 Now, thank you. Thank you very much for the invitation. It's a great privilege and an honor for me to be speaking at this conference in honor of Gauver. I just want to second things that were said yesterday that have benefited a lot from various things that Gauver has done, including things he's written, but also conversations and perhaps the most productive conversation I've had with him was on a hike. So thanks. OK, so I want to discuss this theory of prismatic comology, which, so everything I'm talking about is joint work in progress with Peter Schultz. And the idea is to somehow try and define a mixed characteristic analog of crystalline comology. And so I should presage that by saying, so there's this analogy between piatic rings and characteristic P rings. And in this analogy, some of our perfectoid rings in the piatic setting seem to correspond to perfect rings in the characteristic P setting. But this analogy somehow, you can't use differential forms in either of those settings, either in the perfectoid setting or in the piatic setting. And so this notion of prisms is some attempt at defining a thing which is capable of seeing differential forms. OK, but let me begin with the motivation for what this prismatic homology theory is or where did it come from? So it comes from integral piatic hot theory, which, and the result we proved recently with Matthew. So let me recall the setup. So you fix a prime P once and for all throughout this talk. C over Qp is a complete and algebraically closed non-archimedean extension. And then OC is the valuation ring. Yes? Yeah, yeah, OK. This is a piatic valuation ring of rank 1. And C is its fraction field. Right. K is the residue field. This is the perfect field of characteristic P. And then the starting point is this construction of 110 of a-inth. So one defines this ring a-inth of OC. And I'll write down the definition, but I won't say much about it because it's not so relevant for us. So you take the ring of integers OC, reduce it mod P, pass to the inverse limit perfection, and then take the vid vectors. And since it's the vid vectors of a perfect ring of characteristic P, it has a Frobenius automorphism. And it comes equipped with this map, usually called theta, going to OC. And the defining property is that kernel, so theta is surjective. The kernel of theta is generated by some element D, which is a non-zero divisor. I'm calling it D now for reasons that will hopefully become clearer later in the talk. The geometric object we're interested in in this context is, sorry, it's a math fact, x. It's a proper smooth scheme or proper smooth formal scheme over OC. There's two of them here. So the theorem that we proved about this context was a comparison between the, or comparison in quotes, between the et al and the Dirac homologies associated to x. So sorry, I jumped ahead of myself. Let me first do the affine version. So R over OC is a formally smooth ring. So you can just think of it as the piatic completion of a smooth OC algebra. And then the theorem is the following. So in this context, we can define a homology theory attached to this algebra R, which has certain nice relations to Dirac homology and et al homology. And so there exists a complex a omega R of a nth modules with some completeness. So it has to be p, diatically complete, plus a map, which is a Frobenius map, from the complex to itself, satisfying the following two properties. So it's supposed to be a defamation of Dirac homology. And so you have the Dirac comparison, which is just the statement that if you go mod this ideal kernel of theta, so you go down to OC, then you recover Dirac homology. So this is the continuous Dirac complex of this algebra R over OC. And if instead of killing d, you invert d, you get something that is essentially et al homology. So more precisely, you take a omega R, you invert d, you piatically complete. And by virtue of, I mean you have to think a little bit, but this object, again, acquires a Frobenius from the Frobenius over here. And so what I'm allowed to do is pass to Frobenius fixed points. And this is exactly the et al homology of the generic fiber. Yeah, everything is derived. So this is like homotopy fixed point. So it's the cone shifted by minus 1 of phi minus 1 acting on that complex. No, no, this is true in the affine case. If you want to go from this statement to a statement, that doesn't involve taking Frobenius fixed points. You need the properness. So it's saying that somehow there's a defamation between something that's pretty closely related to et al homology on the generic values of the defamation to something that's related to Dirac homology when you specialize. And so if you globalize this construction, you get the corollary, which is fx over OC is a proper smooth formal scheme. Then there is a relation between the et al and the Dirac homologies, namely one is a specialization of the other. And so it's not so hard to go from the theorem to this dimension inequality. If you pass to the generic fiber of x, that's something that losing over this algebraic equals field C of care 6, 0. You look at et al homology with a lot p coefficients. And this is bounded above by the Dirac homology of the special fiber of x. And so now everything is integral. And so what is this somehow saying is that if you have somehow extra integral homology in care 6, 0. So if you have some p torsion in your singular homology in care 6, 0, then this object would be somehow larger than expected. Namely, it would be larger than the value with care 6, 0 coefficients. And then what this tells you is that this object is also larger than expected. So you will see pathology in Dirac homology for the special fiber every time you have torsion in singular homology for the generic fiber. But you need some finiteness properties of the complex of this. Actually, the finiteness follows from this. Because if you, yeah, OK. And so let me, I'm not going to explain the construction, but I'll just say what I think I'd be able to fit it here. So the vague idea of the construction is as follows. So the way you define this is some kind of a carefully controlled modification. The following object, which is you take an inverse limit over all maps from R to S with S perfectoid. Sorry, inverse. Thank you. Of A in for S. So if one tends construction of A in makes perfect sense for any perfectoid ring. And so for every perfectoid ring that receives a map from your algebra R, you can contemplate A in for S. This forms a diagram and you take the derived inverse limit of this diagram. And so what's not so hard to see is that this object is something that will essentially have the etal comparison isomorphism built into it. And then what this modification does is that it produces something which also does the Diron comparison built into it. So somehow in this picture, you're starting on the etal side and then doing something to get the Diron theory to work. And in this talk later, I want to explain how to do it kind of in the other order. But maybe before I do that, I want to make some remarks. Actually, I think the most recent is. If you choose coordinates, there are more explicit descriptions of these complexes. So I would like to view this construction as some analog of the construction of crystalline cohomology. So the input data is you have this ring A in. It's mapping down to OC. And you have a smooth algebra R over it. And the output data is this complex A omega R with its Frobenius action. And this is somehow analogous or feels analogous to what you do in crystalline cohomology, which is you start off with the perfect field of Fierce P. For example, you look at this ring of bit vectors, which is a one-parameter defamation of k, just like A in flows of OC. You start off with a smooth algebra over OC. And then you think about the crystalline cohomology of R relative to W with its own Frobenius action. And there is a precise sense in which these two are analogous. So we have a different construction of this functor, which uses the theory of topological Huxchel homology. And that construction is not sensitive to working over a base like this. It makes sense for a much wider class of rings. In particular, it specializes to both this one and this one. So there is a sense in which the two are literally examples of the same. But at least this perfectoid construction is quite different. Like, it wouldn't make sense in characteristic P. Here, you're trying to do something with differential forms and divided powers and so on. And here, you're doing something else. For you to protect it, you use our p-torch in play or doesn't matter. It actually turns out it doesn't matter because this carefully controlled modification tends to kill almost zero stuff. Maybe a warning is that this is something that lives in mixed characteristic and has a Frobenius action. Now, based on intuition from crystalline homology, the natural source of Frobenius actions is Frobenius on a ring of characteristic P. You apply some functor to it and you get a Frobenius on a mixed characteristic object. But that is not true over here. So the phi action on a omega r, it definitely does not come from characteristic P. So let me actually make a precise statement. So i.e. what I mean is that this pair is not a functor of r mod P. So r mod P would be the natural object associated to r, which has a Frobenius on it. But the output is not a functor of the r mod P guy. So there are concrete examples of things which are the same mod P but have different et al homologies of the generic fiber. And so this is just completely false. OK. And then the last remark, which I want to elaborate right now is that it's over pointed out. So if you choose coordinates on your algebra r, there's a very explicit description of these complexes in terms of q-derom complexes. So it can be computed as a q-deformation of a derom complex for a specific element q in the string gain. It's q. And so that's extremely useful in computations. OK. So right. The goal of this talk is to give a side theoretic construction of this homology theory. And in the process, we also end up getting a construction of crystalline homology, which is actually different somehow from the classical construction. There's no mention of divided powers in the definition. And it has some other applications that I'll try to explain at the end. But that's the plan. Any questions? OK. So in order to define this, I need to define this notion of a prism, which is essentially going to be some generalization of perfectoid rings. This definition of prism is sort of in terms of these things called either delta rings or p-topical lambda rings. So this is in the sense of Hu-Yum and Jo-Yal. We studied them extensively. And the definition is extremely elementary. So that's the following. So our delta ring is a pair, a delta, where a is a commuter ring. And delta is an operation that goes from, so it's a map that goes from a to a with some properties. It's a map of sets, which behaves in a predictable way with respect to addition and multiplication. So delta of a plus b looks like a to the p plus b to the p plus 1 over p times the quantity a to the p plus b to the p minus a plus b to the p. So I mean, you think of that as a polynomial. The coefficients are divisible by p. Thank you. Yeah, that's much better. Sorry. So you think of this gadget as a polynomial in a and b, and then it has integral coefficients. So you can evaluate it there. And then delta of a b is a to the p delta of b plus b to the p delta of a. And then there is a Futch term, which is p times delta of a delta of b. I'll explain where these come from. And then delta of 1 and delta of 0, 0. So these things are also called p derivations, because they tend to lower the periodic order of vanishing. So I'll give examples. But let me first just make a remark about where these formulas are coming from. So if this guy is a delta ring, then it comes equipped with a canonical liftoff for Baynes. So I can define phi as a map from a to a with the property that phi of x is x to the p plus p times delta of x. And then this guy is a liftoff for Baynes. I mean, it's obvious that it's a liftoff for Baynes. The fact that it's a ring homomorphism is exactly what's being encoded by the identities on delta. And the converse is also true, at least if the ring is p torsion free. Because if the ring is p torsion free, you can recover delta from phi just by the formula. OK, and so now you get a lot of examples. So the initial object in the category of delta rings, so I guess I didn't say this. But everything in my talk is implicitly going to be periodically complete. So zp is the base ring I work over. And so zp is a delta ring with phi equals the identity. Of course, the identity is a liftoff for Baynes in this case. And it's p torsion free, so I get a delta structure. And note that in this delta structure, if I do delta of p to the n, you get, well, for Baynes of p to the n is p to the n minus p to the n times p, the whole thing divided by p. So this is p to the n minus 1 times a unit. So it's lowering the piatic order of vanishing. OK, OK. So if the order of vanishing is non-trivial, then it lowers. Is there a formula for delta of p times something? I mean, other than where you get from that? I'll call you a polyp in the right one. There is something that's exhausted. I don't know the top of my head. And so in particular, delta of p is a unit. And this fact will be relevant multiple times later. So another example is you take zp. I mean, you can just write down these rings with the liftoff for Baynes without any trouble. So consider zp power series q minus 1, with the for Baynes given by phi of q equals q to the p. And so this is a delta ring. This will somehow be related to q to romcomology later. There are also sort of general constructions that produce delta rings. So it turns out that there's a forgetful functor from the category of delta rings to the category of rings, and it has both sided adjoints. And so the right adjoint is the vidvector functor. So the ring of vidvectors, p-typical vidvectors, is always a delta ring. And the left adjoint is sort of the thing that produces free algebras. And so my notation for that is going to be something like this. So this is the free delta algebra on a variable. And so you can describe this very explicitly. As a ring, it's just a polynomial ring, uncountably many variables. But the idea is that delta of xi is xi plus 1, and x is equal to x0. And so since you have free objects, and the category has enough limits and co-limits, you can sort of do all sorts of constructions, like adjoin variables, or set some equations, equal to 0, and so on. Of course, those things don't look very good. You said you have only the adjoint. Very good. No, no, you can develop the theory without assuming that. At various points, I'll say something. I'll check that something is a unit, but I'll only check it mod p. And I would like it to be true on the ring. OK, so I mean, it's pretty easy to work with these objects, because it's sort of played with these identities over here. And I don't want to sort of do all of it standing here. But let me just give one example for why for the kinds of things you can prove. So here's kind of a funny lemma. So let's say a is a delta ring, and x inside a is a p-torsion element. So p times x equals 0. Then it turns out that x is killed by Frobenius. So there is p-torsion, but somehow if you work in a setting where Frobenius is harmless, then there really isn't. So let's actually prove this, because it's very easy. So we have px equals 0. So we apply delta to both sides. And if you think about it, I guess I should have said this. So in this identity, once I've defined something called Frobenius, I can combine these two terms into a single term, which is phi of the Frobenius times delta of A. And so if I do that over here, I get phi of x times delta of p is equal to 0, if I did it right. OK, and so we want to show that this guy is 0. Now this guy is a unit, as I just explained. So it's enough to show that this guy is 0. But why would this be true? Well, we just do it. It's because p is at least 2 to prime. So p to the p times delta x. You can write as p to the p minus 1 times p delta x. Now p delta x has a nice formula. It's the Frobenius of x minus x to the p, which I can then pull out another power of p inside. So I get p to the p minus 2 times Frobenius of p times x. Frobenius of p is always p minus p times x times x to the p minus 1. And then both these guys are 0, because p x is 0. And so OK, it's completely stupid. But it's quite useful. But so you use the fact that delta p is a unit in z localized to p. And so what I suppose is that y from p there shouldn't be a problem. So this level should go to the south. Yeah, they could complete it. Well, if you're away from p, then p x equals 0 in place. x equals 0. Yeah, right. But you have to know that they're not supposed to be localized. It localizes from the negative sets. It localizes from multiplicative sets, which are Frobenius table. So I mean, if you invert x, you also have to invert phi of x in order to get a delta search. Ah, OK, the rest of this is in Frobenius, what I said. This is automatic. I'm fairly sure it's automatic. But don't ask me why. I mean, because of this identity, right? It forces delta of any integer on you. Delta is periodically continuous. Yes. Sorry? I thought you. Delta is periodically continuous. Delta is periodically continuous. You have to prove it. But I think it like, since it lowers the order of periodic order of vanishing by 1 by a generalization of this, you can prove that it takes the periodic topology to the periodic topology. OK. And so let me record a corollary of this llama, which is the following. So let's say I have perfect delta rings, by which I will mean delta rings where Frobenius is an isomorphism. So this is, by definition, delta rings A, such that phi is an isomorphism. So this category is equivalent to the category of just perfect rings of characteristic P. And the function is the obvious one. A goes to A mod P. And R goes to the vectors of R. It's not so hard to prove this given this llama over here. Because the Frobenius being an isomorphism forces everything to be peach-origin-free. And I have a blanket-piatic completeness assumption. So this is, once you have that, it's not so hard to prove this. And so this provides some kind of a mixed characteristic way of probing what perfect characteristic P rings are. Of course, you could also have done this just using bit vectors. But in a moment, I want to generalize this, where here I allow perfectoid rings. And here will be a slightly different notion. Is that OK? So this is kind of the language we need. And I guess I should have said one more thing. So I said that the converse is OK. Over here, A is peach-origin-free. Namely, that Frobenius lift gives you a delta structure. If you interpret everything in the derived sense, that's always true. So a delta structure is the same thing as a derived Frobenius lift for A, meaning a map from A to A, which agrees with the Frobenius on the derived reduction of A mod P. In particular, if you think about what that means for pi 1 of A mod P, you get another proof of this lemma over here. So there's various ways to do this. So let me now actually discuss what prisms are. So here's the definition. So prism is a pair, A comma I, such that A is a delta ring. I, inside A, is an ideal that defines a cariat advisor. I need some completeness along the ideal. So A is P comma I, ethically complete. And then the crucial condition is the last one. So let me write down the easiest way to write it in a line. So it says that the prime number P is contained in the ideal generated by I, plus its Frobenius translate. So geometrically, it says if you look at the cariat advisor defined by I and pi of I, there's somehow only meat and characteristic P. Do you know that pi of I is a cariat advisor? I don't know that. But there will be a kind of, somehow, pi infinity of I will be a cariat advisor. I'll explain that in a second. So yeah, let me make some remarks about this. So let's say I have a pair that satisfies the first three conditions, which are kind of innocuous. The last condition is equivalent to something that can be tested after going to the perfection. So P being contained in I plus pi of I is, first of all, you can always test it after going to the perfection. And after going to the perfection, it actually means something very simple. So it means that I, if you look at I times Aperf, then this is actually principal. It's generated by an element D, which is distinguished, which I'll explain. So IE, it means that the pejorative applied to this element D is a unit. Aperf is the direct limit of A mapping to itself along copies of Frobenius. Yeah, I think you can finish it. Yeah, it doesn't matter. But yes. So if you're only willing to work with these perfect delta rings, then this is just the same thing as saying ideal is generated by elements whose pejorative is a unit. And it turns out you can prove a lemma that as soon as you have an element in a perfect delta ring whose pejorative is a unit, it's automatically a non-zero divisor. So after you go to the perfection, at least you know, it gives you a car chair divisor. No, there's no delta ring where p to the n equals 0. Because if p to the n was 0, then by this thing, p to the n minus 1 would also be 0. Yeah, I'll explain. I'll explain. So if I give you a prism, then I was trying not to lift this. Is that OK? OK. So for a, i being a prism, somehow any local generator of i has to be distinguished in the preceding sense. So if you trivialize this car chair divisor, then it has to be generated by a distinguished element, except with the caveat that you have to trivialize it in the sense of delta ring. So you're not allowed to invert arbitrary multiplicative subsets. You're only allowed to invert multiplicative subsets, which are Frobeni's table. But if you allow some kind of a pro-Zerisky localization, then that's OK. Which board I'm supposed to raise. OK, so what are the main examples of prisms for this talk? So there are four examples corresponding to kind of four different comology theories. So the first one is zp with the prime number p. And this will give sort of a corresponding to crystalline comology. And it's a prism because I explained earlier delta p is a unit. The example from the start of the talk is also a prism. So I can do a n of oc, the kernel of this map theta back down to oc. And this is also a prism. So you have to check that a generator of the kernel of theta is actually distinguished. But this is a classical thing. Here's an example of relevant to q to romco homology. So take zp power series q minus 1 with the delta structure I described earlier. So phi of q is q to the p. And the distinguished element is the q analog of p, which means it's q to the p minus 1 divided by q minus 1, which means it's 1 plus q plus q squared up to q to the p minus 1. And then you have to check that this is a prism. But it's not so hard. It specializes to, I mean, both 3 and 2 essentially specialize to 1. So it's easy to check at using that. And then the final example is kind of an imperfect variant of example 2. So it's relevant to the kind of Broglie-Kissen modules. So you take w power series u, the ideal generated by an Eisenstein polynomial, where w is the width vector of a perfect field of kercic p, e is the Eisenstein polynomial. And I guess I have to specify the delta structure. But phi of u is u to the p. So that's the lift of Frobenius I use on u. On the width vectors, I use the obvious one, the width vector Frobenius. So that defines the lift of Frobenius on this ring, giving me a delta ring. And the ideal generated by any Eisenstein polynomial gives you a primitive distinguished element. How does the degree of the Eisenstein polynomial allow you to do this? It doesn't enter. Because you can somehow specialize to u equal 0. And then you get something which is p times a unit. And that immediately proves it's primitive without actually worrying about the higher terms. In a local generator, it's too much. If it's a local generator in the sense of delta rings, meaning you localize at a multiplicative subset, which is Frobenius stable. I mean, I wonder if you have the product of two rings. And the divider lives in one vector and c is one of the. I need to complete. So now this is all. And so what is this? You say you verify something with specialization in another form. Yeah. So setting u equal to 0 gives a delta map back from w to zp. I mean, from w power series u to w. And it sends the Eisenstein polynomial to p times a unit. And p times a unit is distinguished. And checking that, I mean, the map is compatible with delta. And checking that it's a unit can be done after the specialization. And you send it i to the particle k divider. But to principleize it, can you do it locally in this version of this risky topology where you have delta? Yeah. OK. Yeah. Yeah, there's always a pros and risky cover where delta survives. And the ideal is generated by a distinguished element. All right, so here's the proposition relating this notion to perfectoid rings. But you said that the fact that it is generated by distinguishes the lemma just in the delta case of that is the. No, no. So it's a general statement. I mean, that implies this, which is if you have such an i which is generated by a single element, that single element has to be distinguished. In general. In general, yeah. OK, so the lemma is if you work with perfect prisms, so prisms a comma i with a perfect, then this category is equivalent to the category of perfectoid rings in this kind of generalized integral sense. And the function is a comma i goes to a mod i. And conversely, r goes to a in for far comma the kernel of theta. The fact that it is the periodic performance. OK, and then there is, I just want to mention, one useful lemma which I've already implicitly used in answering questions. So it's that this notion of distinguished elements or prisms, it's kind of primitive. Like you cannot divide it further. So here's the lemma. So let's say you have two prism structures on the same ring. So any in file is well for with vectors of the tilt. So let's say I have two prisms structures on the same ring and i is contained in j. Then actually i equals j. So these ideals generated by distinguished elements, you cannot somehow divide them anymore into distinguished elements. And so as a corollary of this, you can describe the category of all prisms that live over a given prism in a very easy way. So well, let me just say it without any category theory. So if I have a map of prisms, meaning a map of delta rings that takes i into j, then actually j is generated by i. And so once you're going to fix the prism you're working over, you can essentially ignore the ideal. OK. So now I want to explain how to use this notion of prisms to define prismatic homology. So this is section four. So the root of this relation is the non-distinguished elements. So what is the, so if you have a distinguished element, does it give you a prism? In a perfect delta ring, yes. Because any distinguished element in a perfect delta ring is a non-zero divisor. And generally, you have to check that it's a non-zero divisor. And then it gives you a prism. But there are examples where it's not a zero divisor. I believe there are, but I don't know what they are. Not any delta ring is p-torsion free. Oh, OK, OK, yeah. Take grid vectors as something non-reduced. This has p-torsion, but p is a primitive distinguished element. And it is not the prism of the ring. OK, so actually from now on, let me make one assumption. So all prisms that appear from now on are assumed to be bounded. By this I mean that if I look at the ring, a mod i, which I sort of think of as the face of the prism, then its p-power torsion is bounded. And this is just for technical reasons. I mean, we have to deal with various piatic completions. And if you don't have this assumption, it kind of messes up the completion. I mean, it's true in all of these examples. So OK, so fix a prism, implicitly bounded. And so in this theory of prismatic homology, you're supposed to take, like in the example of a-nth homology, we started off with something that lived over the perfectoid ring over OC and outputted something that lived over a-nth. And so we want to take something that lives over a-mod i and produce something that lives over a. So the setup is going to be x over a-mod i is a formally smooth scheme for the piatic topology. OK, so the prismatic site is defined as follows. So it's the prismatic site of x over a. So it's the following category. So the notation is going to be x over a subscript of prism. And the objects are going to be essentially prisms that live over a together with a map from the face of the prism into x. Yeah, I'll tell you. So I start off with a prism over a, which by this lemma or corollary has to be of this form. The ideal is just i times b. And then I have a map from spoof of b-mod i b into x over a-mod i. And then I'd also specify how the topology works, but the topology just comes from this part. So I mean there are choices, but let me stick to one. So the topology defined by the etal topology on spoof of b-mod i b. So there is no prismatic condition on this? B-mod i b is a prism. No, but the etal topology, you said it for this risky topology, you have some. For etal topology, you also have this unique liftability of delta structures. If you have a map of p-complete rings a to b and a as a delta structure, then there's a unique delta structure and b lifting it, just like in the risky case. No, but when we talk about localization and this risky topology, you said the multiplicative set should be in value. I p-complete it. I said p-complete. So that forces that. OK, OK, then everything is OK for that topology. All right, so that's the site, and then there are sheaves on the site. So we have structure sheaves. There are two of them, which will be relevant. O prism and O prism bar is the bar on the O or on the whole thing. That probably looks uglier because of the erasing. OK, and so how are these defined? So if I have an object of the site, I can think of it as a diagram like this. So I have spoof b containing spoof b-mod i b with a map to x. This is a typical object of my site. And I send it to either b or b-mod i b. It's the stupidest thing. And then, of course, you have to check that these are sheaves. OK, and so the theorems are going to be theorems about the homology of these sheaves and how they're related to certain other things. And so let me give those things objects a name. So notation. So I mean, what I'm about to do is similar to what one does in crystalline homology. So if you're familiar with crystalline homology, there's this projection from the crystalline site to the Zariski site. And usually you study crystalline homology by studying its push forwards. So there's a similar thing here. There's a projection down to the etal site of x. And I will study that. So prism x over a is the notation. So this is going to be an etal sheaf on x of a module, or a algebra is even, is the corresponding homology complex. So I'll say what I mean. So roughly, if I want to describe a sheaf, I have to describe its values on opens. And so its value on some etal open u is just the prismatic homology of that u. And there's a phi action. Right. So there's a map of sites. Right. It reduces to this observation that we were just discussing, that if you have an etal map and a smaller guy has a delta structure, there's a unique delta structure upstairs. I guess I should, yeah, so all my objects, these b's, they all have a Frobenius action, because they're prisms. And so just by functoriality, I get a Frobenius action on this. And that's going to be the Frobenius action relevant to a infcomology later. And then likewise, you can also do the thing for the bars. So prism x over a bar is defined similarly. I don't want to write it out. But this is actually a sheaf of OX modules on X. And the reason is that in my site, these b-mod IBs have a structure mapped to X. So the corresponding rings are really algebras over OX. And then when I take the homology, I still get an algebra over OX. So this thing lives over here. But this object has no Frobenius action. This one does, because the ideal I is not stable under Frobenius. OK, and so here is kind of the construction of a comparison map. It's kind of the most primitive comparison in this business. So this prism bar gadget is, by definition, I take the prism thing, and I mod it out by I in the derived sense. Like if I take this guy and I reduce mod I everywhere, I get o delta bar there. And so I get prism bar. But that means that when I look at the homology of this, I mean, this is essentially generated by one element. If I have a ring and I mod out by a non-zero device, if I have a complex and I mod out by, sorry, if I have a complex with a commutative algebra structure and I mod out by a non-zero divisor, then there's a box sign differential on the modded out thing. And that's what is going to be relevant here. So therefore, we have a box sign differential on the graded ring, 8 star of prism bar, except I have to twist everything because my ideal need not be principal. I mean, you can ignore this twist for the moment. If you just think about one of the relevant examples, the ideal is principal. And then it's just the usual box sign construction that goes from HI to HI plus 1. So what is the precise relation with the construction of the crystalline cobalt? Yes. Yeah, yeah, I'll explain. So I also wondered about this when we asked you, do you have this conditional boundedness? And there is a problem, because when you compute crystalline cobalt, you use PD, I mean, you've got some PD and the loss. Then you can have a lot of problems. So let me use this prismatic thing to compute the limit. Well, I mean, the comparison is in good cases. So like smooth algebras with crystalline. So you just make the size of those things which is bounded between those. Yeah, so that's what I'm doing over here. Yes. Yeah. Right, so sorry, I was saying something. So I have this graded ring. It's a, so each term is a coherent, is a quasi coherent, it's an OX module. And I have this differential. So I get a commutative differential graded algebra with terms in this category. And then there's this general principle, the universal property of the Dirom complex implies that you get a comparison map. Is this quasi coherent? It will be. So I will call this the hostage comparison map, which goes from omega star of X over OX into this graded ring that I just wrote down. Oh, sorry. Thank you. Again, let me just put a twist over here, bracket star. It's a Braykisson twist. So that star over means I do this tensoring operation. Again, this is just a stupid fact that if you have a commutative differential graded algebra, then it automatically receives a map from the Dirom complex of the zero term of the algebra. And then the theorem is the hostage tape comparison, is that this map is an exomorphism. Is this strictly commutative? Yes. Yeah, I assume X is smooth, didn't I? Yeah. Yeah. And then it is. Well, it's OK by the comparison somehow, but it's OK. Right, so what it says roughly is that if you're interested in computing the homology of this guy, when you reduce mod i, you get something whose homology groups are given by differential forms. So it's like what happens with the crystalline homology, except in crystalline homology, you have to go to Frobenius twist in order to see the differential forms occurring as the homology groups. So some corollaries of this are. What is the standard there? It's just the twist by powers of the ideal. Ah, the twist by the corresponding power. No, which power? Well, if you do hi, you twist it by distinguished element to the i. The star is the same, yeah. Star is a number. OK, and so now you can specialize. I mean, once you have this, it takes a little bit of work. But what you can deduce is that if the ideal is actually generated by p, so somehow you're in the setting of crystalline homology. So a mod i is just a mod p. Then this is actually recovering crystalline homology. So then you get a comparison isomorphism between Frobenius pullback of prism x over a and the crystalline homology complex. So if everything was athine, I would just write down the crystalline homology. In general, it's the corresponding sheafy thing by other similar construction. And so note that there's a Frobenius pullback. So I mean, somehow it says that the crystalline homology always has a canonical Frobenius descent if you're working in this setting over prism. And if you're in the setting from the start of the talk, so in the setting of algebraic closed perfectoid fields where we had these omega complexes, then these guys recover those again with a Frobenius twist. And everything is Frobenius compatible. And so it gives some kind of a common construction of crystalline homology, or a the same homology, which doesn't make any reference to the notion of divided powers. And I'm essentially out of time, but let me write down the key lemma that makes everything work. And it's again one of these elementary lemmas about delta rings. So the lemma is if a is a p torsion free delta ring, and f and a is some element such that its p power is divisible by p. So it has the first divided power already lives in a, then in fact, all divided powers live in a. And so the notion of, I mean, even though divided power objects were not in the constructions, I mean the definitions, they come up when you try to compute homology, which is why the comparison with crystalline homology ultimately comes in. I mean delta of f to the p over p essentially gives you the first non-trivial divided power, right? Yeah. OK, so I wanted to say something about an application to the perfectoid theory, but I think I'm out of time. So I'll stop here. Thank you. Thank you very much. And Christian, is there a specific property of this product studying some kind of a little bit? Yes, we haven't completely checked this, but I believe it is true that this also comes equipped with an LADA styloisomorphism. So the Frobenius pullback of prism, mapping to prism, identifies with LADA of the prism, mapping to the prism. But LADA is computed with respect to the ideal i. Very vague question, but where does the terminology which I find appealing prism come from? Dark side of the moon. I can explain later. Sorry? Yeah. Maybe Peter will explain a more mathematical reason in the next talk. Yeah? You just mentioned some application. Right. I can say more than a word. So the statement is maybe I can take like two minutes. All right, so let me. I mean, somehow what this prismatic theory tells you is that it gives you another way of computing perfections or perfectoidifications. And so here's one application to the perfectoid theory. So say I give you r, which is a perfectoid ring, and s is a semi-perfectoid. So it's a quotient of r. And a comma i is going to be the prism attached to this perfectoid ring. So it's a inf of r comma the kernel of theta. So in this situation, you can ask if there is a universal map from s to some perfectoid ring. And the answer is yes. So sorry, this i is a relevant something. So you can look at the prism of s relative to a. And you can perfectify it. So I mean, this is an object which has a Frobenius. So you can talk about its direct limit perfection just by iterating Frobenius. And this is the universal prism over s, which means that if you reduce mod i, then this is the universal perfectoid ring, a perfect prism, sorry. So this is the universal perfectoid ring over s. But this prescription of this universal perfectoid ring as a direct limit of prism mapping to itself along Frobenius gives you a way to actually compute it. Because you have this hostage state comparison. So you can essentially compute the terms in terms of differential forms. And so when you pass to the direct limit, you still get an expression of the direct limit object in terms of differential forms. And so you can use this, for example, to re-prove your flatness lemma. That if you formally adjoin all p power roots to a perfectoid ring of a given element in some setting of integrally closed rings, then you actually get a perfectoid ring, which is faithfully flat over the original ring. And this proof does not use anything about attic spaces. And so you can then actually go one step further and you can re-prove the almost purity theorem the same way, without attic spaces. Is it to recover the etal comparison from that estimate? No, I mean, for the etal comparison, I would still say that this thing is somehow the limit over all prisms over your ring. It maps to the limit over all perfect prisms over your ring. And the latter is very closely related to etalcology. But I think that's essentially the same proof as the previous one. And like, when x is not smooth, how can x have something? Is the comology as bad as the constant? The comology, so if x is not smooth, I would propose defining this slightly differently. I would not use the site. I would do this thing where it's in place for resolutions, as in the definition of the cotangent complex. And then it's about as bad as just line comology. So you would still have the statement like the hostage comparison isomorphism using cotangent complexes. And that's actually what I am implicitly doing when I talk about this. Can you use the traditionalism of the algebra? Yeah. OK. So any recursive? OK. Can you? Sorry. Do you know or do you expect if the biocomology is related to THH in general? Yeah, it should be. There should be some Nygard filtration in this business. It's just kind of obvious to define. It's all elements of B, so that probanus of the element is divisible by i to the n, is fill n. And then I would kind of expect that that relates to pi 2n of Tp or Tc minus. OK, so let's start with our speaker again. Thank you. Thank you. Thank you.