 But let me begin by thanking the organizers for inviting me and giving me the opportunity to pay tribute to Luke Izzini on the occasion of his 80th birthday. So I have to say Luke has been a guide throughout my career from the earliest days when he first guided me physically to the IHES to the most recent days going back to the very last few weeks when I'm trying to struggle with the certain problems I'm stuck on. He's been a consistent source of ideas both conceptual and technical. He's also provided enormous friendship to me and to my family. We've been able to discuss mathematics, music, literature, astronomy, and I realized that he applies this friendship not only to me but also to my students and effective mathematicians worldwide. A remarkable presence in mathematical community. So my goals for today are the following. I want to try to give a very explicit description of prismatic homology, of sleuth, scheming, characteristic, and also the category of prismatic distance. I'm following some ideas suggested to me by Bargo Fatt. I think it was in the fall of 2019. He told me at that time that I was very much like these explicit descriptions and he was correct and I waited eagerly for them to appear in the paper, the huge manuscript about prismatic homology to the hidden Schultz. But it did not appear there. And so I wrote to him practically and he said, well, in fact, those ideas never got worked up between me. It encouraged me to do so. And that's what I've been trying to do since then. I also want to try to relate this new prismatic theory to previous work, in particular the Christen homology, the convergent homology, worked up by Berlio and myself. To the theory of the Piata Cartier transform, again, me and Volodotsky, that was just a Mon-P version and also a Yama and then later a Piata version due to Ksu and Shiho, as well as others. I also want, if there's time, I'll discover two applications of this simpler work was that description, which gives simpler and more explicit description, proofs of some of the results in Barton Schultz's particular really direct relationship between prismatic and pristine homology, a relationship between prismatic homology and hutch-hake homology, the fact that prismatic or vanus action is an isogeny, and the relationship of the co-tension complex. So here I'll only get to discussing chrysaline prisms in a way this is too bad because I think the main value of the prismatic theory is that it gives a unified approach to many different kinds of Piata-Cogs theory. However, it is true that chrysaline prisms play an absolutely key and simple role in theory, in particular I think many of the proofs of the general cases reduced to the chrysaline case. So I think that's an justification for just concentrating on that, beside my own limited ability. I also have to note that this is work in progress, sub-technicalities have not been worked out completely, and I'm surely not going to describe any of the technical points that often cause me great difficulty. I want to thank Bargol Vot, both for his original suggestions and also for continuing conversations throughout the course of this work, also Martin Olson, Matthew Moro, and of course, Niki Newsy. I also want to thank Ahmed Abdesh in particular because he was the very first person, as far as I can remember, to alert me to this new theory and to his possible links to previous work. So let me begin by recalling what chrysaline prisms are. I'm only going to be discussing this first, but it's okay. Questions? Anything okay? No problem? A sound problem. What's the sound problem? People can hear anything. You can't hear anything I should say? It's okay for you, don't worry. What? It's not your fault. It's going to go ahead and depend on your own practice. So instead of using the language of delta schemes, I'm going to be using the equivalent language in the case of p-tortion-free schemes, which is all going to be in the case here. So I'll say a p-scheme is going to be a p-tortion-free, peatic-formal scheme endowed with an endomorphism is a restriction to the closed sub-scheme of p defined by p, that sub-scheme of definition, is the absolute provenance endomorphism. That's the absolute provenance endomorphism of p-1. That's what I mean by p-scheme. Always p-tortion free. Morphisms of such objects must be compatible with the provenance lifting. This is a very rigid and difficult restriction, which makes the whole theory much more complicated, but this is very important. Well, I don't completely understand exactly when this compatibility is really crucial when it isn't. So I will fix a base p-scheme s. It might be, for example, the spectrum of formal spectrum of the wit-ring of the perfect field. But we want to consider more general basing as well. So I'll begin with a spoon-s-1 scheme. This is scheming characteristic p, s-1, if you recall, the closed sub-scheme that's defined by p. And now I'm going to say what an x over s-prison is. It's simply a pair, t, comma z-t, where t is a free-scheme over s. That's the endomorphism of free-scheme from p to s. And z-t is a map from the closed sub-scheme that's defined by p, p-1, to x. And of course, it's morphisms to be an s-1 morphism. Okay, so we're going to consider the category of all x-prisons, endowed with a suitable topology. I'll say more about that perhaps later. Maybe it's risky topology, maybe it's natal topology, maybe it's some sort of flat topology. And I'll use this notation that the rest of delta is the topos of sheaves on this site. And this site is endowed with a sheet of rings, making it giving us a ring topos. If you have a full scheme here, just look at the global sections of the spectra field, and that defines a sheave on this site. Of course, if we use some exotic topology, we've got to check if this guy really is a sheave. And that's an issue which is partially addressed in the picture by Barton Schulte. There's another sheet of rings on this thing, which you get by simply reducing module OP. That's the sheet of rings, which takes a full scheme p to the global sections of the structure of its spine. That's a productive module of p. Okay. And the prismatic topology of x over s is by definition simply the structure of the cosmology of the structure sheave on this topos. One can work locally on x either using the etal is risky topology. There's a morphism, which I call here vis of x over s that's sort of distinguished from use of x over s by Bertelow. In this case, it's a map from the prismatic topos via topos. And it's quite straightforward to construct this thing just by localizing everything in the etal topos of x. And then of course, you get the fact that you can calculate prismatic topology of x by taking the drive push forward the structure sheave by means of the morphism v and looking at the topology of that on the topos. It's probably working mostly with this thing on the drive category. And now here, let me try to explain the so-called explicit formulas, which describe this prismatic topology. So let's suppose we're given a fee scheme y over s, which is a smooth lifting of x over s one. So it's a fee scheme. So we have lifted probes as well. In that case, there are canonical isomorphisms from this prismatic topology, this derived object here, and the p-deron complex of y over s. So that means I just take the ordinary deron complex. Of course, I'm using the periodically completed differentials here, and I multiply the differential by p. You know, you're very striking. And if you reduce modular p, what you get is the same complex with this modular p. This guy has no differential at all because the differential had been multiplied by p. So this homolog is just the direct sum of the hodge groups or the scalar differentials. Are you assuming it's smooth or what are the assumptions on when you do the differentials? Is there any smoothness assumption or? Yes, yes, everything. Yeah, it's smooth and why the lifting? Yes. Okay. So yeah, these isomorphisms depend on y, the lifting, and they also depend on the lifting of Albanias. And I haven't really found a really nice way of expressing how this dependence works. So that's something perhaps someone else can help me with at some point. And now a description of prismatic crystals. I claim there's a natural influence of categories between some sort of quasi coherent condition. I'm not going to make explicit here. It's important though, quasi coherent crystals of OX over S modules on this prismatic topo on this prismatic site. And quasi coherent Y modules endow with an integral and quasi nilpotent p connection. Okay, so what's a p connection? You can probably imagine that it's a map novel from E into omega one tensor E satisfying the following twisted legend rule. You have a section E of capital E and a section A of O or Y, the novel of A times E is A times novel E plus P times V A tensor E. There's a P there, that's what I'm thinking, connection. Quasi nilpotent means that if you reduce module O P, well what you get then it's a linear map, it's really a Higgs field as a matter of fact. And Higgs field should be locally nilpotent. Okay, so these are very striking, but in practice you can't really use these results globally because in practice you can't really expect to find the lifting of this game together with the probanias globally. That's quite rare. So one wants to generalize these results in some way that it is more globally useful. So let's assume only that X is embedded as a closed sub scheme of Y over S. For example, if X is quasi projected, you can put it in projective space and that has a nice global lifting with its probanias. What did you mean before by integrity already raised in this slide, but what is integrity? I think you said this way for a P connection. Yeah, there's a natural way to extend NABLA to a map of E into E tends to only go 1, then E tends to only go 1 to omega 2. It's the usual thing. It's the usual thing, yeah. Okay, so then one can form the prismatic envelope of X and Y, it's sort of too good a neighborhood. I'll try to explain more about that later. You can imagine that it might exist. Then this guy has a structure key which you're going to view as an OY module and turns out that structure key has, you know, as an OY module, a canonical integral and quasi integral P connection. And you can take the P interarm complex of that guy. I have to sort of imagine how to construct that. And it turns out the P interarm complex of that guy, I write D prime there because it's not D. It's a P connection, not a connection. And so the drum complex of that guy, repeat drum complex of that guy, calculate again the prismatic homology of X. And there's an analog of this for crystals of OY modules, OY suggest modules, which I won't make explicit. Yeah. Okay, so those are the main comparison terms I want to describe, but I also want to take some time to describe some antecedents here. Yeah. So let's say I've got, okay. So first of all, just for general remembrances here, we're going to do things past. There was very first thing I learned about this kind of theory was theory of thickenings, work of hearts on and other people on algebraic drum homology characters to zero, which one studied a millimersions. So we have a scheme X that we have a millimersion and T1 to T and a map from T1 to X. In those days, T1 was equal to X or maybe an open subscript set of X, but I'm going to be forced to work with a bigger sort of topos, a bigger site, and so we have a more general morphos in T1 to X. Later, of course, came the crystalline version of this in which instead of just taking an arbitrary millimersion, we take a PD immersion. However, to make things work in my context, we're going to take T always to be pre-tortion free. And again, T1 to X would be some sort of map. Okay, that's sort of ancient history in many ways. More recent is the theory of periodic enlargements of X. So here, T is a pre-tortion free periodic formal scheme. And we have a map from T1, again, the closed sub-scheme would be defined by P, X, morphism over S1. And this occurred a long time ago in work, really, of my own in Beralos as a sort of very first step. We didn't do much with this. It doesn't get too very far. And here, the difference between this and what comes up in the prismatic theory is there's here no Frobenius left. Otherwise, it's exactly the same. And this was also considered later by Ollama and also Tsuji. And Ollama also, if you're studying the Cartier transform, gave a version of a tweak of this, which I call here P gamma attic enlargements of X over S. Again, we have T a pre-tortion free periodic formal scheme. And now instead of a mapping from T1 to X, you have a mapping from the scheme theoretic image of Frobenius and morphism of T1 to X. So FT1 of T1, that's the scheme theoretic image of Frobenius. And so we got a mapping just from that closed sub-scheme of T1 to X. That's what's given the data. Okay. So Ollama studied the sites that you get using those two things. And he described how to work out how to view the Cartier transform with P using those two sites. Also, I want to mention just briefly what I call enlargements. Again, this is a P-tortion free periodic formal scheme. And now I'm mapping from the reduced sub-scheme of T1 to X. And let me just note here by way of passing, this guy is contained in the Frobenius image of T1 and that's contained in T1. So that's one way to relate all these different kinds of theories. Okay. So each one of these theories has two of the neighborhoods attached to it. And I always thought of these theories as really depending on these two of the neighborhoods. You have some scheme X, which is hard to understand maybe because it's singular. And what you're trying to do is embed it in something smooth, Y, which is nicer topologically and or analytically. And then you look at the two of the neighborhood of X and Y and that two neighborhood should reflect the topology of X you're trying to calculate. So, for example, in character zero, if you have a formal scheme Y and X contained in Y1, then you, well, more generally actually. So TX, Y will be the universal whatever it is that we're looking for mapping to Y. So in each one of these theories, you have a corresponding two neighborhood. So for example, for mill inversions, we've got the formal conclusion. That's not really a mill version. It's just a pro-object. It's okay. In the case of PDE immersions, we have the PDE envelope of X and Y. I'll be taking the P portion free version, which is sometimes easier. It's not as hard to work with. For the pediatric enlargements, universal pediatric enlargement is called the pediatric dilatation. I think I first learned that terminology from Sue's work. And there's also the P gamma articulation, which I won't be discussing much here today. The pediatric dilatation will play a big role later on. And then there's for general enlargements, there's the tube, which is a really a formal, a pediatric analytic space. And finally, of course, the most important to us today is the prismatic envelope. Each one of these theories leads to a different homology theory. And each one of these theories, each one of these kinds of people in these countries give rise to a sort of groupoid. So given a smooth Y over S, let's slide them on here. Yeah. I can look at the tube of the neighborhood of diagonal of Y. So look at the diagonal of Y cross S with Y. Maybe I want to hit the closed sub-scheme of Y defined by P in order to make things really consistent. Turns out by functoriality in each of these cases, this groupoid acts an envelope. So if X is contained in Y1, you get the tube of the neighborhood of X and Y, and this groupoid acts, I like to write it on the right, on this tube of your neighborhood. And corresponding to these group oids, we get rings of differential operators defined roughly as follows. You look at the structure sheet of this groupoid, which I'm denoting by OT1 by 1. You look at a suitable topological comb, some sort of, depending on the context, some sort of pseudo-hom there. And that linear dual, the groupoid structure of this groupoid gives me a hop-out of a structure on O. It gives you a ring structure on this corresponding dual, which gives me a ring of differential operators. Each one of these things own a different kind of ring of differential operators. So for example, if I work with formal schemes, I get the ring of all differential operators. There are a whole lot of those. But I work with the PD envelope because they're usually called hyper-PD differential operators. These are the operator, this ring is generated topologically by the derivations, which makes it much more connected directly to calculus in a way that we're used to. If I work with the Pianic enlargement theory, which I thought you would call capital D there, the voltage D, I don't have a very good description of what that ring of differential operators is. However, if you reduce the modular P, then it's not difficult to see what it is. It's sort of generated by the derivations, that is the sheet of tangent vectors. But instead of taking just the symmetric algebra on that, you better take the divider power algebra on that. And I have a completion there because it's the hybrid. So if I take this D gamma business, then what you get is the ring of differential operators. Again, I only know how to do this modular P. Take the ring of ordinary differential operators, but you recall that as a big center. The center that was generated by the derivations that's using the theory of peak curvature. And if you extend that to having divided powers on the peak curvature, you get this ring of differential operators. And so just roughly speaking, the first guy corresponds to sort of divided power Higgs field. The second guy corresponds to divided power or rather divided power extended differential logical connection. And that's how the theory of the criteria transfer worked out in this context. Okay, so now I want to turn to the prismatic case. And here, a miracle occurs. In this case, the corresponding ring of differential operators is in fact generated topologically by the P derivations. And that's why this whole theory is connected to a P connection. And more precisely, fuzzy milk load interval P connection. This is really quite a miracle. And I guess somehow just the action that's pervading us on T makes this happen in some way. And in particular, if you reduce the module of P, you just get the completion of the symmetric algebra of the derivations. And this just corresponds to fuzzy milk load and Higgs fields. So that's really quite satisfactory. Now, let me now turn to the F-transform. I'm going to look a little bit more detail at this theory. This was, I call it the F-transform is the given different names in the literature. This is used by Faltings, by me and Volodotsky, by Oyama, by Tsu, and especially by Shiho. And Shiho is trying to give an explicit description of a lifting, a periodic lifting, of what I call the local inverse criteria transform about even I worked out. Let me ask you just a small question to clarify, is when you said quasi-nilpotent, are those things which locally are increasing the unions of things which are 7 degree of nilpotency? Yes. But then when you said that this structure shift of what you didn't describe it completely of delta, x, y, is integral by quasi-nilpotent, this looks strange because you would think about those sickening as an inverse limit. And so it's, but here if it is a quasi-nilpotent means it kind of a direct limit. So I mean, I don't, but maybe you can write out. No, you reduce module of P and then it's okay. You reduce module of P and then look, yeah. It's a combination of periodically and yeah. So quasi-nilpotent means module of P, the thing is nilpotent. Okay. So it's, yeah. Okay. So let me try to describe this F transform. It's really in some ways very simple and very, very suggested. So again, I'm going to start with a smooth piece, the y over s. And then I have a relative Frobenius map of y to y prime. Y prime is the pullback of y with respect to the Frobenius of s. Okay. And then there's a funk there from the category of modules with quasi-nilpotent P connection on my prime to the category of quasi-nilpotent ordinary connections on y. And it's very simple to describe this spunker. So if I start with a module with connection P connection, E prime, novel prime, what is the F transform of that? But first of all, you just take, as the module, just take the relative Frobenius pullback, just F or start of E prime. Now I have to tell you what the connection is. So suppose I have a local section E prime of E prime. Then I can pull it back to get a section of E. E is the Frobenius inverse pullback of E prime. And novel of F thing, you get by simply taking novel prime of E prime, which is in omega one of y prime tensor E prime, and then apply Frobenius, relative Frobenius on omega one y prime of s. That's divisible by P. So just divide by P. And then you get something in omega one of y over s tensor E. And because the original connection was not a P, it was not a connection, but a P connection, and because you divide by P here, this turned out to be a connection. And you've got to check that the quasi-nilpotent corresponds to the quasi-nilpotent. That's not so obvious, but it works. And here's the theorem of Shiho. This spunker, which is quite easy to describe, is actually an equivalence of categories. The spunker in the other direction is much more difficult to describe, and Shiho's proof is really not so trivial. You can sort of understand that we will better using prismatic theory, makes everything a little bit more conceptual, but Shiho worked it up before that. And let me just note right away, there's a really striking consequence of Shiho's theorem, which shows that relative Christian homology descends by Frobenius. So I suppose I've got some smooth scheme Z over Y, or Z over Y, Z1 or Y1 even. It's got a Christian homology, going to give you a module with interval connection on Y. And automatically, just from that data, just from those data, we get something on Y prime. So Christian homology automatically descends by Frobenius. That was one of the outputs of the prismatic theory, but here it is, going way back to Shiho's theory. Okay, what's next? I want to make a few more remarks here. If you'd like to hear something. Be quiet, Shiho. Okay, so let's start with a module of P connection, E prime, novel prime, and let E novel be its F transform. Let's compare this to the pullback of E prime, novel prime as a P connection. That's a different thing, E double prime, novel double prime. Well, it turns out that E double prime, E double prime, novel double prime is obtained from all you have to do is multiply the connection by P and that gives you a P connection. That's no surprise for your difference in the formula. However, if there are any L-ata fans still in the audience, you have the following, I think, nice result, which shows that if you take the L-ata functor, which I'm not going to remind you of the definition of, it's not too important here, just for old fans, I'd say, take the L-ata functor, apply it to the Duran, the P Duran complex of E double prime, that gives you the Duran complex of E. Okay, and that's related to following you. Yeah? Question? Can you? No. Okay. So here, this is related to the following result, which almost, but not quite appears in Shiho. Suppose you start with a module which has bounded P torsion. There's a natural map in the P Duran complex of that guy, the Duran complex that was F transform, that map is actually quasi-Azimuthazine. So, let's see, do I have time to give the proof? Not really, so let me just give the proof. It's the reduce of the Carti-Azimuthazine. So, quite simple. Another result that's easy to see from this point of view is the following. Suppose I start with a module with connection, multiply the differential by D. The question? Do you have a problem? Can I go ahead? Do you have a problem? Hello? Okay, so if you multiply the differential by D, you get a module P connection. There's a map from one guy to the other, just take multiplication by P d i and degree i. This is a map of complexes. It's clearly a misogyny because, yep, I'm not back in the other direction, but multiplying by P to the m minus i and degree i, m is the dimension of y, so a misogyny that way. And that gives you a clear understanding of why the mapping from the P Duran complex of my originally prime to the P Duran complex of its pullback as a P connection is an misogyny because the guy on the left is quasi-azimorphic to the Duran complex of its pervanous transform. And we've just seen that guy is quasi-azimorphic to its multiplication by P. And that guy is the Duran complex, the P Duran complex of its pervanous pullback of the P connection. And this, yeah, okay, so this has applications through prismatic homology. Now, here's an example of, a generic example of how this f-transform works. So let's suppose I've got a closed subspeed x of y1. Again, y1, y1 is remember this smooth peace game over s. Then you can take the prismatic envelope of x prime and y prime, which has a canonical P connection, take its f-transform, which is going to be a module over y with ordinary connection. And that turns out to be the structure T of the P D envelope of x and y. This is actually an amplification of the resultant bot in Schultze, which is a very key resultant in their paper, because they prove really that if you take the pervanous pullback of the prismatic envelope, what you get is the P D envelope. And this enhances that result by talking about the connections, more precisely the P connection on one and the connection on the other. They correspond to each other too. Okay. Okay, so that's the end of what I want to say about antecedents. Now I'm going to talk about geometry of these schemes. So when I started to think about prismatic homology and prismatic compos, my main thing was to understand the role of this pervanous and the morphism, because I'd considered these pediatric enlargements before with no pervanism in the morphism. So what does this add to us? That's what I really wanted to understand, and I really wanted to get my hands dirty. And I did a lot of work on this, and here's one result which turned out to be really, really important and in a way surprising. So I recall I've been talking about a smooth P scheme while we're at it. And by that I just meant a formal scheme, which is formally smooth as a scheme. I don't have any condition about pervanous there. So what is going on? What is the role of pervanous there? In particular, a key fact about smooth morphism is the characterization of smoothness. Well, interest has a lifting property. And what happens if you put the fee into the into the soup in that case? All right, so suppose I've got another fee scheme, t over s and a morphism from t1 to y over s. What I want to know is, is there locally at least a lift of that morphism from t1 to y as a morphism from t to y, which is a morphism of fees schemes that is it, which is compatible with the pervanous lift. So what is geometry there? Well, the first surprising result is that such a lifting is unique if it exists, at least if t1 is reduced. This is kind of upsetting, as well as surprising, because you don't expect it to exist globally. There's a local unique scheme, then it's not going to work. And in fact, locally, if you work locally, there's risky pathology or the ethyl pathology, there are obstructions to this lifting. However, these obstructions can be killed by taking peace routes of things. And here's what you can actually end up proving. In the category of fees schemes, such a lifting is in fact possible after what's called a p completely flat, basically flat extension, t tilde of t. So the p completely flat morphism of fees schemes from t tilde to t, and after that, you can't actually find a lifting of the original morphism from t1 to t, which is compatible from t1 to y, which is compatible with the range lifting. Okay, so this, I found this, I was completely stuck. My original strategy to prove bus formula completely broke down after a while. And I was completely stuck until I found a very special case of this result in a paper by Matthew Moro and Suji, and I was able to generalize it to this result. And that was actually key in the rest of my work. The key fact here to spoke to this extension is the fact that t, the forbanus on morphism in fees schemes is surjected to the p completely flat, flat topology. Right, time to say what the, yeah. Okay, Karlie, what's the Karlie? Oh, yeah, important Karlie here in the prismatic theory is the fact that if x is closed in y1, and if you take, again, y over sd is formally smooth, then the prismatic envelope of x and y covers the final object of the prismatic copos, if you work with a p completely flat topology. Otherwise, if you work with a topology, coverings of the final object are very much more complicated. So this is a really big improvement in my work. P complete flat, flatness is something that I struggled with also. There's a theory of i complete flatness in general for, I guess, I don't know, I'm not sure exactly the origin what's explained in papers by Barton Schultz. Fortunately, if we're working in a case in which i is just p, things are much simpler. If you have a p torsion three, not necessarily if they're in ring and m is a p, I think it's separated our module, the following things are equivalent. First of all, m is p torsion three and m module of p is flat over r mod p. Secondly, the same is true module every power of p. And third, these tors all vanished. So basically what this means is when you want to work with p completely flat guys, what you're really doing is working module every power of p. In fact, this is the approach taken by Grinfield in his manuscript on stack versus system from logic. And once you're really thinking about working either with p had to complete things or with systems, suitable systems of modules module every power of p. Okay, so the next thing I had to study pretty carefully was prismatic neighborhoods. And I really want to understand, get my hands dirty and really understand what prismatic neighbors will look like in this special case of prism and prisms. So suppose I've got again a p scheme y close substitute x of y one. And as before delta x y is the universal ex prism mapping for y. And the key fact is the following that if you take, remember, by definition, I've got a mapping from the reduction of this thing module p to x. And the key fact is the fact that this map is affine and faithfully flat provided x is regularly immersed in y one, which will be the case for us. Okay. Now I wanted to really understand what this looked like and constructed by bare hands, rather than using universal properties of the theory of delta rings. And so in order to get my hands on this, I found the following generalization. Instead of starting with a fee scheme, you start with a scheme y and that was an amorphism. It doesn't have to be for various module. Okay. It's enough to be for various when restricted to x. And once you have that, that's enough to create this prismatic neighborhood. It still exists. There was a prism mapping to this guy. This is a nice generalization has applications. But the main reason I do this would support in the way I constructed the prismatic neighborhood. So how did I do it? Well, the first step clearly is going to be to take the periodic dilutation of x and y because the prismatic neighbor is going to map to the periodic dilutation just because you've got a mapping from that guy to the inverse image of x and the prismatic envelope is the thing in p. So in fact, the construction is carried out, at least the way I carried out, the infinite sequence of periodic dilutions. So if dxy is the universal periodic enlargement of x nothing to y, that's what the periodic dilutation is. Let's get a better idea what that really is like. Disappeared in some old papers of mine and also worked out more carefully completely by two. All you do is you take the affine piece of the block of x and y defined by the element p in the ideal of x and y. So you sort of join the ratios of element, all the elements in i divided by p and then you can kill the p torsion. In general, this thing could be empty, but it's okay if x is nicely behaving y. So here's a simple example. Suppose x is defined in y by p and x, the single element of p. Then this dilutation is obtained by taking the formal spectrum of the dilution of the p torsion p quotient of b adjoin t modulo pt minus x. Just adjoin x divided by p, but you have to kill the p torsion. And that p torsion can be hard to understand and consequently kill it and then reduce modulo p or take the p out of the equation to be empty. That's not good. However, you want to check that if px is a b regular sequence, this quotient that I've got up there already p quotient three. And that means when you reduce modulo p, it's faithfully flat over x. So that's good. Now how does the construction of the cosmetic envelope actually work? Well, the first thing you do is you let y prime be this periodic dilutation of x and y. And it turns out that the this anamorphism of y that you started with acts naturally by punctoriality really on this periodic dilutation. Gives you a new anamorphism of this guy. Now it's not even if the original fee was it was a lifting of pervades this guy might not do. That's the that's the key fact. So it turns out there's a natural thing you can do. I won't like the equation. You can just find it if you just do a straightforward calculation. There's a closed set of px prime of y prime one. Just look at it, which is the obvious obstruction to this guy being a lifting of pervades. And so you take the panic dilutation of that. And you do this thing over and over again. And then you take some sort of limit. And what you check is that the original x and y one was a regular immersion, then this x prime you get in y prime one is again a regular immersion. So it's a whole lot of pediatric dilutations along regularly immersed sub schemes. Each one of these is faithfully flat over x. And so you're in good shape. Here's a concrete example before the explicit. So suppose x is just a point and suppose y is the f online. Yes. Yes. So the flatness assertion that you made. Yes. So the project assertion that you can take a period dilutation of a regular scheme x and y one. And the period dilutation maps down to x is reduced module p. That matters faithfully flat. And this implies the flatness for the charismatic envelope. Yes. No, no, no, no, no, no, no, no, no, no, no. What you said something about the universal prism is flat over something that they didn't catch it. Exactly. Yeah, the universal if you take the universal prism reduce module p to map that down to x, that matters flat. But the map from the prison to y is not flat. And the rich condition. Okay. Okay, here's a specific example. So suppose p of x is given by x to the p plus p times delta x. Here is the formula for the universal prism, the prismatic envelope of x and y. We take the original w of x and join a whole lot of variables t1, t2, t3, and so on. Module a certain relation and then take the period completion. And here are the relations. p t1 is equal to x, that's the same thing we saw before in the first step of the period dilutation. The next after that is exactly defined by the following formulas. p times pj plus one is delted to the j of x minus pj to the p. So you have to keep dividing making more and more things divisible by p in order to make the nomorphism of this guy actually equal to lifting it for banias. And these are exactly what you have to adjoin. As a special case, suppose delta is zero. Then what you get, if you see it from this calculation or from direct calculation using other techniques, what you get is the prism is simply the pd envelope of t, t being just x over p. So that's a relationship between prismatic envelopes and pd envelopes. Let me restate that more generally and more geometrically. Suppose I've got, again, x in y1 and let's suppose x admits a v-invariant lifting x tilde. This is not always the case, but sometimes even locally it might not exist. But let's suppose we happen to have something x tilde. Okay, so this is a p-section 3 lifting of x1. So because it's p-section 3 and lifts x1, it defines a map into the periodic notation of x and y, just by the universal property of periodic rotation. So it's a section of dx over y over y, x tilde. And it turns out the prismatic envelope of x and y is the divided power envelope of this section of the periodic notation of x and y. It's a very simple description. And here's a very important application of that. You remember we had this prismatic group void. If we start with the smooth y over s, we have the prismatic group void obtained by taking the prismatic envelope of y1 embedded in y cross y via diagonal. But that y1 has a v-invariant lifting, namely y itself embedded in the diagonal. And so what you see is this prismatic group void can be viewed as the divided power envelope of this section of the periodic dilution of the diagonal of y. And that gives you the following description of prismatic differential operators, namely that they are, as I said before, generated by the p-derivations instead of the derivations. And the p-derivations because they're taking the periodic dilution. After that, it's the p-neval of that section. So that's why you get the prismatic differential operators being generated by p-derivations. And this also leads to what I call the prismatic Poincare lemma. So let me explain this. So let's suppose I've got a morphism of smooth phi scheme over s, y to z. And a close immersion of x and y with a property that if I take the composition of this close immersion of x and y with this not g, it's again a close immersion of x and z. Then I want to be able to compare the prismatic envelope of x and y and the prismatic envelope of x and z. If there's a good homology theory, these guys should give the same homology. And that's what the Poincare lemma said. Well, precisely, you have the corresponding mapping of prismatic envelopes and p-derom complexes. You get a mapping reduced by g from the p-derom couplet of the prismatic envelope of x and z corresponding to the thing of x and y. That's a strict quasi-artism morphism. I view these guys as living on x, by the way, via the natural map down to x. Okay. So that's prismatic Poincare lemma, which is sort of an analog, if you like, of prismatic Poincare lemma. There are several ways to prove this. One way is if you use the f-hence form, you can prove this. After pulling back via s, you can reduce everything as we saw before. You can change prismatic envelopes to p-derom envelopes. That's sort of the technique followed in some sense in botanical theory. And then you can use the prismatic Poincare lemma. That works. You can go back to the original s if your Frobenius on s is faithfully flat, which will be the case in many cases. But I wanted to give a direct proof anyway. And so let's see if we can see how that works by doing the local calculation. So here, x is, again, a point. y is also a point that's now periodically thickened. And z is the affine line. And I embed y and z in the obvious way. And phi of x is just x to the p. And then prismatic envelope, well, this is the dilatation of x and z, just given by taking joint t. And then t is x over p, essentially. And the prismatic envelope of x and z is obtained by taking the p-d envelope instead. So at w, w, t. And what about the p connection? Well, d prime to t, I could have written all but find the t. I'm writing d prime here. d prime to t is, well, it's p times dt. And it's d times pt. So dx. So d prime to t is dx. So that's really what we want because now we've gotten something with differential in x, which is what we need in order to get a Panko-Rey lemma. But now we've got divided powers there as well. So d prime to t divided power n is t to the n minus 1 dx. And you see the Panko-Rey lemma really works. We get a quasi-idomorphism between the ground complex of prismatic envelope with x and z and prismatic envelope with x and y. That's your w on the right. So there's the quasi-idomorphism. Now, this is just the case of a point, but in my experience, a suitable generalized point is enough. And so how do you make this work? So for particular, my map g from y is smooth and admits a section, then we can make this argument sort of work. So, and here's the key fact, we allowed to work p completely faithfully flat locally on the base. And that means we can assume my smooth morphism in fact has a section because of the fact you recall at the start of the beginning, a smooth morphism of these themes satisfies you if it has a lifting property, if you work p completely faithfully flat locally. Okay. And now you can reduce the case of a smooth morphism by considering the graph of g. And that gives you the general result. And let me just mention that once you have this result about different paedic, this general form of the prismatic Panko-Rey lemma, you can use it to simply prove that taking the p-derom complex associated to a prismatic envelope gives you a well-defined functorial cosmology theory without actually constructing a topos. However, of course, it is nice to construct the topos. And let me just say a few words about how I actually went about proving the prismatic cosmology comparison to you. So again, we've got this prismatic groupoid. I'm just going to go over this very rapidly because I ended up just following by my nose the method that Berlow and I wrote up in our book on prismatic cosmology. So we have this prismatic groupoid. Actually, this groupoid corresponds exactly to a prismatic stratification as you can imagine what that is. And that corresponds also to an action of prismatic differential operators. And we've seen that corresponds really to a plausible plot of internal P connection. And as I said before, this prismatic groupoid acts on prismatic envelopes. And now I just preceded it as in Berlow and August. We have the linearization of prismatic differential operators functor, which takes OI modules, the prismatic OX modules on the prismatic topos in this obvious way. I'm not going to make, let's just skip over the details here and save you the time. And one checks again that this, if you take the linearization of an OI module, the crystal you get is acyclic for the push forward bunker. And you apply this to the Drone complex of Y and it gives you a resolution, in fact, of the structure C of the prismatic topos. And it should push it forward. That defective resolution follows from this boundary limit described. And consequently, you take the push forward of this linearization of the Drone complex, you get, well, you calculate this calculates just the Drone complex of the prismatic envelope. And that, that gives you the theorem. And of course, the same thing works in Cook, you're doing crystal just about as well. Okay, this will be a little bit of time for a few applications. I'll go over them pretty, pretty rapidly. First of all, the Hodge-Tate comparison theorem. This is very simple to see now. So this is the same as the following way in the paper by Bobby Schultz for more general prisms, but they reduce the more general case to the case of crystalline prisms. And the crystal prisms, this explicit description we have makes it very simple to prove the result. Here is the statement. The statement is that the canonical isomorphism of different graded algebras. First of all, you take the sheath, the ordinary sheath of differentials, the simply Drone complex of X over S, the underlying means of the sheath, with its usual differential D, its Dirich differential. On the right-hand side, you take the direct sum of the cosmologies of O bar, X over S, and now that guy with a box-side operator coming from the fact this is the responsible P of O, X over S, along with O, X over S. And there's a natural map from the left side to the right side just by universal property of Kali differentials. One, they construct that. And the question is, how does one show this map is actually an isomorphism? That's a local condition. And so to prove this isomorphism, we can assume that locally X admits a lift together with its remaining isomorphism. Now, let's add these to the box-side differential. Okay, I said all that already. And oh, I'm sorry. So, and now, what's the right-hand side? The right-hand side, you recall, was given by omega with P times D, but now for this module of P, and the differential there is just zero. So the right-side hand side, as I said before, that just gives me the direct sum of the Kali differentials on the right-hand side. So it's obviously an isomorphism. I don't have a good analog of this for crystals. One can make a map, a coefficient of crystal map on individual homologies, but I don't have a box on different from the right-hand case. Okay, in relation to crystalline homology, we've already seen this from the F-transform. Let me state it, though. There's a canonical isomorphism between the pullback by probanias on S of the prismatocomology of S-organisms S and the prismomology of S. And this follows quite easily from the theory of the F-transform, as I said before. So I'm not going to say more about that. The fact that probanias is an isogyny, again, that follows from the theory of the F-transform. I explained that also in fact that probanias on prismatocomology is an isogyny that also both in the S-transform on the level of complexes that I explained before. And now I want to explain in relation to the cotangent complex, I find it's extremely striking that the prismatocomology is so directly related to the cotangent complex. And this, of course, ties into the work that we do in the Museum. So here's the statement. There's a canonical isomorphism between the cotangent complex on the left. Again, I've got, remember, I've got X over S1. S1 is the close subspeed of S defined by P. So this cotangent complex is not smooth over it. So the cotangent complex lives in degrees minus 1 and 0. And so this cotangent complex is obtained by taking the truncation in degrees less than or equal to 0 of the prismatocomology of X over S of the reduced structure of OX modules shifted by 1. So how does this come about? So this statement occurs in Bonchaltzi or appears in literature in other places as well. The only proofs I could find were made explicitly in the case in which X is alpha. And I suppose if you're good at superficial techniques better than I am, that's enough to make the statement in a more global case come out. But I wanted a more explicit formula on the level of complexes using an explicit formula for the right-hand side there. And to do that, one can also find the explicit formula from the left-hand side if you're given an embedding of X and Y, where Y is now smooth over S. In that case, one has a very nice explicit formula for the cotangent complex. You get it as follows. Take the Diron complex of Y, you get a map from D from OY into omega 1 of Y, restrict that map to the ideal sheet of X and Y, that's a map from I into omega 1 of Y, tensor that map with OX. Not obvious that it makes sense to tensor because D is not linear, but it does. And the resulting map actually is a linear map. It's the map from I mod I squared if omega 1 or Y tends to OX. And remember, if X were smooth, the quotient would be only the one of X. But anyway, that's what it is. And that turns out to be that complex of OX modules is the cotangent complex in a nice factorial way. Okay, so on the other hand, if I take the prismatic homology of X, how do I do that? Well, you take the prismatic envelope of X and Y, that's O delta XY, reduce the module OP, that's got a P connection. The P connection now is just a Higgs field. So it's a linear map from O bar into omega 1, tensor O bar, linear map. And we've got the whole Diron complex of that. And I'm supposed to truncate it. I'll do that later. That complex gives me the prismatic homology of O bar. And now I've got to map one complex to the other. Okay, there's the cotangent complex. There's the prismatic homology complex. I've got to find maps. Okay, so the map on the right of omega 1 into omega 1, tensor O bar, that's kind of obvious. I've got a map after all from delta XY, reduce module OP into X. Yes? You have two or three minutes? Okay. Yes, I'm almost finished. Yes. Yeah, that map is clear. What about the map on the left? Well, you recall that I've got a mapping from delta XY into Y, and the inverse image of X in delta XY is P times Y1. It is P times delta XY. That means everything in the ideal of X and Y gets divisible by P in O delta XY. And if I, but O delta XY is P plus 3, so I can divide by P. That gives me a mapping from I into O. And I reduce that module OP, I get a mapping on the left there. Okay, so the mapping on the left is essentially pullback divided by P. The top arrow is reduced by D. The bottom arrow is reduced by P times D, but the left arrow has a P inverse in it. So the diagram commutes. So that's the mapping on the level of complexes. And to check that it's an isomorphism, quasi-isomorphism, I can work locally. And that means I can replace Y by simply a lifting of X, in which case the map becomes simply the identity. So it's clearly an isomorphism. Okay, this is a very good place to stop. I've already described these things. This is a good place to stop because in fact, I was only able to find this construction thanks to Ylouzi who generously shared with me just a very few weeks ago, some notes he gave on the prismatic version of the theory of Delene Ylouzi. So thanks again to Luke. Let's thank the speaker.