if you want to, if you are a non-proliferation like getting classification results over like Nucleus and rings, then the proof is such that actually you really need to work with this big category QC, because the argument is by descent. This is very big quasi-regular semi-perfected ring. Okay. Okay, and then I will skip this point. I will skip this point. I just want to finish by saying that there are some natural questions which are left open. First of all, it would be interesting to see what happens for more general baserings. We only prove results about quasi-syntomic rings, but for example, in the work of Tzink, you can find some results for a very general, periodically complicated ring. So, for example, in the work of Tzink, you can find some results for a very general, periodically complete rings. But I have no idea how to do it for more general rings. And then another thing we don't have is something like deformation theory in this setting. So for this prismatic duodonné functor, so for the usual crystalline duodonné functor, you have this gotennik-messing deformation theory, which is very powerful. But here, we don't have any analog of this. And then finally, a final remark is more anecdotal, but it's about the case of perfect rings in characteristic P. I just want to point that in this case, well, because if you live over like an object of the prismatic side of such a ring R, it will be P torsion-free in particular. So you can always map this prismatic structure shift on this side to the same thing where you invert P and take Q to be the quotient of this injectable. And then you check that, well, in this situation, the prismatic duodonné module of any P-zero group G is in fact the same thing as home from G to the push forward to the quadrathymtomic side of this shift on the prismatic side. I mean, this is just obtained by looking at the associated long exact sequence. And then a natural question is whether the, I mean, front end has, was the first to give a general definition of the duodonné module over such a perfect ring of characteristic P. And the definition looks a bit similar, except that instead of this mysterious push forward here, you have this shift of VIT covectors. And so it would be also interesting, and we did not do it to know how to relate this object here to Fontaine's original definition of the duodonné functor with VIT covectors, but without choosing the general crystalline comparison C-ray, which of course implies this comparison, but just directly using the two definition. Okay. Thank you very much. So thank you very much for the interesting lecture. So are there any questions? Yes, Luke. Could you unmute him? Yeah, I'm trying, but it doesn't work. I don't see one. Luke, could you unmute yourself? Luke, can you put the microphone on? Now you can hear me. Yes. Okay. So I think you skipped one of the last slides where you were writing something about classification of finite, locative free community group schemes. You discussed PDVS groups. I presume that you, but presumably you can also classify truncated BTs and maybe more general. So that is the, Oh, I can't hear you well. Okay. So it was just that I wanted to see this slide again. Yeah. But then is it in terms of a module or whether a complex maybe? Okay. So we only do it for over perfected rings. So I think that's the question. I think that's the question. I mean, the idea is the same as the one you find in the paper of kissing. Well, actually you use the fact that you can just like identify the category of final locality free group schemes with the category of like two times complex. Of peace by groups. With an isogenic between them. And the only difficulty in general for general quasi-syntomic rings is I'm not sure by which kind of objects you would classify fine and local if we group schemes. So for perfected rings, as I said, for peace by groups, you can forget the filtration. You just have this minuscule boy kissing fog modules. So it's not difficult to guess what, which can, by which kind of objects you will classify fine and local if we group schemes. So it would just be a module of your ring prism, R, which is just a inf of R in this case of projected dimension less than one killed by your power of P with a four been used on a Vershebu. But in the general situation where you should also take the filtration into account, I'm not sure how to describe the very local if we group schemes. So that's why we only did it for perfected rings. Actually, this was already known in all cases except maybe when peace to my work of law. I had another question you mentioned work by think. So is there a relation between this place and the theory? Yes, this is another slide. I had to skip here. So let me take our game to be one first one reduces to the case of quasi regular semi perfected rings by quite a systemic descent. And then, well, you have this natural map from prism R to R was kernel is the first piece of the nugget filtration on this prison. It's a map of Delta rings. So it's a map of rings. But I said that the forgetful functor from delta rings to rings as a right adjourn, which is the big vectors fun. Yes. So this mapping juices a map of delta ring from prison R to W of R. And using this observation, you can actually check that you have a functor from our category DF of our filter prismatic modules. To the category of displays of our in the sense of. It's not an equivalent. I mean, the classification of sink is both more general and more restrictive. It's more general in the sense that I think he only assumes a ring to be periodically complete. But then he has to restrict to formal feasible groups because when he is to like some difficulties. So this is due to the fact that this functor we get from DF of R to this place of R is not an equivalent. But it is when you restrict to any important objects. The terminology of. Thank you. Thank you. Okay. Thank you. Then offer. Okay. So I remember the concern in this. Welcome to the net. There were some after seeing the way in particular some papers of law where he. He looked at the for example, complete. Mix characteristic. The local rings with. Is it perfect or not perfect. Then he gets some using saying he get, I mean, he gets. He does things which was using windows and friends. So they get a very simple linear algebra. Sinks which classify PD visible groups. Over mix characteristic regular rings, but sometimes he needs to. He needs to get a. To assume it is a. The result is different and the rest of it is not perfect. But any case he has some. Do you get a relation between. What you do and use to, but maybe there are several such references. I don't remember exactly, but I remember it's quite simple. Like you only need to give. Some. Some final free modules and some maps. Composition equal to the equation or something like this. Is it possible to relate it to your. Yes. That's what I mentioned here. So. As I said, in. For such. A P complete regular rings. This the filtration is also not relevant. So you can describe everything. I think, as I said, using currency realms, you can. Get a natural prism attached to the situation. So for example. In the simple case where I just okay. Yes. You just take this break is in prison. So you just take like. Is it your field double brackets. A. U for some formal variable U. And he is some ascent-style polynomial, which is like determined by after you choose a uniform either you. You have a natural map from this track S. Towards. Your ring R and either generator of this. Of the kernel. And then yes, then, then you get a quite simple classification in this case. in this case as modules, find locally free modules over such a ring together with the Frobenius which has the properties that it's, like after linearization it's co-carnalized. After you invert E it becomes, the linearization becomes an isomorphism, the linearization of Frobenius. So, but for checking that the functor is really the same as the one which is used in law. At least we checked it in this particular case, but this is the same as the functor that people usually consider. But in the marginal situation, I'm not sure we checked it. So there was also a technical question about whether the resdiel field is, when the resdiel field is not perfect, I think he needed to work like sync with only formal pt-visual group, but maybe I don't, do you get, do you get easy to, that is classified, well I don't have a, now I'm sorry. When difficulty that low on contours is that, I mean he uses crystalline Jordanian theory, yes, so somehow you have your ring R, some of us you reduce to, to arm of p where you can use crystalline Jordanian theory, but then you need to go back from arm of p to R. So you need to use gotonic messing theory. But the issue is that this, like this divided power structure on the idea and generated by p is not important when p is two. Gotonic messing theory only works well when the divided power structure is important, right? So I think for this reason, you have to do some, usually you have to do some extra work when p is two. But here somehow I think this issue does not appear because in some sense you directly, with this prismatic Jordanian function, you directly land in the correct category. Like again, back to this example where R is okay, in the original work of Boy and Orkissin. Actually you, what you first do is you produce a factor from the category of feasible groups over this ring R towards the category of filtered modules over another ring that's frack S, which is like the p completion of the pd envelope of E inside this ring frack S. And this is usually just called curly S. And then you have to do some semi linear algebra to check that this is indeed the same as the category of minuscule Boy and Orkissin modules. But here we directly get a factor from feasible groups over R towards minuscule Boy and Orkissin modules. For some of we avoid this difficulty and that's why we don't have any assumption on p. I have another small question. Maybe what is, so you said that in the definition of a prism you have a Cartier divisor. You can, are there examples where it is not globally principle? I know I don't know any example writes, writes not principle actually. I think all the examples I'll know the ideals are principle. I see basically one question. Can you hear me, Ophar? Yes. Okay, so the question was, does there exist any coincidence between the filtration of prismatic Jodonaic crystal and the filtration on bun G on the farfontaine curve since Fark proposed that using chromatic filtration one can correspond perfect with BT1 with bun G. One can what? Correspond perfect with BT1 with bun G. I don't know what that means. I'm just reading what it is. BT1 is like Berserk's detail of level one. I guess so, but I guess what is perfect with BT1 is, BT1 over perfect with bun G. I don't know what it means. But maybe BT1 over perfect, okay, I don't know. But I don't know any relation with bun G. I mean, of course, Burkis and Fark modules are related to modifications of vector bundles on the Fark-frontaine curve. But bun G is like the bun G bundle, so. Yes, so I don't see. For a group, they do it with a. I guess, well. Yeah, so front-end. The G should be. G should be reductive group or something. So I don't see any relation. Okay, well, if I, so it's not, I'm not, I will need more to see it in more precise form. So, yeah, so as far as I understand, okay. Yeah, okay, so by any case, you prove that the same construction of display is canonically correspond to what you do, using these functors to display, you get exactly things construction, which is defined for more general rings. Right, yes. Okay, all right, so. Yeah, so then this partly answer, because the work of Lowentz-Holm is kind of trying to use this display to do some complicated tricks to which I don't, okay, so it's, and of course you get also the crystalline deodorant theory from your. Yes, as a. You need to use some morphism between relating the crystalline, okay, but you wrote it, I don't remember if you put it in the slides. I just put a remark, I mean, yeah, basically you can check that the usual, I mean, you have this prismatic side, you have the crystalline side, let's say you just push everything to this color-syntomic side and make the comparison there. And then the claim is that the two categories you get are actually the same, so. For the ring. So the ring is killed by P. Exactly, yes. So you get a, okay, I have to, this was a. I mean, basically this follows from this crystalline comparison theorem. Okay. Okay, thank you, so I will have to think. Okay, not there. The slides will be online at some point. Okay, thank you. You're welcome. Okay, goodbye. Bye. Thank you very much, so goodbye. Thank you.
the recent theory of prisms on prismatic homology, which has been developed by Bat and start speaking about the joint work with Uranus. structure. This is one of the key inputs to in piece of the Niagara filtration. I would just results for a very general, periodically