 Yes, thank you, thanks a lot for the invitation to give these lectures, it's a great pleasure to be back at the UHS. And thanks a lot for the introduction. So the title of this series of lectures is On the Local Languards Correspondents for Reductive Groups over Periodic Fields. And so maybe I'll start with some kind of general introduction to the whole series. So I guess throughout the course the field QP will be fixed, in particular there's a fixed prime P somewhere, and say at least in the second half there will be a reductive group G over QP. You can already fix it now, but it won't appear until next week. So let me say that everything I will do actually works for a general local non-archimedean field. So for a general local non-archimedean field. E, I will just specialize to the case of QP to have one variable less. And I mean also in case E is an extension of QP and G is over E, this also reduces to the previous case by considering the very restriction of skaters. So in this sense it's not even really a restriction, except that I could also consider the characteristic P local fields. But if E is actually of characteristic P, what I'm doing is at least should be very closely related to this work in progress of GST and Vincent Laforgue, who also do something very similar. So in this sense I'll just restrict the QP. So the goal of the course is the following. It is to construct a map from irreducible smooth representations. And let's say they were in QL bar of this periodic group, where of course L is not equal to P. So to any such irreducible smooth representations, the local annuals conjecture would predict that there is an L parameter and the goal is to construct such an L parameter. To construct a continuous semi-simple map and wave parameters are maps from the wave group, L parameters are maps from the wave group of QP to the L group of G. And I will denote this map by pi maps to pi pi. So I mean, this should be L parameters, so in particular I should commute with the projection to the wave group. So I guess this goes some way towards the local angle correspondence. Sorry, what? I mean, I wanted to say that it commutes with... Is this map? No, it's a map. That's the identity. Sorry. Towards the local angle correspondence. But with the following caveats. So there's a whole number of them. Notice that whatever we're getting here is only a semi-simple object. And in particular, the monotomy operator is necessarily trivial. So this procedure will not give us the monotomy operator N. So we do not get a valedoline representation, we just get severe representation. Secondly, I fixed somewhere here a prime L and there's a category of irreducible, smooth-scale bar representations and of continuous semi-simple maps here because of some incompatibility between the topology here and here. These things are actually independent of L. But it's not clear that the map is independent of L. So this may depend on L. Also, it's just a map and so far I'm not able to say anything about the fibers of this map. So you would expect that at least. The group is quartile split and the map is surjective and the fibers have some internal description related to the centralizer group of this parameter phi. I don't know whether it's surjective. For GLN it is what's known where here is Taylor and Young. Other than GLN, I think it's not so clear. So I mean it's not clear how this relates to other correspondences. So say there's a work of Arthur for classical groups using the endoscopic transfer relations and then for not too ramified groups there's work of Debaker and Rieder and Caleta for not too ramified representations. And maybe there are others who have worked on this. I mean so far you could map all elements to just one. Good point, right? So I could map everything to. To make it dependent of L and maybe one for one L. Yeah, right. So this is pretty stupid so far. I mean one can prove some things and I mean in particular one can say a few things about the compatibility with the co-mology of some reporting space and generalizations of this which gives some control. In particular for GLN you can prove that it agrees with what you think it should be. But I mean it is a valid comment. There's still a lot to be done until. I mean it's not clear what, I mean to ask whether it's functorial but first of all need independent automorphic functoriality, right? No, but if you have H mapping to G. Well but I mean do you have a map on L parameters then if you have a map of L groups you probably have a map on the representations if you don't have an independent construction of functoriality. For example with the center of things. So I think you should be able to identify the central character and you should be able to prove compatibility with twisting and parabolic induction, things like this. So maybe I will want to start with a brief outline of how the construction is supposed to go to get some motivation for. A lot of technical stuff this week. An unified base change. In which generality do you have unified base change? Would you QP to some unified extension? In which case do you know that this exists? You take your E unified. Do you always know that you can do unified base change on the old one? I don't know. A geometrical M maybe. Do you think? Okay, Laurent says it's okay. Okay, so let me give some brief outline of the construction. So for some reason I want to work with torsion coefficients as it's usually easier in the Cittal series. So let's say my coefficient ring will be some O mod L to the N O where O over the L is finite extension, finite normal extension. So the crux of the matter will actually be to define a certain category of something like constructible sheaves on the stack of G bundles on the 5 from 10 curve. But actually you have to be careful with the constructability notion. And so instead there will be a slightly different condition called reflexive. For reflexive sheaves on the stack of G bundles. So reflexive, this just means that F maps isomorphically to the double via DA dual. This is something that's usually satisfied for constructible sheaves and is clearly a kind of finiteness condition. And it turns out to be exactly the right kind of finiteness condition that does what you want in this situation. So why do we care about bun G here? The stack of G bundles on the 5 from 10 curve has this funny property that the automorphism of the trivial G bundles is not the algebraic group G as would usually be the case on usual smooth projective curves. But instead as a periodic group G of Q P. And so this means or one can show that there actually is an open immersion of like the classifying stack for this periodic group G of Q P as an open substack of the stack of G bundles. And so this means that if I consider sheaves on bun G then if I restrict them to this open substack that they will precisely correspond to representations of this periodic group. So F is a sheave on bun G and this will actually correspond automatically by the formalism to a smooth representation. Okay so in this way we want to embed this representation series of this periodic group somewhere into the series of sheaves on the stack of G bundles. But we see that if you want to do this we necessarily have to consider sheaves with infinite dimensional stocks because these smooth representations here they are usually infinite dimensional. And so we need some kind of constructability notion for these sheaves on bun G which takes into account infinite dimensional stocks. But it turns out that you can do this. And so the three key ingredients and will be the following. The first is a classification result for these reflexive sheaves. So this will be this reflexivity condition which is this purely abstract definition that you can just easily write down. Turns out to be related to the admissibility condition of the representation. So it corresponds to the admissive representations. The second thing is that what some of FARG's idea is that if you want to study the local language correspondence you can try to do it by doing some kind of geometric language correspondence on the stack of G bundles. And so we have this suitable category of constructable sheaves now. And so we want to show that the hacker operator preserves this category. So we need to show that... By one you mean that they are determined by the restriction to the open source? No. So in general this has a countable number of points given by the Scottwood set B of G and whenever you restrict to one of these points you automatically get a smooth representation of this group JB. And what you ask is that this is always admissible representation. The second thing you need is that reflexive sheaves are stable under hacker correspondences. And so there is a picture that you maybe have some hacker stack, some mu, then maybe times a curve. You have two projections. And you want that if you take P2 lower streak of P2 one upper star that this gives you a function from the reflexive. And I mean related to the second you actually need a classification of HEC operators in the usual geometric Satake way. So that you won't need that HEC operators correspond why some version of geometric Satake. So let me now briefly proceed assuming that we had this kind of formalism set up. How one would go about defining this alpha meter here. So the strategy is to define what Vincent LaFau called basically called excursion operators. So you can do the following. So whenever you have representations of the L group you get a function just some of the HEC operator for all these guys which will go from the category of reflexive sheaves on bun G towards itself. But actually if you apply HEC operator there's an extra copy of the curve on the target. And the way this works out is that after you apply HEC operator you actually get a wave group action on the sheave. And so actually if you apply N HEC operators which is some of the composite of them, you actually get N commuting wave group representations. And the formalism works in such a way that if you restrict to the diagonal action then this is also the same thing as the HEC operator for the tensor product of this representation. And so the other data that you have when you try to define excursion operators are the following. You have a map from the trivial representation to the tensor product and you have a map from the tensor product back to the trivial representation and you have elements say tau 1 up to tau n wave group. Then you get the following endomorphism of any reflexive sheave. So via alpha, so for the trivial representation this HEC operator is just the identity. And so in particular this map alpha will give you a map from f into the guy for the tensor product. But this is the same thing as the operator for this tuple of representations. But if I have such a tuple of representations I have these commuting wave group actions and I can apply these elements and go back to itself. Which then again is just the HEC operator for the tensor product. And then via beta somewhere this is. So this formism tells us that we can produce a lot of non-trivial endomorphisms of our sheave. And so if f is irreducible then these are scalars. And a lemma of Vincent LaFauque and also they satisfy a bunch of compatibility equations. So what is irreducible? Is it like irreducible perversive, irreducible sheave? I think let's say it sits in the usual t-structure and it's irreducible. And lemma of Vincent LaFauque essentially tells you that these scalars determine a unique continuous semisimple. And I guess in this step I'm implicitly also some passing from my lambda to ql bar in some limit procedure. I mean if I see scalars for all such lambdas then I can go to inverse limit and get l-addict numbers. And so if he applies this to a sheave of the form the extension by zero of a sheave for some pi where pi is an irreducible smooth representation in particular admissible. And j was this inclusion from the point where g gives qp into bun g. And maybe I should also say what's the relation that's between this. So whenever you have an l-parameter you can also produce a scalar out of the same data. Namely you have a map from what if he says this. If you have an l-parameter then you have a map into the l-group and if you have representation of the l-group you can compose it with this. So this is some representation of the way qp. So this is a representation of n copies of the way group. And so then you can also wire alpha you can embed it into there. And then again you can apply these elements to 1 through 2n to go to itself. Sorry, yeah. And then via beta you can go back to ql bar. What was this n that anything to do with f? No you do this for all possible n. Why do you need to do this? Because otherwise if you just have one representation I mean there's not so much interesting data if you just have one representation. To recover the representation category of gl. Right, I mean basically you want to say that if you look at, I mean if you want to construct this continuous semisimple l-parameter you somehow have to know all the simultaneous conjugacy classes of elements. And then these are determined by certain rational functions on some simultaneous orbit space. And then what Winston LaFoq proves is that the functions from this kind of data you can produce functions on these kind of simultaneous orbit spaces and they generate. If you have a classical group you probably need fewer representations but I'm not exactly sure how this works. I mean for gln I guess you only need two right. And you basically take the standard ten to the standard duals and you have a natural maps into and out of it. And then for any two elements you get something. And I think this basically gives you the trace of what happens for the other stratafmanji. Right I mean then you also get, right then this formalism applies as well and you get some l-parameters there. But I mean all the groups you're getting there are some kind of inner forms of levies. At least if the group is called a split. And then the l-parameters you should get is just, well inner forms they don't really care. And then you can just embed the l-group of the levy into the big l-group. Then of course there's a question whether some kind of compatibility with parabolic induction question there. Which one can address I think. Okay so to make this work we need solid foundations on the talk homology of the intervening objects. And so in particular we need some kind of proper base change, smooth base change, concurrent duality stuff like that. And so I didn't really say what kind of geometric object the spanji is so far. So in the setting of the fog front end curve the spanji was usually on this usual smooth projective curves this would be some smooth outing stack. In this fog front end case this turns out to be a stack on the category of perfectoid spaces. So it turns out that you can always define some kind of relative fog front end curve. When you have a perfectoid space of characteristic p and well then these are your test objects and then some kind of stack on this category. And so usually we have this diagram, we have schemes and then these are maybe generalized to algebraic spaces. Which we can then further generalize to arting stacks. And we will need a variant of this picture where we have perfectoid spaces of characteristic p which sit fully facefully in some category of so-called diamonds. So there is the analog of algebraic spaces in this world. And then there are some kind of good stacks, stacks like banji, which admit some kind of smooth cover by a diamond. And so actually for the most part you need to develop some formalism on this category here. Extending to stacks is then rather formally by some kind of smooth descent and so on and so forth. Still technical but ok so the main thrust is here so. First week we'll be about to talk on module of diamonds. So how do I ask you this back there? In algebraic geometry classical one most of the arting stacks which are occurring you don't need really algebraic spaces. Because you have projectivity you have some... Right so in the usual case the difference between schemes and algebraic spaces is not too big. Whereas for us the difference between these perfectoid space and diamonds will be really big. And so in particular there are essentially like the usual kind of smooth spaces like the A1 over Qp or something like this. If you would like to consider this in this world it would really just be a diamond not a perfectoid space. Because it only has this perfectoid huge perfectoid cover. And so for us it's more critical to really work directly with diamonds differences. So here there's a small difference and here there's a really big difference. Ok so diamonds are certain technical objects. Let me try to give some motivation for why one would consider these technical objects. I will define... I guess I will be a break right? Yeah from when to when? That's certainly okay. So after the break I will define diamonds. Let me try to give some motivation first. So I'm purely couch theory. So I want to study say some smooth rigid space like I don't know maybe the torus over Qp. Let me just give you what the classical points are. A set of Qp points of absolute value one. And one often does this by looking at the falling kind of tower always sending x to x to the p. So these are some finite detail covers of degree p. And then the limit you maybe get something called t infinity. Yeah so there's a word missing in the sentence. The pay cut theory one wants to study does this. Sorry. Ok so by looking at this tower. So in this inverse limit this inverse limit will be what's called a perfectoid space. Because you've now extracted lots of p power roots of the coordinate. And then recovering the guy the smooth rigid space as a quotient of this by some zp action. So if I project this down to t of zp then this is some zp cover. And so it's a quotient of perfectoid space by some pro finite group action. So this will be the kind of general thing that a diamond will be. It will be a quotient of perfectoid space by some pro etal equivalent relation. So that's one setting in which someone more or less naturally comes up. Another thing where these came up is in the context of what are so called Banach-Colmes spaces. So in pay cut theory again there's this Font-10 string p-deron plus which is a complete DVR to residue field cp. And so in particular if you consider it modulo the second power of the maximal ideal. And what one would like to do is one would like to consider this as a geometric object which sits in the exact sequence where both of these are a1's. So one would like to geometrize this to a short exact sequence in the a1 over cp to some object x. If you get for p different p's so you will get to different p spaces? For different p's sorry. No I mean the p is the same p as the base. As a limit so if you vary the p's so what do you get? If you vary the p's there is a fixed p. Sorry. I mean you could also take first inverse limits for other primes and then get a bigger cover of say Galois group c hat. And then you could also write this as some bigger guy modulo the action of c hat. But assuming such an object exists in whatever category it lives in someone should be able to say what the something value points are. If something is a kind of geometric object that says x is in particular. But there is this ring like p around plus mod fill 2 can only be defined for perfectoid cp algebra. So this means that the only thing that this x could be is that x is some sheath on the category of perfectoid spaces over cp. But it's not itself a perfectoid space because the a1's for example are not. But again you can write it as a quotient x itself is not perfectoid but it can be written as a quotient of a perfectoid space. So these are the kinds of examples that make you think that maybe one should try to study these kinds of objects that can be written as a quotient of a perfectoid space by pro-tallic equivalence relation in some generality. And some kind of pleasant surprise in this picture is that you can also make sense of if one sets up the theory correctly. You can also make sense of some objects that didn't have any meaning before in this world. Something like specqp times specqp over some deeper base say fp. Do you know the perfectoid spaces are always analytic? Yes, yes they are always analytic. So actually I want to give a very brief reminder about attic space and perfectoid spaces next. But maybe, well let me start this before the break. So as well we probably cleared this point is that perfectoid spaces will be used a lot. And well I gave these six lectures here a while ago and in some sense I just assume that you have all been there. But let me nonetheless try to recap a little bit. So as for schemes, attic spaces are associated with some kinds of rings and these kinds of rings that I want to consider. I mean I don't like this name so much but anyway. A tate hoober ring. It's this way. So there's a general notion of a hoober ring and a hoober ring can be tate. It's a kind of adjective you can put to a hoober ring. And I only want to consider, so there's a closely related notion by the way now of what's called an analytic hoober ring. Which I think is a much better notion. But it's also slightly more general and I don't really need this extra generality and it's easier to think about these guys which are actually tate. And so for this reason I want to stick to this. And it's not always part of the definition but let me just make it part of the definition because I will never consider non-complete guys. So it's a complete topological ring A. Such that there exists some topological linear potent unit. This is the state condition. Such a guy is called a pseudo-uniformizer. And an open subring A0 of A containing pi. Such that A0 has a pyridic topology. So concretely what does this mean? So any such A is of the form. You take some ring B, you take its pyridic completion and you invert pi. Where pi and B is a non-zero divisor. In which case we can take A0 to the pyridic completion. And one can show that any such open subrings A0 are called rings of definition. And one can show that any two different rings of definition differ by a bounded power of pi. And then maybe let me make two other definitions before the break. A ring of integral elements is an open and integral closed subring. Contained in what's called A-circ which is a set of all power bounded elements. Which is a set of all x and a such that there exists some n such that the set of all its powers is contained in pi to the minus n A0. The power bounded means that the set of powers is bounded. And this boundedness condition in this case can be phrased pretty explicitly as saying that there is some n so that it's contained in pi to the minus n times A0. And this is always a subring. So the role of this A plus is slightly tricky to understand what it really means. It becomes really important when one tries to study complex political homologies. And it becomes really critical to always think about what the A plus is. And so edict spaces are associated with such pairs of such a Tate-Huber ring and the ring of integral elements A plus and A. And let me also define perfectoid rings. So in this form this definition is due to fountain. So it's supposed to be of this form such a Tate-Huber ring. Such that there exists some uniformizer pi such that the piece power divides P in A-circ. So let's say throughout the course QP is fixed, some particular P is fixed in A-circ. And the Frobenius is an isomorphism between A-circ mod pi and A-circ mod pi to the P. And A is what's called uniform, i.e. A-circ is bounded. Right, so being uniform is a slight strengthening of the conditions of being reduced. That's some kind of topological reducedness. For much of the course it's actually enough to understand what it means to be perfectoid of characteristic P. So this means that P is zero. And this turns out actually to be just the conditions of being perfect in addition to being a Tate-Huber ring. So there is a general argument that shows that if you're perfect then you're actually automatically uniform. And then I learned this from Yves André. Okay, so maybe it's time for a break now until, for 15 minutes, until 50? Yeah, let's say for 15 minutes until... All right, let's continue. So maybe let's just give some very brief examples. So there are some examples of non-accommodian fields which have this property. Say QP is one of all P power roots of P and then complete it. Where we can take for pi a P-shoot of P. Or you could do a similar thing in characteristic P. Or you could take algebra's over those. But if you take an algebra over those, you better join with any coordinate also all P power roots. Because otherwise you will not ensure this condition. That's for benesus to ejective there. Stuff like that. And again, let me also very briefly recall the process of tilting. If A is perfectoid, then we can define a new ring called the tilt of A, A flat. And as a topological monoid, topological multiplicative monoid, this is the inverse limit over the P's power map, which is now multiplicative but not additive. And then there's this funny formula for addition that X0, X1, Y0, Y1 gives you some new search sequence of elements which are compatible under the P's power map. Where Zi is a certain P-addict limit. The limit as n goes to infinity. Where you first take further P power roots and then raise to the P to the n's power again. And then this turns out to converge. And this turns out to have the property that it always produces algebra's which look kind of, it takes this completion and tilt, you get this kind of similar algebra characteristic P where T is the sequence of P power roots of P. Or if you do an algebra like so and tilt this, you get a similar algebra characteristic P. So A tilt is always of characteristic P. And if A is already of characteristic P, tilting does nothing. So yes, and some kind of general intuition that will be in some sense justified by the theory of diamonds is that tilting preserves all topological information like the underlying topological space or the etalside. But it forgets the structure morphism to I don't know, spec ZP. So of course this A always has a unique map from ZP because P is topological independent. But now if you pass the characteristic P somehow in some you forget in some sense is morphism in some sense. That's precisely what you forget. And so now I need to talk about edict spaces briefly. So let A be such a T-tuba ring and A plus an A, a ring of integral elements. Then the edict spectrum of this pair is defined to be the set of all continuous variations, usually how I've written as an absolute value sign from A to some totally order to be in group union zero. Such that they are less or equal to one on the suffering of integral elements up to an appropriate notion of equivalence. And so for any x and x I write f maps to the absolute value of f at x for the corresponding variation. So this edict spectrum is always what's called a spectral topological space. So spectral space, well one stupid characterization of them is that it's one which is homomorphic to the spectrum of some ring. Another characterization is that it's an inverse limit of finite T-naught spaces. And there are others. They have some very good quasi-compacity constructability. So I have this long manuscript about the telco module of diamonds and some sense half of this manuscript is about the points of topology. It's about the points of topology of spectral spaces. Anyway, this is a spectral topological space for the topology. Generated by rational subsets. So if these rational subsets take the role of the distinguished open subsets in the world of schemes. And they are defined as follows. If you have any elements generating the unit ideal, then you find a subset, let's call it U, where you invert G. But you also insist that certain inequalities are satisfied. So it's a set of all X and X. That's it for all I, the absolute value of FI at X. Which in particular implies that the absolute value of G at X must be non-zero. Because if they generate the unit ideal, then they cannot all have absolute value zero. So one of them is non-zero. But then also G is non-zero. So indeed G is non-zero. G is non-zero on the subset. But in fact it's in something that's bounded below. So these are certain basis of further compact open subsets which are in fact stable under finite intersection. And in some characterizations of spectral space, having such a basis is part of the definition. And something's the hardest. Well, I don't know, it's part of the definition. Okay, and so then you can define a structure pre-chief. So you set OX of U to be what's written as you take A and then you join convergent power series in all these quotients FI over G. Well, precisely, this is defined as the pi at a completion of the sub-algebra generated by all these quotients inside of A inverse G. And then you invert pi again. And you check that this is independent of the choice of such an open suffering A0 and absolute uniformizer pi. And it has a universal property. I will say this in a second. Okay, so I said it's OX of U where so far this really depends on the elements F1, N, G. It will actually only depend on this open subset U in a second. And you also define a certain sub-algebra OX plus of U. One way to say it is that it's a topological closure of the integral closure A plus and then you invert all these FI over G. So this OX plus should always consist of functions which are bounded by one on the subset. And FI over G is such a function on the subset. But also it's always supposed to be open and integrally closed. So you massage this a little bit to make it open and integrally closed. Okay, so the proposition that Oprah Gabbard was referring to is the following. It is set. Well, first of all, this is again a T2 bar ring. And OX plus of U is an integral ring of integral elements. The map from this edX spectrum to the edX spectrum of A A plus, it factors over U. And so recall that this was this X and U was this open subset in there. And OX of U or X plus of U is initial A A plus algebras with these properties. In particular, this implies that indeed it depends only on U. And in fact, it doesn't just factor over U, but instead it really, the edX spectrum is homomorphic to U. Not just that, it's not just homomorphism, but actually it also preserves rational subsets. Maybe I said also the following lemma, that actually one can always recover this plus sheaf from OX. Sorry, I should say something else first. The fact that any, that you have this homomorphism means that whenever you have a point of view, meaning a continuous variation on an A, which happens to lie in U, then it automatically extends uniquely to a continuous variation on OX of U. And so this implies, right, so this implies that the variations at X, in fact, extend to the stock of this presheaf. And now I can state the lemma I wanted to state. OX plus of U is in fact the set of all F and U, OX of U, such that for all X and U, the absolute value of F at X. Okay, and now everything would be really nice, except for this, there is this one problem that OX is not always the sheaf. This will not concern us very much because in all cases that are relevant to us, for example perfectoid spaces, this will be a sheaf. And actually I don't know any natural example of a pair A plus versus it's not a sheaf. So for anything that occurs in nature, things seem to be good, but there are no natural examples. Yeah, so there is, you can find examples in the paper of Buzzard, Baebaek and others in the paper of Mihara. Okay, so you can make the following kind of definition that such a pair A A plus is sheafy, which is a nice word, but if OX is a sheaf. And there is a remark due to GABA, I think, that in fact this depends only on A. And also if it is a sheaf, then it is automatically a topological sheaf in the category of topological. Yeah, so it turns out that, you might think that if you just ask it's a sheaf, you probably gain nothing because you just know it's a sheaf and then you can't really go on from there, but actually the contrary is true. So as soon as you notice this is a sheaf, you get a lot of other properties. So this was, for me it was a surprise, something proved by Kid Lya and Liu. If A A plus is sheafy, many good properties follow. So yeah, it's a topological sheaf. You have that if you take the higher-core module is zero for I because it's zero. It's just a cyclicity. Also you can glue vector bundles, so there exists a good serial vector bundles. Meaning that the category of finite projective A modules is equivalent to the category of finite locally free or X modules. Or locally finite free. Well, if your rings are very non-Syrian, of course you wouldn't expect a nice series of coherent sheafs, because not even for schemes you have that. On the other hand, because of some topological issues, you can't expect a good series of quasi-coherent sheafs, even if everything is as Syrian as you like. But Kid Lya and Liu observed, however, is that you have a good series of so-called pseudo-coherent sheafs in general, pseudo-coherent meaning that you have this possibly infinite resolution by finite free modules. And so all these good properties follow once you pass this first problem, which I find a bit surprising. But it's very nice. And so the serial vector bundles will actually be important to us. Finally, I think I can give the definitions. So in case you really get such OX as a sheaf, what you get is a topological space X equipped with a sheaf of complete topological rings OX and with a choice of continuous valuation on each stalk. And so this will be the kind of object that an attic space is. And because I was only working with these guys which rotate, I only get what's called an analytic attic space. So this is a locally topological ringed space X or X equipped with a choice equivalence class. Continuous valuation for all X and X. It's such an object that it's locally isomorphic to some spa AA plus for some AA plus as a buff, which is sheafy. So continuous on X, X means continuous on all U for every U. Yes, I mean, the krypsis was a direct limit apologies to say. Direct limit. Yeah, I mean, it's a problem to saying it's continuous when you restrict to X of U for any open subset U. It's not clear directly with the quality of the ring, the topological ring, the topological ring. And a perfect art space is such an analytic attic space covered by spa AA plus with a perfect art. So there is a small question where there is even spa AA. Right, so it might be that you have an affinity attic space, which is perfect art space because it's covered by such guys. But it's not clear that this A itself is perfect art. This is still not resolved. Okay, so 10 minutes left. So maybe I say a little bit about tilting on spaces. So if A is perfect art and A plus is a ring of integral elements, then A plus is a tilt of A flat, a tilt of A. And this also contains automatically a plus ring corresponding to A plus, which you define to be the similar inverse limit where you just take A plus. It actually turns out that rings of integral elements in A flat and in A are a natural bijection under this correspondence. Then you can relate the corresponding attic spectra. This maps homomorphically to X flat, wire, so if X maps to X flat, given by the rule that the absolute value of F and X tilt is the absolute value of this F up to 0 at X. So this is a homomorphism preserving rational subsets. For all rational subsets U, or X of U is again perfect art. And if you tilt it, it's the same thing as the structure sheet for the tilt evaluated on the same subset U. And so this tilting procedure shevifies. So it's a category of all perfect art spaces. It's a category of perfect art spaces of characteristic P. And so something we will need to know is how the etal site works for perfect art spaces. So I'll do this in the last minutes. So one issue with defining this in some naive way, say using some finite presentation plus unique lifting property is that there are no neopotents, no perfect art algebras. One could try to define some kind of very general notion of etal morphisms for all attic spaces. And this might be possible. And it might have a characterization of this form. But let me just give a very concrete definition of what an etal morphism is. X to Y of perfect art spaces. And the first thing you define as a finite etal morphism. The condition there is that if all perfect art affinoid subsets are V, which are some spies as plus in Y. Perfect art affinoid meaning it's such an affinoid subset, but as Ofer Gaba pointed out, it's not in general known that this S will automatically be perfect art in this case. So I asked that it is. The pre-image of inverse of V is again of this form. And the map of algebras from S to R is finite etal in the algebraic sense. And on the plus algebras, you just get the integral closure. What you need to know about this definition is that this can be checked locally, which is not upper clear because I asked it for all affinoid perfect art subspaces. This happens. Also, if S is perfectoid and R over S is finite etal, then it's always true that R is again perfectoid. This is related to faulting this almost purity theorem on Y. I mean, it's enough to find a cover of Y by finite perfectoids of this form. And so, in fact, if you look at finite etal spaces over the edict spectrum of some SS plus, then this is really equivalent to finite etal S algebras. So this notion really just locally reduces to the notion of a finite etal algebra over the ring. In particular, the patching of vector, but this is one way to do it. You could use patching of vector models. I mean, I already proved this in my first paper where I didn't know this patching of modules. So then I reduced it to characteristic P and characteristic P I reduced it to in the Syrian case. I guess it's easier if you just use the gluing for modules. Okay, so this is the notion of finite etal morphism. And then there's the notion of an etal morphism. If locally on X, it's of the form as follows. So where this is an open immersion, this is an open immersion, and this map here is finite etal. So it's clear that anything which is of this form should be considered etal. It's not clear that anything that should be considered etal admits such a description locally. And this is actually something that fails for schemes, but is true in this analytic world. So the usual counter example is if you downstairs you have the A1 and upstairs you have some etal map which winds itself like this, but you take out two ramification points. And then if you have this point up here, which maps to the same point as this ramification point, then however small you try to go into a neighborhood of this point, and still if you try to compactify it into something finite, it's how you would still have to compactify over this point and it would not be etal. But in the analytic world what you can do is you can just pass to some small open neighborhood of this point. I'll make it finite anyway, time's up. So let me just state from the theorem that if X is a perfectoid space, let's hold X flat, then under tilting the finite etal and the etal sides are equivalent. And I also need the following property, that if X is the side. So the categories are clear and the coverings are just given by families of maps which are jointly surjective on the topological spaces. And if X is a finoid, I also need the following property that if I look at the etal homology, there's an easier statement if I remove the plus here, and let's be true in general that this should just be R in degrees 0 and 0 in positive degrees. But in this perfectoid situation you have something much better, namely even on the integral level, well it's still R plus if I is equal to 0, but it's almost 0, if I is bigger than 0, where almost 0 means that it's killed by all fractional powers of obsidian and this property will be critical. And so this kind of property that you have is almost vanishing here is actually something that for example doesn't hold true for general rigid spaces. So if you have the cusp say as a rigid space, then you could have unbounded torsion in the H1. Right, so next time I will introduce pro etal maps of perfectoid spaces and then I can define diamonds and so on and so forth.