 OK, before starting my lecture, I would like to give a few personal remarks. So I first met Professor Li Qilusi 17 years ago in Tsinghua University, Beijing, China. So at that moment, I was an undergraduate student. And he gave a course on algebraic geometry. And that opened the door of modern arithmetic algebraic geometry to me. OK, that was a fantastic experience. And after that, with the help of him and also other French professors, such as Jean-Marc Fong-Dan, Michelin-Hélo, I went to France to have my PhD. So Li Qilusi played a special important role in my mathematics career. And after that, he has always been a good teacher in mathematics for me. And also in music, in French language, and many other things. So it's my great pleasure and honor to be able to give a lecture in this conference. And I also want to express my sincere gratitude to him. OK, so the title of my talk is Co-Homology of Prismatic Crystals. Let me start with recording some basic lotions on prismatic sites. So I will fix a prime number P. So record that a prisp for me is a pair A i. So where A is a delta zp algebra. So here delta algebra just means that, roughly speaking, you have a lift of Frobenius and the morphism on A. OK? So basically means that phi x equals to x to the p p times delta x is a lift of Frobenius. So delta will have to satisfy some special properties according to this condition. And i is a locally principal ideal. And locally generated by so-called distinguished element generated by a non-zero divisor dc with delta d invertible. OK, there is one more condition. So it should require that A is, well, derived p i complete. OK, so if you are not familiar with derived completion, so you can forget about this word. Because for our case, the derived completion is the same as p i audibly complete. At least for bounded prism. A i is bounded. I think you need that p belongs to i plus phi of i. But I mean, this condition will imply that. No? No, no, no, I think so. Yeah, if you apply phi to the d, then you get d to the p. Yeah. OK. It's called bounded. So if A modulo i has bounded p power portion. So I will copy the examples in the paper of Bart Shosa. So the basic examples that we should keep in mind is the following. The first example is the crystalline prism appearing in the talk of Professor August on Monday. So you can take A i to be w and is the ideal generated by p. So k here is a perfect few in characteristic p. OK, the second example is the favorite for the people working in periodic logic theory. So A i, you can take A i to be A inf. And this guy takes to be p minus partial p. So where A inf is, according to Shosa's notation, is the width vector of ocp flat. So here ocp flat. The definition is ocp modulo p inverse limit ocp modulo p with transition map given by the Frobenius. And the p underline is the element p and p power square. So on in ocp flat. So this is the second example. Well, also this third one is a Borel-Keithing example, u. And i is generated by an isenthalinomial. OK, so in the previous two examples, there are lecture delta structures. Ah, right, thank you. OK. And here you give the delta structure with delta u equals to 0. Here, e u is an isenthalinomial. This is the equation example. OK, there is a fourth example, maybe. It is zp minus 1. And with the delta structure, delta q equals to 0. And i is generated by the q analog of p. So here, my definition is q to the power of p minus 1, so when q equals to 1, then this is just p. OK, so this is the definition of prism and bounded prism. By the way, all these examples are bounded prism. The definition of a shorter prismatic set is the following. So you have to consider just the relative prismatic set. So let's fix a bounded prism. And x be a smooth, periodic form of scheme over a modular i. So here I equip AI with the natural periodic topology. Then the prismatic set, which is delo to x slash a prism, consists of flowing data. First of all, the objects are given by bounded prism, called b ib over ai. And together with a diagram like this, so b modulo ib plus immersion f over respect a. So here it's x respect a over i. That's a committed diagram like this. So I'm going to need the coverings that I consider is the physically flat map. So map f, OK, so it's slightly complicated to another object called fc, x is a covering for this set. If the underlying morphism of delta range from b to c is so-called pi completely physically flat, we will consider the flat cover. The map from spoof b to spoof a are supposed to be a mapping of delta range from a to b, right? Yes, this is compatible with delta structure, right, with delta. So this looks like so- Is it b to c or c to b? Ah, sorry. Yeah, this is a loin part of this thing. OK, so I really want this. In the case where the i is generated by p, this gives something like what August explained in his lecture. Yes. It's essentially this. Yes, I think so. But here the function is considered as a flat topology. OK, so this is the definition of prismatic site. This just means a physically flat modular power of Pi. OK, so this, yeah, it's a bit of a question. So this completely physically flat means that if you take the derived tensor, Pi, b, this guy is concentrated in degree 0, 0, and is a physically flat v-modular Pi module. OK, so for the codendic, I will denote the associative topos by to the associative topos, OK? There are two structure sheaths on this site, right. The first oprisma sheath send an object b, b-modular i, b, x to b. The second one, I will consider, is oprisma bar, which sends such an object to b-modular i, b, OK? If they use only bounded prisms, why do they have to define unbounded? So I mean, this is just for some tackles. Yes, but for the moment, I haven't considered any unbounded prism. I don't know much about it. So in the condition of prisms, there is this derived Pi complete. But in the bounded case, it is simpler. Yeah, yes, yes, yes. So in the work, where do they need to go through some unbounded prisms in some, oh, it's just for a nice generality of the final unbounded ones? Or is it, OK, I'm not. I'm not sure. Sorry. OK, so that's when we gave the definition of a prism. In my own experience, when we're trying to do some operations by taking private products, things become unbounded sometimes. So you have to be careful. OK. OK, sorry. For me, oprisma crystal or, respectively, oprisma bar crystal is a shift, say, f of oprisma modules or, respectively, oprisma bar modules such that for each object, I'll say, well, maybe, the ix for each object here. Its valuation, well, I would just denote it by fb, its value, is ip, completely flat, and ip, complete b module. OK, so in the case of opr module, what just requires is to be p, completely flat, and p, completely flat, and, periodically, complete, you model everything by i, OK, module. So when you say ip complete, is it the same as derived complete in this case or not in the same? It's the same as derived completion. I mean, it's the same as ip artically complete in the classical sense and also in the derived sense. In this case, these two lotions coincide. So that's why I just simplified this. Concentrated degree zero. Yes. Maybe i is kind of an n0 device, and so it's twice as much low i, and then precaution is found there. OK, OK. So modulo i, and then because this two lotions coincide, OK. So there's no difference to consider derived completion or the euro completion. OK, so the second condition is for animorphism, for prism, like a bath. I will just say, expect b, maybe, pair, for animorphism, OK, morphism in this. I mean, I should write it completely like those, but it's too complicated. For animorphism f, I require that f star fb, my definition is fb completed tensor over b with c. This is a transition map, fc is isomorphism. This is, in the case of opry's crystal, in the case of opry's bar crystal, you have some similar, but in that case, you have tensor just over g modulo i, here is modulo i, OK. Yes, completed tensor product, both in the euro sense or in the derived sense. In the euro sense or derived sense, because we impose the flat list condition, then these two lotions coincide. Well, it's not a completed trivial, but. OK, because you can define the universal focal to derive from plate OK, OK. I mean, strictly speaking, you have to take the derived tensor first and the derived completion, and then you will find that, in this case, the derived completion is concentrated in degree 0 and coincided with the euro completion, so. This is completion. Derived completion. coincided with. The euro completion. Then it's a question of euro. The euro completion. User. The classical completion, I mean. Sorry. Here, so I insisted to work with a flat object. This is because for flat objects, we have FPQC descent for such objects. OK, so a flat list assumption is, in order to use, is for FPQC descent in this setup. OK, so from now on, I will suppose. So I would suppose that we should work more generally. You think the flat list assumption, without the flat list assumption, the set doesn't roll? I don't know whether it holds a lot. At least, I cannot prove it. The problem is, if you take the completion or inverse limit, this is not an exact factor. Even our limit does a lot of work so well, so you should really take the derived completion and so on. So you mean FPQC descent in the sense of pre-completely flat maps? Yeah. At least over a base of a prism. No, but then, so your classical FPQC descent model of power. Yeah, yes, exactly. And then you want to take some inverse limit? Yeah, yes. You want to get the FPQC descent for complete objects. But in general, there is this problem of, exactly, of take inverse limit. How is this a problem? I mean, the algorithm would be, like, if your objects aren't derived complete, then you would just be able to check multiple powers. It looks like that, well, here you are working with the derived notions. But since you usually, your conditional equivalent or more classical ones, it looks like there is no problem with. But we want to really work with the module, not an object in the derived category. It's a module, yeah, so if you have. If you just take the derived completion, then you have to take, somehow, H0. And then this H0 is not exact anymore. I mean, if you want to come back to a concrete module, instead of something in the derived category. Modules. So I get a. Yeah, maybe we can discuss later. It's slightly technical. OK. I apologize. Thank you. Excuse me. Sorry, that's the question from Sifo. So he asks if there's no crystal assumption. Sorry? What's the question? I mean, you read? The question was from it, so his question is, is there no crystal assumption? Crystal assumption. At least as a morphism. I mean, this is, for any morphism, this transition map is as morphism is a crystal property, the Euro crystal property. I don't understand what's. So assume A is p total of 3. OK, now I can state my main theorem. OK, so theorem A, this is the finalist result in the abstract. So assume x is proper and also always smooth as 0 of relative dimension n, say, over A modulo i. So let f be i o prismal crystal. Well, you need also some final assumption, locally free or final rank. Then if you take the prismatic cohomology of this crystal, this guy is the perfect complex of A modules concentrated in degrees 0 to n, without too much surprise. OK, moreover, this prismatic crystal commutes with arbitrary base change in A. OK, moreover, if, say, A i to B i B is a map of prism, a bounded prism, then the natural base change map from the cohomology of f over x derived tensor with B. I think this is over A. So you have to base change x to B modulo i B. So I just write it as x B, since the notation is self-evident. And also you have to restrict your crystal to this prismatic set. OK, I mean, you take the probe back. That's just a big prismatic set. This is restriction. In degrees 0 to n, it's a perfect amplitude there, or? It's locally represented by a perfect complex in degree 0 to n. Yeah, yes, yes, OK. So this is the mesuring. But is it a theory in here? No, no, no. Is it? No? What? No? It's not a theory in here. No, it is not an assasin in a theory. Yeah, so this is like the final skill for crystal in cohomology. Right, OK. Where's the next page? OK, so. So this is the first theorem, and not to state. So this theorem is a consequence of a corresponding theorem for all bar crystals. OK, so actually for all bar crystals, we can see something more. So let's consider the natural projection from the prismatic set of x to the apatopos. So if you have a shift here, the push forward to an object in the etosite is given by the prismatic cohomology of f with respect to u. OK, theorem b says the following. So let's say f will now be the total bar crystal, or still locally free of finite type, a finite rank. Then our new f is actually a perfect complex, perfect complex of OX modules in degree 0n. Also a degree, a perfect amplitude. And also comes with arbitrary base change. With arbitrary base change in A. OK, so it's clear that theorem b will imply theorem A, because if f is an O crystal, then of course its reduction module i is an O bar crystal. So if you take a prismatic cohomology of f tends to be with i, then it's kind of isomorphic to over i, please. And this guy is going to be x at all of r new. Then r new f is a perfect complex in degree 0n. Yeah, but if f is just an O bar, then you would get the wrong complex. The wrong complex? Not exactly the wrong complex. Not exactly. It's a hard complex. No, 0s. It's just a hard complex. In general it's going to be a hard complex. Yeah, it's a hard complex. I will explain that later. So if you apply this, you get a theorem A. OK, so I was remarked. So actually we will give a more concrete local description. So the local description of this r new f, depends on the elevation. Well, then maybe it's better to delete a bar to distinguish to an f in terms of peaks modules. Peaks, OK. So far for any questions? Good. For the moment I don't use box type operators yet. OK, I will just use these two words. Maybe keep the statement of theorem for the moment. OK, so let's come to the second part, local description. So in this part I will consider just a fine object such that r admits lift say r frame over A. That is a tau over A modulo i converging power series in n variables. Oh, sorry, it's not A. OK, so you have r frame. Everything is a foam. Yeah, yeah, yeah. IP completely flat or completely atar. OK, so now here you can choose any data structure you like to extend the natural data structure on A. So for instance you can take a ti equals to zero or delta tx equals just with ti and so on. And this data structure extends uniquely to r frame. So this is a lemma in the paper of Bart Schurzer. OK, so now you have a lift of x and equipped with a delta structure. The assumption here is just that there is these kind of coordinates that A mod i with these t1, tn goes by and that tau map be completely atar map. Yes, yes. Because I suppose r is smooth over the base. So atalically there exists such coordinate. OK, and the left exists because you can just. Yeah, yeah. OK. So a simple lemma which is crucial for our discussion is that such a lift defines an object in the prismatic side. When you modulo i this is just x, right? This object is actually a cover, a flat cover of the final object in this prismatic purpose. Well, the proof is actually simple. Just one word. OK, for any object u in this prismatic side, what you have to check is that the product of u with x frame exists and is a cover of u. I mean, it's a flat cover of u. OK. Now I will have to consider some official object or... Well, it's... So for any n greater than 0, let me denote r frame m plus 1 to be the delta envelope of m plus 1 copies of r frame subject to r. OK, so geometrically it takes the diagonal embedding of x into m plus 1 copies of product of r frame. OK, so this is delta envelope. This defines an object in the prismatic side. So, let r frame m plus 1. So the delta envelope is compatible with the previous delta on all the... Right, right, right. So this delta on the factor does not define uniquely a delta. I may be confused. At each copy you have a data structure and there is a canonical data structure on the tensor product. So delta envelope, not prismatic envelope. Oh, sorry, sorry, prismatic envelope. Yes, yes, yes. Thank you, thank you, thank you. Yes, yes, exactly. Prismatic envelope, sorry. So it's difficult to explain. So you have such an idea and it's generated by some regular sequence. Well, say xr and then somehow you take... Well, you suppose di is generated by one element d. So you have to first do some blow-up. So you add such elements and also all the deltas of such elements. That's roughly the prismatic envelope. Okay, not r, but I mean nk. So this guy is an object in the prismatic side. The prismatic side actually suggests the n plus one product, product of x squared. So you get some tissue object. So x frame zero is just x frame. We have two projection, p1, p2. We have one, two, p12, p13, p23. Okay, I have 15 minutes. Okay, so let m be p completely flat and p periodically complete. I just take p complete object, r module. Prismatic stratification on m relative to the base is an isomorphism epsilon. So r frame one, module i. This is a tensor, p2 to module i. Okay, so it's approved back to this object such that the ural cross-cycle condition is satisfied. So this form is naturally a category. I will denote a strat. Okay, so what is my notation? p bar x frame. So it's a category of such pairs. As usual, you have a description of o bar crystals in terms of stratification. Okay, so there exists an evaluation function from the category of o bar crystals to the category of stratifications over x frame. Well, the proposition is this evaluation function is actually equivalence of categories. Okay, and there exists also a version for o crystals. Well, let's just say one word for the proof. The essential point is the fqc descent in this setup. As I mentioned earlier, that's why we need the flatness assumption. Okay, so similar ideas appears in the work of Schatzi-Mathieu. Okay, lots of plans to work on. It's called q-crystals and q-connections in a similar setup. Actually, I got inspired by his paper. Okay, so this is stratification. There is some work also by Grohl's team. Ah, right, right. Yes, this is also the work of Grohl and his two heroes. I think it's like this. Okay, so I would consider them to relate this stratification to Higgs modules. So, well, maybe many people it's just a flat Higgs module. Higgs module. R is a P completely flat and a periodically complete R module, together with morphism theta from M to omega-1. So, here you may consider just the completed omega-1. Well, since everything is not necessarily finite type so you need to take completion such that theta-theta-veg equals to zero. Okay, and I will denote Higgs R hat to be the category of Higgs fields such that which is topologically quasi-lippant. Okay, yeah, I just ignored it. I will just work with the case where I is generated by one element. The twist disappears. But you're right. Canonically it should be some twist there. But I haven't figured out how to define Canonically such an object. For the moment I choose to ignore it. So, as you can see, I assume I is generated by one element. Then, first of all, there exists an equivalence of categories between the O-bar stratification over X3 with this category of topologically quasi-lippant Higgs modules over R. And so, if you combine this statement with the previous proposition you get the equivalence of categories between the O-bar crystals with the Higgs field, with the categories of Higgs modules. Okay, so maybe just write it here. Okay, so now let's take a crystal. Okay, say it's with the associated Higgs module and data. Then we have an isomorphism which computes our view of F in terms of the drama complex associated to this Higgs module. So, let's just write M tensor with omega as you already take. Okay, so this implies directly a theorem B, which just says that our view is a perfect complex. So, locally, you can write this complex in this way. This is the drama complex defined by the data. Yes, but here data is R linear, so it's just linear version. Now, when you change the generator of I, can you describe what happens to this equivalence or is it complex? Generate. No, I haven't considered. Ah, okay. You have two generators of I, I think it doesn't matter. You just twist the connection by the ratio of these two generators. Okay, but then you can glue. Okay, but for I, the gluing is not so... Yeah, you can glue, right? It's risky, just for the risk localization. Right, right. Unless there is some... Yeah, the gluing for problem I is simple, I think. The real problem is how to glue this equivalence categories for different choices of liftings. So, you see, in the first statement, this category and this category, these two categories are independent of the lift. Yes. But then in order to establish such equivalence, you need to use the lift. So, this gives rise to the natural question is whether it's possible to glue this equivalence. But you said that the equivalence of categories is independent of the lift. Is it well-defined? I mean, the category itself is independent of the lift. But in order to establish the equivalence, you have to use a lift. But when you change the lift, the equivalence remains canonically isomorphic to the previous one? No. No? No, no, no. But even if it's isomorphic, is it uniquely? How do you identify that? But Suzy has a category of Higgs, crystal. Yes, yes. It's a new appropriate to this context. She's, which in fact, inspired the impact. So, in some sense, this should be a new appropriate to this context. Yes. You mean, you mean the notion of Higgs modules in Higgs crystals? In some sense, it unifies all the liftings and then… Right. But in this setup, there's no delta structure. I mean, in my opinion, the version with delta structure is the correct one. Because you will see why it's correct. Sorry. Excuse me. There are two questions from Suzy. The first one is, is there a version of 7C for all the delta crystals? And the next question is, is there a version of 7C for the lift x square replaced by regular identity? No. I don't consider liftings over just over i square. I don't… Yes, yes. Instead of lifting, you already did something. I mean, I need a lifting to RA, not just a modular i square. Okay. Does that answer your question? This is the second question. I don't quite understand the first question. Can you repeat? What question is, do you have something with O, not with O? Ah, with O? Not yet. Okay, but I think Moro and Suzy had a version for all crystals in a special case with so-called the cube prism. The fourth example, or the second example that I gave. Okay, so the advantage of this theorem is it works for all crystals. Yeah, and the second question is, I think it was whether instead of a lifting, you embedded something. Ah, yes, I think that's still possible. Yeah, yeah. I could say something about that in the case of crystals, which is, yes, if you take it embedded, then you shouldn't do this now. Can we discuss it just after? Okay. So maybe I just finish my talk with one more key name, which explains roughly why this is true. So actually, there exists an esomorphism between r-frame-1 modulo i. So r-frame-1, the prismatic envelope of the diagonal embedding is quite complicated, at least in general case. In the crystalline case, it's relatively simple. That's what explained by Professor August Armandi. But in general, I think it's quite complicated. But however, when you modulo i, it's esomorphized to something quite familiar. So it's just the PD divided power polynomial run over r and then take periodic completion. Okay. And right, there are two maps. The two maps from r to this guy actually coincide with the natural inclusion. So it's a little bit of a word weird to have two identical maps, but that's true. Okay. Right. I think I have no time. Yeah. I think I should just stop here. Right. Okay. So are there questions from the Zoom? Do you have a question? I have a good one. It's just a very good question here. I have just a very good question. So appear in this equivalence. Phi. Yes. You mean the Frobenius? Yes. Oh, no. Usually, there is some Cartier thing. Phi appears in the proof. It appears in the proof. Yeah, in the proof. For example, if you want to prove such things. Okay. What are those variables? Those variables i, xi corresponds to something. I think the notation is self-evident, right? Divided by d. Right. Okay. And then when you try to prove such kind of a sense, the essential point is to prove that phi of xi and also all the deltas are contained in the ideal generated by d. Right. But also in the statement, in the equivalence, is there some phi? You didn't define, really. You said there exists some equivalence. Where? Here? Above. Above? The evaluation is well defined. It's canonically defined. Yes, I was not so precise here. Classically, in the hostage component here, there is some phi which appears. Yeah, but you fix the lifting with the problem. You fix the lifting with the problem. Yeah. And it works with every... It's zero. It's zero during complex and then h1, h2, and then the book time. You said that you didn't... Yes. I mean, I deduced this equivalence starting from here. It appears there, but it's not in this setup, maybe. Maybe not in this setup. It appears in the proof, but phi is important to get this. What was the meaning of the word crystalline prism? Crystalline prism? Maybe you didn't make that mistake. That's the case, i equals p. Yeah, yeah, yeah. In the crystalline prism, i equals p. Yeah, yeah. Maybe one more word. The same kind of argument for the KNM allows to prove that if we are in the crystalline case, then there is an esomorphism of r squared 1. It's just esomorphic to this. The same proof, I think. Not only modulo i, but the whole r squared 1 is esomorphic to this guy. So any application corollary of this generality onto the examples you mentioned before? Corollary, I think, is a finalist theorem that I stated about it. Yeah, yeah. I have a comment. So in some sense, you did not discuss in this prismatic picture at all the Italian realisation. But if you combine with this equivalence, do you see a connection with the Bialik sensor? So in some sense, you should see two connections, one with the... Yes, yes, yes. And also the Bialik. Right. Yes, that's what explained by Professor August Armande. So he did not discuss the etal, but he discussed the connection with Oyama and Shu. You want to say some connection with etal side? With etal side, I'm not sure. Maybe that's already contained in the paper by Moro and Tsuchi. They discussed some generalities on generalized prismatic crystals or generalized representations. And then in some sense, the question is, do you have all... In some sense, your ricks still satisfy some nil potential. So this should correspond to the smallness, maybe, condition. Smallness. I'm not sure. From the point of view of proof, the constant impedance condition comes from the... You take the PID completion. Okay. You have it until on the direct image, right? So in your direct image on the prismatic side and also direct image on the etal side, then the correspondence should be compatible with the direct image. Is it the case? I haven't considered etal since, sorry. Yes. When we ask whether this equivalence of categories is compatible with push forward and so on. The PID object corresponding to direct image on your crystal, then it should be the direct image on the... Yes. I think, yeah. Yeah, probably. Any more questions? I have a comment or question. So there was a question about what to do to make this more global without working with the lifting instead of working in embedding. I think that what I try to do, perhaps we adapted to this setting, is show that if you look at the prismatic envelope of actually embedding, it should have a crystal structure. It shouldn't be flat, but nevertheless it should still have a crystal structure. In particular, when you're working modular life, you have a Higgs field, a locally built Higgs field. And you can describe, I believe, crystals as modules equipped with modules over the structure shape of this prismatic envelope together with a Higgs field compatible with a Higgs field on the prismatic envelope. I think that should be the equivalence. And then you could use a Higgs complex of the value of this crystal on this envelope, on this prismatic envelope to calculate the collology. That will be fine because this envelope is really big. So in order to find this, you've got to work well with the lifting. So are you thinking of just a variant of the Piedoram complex? Well, this will be a Higgs complex. It will be a Higgs complex instead of a Piedoram complex. But is there a variant of this Piedoram thing in this non-crystalline prism, base prism? No, we don't know. Because as I said, the envelope is not for people to calculate. There's a huge deformation of the eurocrystalline side which may give you a lot of Piedoram complex in more general prism. I don't know. The main point is to calculate the structure shape of the groupway. That's what this R1 module is. The language is... It's very hard to calculate if you don't divide by R. So if there's no other question, please send again a speaker.