So, welcome to Faripek in 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 Seju-Lublar. So, he will tell us the prismatic dual 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 De Donne theory. And so, everything I will discuss is a joint work with Joannès Anschutz. And for all this talk, I will fix a prime number p

once and for all. So, the goal of our work was to prove some classification results for feasible groups over various kinds of rings. And the main tool we use to do this is a recent theory

of prisms and prismatic cohomology, which has been developed by Batt and Sholtsu. 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 and prismatic cohomology. So, recall the basic definitions and constructions.

And then I will start speaking about the joint work with Joannès. So, first of all, I need to tell you about which kind of rings we want to classify feasible groups. And these rings are called quasicintomic 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 feasible groups over such rings. And we call them filtered prismatic Jordanic 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 and prismatic cohomology. And I should make it clear, like everything in this part is true to Bat and Schulze.

So, the theory of prismatic cohomology 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

lives over the 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, and which has the following properties. So, first of all, it maps zero and one to zero.

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 some 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 canceled here, are actually divisible by p. So, this expression makes sense in any ring.

You don't need to assume that the ring is Peterson free or anything like this. Okay, 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 the map phi, which goes from A to A and which sends x to the p plus p times delta x. And then you check and 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, just a map of sets from A to A is actually a ring morphism. And moreover, just by definition, it's a lift of Frobenius.

So, if you kill p, then this term disappears and it just becomes the Frobenius, 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 Frobenius 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 to delta rings as rings together with a lift, with a ring morphism, which lifts Frobenius. Right, but the two notions are not exactly the same thing when you have p torsion in your ring.

Okay, 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 two bit vectors 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 comma delta x. And once again, you can check that the axioms of a delta, which tells you that you have a delta ring, is exactly what you need to verify that this is a ring morphism.

Okay, and this point of view is useful because this allows you to prove that this category of delta rings, contrary to the category of rings with a lift of Frobenius, has all limits and co-limits, which you can just compute on underlying rings. And so, in particular, this forgetful functor

from delta rings to rings, we have both the left and the right hydrant. And the right hydrant is given by the bit vectors functor. Okay, so that's the first remark. And then another remark, which is more an exercise that you can do.

I said that, I mean, 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 zero for some a. Okay, this is something you can check just using the definition of a delta ring.

Okay, very good. So, we will see some examples later. I'll wait a second. No, it's a trivial remark that zero is a delta ring, so p to the n. Yeah, okay, I assume that zero is different from one, but okay, yeah, if you want.

No, but if you want all limits and co-limits, then... Okay, yeah, then I should have said that there is no non-zero delta ring in which p to the n can be zero. Thank you. Okay, so as I said, we'll see examples soon,

but now we'll come to the next definition, namely the definition of a prism. So, what is a prism? It's just a pair a comma i. Well, a is a delta ring. So, usually in the notation, I will just forget the delta. 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 defines a Cartesian divisor on spec a. So, it's just like locally principled generated

by a non-zero divisor. Then, you also require that i is p i-addicali-complete. And for technical reasons, you should mean this in the derived sense. Also, soon we will make some assumption on the ring,

which ensures that in practice, like the derived and the classical completions agree. So, first approximation, you can just think that this is classically p i-addicali-complete. And then, the last condition, the most important one.

So, i has to be prosariski-locally generated by a distinguished element. And what is a distinguished element? By definition, it's just an element of d, which has the properties that delta of d is a unit. Okay, and once again, in practice, i will always be principled.

So, you should just remember that a prism is a pair a comma i, where a is data ring. And let's say i is principled generated by a non-zero divisor, which is distinguished. In the sense that it's image by delta is a unit. And moreover, the ring should be p i-addicali-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 and 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 there is no p-torsion, so p is non-zero divisor. Then, 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 that I gave before that it cannot happen that p to the n is zero. And the way to prove this is just to check that delta p has to be a unit in any delta.

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'll say that the prism is perfect

if it's Frobenius phi. So, remember, whenever you have a prism, sorry, a delta ring, you have these deltas and you can define a phi which leaves Frobenius by the 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 functors in both directions, which are quasi-inverse of each other. And let me tell you how. So, in one direction, assume you start with an integral perfect to a ring, R. Then you can do this classical Fontaine's construction.

You can consider a inf of R. So, what you do is first you tilt your perfect ring. So, you get a perfect ring of characteristic p, R flat. And then you take its ring of bit vectors. This is what is called a inf of R.

And a inf of R comes with a natural map, theta, which goes from it towards R. And I mean, part of the definition of or at least consequence of the definition of an integral perfect to a ring 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 periodical theory, is in fact distinguished in the previous sense. So, namely, sorry, I should have said first that, okay, this ring is p torsion free, R flat being perfect.

And so, it has, it comes with a natural Frobenius. And you just take the delta structure attached to this Frobenius leaf. And then my claim was that the generator of this ideal is actually distinguished. Well, the way, I mean, you can check this by proving more generally that

in such a delta ring, in the delta ring of this form, bit vectors of some perfect ring, an element is distinguished only if it is primitive of degree one, which means that when you write the expansions of

as sum of Teichmann's 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 functor. And then if you want to go in the other direction, it's even more simple. You just mod out.

You have a prism ai, which is assumed to be perfect. And then you just look at a mod ai. How do you recover delta? How do we get delta? Well, okay, I said that if the ring is Peterson free,

which is the case for this ring aint of R, then having delta structure is the same as having a Frobenius field. And you have a Frobenius on the ring of bit vectors of R. So you just take the delta structure attached to this Frobenius. Okay, and so this proposition is, I mean, this example is the reason why

I guess Batten should describe prisms as some kind of de-perfection of the category of perfectoid rings. Because you see that inside the category of all prisms, I have this subcategory form by perfect prisms. And these are exactly the same thing as perfectoid rings.

And ai is like, choosing ai, my ideal is the same as choosing some un-tilt of my perfectoid ring. Okay, so once you have done this, namely introduce delta 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 R.

There's a question.

Could you unmute the person who asked the question? Yeah, I did not hear anything. Takeshi, could you unmute the person who asked the question? Yeah, so I'm trying to do that, but it doesn't work.

Maybe there's a question on the microphone. There's a problem on the microphone. Yeah, I cannot do it.

Okay, so maybe he can ask his question by chat, okay? Okay, so I defined now the absolute prismatic side. So let me fix the ring R, which is assumed to be periodically complete.

Then the absolute prismatic side of the ring R, which will be denoted by R prism. Okay, so this symbol is supposed to be a prism, even if it appears as a delta here. So as a category, it's just the opposite of the category of all bounded prisms, B, J,

together with a ring map from R to B mod J. Okay, so here there is one adjective which I did not define yet, bounded. So it's just, again, a technical condition about which you can forget in first approximation.

It's telling you that you have your prism B, J, and what you require is that B mod J here has 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 a power of P in this ring, B mod J, is actually already killed by P to the n. This is a condition you put for technical reasons, which have to do 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 B modulo J.

And then you put a topology on this category. So you define covers to be morphism of prisms. B, J goes to B prime J prime. So morphism of prism is the obvious notion.

So it's just a morphism of delta rings compatible with a delta structure, which sends J into J prime. And you say it will be a cover if when you just look at the underlying ring map from B to B prime, it's P, J completely face fully flat.

So this means that if I take the derived answer product of B prime with B modulo P, J over B, I mean, first of all, it has to be concentrated in negative zero. And it is then face fully flat over B modulo P, J in the usual sense.

So it's just a slightly weaker notion that the notion of face fully flat ring morphism, someone would just require a condition modulo P, J for the reasons that again, everything is assumed to be complete and you want a notion which is stable under completion.

So instead of looking at face fully flat morphism, you just look at P, J completely face fully flat morphism. Okay, so first for the prism, there are several technicals. So in the morphism of prism, you require that J goes to J prime,

but I suppose it should follow that J generates J prime, is it correct? Yeah, this is true. Okay, 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 N,

like N opens. Here you just have covering by one thing, of course you need to add like the disjoint union of N level, well, it is a product of situations,

then it's taking all of them will be a covering by N thing. Okay, so yeah, I agree. Okay, so it should just be generated by these covers. Okay, yeah. Okay, no other question.

Okay, so this is the definition of the prismatic site. And then you have two natural pre-shifts on this site. So one is denoted by O prism, and the other one is denoted by O prism bar. And this has a functor which defined on the prismatic side,

which send a prism BJ on the prismatic side to B. So this is for O prism. And O prism bar will send BJ to B mod J.

And something bad should also check, is that these two functors are actually shifts on this prismatic site. And they both have a name. So this shift O prism will be just called the prismatic structure shift. And the other one O prism bar is called the reduced prismatic structure shift.

Okay, and then I mean, okay, something you can deduce from this is that you could also consider the functor I prism, which sends just BJ to J itself. And then this is also a shift on this side. Okay, so you have these two shifts,

these three shifts on this prismatic side. And this is the one we will use later on. But before doing that, I want to mention that, okay, I defined the absolute version of the prismatic site. 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 living over ai. So R is a P complete a mod i algebra. And then you can define a variant of the absolute prismatic site where everything lives 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 and as before, you want that b mod j receives a ring morphism from R,

which now is required to be a morphism of a mod i algebra. So everything lives over my bounded prism ai, which I fixed at the beginning. And the topology is as before. And this is the version that Bachelors will use and what they do, I mean, one of the main objectives of the paper

is to compare what you get using this notion with other classical, more classical periodic cohomology series. And so to do that, I mean, once you have defined the site, you can define prismatic cohomology.

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 cohomology of R over a, which will be denoted by prism of R over a, is simply the cohomology of my sheaf 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 formally smooth over a mod i. You could do it always, but it's not well behaved.

And the way Bachelors would define prismatic cohomology for general a mod i algebra is using left kind extension from the smooth case in the same way as you would define the cotangent complex from the sheaf of degree one differentials in general.

But at least when everything is smooth, then the definition is just the cohomology of the sheaf on this side. And as I said, what Bachelors would do is they compare this new cohomology theory with other periodic cohomology theories. I just want to mention two comparison results I proved.

There are many of them. Namely the Hodge state and the crystalline comparison theorems. 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 Hodge state comparison. So it tells you that, I mean, let me start with the right hand side. So here you have your prismatic cohomology complex, which I defined before, prism R mod a, and you take its derived tensor product

with a mod i over a. So it would be the same as considering the cohomology on the prismatic side of my reduced prismatic structure shift or prism bar. And I consider cohomology of this in some degree i. And then the claim is that this is canonically isomorphic as an R-module

with the module of degree i differential forms on R over a mod i. And here, I mean, it's implicitly assumed to be periodically completed. Up to some small twist, which is denoted by this symbol with an i.

It's a bolekesin kind of twist. So I recall the notation below. If you have some a i-module, m, you will denote by m twisted by i, the tensor product of m with i mod i square, i times over i mod i.

So this is a rather surprising result when you think a little bit about it because you have made this definition of prismatic cohomology just using delta rings and prisms. And you see that naturally when you compute it, when you compute the reduced prismatic cohomology,

the cohomology groups of this complex, then you see differential forms showing up. Okay. And as a remark, I said before that... There's a question. Yes. So you didn't define what is P completely smooth.

I imagine that completions of smooth things or direct limits, fitted limits of those are P completely smooth, but what is the exact definition? So I would say that I have a map a to b. I would say it's P completely smooth if I take the derived tensor product

of b with a mod P over a and I require this to be sitting in degree zero and being smooth over a mod P in the classical finite type sense. Yes. Okay.

Other question? It's okay. Okay. A remark that I said before, if you want to define prismatic cohomology in general,

you don't do it just by computing cohomology of the structure shift on the prismatic side. You do this process of left-hand 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, R, P complete, but not necessarily smooth, then it's reduced prismatic cohomology, so this complex over there, the base change to A mod I of prismatic cohomology. It actually comes equipped with a natural filtration, which is increasing,

and which has the properties of the graded pieces of this filtration, which is called the conjugate filtration, R given just by wedge powers of the cotangent complex of R over A mod I. And suitably shifted and break is twisted and periodically

completed. So, this is something you can directly deduce from the previous hot state comparison plus the definition of both sides in general. I wanted to point this out because we will see the cotangent

complex appearing later once again. Basically, what you can remember from this statement is that hot state comparison gives you a way to have some control on prismatic cohomology or at least its reduction modulo 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 cohomology. And then, the next statement is the so-called crystalline comparison. So, this is a case where you assume that in your fixed prism A

I, I is generated by P. Then, in particular, because R lives over A mod I, it means that P is zero in your ring R. Then, you could ask the question, how does this prismatic cohomology relate to another interesting

cohomology theory in characteristic P, namely crystalline cohomology? And the answer is that actually they are almost the same. So, if you compute crystalline cohomology of R over A, then, well, it is the same as prismatic cohomology, except

that here on the right-hand side, you have to twist by Frobenius. So, you take pullback along the Frobenius of A. So, in particular, if you know what prismatic cohomology looks like, you recover crystalline cohomology. You cannot necessarily go the other way because A is not assumed to be perfect. So,

phi of A is not necessarily an isomorphism. But, at least if you know prismatic cohomology, you recover crystalline cohomology. And this is compatible with the Frobenius structure on both sides. This is also quite surprising because this way you get the definition of crystalline

cohomology without choosing divided powers and anything like that. And the key technical statement which exists is a funny exercise that if you have a repetition-free delta ring and if you have some element in this delta ring, so that its 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 on the ring. This is one of the key inputs in the proof of this crystalline comparison theory.

Okay. Ah, there's a question. Yes, just a clarification in the theorem. You wrote phi A upper star as a kind of pullback. But since you always work in derived, complete things, is there some completion there? Or is it just algebraic?

Maybe I'm confused. Wait. I'm not sure now.

I don't think you need to complete. What you say is that the delta Yeah. No, go ahead. As far as I understood, you characterized

your things like the relation with forms is only after modding by i and also maybe p to p complete. So it seems that everything

is in some sense derived complete relative to your ideas. And so you and then you take phi A upper star maybe destroys this. I'm not. Yeah. Okay. I'm not sure I should I should check again. Okay.

Okay. Good. So, okay. That's all I wanted to say about prismatic homology in general. No, I turned to prismatic due to the theory 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 we the rings we will consider are called quasi symptomic. So ring R

is said to be quasi symptomic if it satisfies the following conditions. So first of all it's P complete. So this 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 a 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 Z P to have P complete tau amplitude in degree minus one zero. So this means that you take this cotangent complex and you take its derived

product with N for any armored P module N and then you want that this object lives in degrees minus one zero as a complex of armored P modules. Okay so this is the absolute notion somehow and then you can also define what the quasi

symptomic 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 the relative cotangent complex of R prime over R as P complete tau amplitude in minus one zero. Okay and you can also define what the quasi symptomic 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 phase-free flat. Okay so 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 Bat-Moro and also on topological orchid homology. And the idea is that it should extend in the world of periodically complete rings. So

there's some trouble. We lost our speaker. Ah, yes. Hello. Okay, does it work? Sorry, I think it

recovered our connection. Okay. Is it okay? Yes. Sorry, so is it good now? Yeah, it's good now. Okay, sorry, I think the connection is good. Okay, that was, I don't know. Okay, so I was just saying that

this definition is an extension of the classical notion of LCI ring and syntomic morphism. But you don't make any necessarily type assumption in this definition. So,

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, maps which are quasi-syntomic and which are p completely facelift. Sorry, p completely facelift. Okay, and then the notation if r is an object of this side,

I would just denote by r with small letters the sub-site which is formed by all rings which are quasi-syntomic over r. And again, end out with this quasi-syntomic topology. Okay, so I give examples of such rings now. The first example

justifies the claim before that. This generalizes the classical notion of LCI ring. So, the claim is that any p-complete n-syntomic ring which is locally complete intersection is quasi-syntomic. And well, this is checked using 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's theorem to prove that any such ring is quasi-syntomic. Okay, so that's one first class of example but you also have huge rings in this category

of quasi-syntomic rings. Namely, I claim that any integral perfected ring is quasi-syntomic. And the reason for this is okay, well, remember you have to check first of all, it's p-idecally complete by definition. Then you can also check that

for a perfected ring, there could be some p-torsion but the p-infinity torsion is just the same as the 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 the canonical map from zp to r it actually factors through, you can factor it through the theta map. So, I should have written this second map here on the right is fountain theta 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 out 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 yp 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. Because in this triangle the other term will vanish after p-completion. But so then, once you know this, you are reduced to describe this cotangent complex of r over a-inf of r, but then this map is subjective and the kernel

theta is by properties of perfect two rings is principle 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 comological degree minus one.

So, this way you check that in fact in this case the cotangent complex even has tau 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 over a perfect two

ring and take its p-completion or you can take a perfect two sorry, yeah, take an integral perfect two 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- asymptomatic 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- asymptomatic rings. Okay, so I think that's basically all I wanted to say about quasi- asymptomatic rings. But I just wanted to try to convince you that many interesting examples of rings are actually quasi- asymptomatic.

Okay, and one good point of this Badmoor-Schulze definition is that it's purely a definition in terms of the cotangent complex. And we have seen before this was a hard state comparison theorem that if you have some control on the cotangent complex

because of this hard state comparison, you can usually deduce things about prismatic convergence. Okay, now I turn to self-parts of filtered prismatic duodonic crystals. This will be the objects we will use to describe our feasible 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- asymptomatic. And then the claim is that

you have a natural morphism of topos which goes from the category of sheaves on the prismatic side, the absolute one, towards the category of sheaves on the small quasi- asymptomatic side of my ring R. And if you wonder how this

morphism of topos comes 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 co-continuous functor from the prismatic side of R towards the big quasitomic side of your ring R.

And then you just restrict it. The only difficulty is checking this co-continuity of this functor. But if you are familiar with crystalline homology, it's very similar to what you do when you go from the crystalline side to the

et al. or the risky side. Except that here we work with something a bit more general, we work with the quasitomic topology. Okay, and now, what I will do is I will take my prismatic structure shift, or prism, and 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 quasitomic side. And then, I claim that, okay, actually you have a natural subjection from

O-pris 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 quasitomic side. This kernel is denoted like this, so it's what is

called the first piece of the Nygaard filtration on the prismatic on this prismatic shift O-pris. Well, the reason for this notation is that there is this notion of Nygaard 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 Nygaard filtration. I will just take as a definition that it is the kernel of this subjection, which I did not explain. And then, a property of this is that well, because any delta ring has a Frobenius,

so this shift O-pris will come with a Frobenius morphism, phi. And one can check that this kernel has a property that phi of the kernel, so phi of the first piece of the Nygaard filtration of O-pris, actually lies in i-pris times O-pris. So, more or less, on this first piece of

the Nygaard filtration, the Frobenius is divisible by, well, let's say the local locally i, the ideal is generated by a non-zero distinguished element. The idea is that on this first piece of the Nygaard filtration, phi is divisible by this

by this distinguished element. Yeah, here's a question. I cannot hear you. Ophon, can you put the microphone on? You should activate it. Takeshi,

can you unmute him? Ah, yeah, now his microphone must be on. Internet, I don't...

Ah, yeah, now you can speak. So, why do you write i-pris times O-pris? I think i-pris is an ideal in O-pris. Ah, yeah, sorry, sorry, yeah, it's a typo. Yes, thank you, sorry. Yeah, it's just a problem, because later it will appear. Yeah, sorry.

Okay. Okay, so then, the definition of a filtered prismatic deudonic crystal is... So, let me again fix the quadratic ring R, and then

a filtered prismatic deudonic crystal is by definition of a triple M, field M and phi M. So, M is a finite locality-free O-pris module. First of all, then field M inside it will be some O-pris sub-module.

And finally, phi of M is morphism, ring morphism from M to M, which is assumed to be phi-linear. Okay, where phi is a Frobenius of O-pris.

Okay, very good. And then, you ask three conditions on this triple. So, first of all, you want that this Frobenius phi M sends field M to I-pris times M.

Okay, but then, there is an obvious O-pris sub-module of M, which also has this property that phi phi M of it lives inside I-pris times M. Namely, consider the first piece of the nagat filtration of O-pris

times M. Then, because of the failure slide, we know that this is contained in I-pris. And so, this sub-module, first piece of the nagat filtration times M, also has a property that phi of it is contained in I-pris times M. And then, the second axiom you have

is you want that this sub-module is in fact contained in field M. And once you have done this, then you know that M-module of field M will be a module over O-pris modulo the first piece of the nagat filtration.

But remember that O-pris modulo the first piece of the nagat filtration by definition is just the structure shift of the quasi-syntomic site O. So, M mod field M will be an O-module, and you ask that it is financially 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-pris times M as an O-pris module. So, if you are familiar with crystalline duodone theory, again, it's very reminiscent of the usual notion of a duodone or filter

duodone crystal. And the third condition is just this setting would just be a formulation of the condition that the filter crystal is admissible in the sense of quotientity. So, that's somewhat just the obvious extrapolation of

this classical notion to the setting of the prismatic site. Okay, and then, a notation is if R is my quadratic ring, I will denote by DF of R the category of all filtered prismatic duodone crystals over R.

And the morphisms are the obvious ones, namely, they should be O-pris linear and they should be compatible with Frobenius and with the filtration. 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, a quadratic 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 a G by O-pris. And here, when I write this curly X1, it's supposed to be like the local X groups in the category of abelian sheaves on the quasi-syntamic side. So, you check that your feasible group G, the reduction to the

case of finite locally free group schemes, defines an abelian sheave on the quasi-syntamic side. O-pris is also an abelian sheave on this side. So, you can take X1 in this category. Then, you also define fill M prism of G to be the same

thing, except that you replace O-pris by the first piece of the Nygaard filtration. And then, the first main result is that if G is as before, then this triple M prism G fill M prism G and

Frobenius of M prism of G, which is just by definition the Frobenius coming from the Frobenius on the prismatic sheave here. Then, the claim is that this is actually an object of this category, DF of R.

So, this is a filtered prismatic dodonic crystal over R. And in what follows, I will denote it by M prism G underline. As a remark, if P is zero in my ring R, so in the characteristic P situation,

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 functor you will find. People have studied in crystalline dodonic theory, which you find, for example, in the book of Bertolot, Brin, and Messing.

So, this is nothing new in characteristic P. Okay, and then our second main result is that, well, this filtered prismatic dodonic functor 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 of R, and the category DF of R, which I defined before. And as a bonus, as a byproduct of the proof, you actually obtain that

the prismatic dodonic functor G goes to M prism of G is already fully face-free. But if you really want to get an equivalence of categories, 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 these two results, and about the way we proved these two results. So, first remark is

that, as I said before, in characteristic P you just recover the usual functor from the dodonic theory. So, in particular, from the theorem 2, the classification result, well, you can deduce that the crystalline dodonic functor is an equivalent 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 LCI rings, which are also excellent, then fully facefulness was proved in the end of the 90s by De Jong and Messing. And more recently,

Hecker's law has proved that this functor is actually an equivalent when the ring is, okay, necessarily an LCI, and moreover, F-finite. So, this means that the full is finite. And this particular implies Xl. 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-perfectoid if it is quasi-syntomic, of course, first, and then you also want that there exists 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-perfectoid, but also like a quotient of such a ring by a finite regular sequence, which has bounded P, P torsion.

These are examples of such rings. Well, for such rings, it's like for perfected rings, in the example we saw, in fact, the cotangent complex after P completion is just sitting in D minus one. For this reason, you can check, using Hodge-Tec 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.

Then also you can check in this case that if you take the first piece of the nagat filtration on this prism to be just the inverse image of I by Frobenius, then the quotient is isomorphic to R itself.

So, two examples of computation of this initial object or some final object. The first case is if R is perfect to E, then this final object is just given by the ring A inf of R together with the kernel of the theta map.

So, 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 constant theta tilde.

It's just a matter of convention. Okay, so that's the first example. And then another example, if you take a ring which is quasi-regular so may perfect to it and with P0 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 filtered prismatic Newtonian module over R will be a collection M fill M and phi M. But now I just have, instead of having finite locally free modules over the prismatic shape, I just have a finite locally free module over this ring, prism R, a certain sub-module of it, fill M, and a phi in our map, phi M.

And you just ask the exact same actions as before, but now you only work with modules over this ring, prism R. So, I won't 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 so may perfect to it, I can in fact just evaluate all the objects I have on this – sorry, I should have written final – on this final object. And the claim is that this gives an equivalence between the category of filtered prismatic Newtonian crystals over R

and the category of filtered prismatic Newtonian 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 subject, this category DF of R.

And moreover, if the ring is perfect to it, then you can do even better. Then you can even forget the filtration. So, just look at the forgetful functor for this category of filtered prismatic Newtonian modules over R to simply what you can call a prismatic Newtonian module.

But, in this case, it already has a name. People call them minuscule breaky sympharg modules. And then the claim is, in this situation, this forgetful functor is in fact an equivalence. So, in other words, as a special case of the theorem 2, you see that you recover the fact that for a perfect-to-it ring

piecework groups of such a ring are classified by minuscule breaky sympharg modules. But this is a bit cheating because, in fact, we need as an input for the proof of theorem 2, we need a special case of this. Namely, we need that piecework group of a variation ring,

which is perfect to it and has algebraically closed factoring field, is the same thing as a minuscule breaky sympharg module. And actually, it's not difficult to – you can deduce a case of all perfect-to-it rings from this special case using a v-decent argument.

So, we need this as an input. So, can I just take five minutes to finish, or should I stop now? Okay, go ahead.

Sorry, I tried 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 duodenum functor. But I just said before that in this remark that for a perfect-to-it 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. As an example of this, just take the ring of integers in some discretely 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 is a group over this such a ring are classified by so-called minuscule boy-kissing modules.

And this has been done by Broy and Kissing, at least for all P, and then extended by Kim, Lau, and Liu to do 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. Well, there is a natural prism attached to such a ring once you choose a uniformizer. 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 an equivalent. And then we check that the functor we have is actually the same as the one which has been studied by Broy-Kissing and 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 true. Okay, and then just two words about the proof. So, for theorem one, it's not surprising.

So, you just follow the strategy of Berthe Lebrun and Messing in their book. So, you have this definition of this triple. You want to check that it is a filtered prismatic deusonic crystal. And the idea is, well, you have to understand what this x1 looks like.

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 Berthe Lebrun and Messing. And you are reduced to compute some prismatic cohomology groups.

So, first step, you use a theorem of Reynolds to reduce to the case of P0 groups which comes from some abelian scheme. And then, via this Berthe Lebrun-Messing partial resolution of any group object in the topos, what you just need to do is understand precisely prismatic cohomology of abelian schemes.

And a key tool for this is provided by the Hodge state comparison theorem for prismatic cohomology. So, the ideas are really similar to the one of Berthe Lebrun-Messing. And then, theorem 2, 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 semi-perfective ring. And a very nice feature of this site, this quasi-syntomic site, which was observed by Bait, Mauro, and Cholsu,

is that these quasi-regular semi-perfective rings, they actually form the basis of this quasi-syntomic topology. Any quasi-syntomic ring, by extracting enough piece roots, you can always make it quasi-regular semi-perfective. And this means that once you have defined this functor, so once you have proved theorem 1,

then you can, to prove that it's an equivalence, you can somehow reduce to the quasi-regular semi-perfective situation. And then, everything is more concrete, as I said. Instead of having crystals, you just have modules over this ring prism R. 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 only interested in getting classification results of all necessarily rings, then the proof is such that actually you really need to work with this big category QC.

Because the argument is by descent from this very big quasi-regular semi-perfective ring. Okay, then 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 Bayes rings. We only prove results about quasi-syntomic rings, but for example, in the work of Tsing, you can find some results for 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 Dieudonné functor. So, for the usual crystalline Dieudonné functor, you have this cotenic messing deformation theory, which is very powerful.

But here, we don't have any analogies. 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 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 injective map. And then you check that, well, in this situation, the prismatic Dieudonné module of any p0 group g is in fact the same thing as home from g to the push forward to the quadricymptomic side of this shift on the prismatic side.

I mean, this is just obtained by looking at the associated long exact sequence. And then the natural question is whether, I mean, Fontaine was the first to give a general definition of the Dieudonné 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 Vidco vectors. 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 Dieudonné functor

with Vidco vectors, but without using the general crystalline comparison theorem, which of course implies this comparison, but just directly using the 2D finish.

Okay, so I'll stop here. Thank you very much. So, thank you very much for the interesting lecture. So, are there any questions? Yes, Luc?

Would you unmute him? Yeah, I'm trying, but it doesn't work. I don't see why. Luc, could you unmute yourself?

Luc, can you put the microphone on? Now you can hear me. Yes, yes. Okay, so I think you skipped one of the last slides, where you were writing something about classification of finite locally free community group schemes.

You discussed TD visible groups. I presume that you, but presumably you can also classify truncated BTs and maybe more general.

Yes, so that is the Oh, I can hear you well. Okay, so it was just that I wanted to see this slide again. But then, is it in terms of a module or rather a complex, maybe?

Okay, so we only do it for over-perfected rings. Because, 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 identify the category of finite locally free

group schemes with the category of two terms complex of visible groups with an isogeny between them. And the only difficulty in general for general quasi-syndemic rings is I'm not sure by which kind of objects you would classify finite locally free group schemes.

So for perfected rings, as I said, for visible groups, you can forget the filtration. You just have this minuscule boikies and Farg modules. So it's not difficult to guess by which kind of objects you will classify finite locally free group schemes.

So it would just be a module over your ring prism R, which is just a inf of R in this case, of projected dimension less than one, killed by a power of P with a Frobenius and a Versibue. But in the general situation where you should also take the filtration into

account, I'm not sure how to describe the finite locally free group schemes. So that's why we only did it for perfected rings. Actually, this was already known in all cases, except maybe when P is 2 by work of Lao.

I had another question. You mentioned work by Tsing. So is there a relation between displays and this theory? Yes, this is another slide I had to skip here. So let me take R again to

be, first one reduces to the case of quasi-regular, so my perfected rings by quasi-syndemic descent. And then, well, you have this natural map from prism R to R, whose kernel is the first piece of the nagat filtration on this prism. It's a map of delta rings. Sorry, it's a map of rings. But I said that the

forgetful functor from delta rings to rings has a right adjoint, which is the bit vector's functor. So this map induces a map of delta rings from prism R to W of R. And using this observation, you can actually check that you have a functor from our category DF of R,

of filter prismatic Dieudonné modules, to the category of displays over R in the sense that it's not an equivalence. I mean, the classification of Tsing 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 T is two, there are some difficulties. So this is due to the fact that this functor we get from DF of R to displays over R is not an equivalence, but it is when you restrict to new potent objects.

In the terminology of Tsing. Thank you. Thank you. Thank you. Okay, so I remember the concern in this work on Dieudonné theory, there were some after Tsing, there were in particular some papers of Lau,

where he, I think he looked at, for example, complete mixed characteristic, notarian local rings with residue fields are perfect or not perfect. Then he gets some, using Tsing, I mean, he gets, he does things with using windows and frames.

So we get a very simple linear algebra, things which classify pretty visible groups over mixed characteristic regular rings. But sometimes he needs to assume it is, the result is different and the residue field is not perfect. But in any case, he has some, do you get a relation between what you do and his theory, 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, I mean, you write the R as a quotient of formal series over the coin ring and you just have to give two,

some finite 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, for such P-complete regular rings, this, the filtration is also not relevant.

So you can describe everything. I think, as I said, using current theorems, you can get a natural prism attached to this situation. So, for example, in the simple case where R is just okay, you just take this poiocasin prism.

So you just take like the vectors of the residue field, double bracket U for some formal variable U and E is some Eisenstein polynomial,

which is determined by, after you choose a uniformizer, you have a natural map from this hack S towards your ring R. And E is a generator of this, of the kernel. And then, yes, then you get a quite simple classification in this case.

As modules, finite locally free modules over such a ring together with a Frobenius, which has a property that after linearization, after you invert E, 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 residue field is, when the residue field is not perfect, I think he needed to work like, I think, with only formal PD-Brisseau group, but maybe I don't.

Do you get, is it true that there is, well, I don't have an outside there. I mean, I think when difficulty that law encounters is that, I mean, he uses crystalline Newtonian theory.

Yes. So somehow you have your ring R, somehow first you reduce to R mod P, where you can use crystalline Newtonian theory, but then you need to go back from R mod P to R. So you need to use Grothendieck messing theory. But the issue is that this divided power structure on the ideal generated by P is not important when P is two.

Grothendieck messing theory only works well when the divided power structure is not important. 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 Newtonian functor, you directly land in the correct category. Like, again, back to this example where R is okay. In the original work of Boyer and Orkizin, actually, what you first do is you produce a functor from the category of feasible groups

over this ring R towards a category of filtered modules over another ring that's like S, which is like the P completion of the P envelope of E inside this ring like 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 Boyer and Orkizin modules. But here we directly get a functor from feasible groups over R towards minuscule Boyer and Orkizin modules.

So somehow we avoid this difficulty and that's why we don't have any assumption on P. I have another small question.

So you said that in the definition of a prism, you have a Cartier divisor. Are there examples where it is not globally principal? No, I don't know any example where it's not principal, actually.

I think all the examples are no, the ideas are principal. I see basically one question. Can you hear me, Ofer? Yes. Okay, so the question was, does there exist any coincidence between the filtration of prismatic duodenum crystal and the filtration on BUNG on the Farke-Fontaine curve?

Since Farke proposed that using chromatic filtration, one can correspond perfectly to BT1 with BUNG. One can what? Correspond perfectly to BT1 with BUNG.

I don't know what that means. I'm just reading what BT1 is like, but so detailed of level one. I guess so. I guess what is perfect with BT1? I don't know. I don't know what it means. Maybe BT1 is perfect.

But I don't know any relation with BUNG. I mean, of course, breakouts in Farke modules are related to modifications of vector bundles on the Farke-Fontaine curve. But BUNG is like the G bundle? Yes, so I don't see... For a group, they do it with...

I guess, well... Yeah, so Fontaine... This should be a reductive group or something like that. So I don't see any relation. Okay, well, if I... Sorry, 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 proved that the same construction of display is canonically correspond to... Using these functors to display, you get exactly things construction, which is defined for more general rings. Right, yes.

Okay. Alright, so... Yeah, so then... This partly answer because the work of Lau and Son is kind of trying to use this display to... complicated tricks to...

Which I don't... Okay, so it's... And of course, you get also the crystalline deodorant theory from your... Yes. You need to use some morphism between relating the crystalline...

Okay, but you wrote it... I don't remember if you've 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 quadratic side and make the comparison there.

And then the claim is that the two categories you get are actually the same. So... For the T by P? Exactly. Okay, so you get...

Okay, I have to... This was... I mean, basically this follows from this crystalline comparison theorem. Okay.

Thank you, so I will have to think... Okay, very good. The slides will be online at some point.

Okay, thank you. You're welcome, okay. Goodbye. Bye. Thank you very much. Goodbye. Have a good day.