 So today's goal is to define diamonds and do some basic stuff with them. And so let's start right away, defining proletar morphisms, which play an important role. So a morphism from y to x of a phenoid perfectoid spaces is, let's call it a phenoid proletar. If it can be written as an inverse limit of a co-filtered system of the tau maps fi from x i to x, y i again, let's wait a second. Okay, so you just take any inverse limits on the inverse limits that you have to say co-filtered of such a tau maps, where I should maybe remark that the category of a phenoid perfectoid spaces has all inverse limits. Well, I mean you also have fiber, it's all connected, connected, I have to say, right? I mean the indexing category has to be connected because, sorry. So in other words, it reduces to co-filtered guys and fiber products. And then maybe you want a notion for all spaces. And let's just say that morphism from y to x of general perfectoid spaces is proletar. If it is locally on source and target, a phenoid proletar. I should warn that these notions are really not so well behaved in general. So say if you have such a map of a phenoid perfectoid spaces and it's maybe locally on x, it is a phenoid proletar and it's not clear that it's globally a phenoid proletar because it's not clear if you have local choices for these tau maps, how to build global choices. So for schemes, there is a counter example to this and I believe it adapts to this case. Similarly this notion of proletar for perfectoid spaces has also some caveats that you can't really check it locally on x and some in the proletar topology. There are some small caveats to this, but they will be addressed in a second. But let me first state, so you can define a proletar site where covers are generated by, say, open covers and subject to a phenoid proletar map. And although the site somehow may not be so well behaved, but the associated topos is actually a pretty canonical object. So anyway, so you have the following theorem that this site is sub-canonical. Also if x is a phenoid perfectoid, the proletar homology of OX plus is almost 0 for bigger than 0 and in degree 0 it is what you think. I should say that OX is a sheet. Let me very briefly sketch why this is true. So last time we had a similar result for the etal site. And so this implies that if you send some x which is some r plus to r plus mod pi for some fixed pseudoniformizer pi, you really can't really choose this globally on the whole category of perfectoid spaces. But whenever you work over some fixed x, you can somehow choose it on x and then it takes a pullback to everybody above. And if you want to check that something's a sheaf, you can always work over some base space. So this thing is almost an etal sheaf. Meaning that if you pass to the almost category, it becomes an etal sheaf. And it's almost a cyclic. But now if you want to pass to the pro settings, then if you have some inverse limit of some spa r i, r i plus. This is some spa r infinity, r infinity plus. Then actually this is co-filtered. Then actually the r infinity plus mod pi is just a direct limit of r i plus mod pi. So this is essentially how you compute the inverse limits. You just take the direct limits of the r i pluses and then pi are completely complete to get the r infinity plus and then invert pi again to get r infinity. But this means that this sheaf is somehow on the pro etal side, you get it by just somehow evaluating it on. So I assume that this is some kind of pro etal thing over your base space over spa r r plus. Then if you value it on this pro etal thing, you just get the direct limit of all the values on the etal things. And then the pro etal sheaf property is just forced by taking filtered direct limits from the etal sheaf property. What about composition of a phenoid pro etal mass? They are still a phenoid pro etal. But just to not clear how to, when you take the limit, it's not clear how to get to descend on a tarma, so if you have got z to y, which is a tau, how do you slightly descend it to something of a yi to get there? You just can certainly descend it to something etal, the question is whether it's still a phenoid. You can certain. So there is a statement that if you look at quasi-compact pro etal separate etal maps to some limit, then this is a two-category direct limit of the quasi-compact pro etal separate etal maps to some. Like what you said, like what is explained in the entire previous sense of, in the sense you explained last time as you locally can write it, okay? But then you have to show it's a phenoid. Then I have, wait. That is, sorry, maybe I only, do I claim that in general, it's these compositions of phenoid pro etal? Sorry, I don't think, sorry, I take back that claim that in general, a phenoid pro etal maps are composed of phenoid pro etal maps. But you could assume that these are some kind of, it doesn't really matter. You can assume that these are composites of rational embeddings and finite etal maps in practice, in which case you can get through what you want. I mean, I don't really care so much about this very notion of a phenoid pro etal, I more care right now about the notion of the topos it generates. And for this it doesn't really matter. All right, so there is this problem that notions are slightly tricky to check. And I want to address this in some sense locally in the pro etal topology. So this can be resolved, so this is based on the following definition. I say that a perfectoid space X is totally disconnected, respectively strictly totally disconnected. So I need that it's quasi-compact and quasi-separated, but more crucially. The condition is that any open cover splits in the totally disconnected case. Or any etal cover has a splitting. So meaning that, I mean if you have some UI to X, some etal or open cover. Then I want that the map from the disjoint union to X has a splitting, a section. And it follows from essentially arguing like the existence of algebraically closed fields or algebraic closures that whenever you have say QCQS perfectoid space to start with, you can build one which is pro etal over it and which is strictly totally disconnected. So if X is any, let me actually say this as a phenoid. We can always find a surjective phenoid pro etal and X tilde to X, this X prime strictly totally disconnected. So it just somewhat takes the inverse limit of all possible etal maps to X in some sense. X tilde, yes, thank you. What will be important later a little bit is that you can even arrange this map to be open and even open after any base change. But I think this will only be important for some arguments that I'm not going to explain. So this means that locally in this pro etal topology, your spaces are such strictly totally disconnected guys. So how do they look like? So you can give a classification result for strictly totally disconnected and also just totally disconnected guys. They are pretty simple. If and only if first of all X has to be a phenoid and every connected component of X is extremely simple. It's of the form in the edict spectrum of some KK plus, where K is a perfectoid field and K plus and K is a bounded and open. So these are the kind of spaces that takes the role of points in the world of edict spaces in some sense. So whenever you have a point of an edict space, you always get a non-accommonian field K, which is some of the residue field at this point, plus such an open and bounded valuation suffering, which corresponds to the valuation on K corresponding to the point you're taking. And then you always get a map from Spark KK plus into your space. But this may, this is a totally ordered chain of points. But I should finish the statement first. So this is in the totally disconnected case. In this strictly totally disconnected case, you also need to assume that K is algebraically closed. This is a totally ordered chain of points. Okay, so you've essentially completely torn apart your space that it's just. So let's make a remark that you always have a map from X to pi not of X, where pi not of X is some profinite set of logical spaces. That's true for any spectral space. So you have this profinite set of connected components, and also connected components are really simple. And so you've essentially lost all geometry after you did this. The underlying topological space, yeah. In defining it is an epic special to restrict the structure shape and then do some completion probably, in general, for connected component. That's right. Yes, yes. So I mean by connected component I mean somehow the, they're closed, yes. Yeah, but they're in this limit of open and closed subsets. Each of these open and closed subsets is itself in a phenoid perfectoid space. And then you take the inverse limit in the category of a phenoid perfectoid spaces to see what this means. Okay, so, in particular, if it's strictly totally disconnected, then I would write these guys as bar C C plus, so it's just a lot of C's. And then if you crush a lot of C's together, it's a diamond. Did anybody get this joke? Okay, so, so locally you are of this form and if you are of this form then actually can classify these pro etal maps more geometrically. So if X is strictly totally disconnected and the map from X prime to X is a quasi-compact and separated map. But I should say what separated means is the easiest way to say it is that separated just means that the map from X prime to the underlying topological space of survivor product has closed image. So for schemes that's one possible characterization, you can take it as a definition. Here there are several equivalent characterization. So you can also prove a variative criterion for separatedness. Okay, we need to separate in this condition here. Okay, so then for such maps you can characterize when they are pro etal. So then F is pro etal, and only if the following happens that for all ranked one points of X, let's say for all generic points of X. So this corresponds to some bar C plus C mapping to X. So I should say that the connected components are always such bar KK pluses, which are a totally ordered chain of points. And in every such connected components there's a unique generic point and also unique special point. But as considered unique generic points, this corresponds to the maximal generalization of this valuation, which just is the one which forgets essentially about the K plus and replaces it just by the ring of integers, by the ring of power bonnet elements inside of the field K. So you'll only look at those points. You want it for all these generic points. If I look at the fiber product, this is at expected of COC. That this is just some profinite set, S times plus COC for some profinite set S. What does this S underline times mean? So if you have a profinite set, you can write it as an inverse limit of finite sets. And this product is the inverse limit of the product with the finite sets. So if you have something which is a phenoid proital over just a geometric point like so, then all the etal maps are just given by finite sets over it. And so the kind of a phenoid proital things should just be given in this way by profinite sets. And so that's the only thing you have to check. You only have to check that the generic points, you are such a profinite set. And in this case, if you are proital, you're actually a phenoid proital. Also, without strictly but taking the algebraic closure for every generic point. Is it true if I drop this strictly and then ask the same thing just for algebraic closures? Good question. I wasn't able to verify this when I did this. And one problem I had was that if you have such a spark kk plus where k is not algebraically close and then you pass to finite etal cover, then in general, this is not somehow local anymore. You may have different close points. If you take such a local space, spark kk plus, which has a unique close point and then pass to finite etal cover, you pass to a finite etal cover of it, then it may have different close points, several different close points. And this was slightly complicating the arguments. So for this reason, it was easier to really assume that you're strictly totally discussed. So another question, a small one about the closed image in definition of separated. Is it like for schemes that for affine or a phenoid, you have a kind of a closed immersion that is a topological quotient of the... Yes, yes. So for a phenoid, it's always true. And then this is a closed immersion in every possible sense. So it's even defined by equations. Right. And so in general, the diagonal map will be some kind of locally closed immersion as for schemes and then asking that it has closed image implies that it's a closed immersion in some sense of the word. So in this secret manuscript, there's a discussion about these notions. Okay. So this means that over such a strictly totally disconnected base, these notions of pro etal and a phenoid pro etal, at least under these small hypotheses, especially the separation hypotheses, the equivalent to the minimal thing you would certainly ask for a pro etal map to be satisfied, namely that over geometric points, it's given by profite sets. And so this means that in this case, you're satisfied with the notion. And then in general, maybe you'd like to redefine the notion. So a notion that becomes more important later than a pro etal is what I call quasi-pro etal. So if after any base change, two is strictly totally disconnected, it's still there. And then at least for morphisms, which are in addition quasi-compact and separated, you can just check this by checking that for all such geometric rank one points mapping to X, the fiber is profite set. And then if the morphism is not separated, I mean, there are again some subtleties to this definition, but at least in the separated case, which is, one can often arrange things to be separated. This is a good notion. Okay. So finally, we can define diamonds. A diamond is a pro etal sheaf on the category of perfectoid spaces of characteristic key. So this is important. Such that Y can be written as a pro etal sheaf. Y, such that Y can be written as a quotient, Y is X mod R. Okay, so I'm here implicitly identifying the space with the sheaf it represents. So I'm using no notation for the Yonindal embedding. As a quotient Y, X mod R, where X is a perfectoid space and R in X times X is a pro etal equivalence relation. Meaning that the maps, the source and target maps from R to X are pro etal. So also the R itself should be a perfectoid space. Maybe I should write representable somewhere. Okay, so upper R, you might fear that this definition is sensitive to these subtleties about the notion of a pro etal morphism. But actually it's not sensitive to these issues. So Y, so whenever you can write Y in such a way, you can, without loss of generality, also assume that X is just a disjoint union of strictly totally disconnected guys by passing to a further pro etal covering. Disjoint union. And so then R in X times X is a subspace and this is a separated space and then by some general nonsense R must be separated. Again, strictly totally disconnected. Well, it might not be quasi-compact anymore. I mean, so this disjoint union is only necessary because the space might not be quasi-compact and you need more and more spaces to fill up more of the space. Then the equivalence relation is automatically separated, which also implies that the map is separated. And so now you're in the situation where you have the basis, well, the disjoint union of strictly totally disconnected spaces and the map is separated. And then, well, to check that something is pro etal, you can somewhat check it on quasi-compact open subsets. So we're in the situation where... The equivalence relation on a diamond represented by a diamond where the maps are in the same sense pro etal in the very weak sense, then it's... Right, so, yeah, Ophagaba was asking whether if I now do this procedure again and try to take a quotient of a diamond by some kind of pro etal or quasi-pro etal equivalence relation, is it still a diamond? And this is true, as for algebraic spaces and so on. So this notion of a diamond has very good stability properties. And so, as I said, so one justification for the name is that how do you get a diamond? You take some strictly totally disconnected guy, which is just a bunch of Cs, totally disconnected, and then you smash them together. In particular, this way of looking at a diamond makes it not obvious that they have a nice geometric structure. Okay, so I claim that you can still define smooth maps and so on. That this will take some time and it will only be done maybe at the end of the next lecture. Okay, so let's do some examples. So the first thing I want to do is I want to relate this to the kinds of spaces we're usually interested in. And so the other claim is the following, that there is a natural functor from analytic edict spaces over Zp to diamonds, extending the functor from all perfectoid spaces to perfectoid spaces of characteristic p, which are a full subcategory here and these are a full subcategory here, taking x to... So in this sense, you can define the tilt of any such analytic edict space now, it will be such a diamond. So let me first give you the intuitive definition. So say x is an analytic edict space of Zp, then you just choose some proletar cover x0 to x, where x0 is perfectoid. Let me not make completely precise by what I mean by this, but say an example would be that you just take the torus here, in the example I did yesterday and then take the inverse limit of all multiplication by p-maps or p-power maps, then the inverse limit is such a perfectoid space. So then if you look at the fiber product, this is proletar over a perfectoid space. And so there's some subtleties here, what I mean by the fiber product because I need to do some uniform completion here really. But if you do it right, this R will again be perfectoid, the equivalence relation you get here. And then define x-diamond to be... It takes this perfectoid cover, which you can tilt and you can also tilt the equivalence relation and then the obvious problem with this approach is that even if you can make sense of all the words I said here, it's not clear a priori that what you get is independent of the choice. And so for this reason the official definition goes differently. So let's say x is an analytic edict space over Zp, then you define x-diamond as a following sheaf. And let's say I only define it on a phenoid because it's supposed to be a sheaf you can then extend to all perfectoid spaces. This is perfectoid of characteristic p. So the idea is that it should be all maps from this perfectoid space to x. But this is of characteristic p and this might be of characteristic 0, so there are no such maps. And so the way to correct this is that you don't necessarily ask that you map this given perfectoid space to x, but maybe instead some untiled, which has some of the same information anyway, in some sense. So the definition of the following is that it's centered to the following data. Does this here some perfectoid algebra, which is an untiled of R in the sense that if you tilt R sharp again, it is R. And then any such untiled automatically comes with a distinguished choice for a plus hub ring. And so then you just want the map from this R sharp. You take such data up to isomorphism with the remarks that there are no automorphisms. And so the theorem is that this is really a diamond and can be calculated as above. Some rises but can be calculated intuitively. Do you mean that there are no automorphisms? Well, I mean if you define some sheaf on some category, some modular problem, then you do divide by isomorphisms. Well, you usually want to divide by isomorphisms. But usually this is not so well behaved if there are automorphisms of objects, right? Like the modular space elliptic curves and so on. And so, but this is not an issue for us because there are no automorphisms of these objects. So it's really a well behaved thing to just divide by isomorphisms. Okay, let me briefly say something about the proof. So first you need to check that this is actually a pro et al sheaf. And that's actually a little bit of work in making in the tilting and un-tilting procedure because I mean it's even the sheaf if you forget about this map to X. So just I mean adding this map to X back in is easy. So the question is whether sending a perfected space to the set of un-tils is a pro et al sheaf. And you can prove this by some work to do. And okay, so then if X happens to be perfected, the tilting equivalent says that perfected spaces over X are equivalent to perfected spaces over X-tilt. So if you have a base space and perfected spaces over Z are equivalent. Without a base space the tilting is of course not an equivalent. And if you think about what this means here, it's exactly saying that this X diamond is equal to X flat. If I'm looking for un-tils which map to my given X, then this equivalent must be somehow the unique un-tilt under this equivalent. So this means that in general for perfected spaces you know what happens. And then in general, you may assume it's a phenoid. And then there is an argument to Fontaine and Cormac maybe, that there exists some sequence of finite et al gi torsors h to ai. I'm going to filter it, sequence. With a infinity which is a completed direct limit, the ai's are completed, which then gets an action of g which is the inverse limit of the gi's such that this is perfectoid. Essentially you just, well one way to do it is to just take an inverse limit of all possible finite et al covers, pass through some universal cover, at least if X is connected it's easy to do. It's also enough to just take some more p-power roots, except if this is of mixed characteristic you have to be a bit careful because extracting p-power roots is not a finite et al procedure, so you have to slightly change the equation and do some out-in-triot covers, but you can sort of extract some approximate p-roots by some finite et al covers and then this means that in this algebra you will have enough approximate p-power roots. Now these are finite, say. And then in the inverse limit you get something. Okay, you use some perturbation of taking p-roots so the grammar book can be slightly, okay, you don't control it. Yeah. So actually the argument I make is in the paper if ai's, I mean I really pass through the universal cover and I make some sense of if ai's not connected. And then you just need to check that after you pass through the inverse limit that it always has approximate p-power roots and for this you just write down any out-in-triot cover and you don't care about the Galois group anymore. Okay, so anyway you can do this. And then you can define some x till infinity which is some spy infinity, infinity plus and then x diamond will be isomorphic to the tilt of this x infinity modulo n action of this provenant group G. Well, I mean this notation is not exactly consistent with this notation because the equivalence relation for a group action is really the group times the space with the projection and the action method. Okay, so that's, okay, so for example, so if I take the at-expection of Qp, say, and I compute its diamond, then this will be the at-expectrum of say the cyclotomic extension which you have to tilt by the action of the Galois group Cp cross. And if you tilt this, you get the at-expectrum of the field Fp around series in t and all its p-power roots which has a certain natural action of Cp cross. Explicitly if you have an element gamma in here then gamma of t is one plus t to the gamma minus one. And then you can also take products. And so here's a fiber product is now over Fp. And why is it over Fp? That's precisely because we insisted that somewhere we only work with perfected space of characteristic p. If we worked with all perfected spaces then the base would still be Cp. So this is the main reason that I want to work with this perfected space of characteristic p because then it's a more absolute theory. Okay, so there's another funny example. So for this, let me make the following remarks that if t is any topological space, one can define a pro et al sheev t underlined which has implicitly occurred a couple of times already. So in particular, we can make sense of this formula above so this is pro finite set also using this definition. I can find a pro et al sheev t underlined by saying that the t underlined of any x is the continuous maps, let's write c0, so this means continuous maps from the underlying topological space of x into t. So you can, this is a sheev because all the covering maps are quotient maps. Continuity can be checked after a cover. And there's a following claim that if t is any compact host of space and k is any perfected field, say, then if I take this sheev corresponding to the topological space, the compact host of space and I base change it to bar k okay, then this is a diamond. So it's maybe slightly surprising that the theory of compact host of space to this world now. So why is this true? Well, there's this funny statement that whenever you have any compact host of space you can always find a surjective map from a pro finite set onto it. There exists a surjection, s to t, where s is pro finite. There's some kind of classical fact. I think the easiest argument for this is it can take for s, of t as a discrete set. So the stone's sheev compactification of any discrete set is a pro finite set. And it's universal for maps. So maps from the set into any compact host of space are the same as maps from the stone sheev compactification to this compact host of space. And there's an obvious map from t into t, the identity. And by the universal property of the stone sheev compactification this extends to a map from the stone sheev compactification to t. And it's obvious to see a surjective because there is a map from t to t was a surjective and so you get what you want. But then the equivalence relation s times ts is a close subspace here. And this is a pro finite set. So r, the equivalence relation is also pro finite. And so this means that then this funny sheev here you can actually write it as this pro finite set times the point which is a perfectoid space and you divide by the representable equivalence relation given by this equivalence relation here. Yes, of characteristic p. Okay, maybe it's time for a break. All right. So actually an important technique for much of what we'll follow is what's called v descent. So the v topology on perfectoid spaces is generated by again open covers and all surjective maps of the phenoids. So it basically allows everything to be a cover. And then the world is nice. So the v topology is sub-canonical. Is it fully faithful? The question was whether the diamond function is fully faithful. It's not just because it gets a structure morphism. So somehow if you have two perfectoid spaces with the same tools identified. Structure morphism to zp. What's more sensitive to us is whether if you say look at analytic spaces of a qp to diamonds over the spark qp diamond. Whether this is fully faithful. And this is known to be true for semi-normal rigid spaces. So under a finiteness condition you need semi-normality because the diamond function will turn universal homomorphisms into isomorphisms. And so in particular the semi-normalization map is always a universal homomorphism. So forget about this. But then on semi-normal rigid spaces one can prove that it's fully faithful. Yeah, right. So maybe I should write this down somewhere. So there's a following theorem essentially due to client view that if you look at semi-normal rigid spaces of a qp to diamonds or spark qp diamond this is fully faithful. Yes, you could take any non-alchemy field of characteristic zero here. But it's critical that it's of characteristic zero because otherwise also the Frobenius map will be sent to an isomorphism. It's an interesting question whether you can prove something like this for some analytic edict spaces or some formal schemes which are flat over zp and have some normality condition. In this case there's also a chance to be true. It's sub-canonical OX OX plus the sheaves and also if X sub-phenoid perfectoid it's a high IV chromology is almost zero. Okay, so in particular any diamond defines a V-sheave, sorry any perfectoid space defines a V-sheave. Part two says that actually also all diamonds are V-sheaves. And also the chromology of OX is Yeah, so the chromology of OX is n zero in positive degrees which is a consequence of just inverting the uniformizer. All diamonds are V-sheaves. So two is an analog of a theorem of Gabber from something like 2012 that all algebraic spaces without any quasi-separatedness condition are FPQC sheaves. Okay, so it may be surprising that you can control all subjective maps here. Like I mean for general edict spaces you can't even control open covers but perfectoid spaces are so nice that they're all subjective and I'm so good. So why does this happen that you can control everything? So by using pro etal descent you can essentially reduce to considering only totally disconnected spaces you might even consider strictly but I think for what I've been saying it's enough to consider totally disconnected spaces. But then there's a funny thing so that if if X is totally disconnected and Y so let's say this is some spar RR plus and Y is any aphenoid perfectoid, it's by SS plus mapping to X then there is some automatic flatness happening. Then the map from R plus mod Pi to S plus mod Pi and it's faithful if that if F Y to X is subjective. And so the the rest of the V descent then follows from flat descent. Again you work on this level mod Pi first and then in some almost world and then go back where does this flatness come from? Well you can check this on connected components but then this is just some K plus mod Pi and the K plus is a variation ring and so K plus to S plus is automatically flat as it's torsion free over variation ring flatness can be checked just by checking torsion freeness and so by base change also the map mod Pi is flat. The connected component is just closed so you check on it. I mean these are both sheaves set of connected components and to check whether it's flat you can check at local rings and then the local rings are somehow given by what you see on the connected component. I mean it takes a little bit of justification to make this precise but that's the essential idea is that once you are such a totally disconnected base you can essentially reduce I mean it's often you can reduce things actually to connected components by an extra argument but this connected component is just a variation ring so you get a lot of properties for free. And then so that's the sketch for one for two you actually need some strong V descent properties okay so I want to state some of those so let's say we are in the following situation let F be a pre-stack meaning just a functor to group or it's without any properties on the category of perfectoid spaces then for a map from Y to X let me denote by F of Y over X the category of descent data meaning a class in F of Y plus an isomorphism in F of Y times XY such that course was a course cycle condition so there is a functor from F of X to F of Y over X and F is a V stack but only if this is an equivalent for all V covers right well you also need to do that this joint union is the fourth yeah but F of X is the fourth otherwise it's not doesn't follow from there okay plus compatible with this joint unions okay and so I've made this a question here because the descent results will be slightly tricky to formulate in terms of that something is a V-shef you can only say that if X happens to have some properties then this kind of equivalence holds and so okay so let me try to say this so if F is the thing which sends X to all perfectoid spaces over X then the functor from F of X to F of Y over X is at least fully faithful for all V covers over X so that's an analog of the separated condition for a sheaf but it might not be the case that whenever you have a given perfectoid space over a V cover and the descent datum that you can descend to a perfectoid space on X and I think that's even false for schemes right for the etal topology I mean there is families of genus 1 curves which are not globally a scheme but it's how locally they are a scheme right, but I know but you might hope that under extra conditions things are better so now if instead I consider let's say F of a phenoid which takes which is only defined on the category of a phenoid X and maps it to the category of a phenoid perfectoid somebody should resolve this question that one implies the other positively if I consider this functor then while this functor is still fully faithful always but it's an equivalence if X is totally disconnected so if you are trying to descend something to a totally disconnected space and your total space is a phenoid then you can do this it fails more generally but this is a failure well you don't see it for schemes and some usual flat topology say because a usual FPQC descent but it fails already in rigid geometry smooth a phenoid rigid space also for the perfectoid case the examples are rather stupid you can find some open subsets of a rigid space which becomes an a phenoid or even a rational subset after a covering of the space but it's not globally a phenoid it's for the two-dimensional ball I just wanted to say that it's a failure which is already somehow in some kind of classical setup visible it didn't cause much trouble there so then you can look at what else do I want to say where am I here if I look at separated and proital maps which say you can again do on all perfectoid guys well again fully faithfulness falls from one the question is whether you can in some sense prove descent the effectivity of descent and this holds true if the base is strictly totally disconnected this is another indication that the category of separated proital maps for such a strictly totally disconnected base behaves well because in this case these things glue even in the V-topology and there is a final theorem for separated to tile maps then it's actually a bona fide a V-stack sorry I should include these then the formulation could be improved I just want to say that this guy is a V-stack and also I think one should have that the category the atal topos of the perfectoid space is again be as well any atal sheave probably you can descend any atal sheave you can also descend yes do you have to do something I think this should follow once I talk about atal sheaves and so on yes let me defer this discussion to later okay so you have some pretty nice descent results you can also prove that if X is an analytic atic space then the atal side of X agrees with the atal side of the diamond where maybe I should say what I mean atal topos site so this means that if you want to construct some atal maps to some rigid space there you can do it by constructing some atal map to a diamond and constructing a atal map to a diamond is equivalent to doing it V locally so you can just cover your diamond by some huge strictly totally disconnected guy finding atal cover there and then applies these descent procedures to in the end get some nice atal cover or atal map to a usual rigid space so as in the site so this gets used to construct local shimura varieties which are well these are again some gadgets which depend on some data of a reductive group, a sigma conjugacy class and a core character and some level and then they should have some period map pi going to some flag variety which depends just on g and u and this period map should be atal but I should make a brief comment what I do I mean by this thing here because so far I didn't define what an atal map to a diamond is so I need to define atal maps of diamonds or more generally of V-shefs so atal if for all perfectoid spaces X the map to Y the fiber product is representable in atal and secondly and here the relevant notion is the one quasi pro etal if again for all X but now I just ask it for strictly totally disconnected representable and then in the strictly totally disconnected base case quasi pro etal and pro etal is the same and then at least if the morphism is in addition separated then these descent results tell you that you can check these conditions locally I mean if you have a diamond you mean I will answer this in one second so if f is separated these conditions can be checked so there was a question whether if Y is a diamond and you write Y as a portion of the perfectoid space by pro etal equivalence relation whether the map from X to Y is quasi pro etal I guess and this is true it's not representable it's not true I I'm not sure I have an example in mind but I don't know certainly I don't know how to prove that after pullback to any X prime mapping to Y the fire product will be representable because I don't have these strong descent results except if the base is strictly totally disconnected so what can be proved is if you have a strictly totally disconnected X prime mapping to Y then the fire product is representable sorry and I should have assumed that X is separated maybe to make this I mean you can always do this by just you can always replace X by this transient phenoid and what's the other side of the diamond I hoped I could come to this today but I'm afraid I'm not well because of some issues with separatedness I don't actually know all the time organisms in this end let me defer the discussion of the tall side of the diamond actually to a little bit later but covers are just given by families of maps which are jointly surjective as maps of sheaves which gives you the correct notion of covering that is part of the question and so maybe another remark in the spirit of why is a diamond and why to Y prime is surjective and quasi-proietar and I'm confused whether I need a small separatedness condition so let me check in the manuscript I can find it no without any then Y prime is also diamond is it any other proietar sheave you know it's enough to assume it's a proietar sheave okay if you want let's assume it's a V sheave already so there's a way to set up definitions and statements in such a way that I could make this for a proietar sheave alright where am I going actually so what are we doing so we have this category of diamonds and we can define some good notions of italomorphisms or proietar morphisms of diamonds because we have good descent properties like here and these are actually V descent properties and so this means this plus some other properties means that actually a few of the basic things don't just work for diamonds they work for something much more general namely any V sheave so there's a following there's a following funny theorem that let's say Y is any V sheave subject to minor set-theoretic assumption so such that there exists this surjective meaning surjective as a map of V sheaves from a perfectoid space X well then you can recover Y as a quotient of X by R as V sheaves where what is the equivalence relation with YX I should say that I refer to this condition by saying that Y is small why the equivalence relation inside of X times X it's a sub V sheave it's always a diamond so any small V sheave can be written as a quotient of a perfectoid space by a diamond diamonds in turn can be written as a quotient of a perfectoid space by a perfectoid space and so on this kind of two-step procedure many statements for general V sheaves to this case of perfectoid spaces which is quite useful sometimes yeah so the word surjective so here it was for epimorphism of sheaves I suppose yes and in the original definition you spoke of surjective maps of affinity it wasn't the level of the political spaces or that's on the level of topological spaces yes so this uses the following proposition that if Y is a diamond Y prime and Y is any sub V sheave then Y prime is again a diamond which in turn one reduces to the following assertion that if X is strictly a totally disconnected space and X prime and X is a sub V sheave and the quasi-compact there is some subtlety quasi-compacity that you have to take care of in the reduction then X prime is actually automatically a strictly totally disconnected space that's the consequence I mean I think what's happening here is that because being a V sheave is so strong it means that you automatically have some kind of geometric structure if you satisfy this V descent in particular what's happening in the proof of this proposition is that if somewhere you have any map from a phenote perfected space into X that lies in the sub-sheave then the map of perfected space has some image and the image by some general nonsense about the topology of spectral spaces will actually be represented by a strictly totally disconnected space so there will be a strictly totally disconnected space over which there's factors but then this map must be a V cover to the sub-space because it's surjective on topological spaces and then the V-sheave property tells you that actually the sub-space lies in the V-sheave itself and then is the sub-space closed under generalization? yes all maps are generalizing anyway so yes it must be closed under generalization so this means that about general V-sheaves can be reduced first to diamonds and then to perfected spaces so for example and that's a following lemma that if f from y prime to y is a quasi-compact and quasi-separated map of V-sheaves so in any topos there's a notion of quasi-compact and quasi-separated maps and I just use this in the case of the topos of V-sheaves then it's easy to check that it's an isomorphism so then f is an isomorphism if and only if for all algebraically closed non-like-median C field C with an open and bounded valuation ring sub-ring C plus and C it's a bijection on points there's some kind of very general statement that there's no non-reduced structure of any kind anywhere here it's all determined by points okay so let me try to let me try to finish this lecture by giving a criterion for a V-sheave to be a diamond it's some kind of analog of artian serum giving criteria for algebraic spaces that's enough to find a smooth cover or something like this and this is actually used to prove that many of the examples that you care about or at least in the original argument this was used to prove that many of the examples you care about are diamonds like the Afro-Grasmanian in this banji setting and so on okay so this is so this uses a condition on the topological space which by the way is a condition that will also be important for ruling out the compact house-top spaces so let me first define the underlying topological space let's say you have a diamond then you define the underlying topological space to be the quotient like so and you check that it is independent of the choice and also by making the use of the serum about small V-sheaves that also extends to small V-sheaves and for perfectoid spaces it just agrees with the usual underlying topological space for example if T is compact house-top the underlying topological space of this funny sheave is T so for usual perfectoid spaces you get some locally spectral spaces here you get some compact house-top spaces and these are some kind of two extremes in the world of topological spaces and the condition you are asking is that you are on the spectral side of things so small V-sheave Y is spatial if essentially you would like to say that the underlying topological space is spectral but you actually need to say a little bit more and you need to say that the way the space is spectral is somehow related to some properties of the sheave and so the correct way to say this is that it has a basis given by the topological spaces of V for V-aquasal compact open sub-functors open sub- these give you quasi-compact open subsets but not if you would just have a quasi-compact open subspace of the spaces would not be clear that they associated ah maybe actually sorry I should have said that Y is QCQS and so it's not clear that if the underlying topological space is quasi-compact then the sheave is quasi-compact the sheave is quasi-compact if it can be covered by well if any cover is a finite sub-cover so it's equivalent to asking that it's covered by finitely many aphinoid perfectoids but I think to be the topological space is underlying V open and closed it's quasi-compact open so if you have a quasi-compact open sub-sheave it will automatically give rise to a quasi-compact open subspace of the topological space so in general open subsheaves correspond by objective fluid to open subspaces of the topological space but for quasi-competities there's only an implication so it is not clear that every open sub-sheave is a union of private contract it's not clear that any no like I mean you might have the underlying topological space might uproarily be a compact host of space so then you don't have any quasi-compact open subspaces I don't understand in the definition do you require that that V absolute value is is what well automatically if V is a quasi-compact open sub-sheave of Y then the V absolute value is a quasi-compact open subspace of absolute value Y you mean just the base of the topological space of Y comes from something of the final object of Y the quasi-compact okay and so then there is a proposition that if Y is spatial then the underlying topological space is spectral and an important example is that if X is a QCQS analytic at X space then the associated diamond is spatial so the ones that come from usual things they are spatial and right and the underlying topological space of X diamond is the underlying topological space okay so I'm already over time so let me just end by stating the theorem so if Y is a spatial V-sheave so you verify this topological condition and then you just verify a very minor statement about points such that for all Y in the underlying topological space of Y there exists a quasi-proletal map from some perfectoid space which you can assume to be some sparse cc plus to Y with Y in the image then Y is actually a spatial you just need to check that the points of the space are not too bad but for example if you are in the situation of the effingers manien where you apply this for example there is already a stratification of the space for which you know that the strata are diamonds and so any one of those points will lie in one of those strata and then you just because it's a diamond there you know this condition and so this means that you only have to verify the spatiality condition here and well that's some points of topology which is fun or not but anyway let me stop here what is the altitude theorem of which it's an analog? well I mean it's roughly an analog of the theorem that if you want to prove that something is an algebraic space you don't actually have to find any tall atlas but it's enough to find some smooth atlas or maybe flat atlas or something smooth atlas is low okay and so here again rather than for a small wishi you automatically have some v atlas so the vague analogy is some of that I consider the quasi pro etals side as being vaguely analogous to etals things in the scheme case and then any v-maps as being vaguely analogous to all flat maps in the in the scheme case but you will find this condition and in our case probably just fpdf yeah yeah yeah yeah