of Scholz's perfectoid rings and perfectoid spaces and of a certain version of the almost purity theorem that has actually motivated this generalization.

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 Andreas said, we always start with a basic setup consisting of a ring and an ideal such that M is equal to N squared.

This is the minimal assumptions that one needs for certain results. One needs more, but this is the minimal setup. Okay, then almost ring theory is 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, f is almost flat.

If M kills the functor 1, then similarly we say that f is almost projective.

If M kills the relevant x, then it's almost unramified.

If the multiplication map, so the diagonal in geometric terms from B prime to answer,

this if it is almost projective in the previous sense, f is almost dull if it is

both almost not.

Okay, then also almost unramified, yes. One also has several kinds of almost finiteness conditions. Maybe I will just skip them, but I will just mention them.

And also almost finite representation. Okay, so one good thing about these conditions is that they localize well, or they globalize well, maybe.

They globalize easily to scheme. So for instance, if you have x in the scheme, and if you have A, a classic coherent x algebras,

then you will say that A is almost tall if you have the same property for all open affine.

This is the natural.

Okay, then the first important definition for almost purity is the following. So suppose you have an A scheme x, z inside x a closed sub-scheme.

And then we say x, z is an almost pure pair.

If the functor, so you have an obvious restriction functor from almost et al.

OX algebras, well with some obvious, well some needed almost finitely presented condition.

So you can now restrict this to algebra 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 definitions. So here you, implicitly I want to invert the almost isomorphism, 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 Faltings proves the first important version of this theorem.

And what does it say? So consider the DVR with perfect residue field of mixed characteristic pick 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 Rm via log-smoth D-algebra for the log structure 1, 2, p.

So what does it mean?

So this means that E is a saturated sub-module, sub-monoid, sorry.

I'm sorry, some z. And we have a map. Actually, we have maps of monoids.

This is just the structure map of the V-algebra R. And so the condition that you want is the co-kernel of the induced map on groups

has no G-torch, has no p-torch.

Yeah, inject here, saturate. And locally, on the eta topology, one has

no p-torch, I think, has no p-torch.

OK, then set this back of R.

So you add all the roots of the elements of m. And z is just the threshold follicle fiber.

And then the theorem in question is just that x, z is an almost pure pair. So maybe the Faltings didn't exactly prove precisely this version.

Maybe it had some other condition. But this is certainly something that one can prove with the methods that Faltings introduced in his paper. These ideas of looking at the local cohomology and the action of Rubinius on eta.

So one little remark is that it is more often stated by starting with the equivalent version.

So start with x0, which is just the spec of the original R. And z0, which is just the closed fiber. And the finite cover, which is eta outside z0.

And then what you do is you take the pullback to x.

And then you normalize. And this is almost eta over.

This is more in the style of Faltings, probably. So in this, if you write it in this way, it looks more like a version of, kind of widely modified version of Abiancar's lemma, where you don't have any tameness assumption.

What did I, oof, y, y, y is 0, yes. But also Abiancar.

Both. I didn't think about it yet. I did. I don't know. So anyway, after Scholz's work, we have now much more general versions of almost purity, of course.

But 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.

The point is that for the application to Piyadi-Kolstyr, one basic step is to relate a talc homology to Galois homology, and then one starts with a given x. One has here some x tilde, which will be the universal covering,

which is a talc 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, b. And then one wants to compute Galois homology of certain modules relative to this group.

And one uses Leres spectral sequence to reduce to a computation 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 p-adic completion of M. 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 log-regular rings. And another reason is that if you think in terms of wild version of Abiancar's lemma,

then one remembers that the usual Abiancar's lemma is not just for smooth schemes over a certain base, but for general regular schemes. So it is natural to look for a version of this purity, if you want,

or Abiancar'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 Catt, of course. So what is this? So it is the data of a ring R, which is R0, M0, which is complete regular,

plus the data of a submonoid which is saturated inside Z to the R,

which can assume it is sharp. And also it includes the data of a map of monoids.

Inducing a surjection, surjective ringomorphism.

Can I put it here? There it is, perfect. Such that...

So you have two cases for completeness though, we are only interested in one case. So if R contains a field,

then phi is just an isa. And otherwise, curve phi is principal and generated by a certain power series,

whose constant term, the constant term theta 0, which is in M0 and not in M0 square.

Okay, this is what for us is like regular ring here.

Complete regular... Also complete. So let's put everywhere.

Thank you. Okay. The definition is... The definition is different. The definition is different and also, of course, it depends if you have the risk or a target.

So they usually deduct the coin rings and all this. But anyway, you can use regular. It's a sufficient condition. Is that the condition on a ring or is it the condition on a log ring? Log ring. Seems like a condition on a ring.

Because the P doesn't seem to have any interaction with some kind of fixed log structure on R. Yeah, maybe it was 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. 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, R divided MP are 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 differentials, and tensor with this,

this is a finite dimensional KA vector space.

One is supposed to do like this.

Okay, now, log regular tower.

Let me replace Falting's tower. So pick a sequence f1, fr of elements of R,

whose differentials gives the basis of this.

You know, you can do it, you find out the many. So then, you get an induced map to the power, this is R, multi times P,

such that if you embed zero times P here, you get the previous map beta. And here you have E1, Er, the basis of 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, differentials are enough, it is not proof of when P is not parameter,

then you have to use a modified differential. Yeah, okay, yeah. Yes, there are some tricky things that I said under the rug, because I don't even remember all this.

So, beta defines a log structure, P tilde on x zero, spec R,

and the locus of triviality, so just maybe P tilde triv, the locus of triviality of the log structure, is a finite union of irreducible divisors,

d1, d, well, it's n where n, oh, I use 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 face.

Ah, and the face could be, so it's not even that one. Yeah, there could be many faces, so there could be many such divisors, it's a bit complicated, but, so. It's the complement of this. The complement of this.

And then, ah, so a log stratum, so the log strata,

define at intersections of some of these,

so each of these sections is what we call a log stratum.

Okay, now, consider,

inside the spectrum of R infinity, x infinity, a closed subset of the form,

well, we just, x infinity, so something on the P locus, union y prime, where y prime is the preimage of some log stratum,

a finite union. Why we have, in general, a positive subset of this, okay, of the diagram of the, okay, so.

Yes, you could even, just, already. Okay, then, let I infinity be the ideal of this I y,

inside R infinity. And then, one thing that one can check, is that, well, of course, it is radically 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, the theorem is,

take z inside of y, any constructable closed subset,

then this is almost pure, course, relative to this almost structure.

So notice that this is an almost structure that is very far from what Andrei called the evaluative type, so it's not at all, not even an inductive limit of principal ideals in general.

Okay. Yeah, so this is what I wanted to say about just the statement of the theorem, to put it on the water. Well, it's cut by, finally, many equations.

Yes, also, yeah, we have all the kind of yoga over almost pure pairs, so you can massage all sorts of pairs to produce new pairs, in particular, you can take limits of such pairs, and then you can remove this condition if you want. Of course, anyway, constructable just

cut by finally. So, yeah, so this was the story, this was the situation before the Hurricane Charles arrived, and so it completely changed the landscape,

and 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 rings that included all what Scholza 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 Frobenius is an isomorphism, and the topology is complete.

Yes, T is the topology. Separate them, and addict for a 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 a general perfectoid ring, so not that it's only in positive characteristic,

is a topological ring of the form

where E is a perfectoid of P-algebra.

And A, so W is the width vectors, so the A is a distinguished width vectors.

So this means that the first coordinate is topologically important, is invertible,

and the topology is the quotient topology.

So I call that for every ring A, we have a natural map,

well, for every, no, 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 a construction due to Fontaine originally, as the inverse limit with Frobenius.

And then what we show,

that if A is perfectoid, then this is perfectoid,

and theta is surjective, and the kernel theta is generated by a distinguished ideal,

is a distinguished element, generated by some distinguished element. What do I do now?

Where do I write E of A? This one, this one,

is the inverse limit of A mod, and well, no, it's not the definition,

so 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.

Then consider the category E of all pairs,

E, where E is perfectoid of P, 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 in E prime, and then it will actually be equal to E 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, 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, well, 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, what is it? It is generated by a distinguished element. So, and then these two functors,

we prove, it's rather easy to deduce from this kind of remark that they are equivalences. They are quasi-inverse of each other's. Yeah, mutually quasi-inverse, mutually quasi-inverse equivalence.

Well, I had some examples of concrete calculations, but maybe I'll just skip.

So this, as a special case, it includes the category introduced by Scholz. Notice also that if you fix a perfectoid,

then for every map A to B, continuous, the induce map, ah, yeah, so if you have any map of continuous,

so both perfectoids, and then the induce map

sends distinguished to distinguished. And so this means that the distinguished ideal

is already determined by the one given by A, if you don't know what it just 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 Scholz has. Okay, so then what we did with... Is there a relation between the distinguished and the distinguished in what we have this morning? Yes, I think so, yeah.

Yeah, yeah, yeah. Precisely? It's the same thing. And we'll be talking about perfecting something as distinguished in our way of the first component? Yeah. So perfectoids in mixed characteristic

can be reduced to perfectoids in characteristic P, that's what you're saying, the last thing. Ah, 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, if you... I don't know. So if you have a given perfectoid, algebra in characteristic P, then there is not just a single way of un-tilt it, there is a whole family of kind of deformations.

The parameter is given by I. Maybe two parameters that are close give the same un-tilt, but in general they are different. Perfectoid in characteristic P to perfectoid in characteristic P. 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. Okay, so then of course we studied

the this category of perfectoid rings and we found that it has many interesting properties. For instance, you have tensor products, completed tensor products, and other type of operations you can do. But I'll just skip. I just mentioned an important criteria.

So proposition. Suppose you have a complete separated topological ring

with tau iadic topology, where I is an idea generated by a regular sequence,

and P is in this kind of modified power, the idea generated by this.

Then C, the Frobenius of A mod P induces an isom.

And then the conclusion is that A then is perfectoid.

We not only have perfectoid rings, there is also the associated notion of perfectoid space. These perfectoid rings are all ephatic rings or Huber rings,

and now they are more called Huber rings. So to every A perfectoid, one can associate the 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 cont A and cont E over A.

How does it go? So if you have the equivalence class of evaluation, you associate to it the map defined in this way, so you have P and modulo P. Then you have the Teichmuller, 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 Huber associates to every space of this type or in generalization, the adiks spectrum of a pair, certain preshives.

And just like in Scholz's theory, these preshives turned out to be sheaves of topological spaces, which is a highly non-trivial fact. And also the check cohomology

for all finite coverings by rational subsets is trivial in higher degrees. And you have similar results for the sub-shift of integral elements inside, just like in Scholz's theory.

Then maybe let 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 ei-dic topology. i, of course, isn't ideally. And then let X be a, Z some closed subset. Yeah, here I don't even need to say

constructible, just like before. The proof goes by reduction to

a constructible case, but it's true in this generality. Then set M the ideal of Z inside A,

and then there are two assertions, M is equal to N squared, and B, X Z is almost pure

for the A-M structure, A-M-A, almost structure. Okay, 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 ei-adically. 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.

And how you check it, I think you want to use a criterion of this type. And a lot of work, but this is maybe the main ingredient. And in this way you prove, you give a different proof of this theorem that we had

already proven with the methods of faultings, but now it is part of a general almost perfectoid, this 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 theta here in this diagram? Theta is, 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's a property of the width rings. Okay, okay. Continuous. So it's a logic in both terms. Yeah, yeah, some conditions, well, yeah, just like Scholz generalized,

I mean, once you have this kind of tilting equivalence, then, of course, even without

the tilting equivalence, I can just decide to patch, to form general perfectoid rings by patching together adiks spaces. No, not variance, special case. They are always special case. Hubert contains everything.

Yeah, because he has done things without any sanction. Just you need to verify that they are adiks spaces, that is, that the presheaf is a sheaf. So the theorem, these kind of theorems just say that certain data are adiks spaces. So you have all the theory of Hubert already.

You have the globalization, because it's a special case of Hubert theory. But then, of course, you are interested in knowing whether there is a tilting for this, but you have it already on these affine faces, so you can piece together things, and if you want, you can do it. We didn't do it explicitly, because we went in some other direction. Also, as you may understand, it's like you generalize both your log tower somehow and

previous work. But in the last formulation, the log structure totally disappear. No, no. Then, I could... I don't see any log structure.

Yeah, no, no. Yes, of course. So now we have a general notion of perfectoid rings. As a special case, as an example of such perfectoid rings, we can take the previous log regular tower. With the periodic topology. With the periodic topology, and you complete... Well, if you don't complete, it's only all performance.

I mean, you have the notion of a log ring. You might have the notion of log perfectoid ring. Ah. Log perfectoid ring. You extract roots of the monoid. And ask for... Logator maps. Log... Hula.

We never... No, no, no, no, no. First, the etal maps are delicate to define. Because of the non-interiality. So there is this... But you can do it. Yeah, no, but there is this thing of Scholz about... So you use rational domains in finite etal and compositional doors. But we also have kind of non-analytic locus

in some definition about it. So it's more delicate to... Non-analytic. 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 peak. And so this is part of the original statement

which motivated by the cases considered by Feltings, where you had some kind of normal crossing device. So the idea is that this is handled by applying usual LaBiancar-Slema argument, plus the... And so we needed some argument with some prime to peak group.

Also, I want to... This survives also in this... Yeah, so anyway, you stated the purity theorem, the log regular case, allowing branch locus... How do the branch locus need to be a union of... Yes, yes, yes. So this argument is still needed,

even in the... But this you get rid of by applying the usual LaBiancar-Slema in CAC 360 and applying some work that we did before about finite abelian peak groups. It's not... I mean, finite abelian peak groups. And also, I wonder,

or maybe this is a question to... You can answer yourself. The definition in present theory was, that was said in the previous talk, in some previous talk, was that you consider a distinguished element of those which are sent by delta to units. So here it is true...

A1 is unit. It is true that A1 is a unit, but we also have the condition A0 is the project new importance. So I don't see how this comes about from the... Ah. Maybe it's a question to... Okay, because... So you think that they are slightly more general? Well, I don't know, I was asked.

I don't see the... Okay, so I think we should probably delay the further questions and discussion until later, because there's an RER problem still, so it might take us longer to get back. So let's thank the speaker again.