 To je nekaj sljedeb vsega vzgovor. Zelo so začelao vsega in generalizacijo deljevnih tukajgov, reznih in tukajgovj spasih. In tukajgov vsega in generalizacija vsega tukajgovje tukajgovje tukajgovje. So, let me start from the almost purity side, so I will just remind a few things about almost ring theory that have been hinted at in the previous talk, so I will just sketch some definitions rather quickly, so as Andres said, we always start with the basic setup consisting of a ring and an ideal, such that m is equal to n square, this is the minimal assumptions that one needs for certain results, one needs more, but this is the minimal setup. Okay, then almost in theory, just a way of systematically neglecting m torsion in a modules and a algebras, so to see the kind of notions that one introduces in practice, let me just give you a sample. So, if V prime is a map of a algebras, we say that, maybe call it f, almost flat, if m kills the fontor tor 1, then similarly we say that f is almost projective, if m kills the relevant, then it's almost unremifed. If the multiplication map, so the diagonal in geometric terms from B prime in tensor, this, if it is almost projective, in the previous sense, f is almost dull, if it is both almost, okay, then also almost unremifed, yes. One also has several kinds of almost finite conditions, maybe I will just skip, but I will just mention them, almost finite, yes, and also almost finite presentation. Okay, so one good thing about these conditions is that they localize well, or they globalize well, maybe, they globalize easily to scheme. For instance, if you have x in a scheme, and if you have a classic coherent algera, then you will say that a is almost total, if you have the same property for all, you open a fine, this is natural. Okay, then the first important definition for almost purity is the following, so suppose you have an a scheme x, z inside closed scheme, and then we say that x, z is an almost pure pair. If the functor, so you have an obvious restriction functor from almost total x algeras, well, with some obvious, well, some needed almost finite presented condition, so you can restrict this to algera with the same presented condition, if this is an equivalence. Yeah, so to be more precise, one has to consider, as also Andrei already mentioned, you have to work systematically inside the localized category to make good definition, so here you implicitly, I want to invert the almost asomorphism, so this is the right definition. Okay, and then with this notion, what is the almost purity theorem? It will be an assertion that certain pairs are almost pure, of course, and fountains prove the first important version of this theorem, so it's called fountains almost purity, and what does it say? So consider the VR with perfect residue field of mixed characteristic peak uniformizer, and then the relevant a here will be obtained by adding all the roots, p power roots of the uniformizer, and the relevant m is just the ideal of all roots generated by all the roots of p. So then consider or m be a lozmos d algera for the log structure, so one, two, p. So, what does it mean? So, this means that is a saturated submodule, submonoids, sorry, inside some z n, and we have a map, actually we have maps of monoids, this is just the structure map of the v algera r, and so the condition that you want is the co-kernel of the induced map on groups has no g tosh, has no p tosh. Yeah, injective, saturate, and locally on the tautopology one has has no p tosh, thank you, has no p tosh. Okay, then set in the spec of r, so you add all the roots of the elements of m, and z is just the special particle of pi, and then the theorem in question is just that x z is an almost pure pair, so maybe the faultings didn't exactly prove precisely this version, maybe had some other condition, but this is certainly something that one can prove with the methods faultings introduced in Spavor, so there's ideas of looking at the local comology and the action of Frobenos on it. Okay, so one little remark is that it is often more often stated by starting with the equivalent version, so start with x zero, which is just the spec of the original r and z zero, which is just the closed fiber, and the finite cover, which is etal outside z zero, and then what you do is you take the pullback to x, then you normalize, and this is almost etal, this is more in the style of faultings probably. So in this, if you write it in this way, it looks more like a version of kind of lightly lernified version of a Bianca Slema, where you don't have any tamed assumption. What did I, why, why, why zero, yes. But also a Bianca, both. I did. I don't know. So anyway, after Schultz's work, we have now much more general versions of almost purity, of course, but it's not, you cannot really say that then these versions or versions like these are obsolete, because for many applications, one still needs a kind of coordinate descriptions of x in this style. So the point is that for the application to piadical stears, one basic step is to relate etalc homology to Galois homology, and then one starts with a given x, one has here some extilder, which will be the universal covering, which is etal in the genetic fiber. And in the middle, you have this, that maybe one could call x infinity, I don't know here, infinity here, which is, well, you have maybe x here, and then here you have x, v. And then one wants to compute Galois homology of certain modules relative to this group, and one uses Lares spectral sequence to reduce to a computation of involving this group and this subgroup. Now, with the almost purity theorem, you will see that the spectral sequence almost degenerates, so that's good enough. So you are left with computing this group homology, but in order to make these calculations, you have to know, you have to be able to compute this delta, and this is what this kind of log geometric condition give you, because basically tell you that you can compute this by taking the periodic completion of m. Ah, v, yes, something like this. Group, group, group, group, group. So, okay. So, yes, so this was an apology for why I want to talk about towers of logarithmic rings, and another reason is that if you think in terms of wild version of a Biancher's lemma, then one remembers that the usual Biancher's lemma is not just for smooth schemes over certain ways, but for general regular schemes, so it is natural to look for a version of this purity if you want, or a Biancher's lemma in the context of log regular rather than regular schemes. Okay, so this is what we try to do. We try to generalize this almost purity to towers of log regular rings. So, just let me remind you what are these classes of rings. So, you have an equivalent characterization, maybe not the definition, but an equivalent characterization of what is a log ring. So, these are notions introduced by Kat, of course. So, what is this? So, it is the data of a ring, R, which is things R0, which is complete, regular, and plus the data of a submonoid, which is saturated to the R, which can assume it is sharp, and also it includes the data of a map of monoids, inducing a surjection, a surjective ringomorphism. Can I put it here? Ah, but there is a cross factor. Such that. So, you have two cases for completeness, though, we are interested in one case. So, if R contains a field, then phi is just an ison, and otherwise, kerfi is principal and generated by certain power series, whose constant term, constant term theta zero, which is m0, m0 in m0 square. Okay, so, this is what for us is not a regular ring yet. Complete, regular. Also, complete. So, let's put everywhere the completion. Thank you. Thank you. Okay. The definition is... The definition is different. The definition is different, and also, of course, it depends, you have the risk, you have a problem. Yeah. So, they usually did that to coin rings and all this, but anyway, you can do this regular. It's a sufficient condition. Is it a condition on a ring, or is it a condition on a log ring? On the log. It seems like a condition on a ring, because a P doesn't seem to be at any interaction with some kind of fixed log structure on R. Yeah, maybe I... That's not properly for me. Okay, so, maybe one should consider the pair R comma P. Actually, the triple R comma P comma beta. Yeah, yeah. You have a chart. So, yes. A sharp chart, and so on. And probably most of the people in the audience know these things better than me. So, I don't have to make too many efforts to be very precise. So, now, let MP be P minus zero. So, this is the maximum ideal of P. Then, as a consequence of these conditions, A R divided R is regular local. And for the theorem that I want to state, I need another assumptions. We assume the residue field greater than zero, and B, the Frobenius, is a finite map. Okay, then B implies that if you look at the absolute differential and tensor with this, and you pick A. This is a finite dimension, kA vector space, and it's supposed to do like this. Okay, now, log regular tower, let me replace faulting's tower. So, peak as a sequence, f1, fR of elements of R, whose differential gives the basis of this. You know, you can do it in the finite menu. So, then, you get an induced map down to the power, this is R. So, it's such that if you embed zero times P, here you have the previous map beta. And here you have E1, ER, the basis of N, R, and you send them to f1, fR. And then, here is the tower, where it's just the natural, you just add the roots. You also need, so this will be the ring, and then you need a normal structure. So, now. So, this assumes for simplicity differential, that's enough, it just wants to look for when P is called a system of parameters, and you have to use the modified differential. Ah, yeah, okay, yeah. Yes, there are some tricky things that I sweat under the rug, because I don't even remember all this. So, beta defines a log structure P tilde on X0, this is R, and the locus of triviality, so just maybe P tilde 3, the locus of triviality of the log structure, is a finite union of reducible divisors, d1, d, well, it's N, where N, oh, I used the same letter, I should not have, so let's say dN here. So, as usual, these admit a combinatorial description in terms of the phase. Ah, and the phase could be, so it's not even that one. Yeah, there could be many phases, so there could be many divisors, it's a bit complicated, so. Is the complement of this? The complement of this, so this is this, and then, so a log strata, so the log strata defines the intersections of some of these, so each of these sections is what we call a log strata. Okay, now, consider inside the spectrum of our infinity, x infinity, a closed subset of the form, would be just x infinity, so something on the P locus, union y prime, where y prime is the preimage of some log strata. Finet union. Why we have in general possible subset of this, okay, of the diagram of the, okay, so. We are, yes, you could even, this already. Okay, then, let i infinity be the ideal of this i, y inside our infinity. And then one thing that one can check is that, well, of course it is radical ideal by definition, but it satisfies our minimal condition, so it is equal to its square. And then here you have the basic setup that we want. Okay, then, theorem is take z inside of y, any constructible, and this is almost pure. Of course, relative to this almost structure. So notice that this is an almost structure that is very far from what Andre called the evaluative type, so it's not at all not even an inductive limit of principle ideals in general. Okay. Yeah, so this is what we need to say about just the statement of the theorem to put it on the watch. Well, it's cut by finally many equations. Okay, yes. Also, yeah, we have already kind of yoga over almost pure pairs, so you can massage also pairs to produce new pairs, and then you can remove this condition if you want. Because anyway, constructible just cut by fine. So, yeah, so this was the story this was the situation before the Hurricane Scholze arrived and so it completely changed the landscape in particular with the techniques that introduced his more general almost purity theorem included some previous versions of almost purity that we had worked out generalizing the method of feltings. So it included all the versions of almost purity except for this one. This one is not immediately a consequence of the perfectoid almost purity that you have. So it was natural for us to try to find a more general theory of almost strings that included all what Scholze did and that would apply to deduce also this kind of almost purity. And this is what we did and this is the subject of the second part of my talk. Perfectoid rings. So it's easy to see what is a perfectoid ring in our generalized sense in positive characteristic. So it's just part A of the definition if you want. So let T be topological free algebra I is a perfect ring so for being used as an isomorphizer and the topology is complete. Yes, T is the topology separated and attic for a finite for an ideal of finite type. Then there is actually also an official definition of perfectoid in mixed characteristic but instead of giving the official definition which is technical and not very illuminating I give an equivalent characterization which actually at this point if you paid attention close attention to the previous lecture you could even maybe guess. So so a general perfectoid ring so not necessarily in positive characteristic is a topological ring of the form where where E is a perfectoid Fp algebra and A so W is the width vectors so A is a distinguished width vectors so this means that the first coordinate is topologically important A1 is invertible and the topology is the quotient topology So I call that for every ring A we have a natural map for every complete ring so for instance so anyway for A like in my definition you have certainly a map like this a continuous map for the topologies natural topology where E is what now is called the tilt which is of course reconstruction due to fountain originally as the inverse limit with Frobenius and then what we show that if A is perfectoid then this is perfectoid and theta is subjective and the kernel theta is generated by a distinguished ideal is a distinguished element generated by some distinguished element what do I do now? maybe where do I write E or A? this one is the inverse limit of A well no, it's not D definition sorry it means that you can recover A if you know it is of this form for some E then you don't actually have a choice for E this is what it is and in this way you deduce that you get tilting equivalences so these are the category E of all pairs F, T and I an ideal which is distinguished so generated by a distinguished element and with obvious maps so these are just the continuous maps of rings such that W, F of I is contained V equal to V prime how do I call this category? let's call it E and then let also B be the category of all perfectoid rings and then what I say here with this remark is that I have a well defined functor well, first of all there is maybe I describe for the until so there is a functor from E to P which associates to E, I the quotient and then you have an inverse which it does exactly what I say here so I take the pair given by for each A perfectoid I take the pair given to the construction E of a fountain and the kernel the kernel which is a distinguished ideal distinguished ideal it is generated by a distinguished element so, and then these two functors we prove rather easy to deduce from this kind of remark that they are equivalences they are quasi inverse of each others yeah mutually quasi inverse usually quasi inverse equivalence well I had some examples concrete calculations but maybe I just skip so this as a special case it includes the category introduced by Schultz notice also that if you fix a perfectoid then for every map A to B continuous the induced map ah, yeah so if you have any map of continuum of both perfectoids then the induced map is distinguished to distinguished and so this means that the the distinguished ideal is already determined by the one given by A if you don't know what it is says so you get in particular that the tilting equivalent the general tilting equivalence that we have here restricts to an equivalence between A perfectoid and E of A perfectoid so if you fix A you don't need to put the data the data of I in the picture so this resembles more the kind of equivalence that Schultz has ok, so then what we did with yes, I think so, yeah precisely it's the same thing A perfectoids are equivalent to E perfectoids for every given A but if you want a general situation you have to put also the data of I so you have families if you want I don't know so if you have a given perfectoid algebra and characteristic P then there is not just a single way you have untilted there is a whole family of kind of deformations the parameter is given by I maybe two parameters that are close give the same untilt but in general they are different yeah you need the I you need to keep the I in the picture to do that but if you fix an A you have already an I and this is the only one you can use we studied this category of perfectoid rings and we found that it is many interesting properties for instance you have tens of products completed tens of products and other type of operations you can do but I'll just keep I just mentioned an important criteria so proposition suppose you have a complete separated topological ring I add topology I is an ideal generated by a regular sequence P is in this kind of modified power the ideal generated by this then C the Frobenius of A mod P induces and then the conclusion is that A then is perfectoid we not only have perfectoid rings there is also associated notion of perfectoid space these perfectoid rings are all aphadic rings or Hubert rings and now they are more called Hubert rings so two A perfectoid one can associate topological space of all continuous valuations and it's interesting subspace if you fix a so called ring of integral elements inside A so maybe I'll not put too many details but anyway we have a tilting homeomorphism between count A and count how does it go so if you have the equivalence class of evaluation you associate to it the map defined in this way A mod P then you have the Taichmuller then you map back to A by theta and here you put V so this is the tilt of V I don't know V whatever so with this operation you get a homeomorphism and then Hubert associates to every space of this type or in generalization of the spectrum of a pair certain pre-sheves and just like in Schultz's theory these pre-sheves turned out to be sheves of topological spaces which is highly nontrivial fact and the also the Czech homology for all finite covariance by rational subsets is trivial in higher degrees and you have similar results for the sub-shift of integral elements inside or just like in Schultz's theory then maybe just me conclude with stating our generalized version of almost purity for perfectoid rings of our generalized type so suppose you have a tau which is perfectoid for some eradic topology i of course is an ideal and then let x z some closed subset here I don't even need to say constructible just like before the proof goes by reduction constructible case in this generality then set m the ideal of z inside a and then there are two assertions m is equal to n square and b xz is almost pure for the almost structure ok, so then no no no, both are true also this you can generalize by putting formally perfectoid here which just means it is not necessarily complete but it becomes perfectoid after you complete it eradically and once you put formally perfectoid in here you can apply it to the log regular tower that I had at the beginning because this will be an example precisely of one of these formally perfectoid rings so how you check it I think you want to use criterion of this type and a lot of work but this is maybe the main ingredient so and in this way you prove you give a different proof of this during that we had already proven with the methods of faltings but now it is part of a general almost perfectoid general perfectoid picture that you have now I wanted to say maybe some other things but I will stop here questions for the speaker what is set up in this diary I said that every time that you have a complete topological ring of this kind of type like and perfectoid there is always such a map because it is a property of the with rings continuous yeah, yeah some conditions well yeah, just like Schultz generalized I mean once you have this kind of tilting equivalence then even without the tilting equivalence I can just decide to patch to form a general perfectoid rings by by patching together adic spaces no, not variance special case there always special case who contains everything yeah, because he has done things without any sanction just you need to verify that they are adic spaces that is that the pre-shef is a shef so the theorem these kind of theorems just say that certain data are adic spaces so you have all the theory of Huber already you have the globalization because it's a special case of Huber theory but then of course you are interested in knowing whether there is a tilting for this but you have it already on these fine faces so you can piece together things and if you want you can do it we didn't do it explicitly so as I understand it's right generalize both love towers somehow and the previous works but in the last formulation the love structure totally disappeared no, no then I don't see any love structure yes of course now we have a general notion of perfectoid rings and as a special case as an example of such perfectoid rings we can take the previous log regular tower with the piadetopology with the piadetopology and you complete but if you don't complete it's only all performance I mean you have the notion of log ring you might have the notion of log perfectoid ring ah log perfectoid ring you expect foods of the monoid yes and I ask for log ah we never no, no first the etal etal maps are delicate to the pie because of the non materiality so there is this but you can do it but there is this thing of Schultz so you use rational domains in finite etal in composition of those but we also have kind of non analytic branch locus in some definition so it's more delicate to no anyway this is not I also want to remark that the purity theorem is said for the log regular case there is a part of the branch locus not in the characteristic p and so this is part of the original statement which motivated by the cases considered by faultings where you had some kind of normal crossing device so the idea this is handled by applying usually by anchor slam argument plus the and so and so we needed some argument with some prime to p group to handle it anyway also I want to and this survives also in this so anyway you stated the purity theorem in the log regular case allowing branch locus part of the branch locus to be a union so this argument is still needed even in but this you get rid of by by applying the usual by anchor slam and factor is 6-0 and applying some work that we did before about finite a billion p groups I mean a prime to p groups is also I wonder maybe this is a question to you can answer yourself the definition in present teori was that was said in the previous talk was that you consider distinguished element of those which are sent by delta to units so here it is true it is true that a1 is a unit but we also have the condition a0 is the project in importance so I don't see how this comes about from the maybe it's a question to ok so you think that you are slightly more general I don't know why we ask I don't see the ok so I think we should probably delay the further questions in discussion until later because it's already our problem still so I think it's longer to get back so let's thank the speaker again