Arthur-César Le Bras: Rational analytic prismatization and syntomification, I

追加: 2026-05-12

[00:00:00]Yeah. Uh thanks a lot. Thanks a lot for the invitation. Very happy to be back in Chicago. U yeah so indeed this is the first first of a series of four lectures I will give on the analytic prismatization. This is a joint work in progress with Johannes Anuk Rodriguez Kamarago and Peter Schultz.

[00:00:21]And so this will be complemented by a series of lectures by Maximillian on the analytics in tomification.

[00:00:28]Uh and so this joint project has somehow had basically like two main motivations.

[00:00:35]Uh one was piic please tell me if I have to write bigger uh padic hutch theory of rigid analytic spaces and uh another important motivation for us was the geometrization of or let's say the geometric aspects of p langland's for them.

[00:01:13]Uh but for some depending on the perspective that you adopt, you can motivate our constructions a bit differently. For time reasons, I decided to focus on this first motivation in this series of lectures. Uh but I'm happy to discuss also this other one uh later. But at least I also try to towards the end of my lectures I also try to discuss something which is maybe in between one and two which is a relation of what we do with the geometric with sensory or geometric sense theory as was discussed earlier today by Louie. So this I will at least try to do. And so regarding one uh to be a bit more specific. So let me first recall that pi whole theory is about comparing different periodic chology theories that we have for periodic spaces and for periodic formal schemes. We now have a very good understanding of the situation because of the work of Bat Schultz and then later Bat Lur and Dinfeld.

[00:02:16]And in this context what we have is uh we have some kind of universal periodicology theory which is called prismaticology.

[00:02:24]And somehow it knows about all other periodic chronology theories that we like. And it also tells us uh how to compare them. And you can do even better than that in the same way that you can geometize durology say by using the dur space or dam sack of Simpson as was realized by Paturian Dinfeld. You can also do something similar for prismatic homology and construct some stacks called the prismatization of the periodic formal scheme or even some refinements of them. And so really the goal regarding one our goal was to get uh similar definitions and constructions uh when you replace periodic formal schemes by rigid analytic spaces.

[00:03:21]And so if you try to do that the geometry that you use is a bit different. It's not modeled on P nil potent or P complete rings but rather on the rings that you encounter when you do rigid analytic geometry. So I want to start a bit informally by just maybe recalling a bit like uh some maybe general stuff about 3D geometry and perfected spaces and then I will try to explain why this geometric framework is some not good enough for the kind of construction that we want to achieve and why we needed to use a more refined setup. Uh so that's the goal for today. So this more refined setup is that's the title for today is a series of gal fun.

[00:04:10]So if I do well with time that will be more or less the first half for today.

[00:04:15]Uh but then I also want to def like in this geometric setup I want to define the construction of the series of dams as some preliminary for the discussion of prismatization which will come in the in the next talks.

[00:04:29]And there I want to apologize if you were there already for my lecture in September or October. I will basically repeat the same lecture today. Exactly the same room. Maybe I even have the same shirt. But uh yeah, I promise that at least from the next lecture on it will be some new some new stuff. But yeah, to to set the stage I need someone to discuss again this this this material.

[00:04:55]Okay. So uh maybe one would be beyond uh perpetuate geometry.

[00:05:06]Okay. So uh to start very basic let me just first say that okay rigid analytic geometry is a p analog of complex geometry.

[00:05:15]uh and of course it took more time to somehat set the theory off the ground because some P topology is totally disconnected but so it's maybe less intreated but thanks to the work of Tate Berovich and Huber uh we now have really a series of rigid analytic geometry which looks really parallel to complex geometry.

[00:05:39]Uh but there is still maybe an important difference between the complex and the periodic worlds which is that if you take complex manifolds they locally look very simple because locally they just look like open discs and so they are like locally contractable.

[00:05:55]But when you work in rigid analytic geometry, if you take a smooth rigid space uh and then you maybe pick a point and look at small aphinoid spaces subspaces around this point, they still have a big fundamental group and so they have still a non-trivial palmology.

[00:06:14]So that's some kind of asymmetry which really is one only because we take some of for granted that uh so open disc are simple from the point of view of Betty or singularology implicitly maybe I'm assuming that bicycomology palmology is some kind of podic analog of beticcomology which is a usual picture uh I would like to argue maybe in the next lectures why this is maybe not the correct analogy from a certain point of view but Yeah, certainly that's a problem in any case whether this analogy is wrong or not that's a problem you have to face that the local structure of regenetic spaces from the point of view of palmology is not is not simple and so the way you deal with this is a very old idea which I think goes back to Kasna then Tate maybe also Fonten van is that things improve after extracting a lots of piece roots.

[00:07:35]And uh to use more technical but more precise terminology, the key statement here is that locally for the poal topology uh rigid spaces become perfect So locally look like like banner rings uniform banner rings with informationer in which you can basically extract a lot of peaceful module when you go mode P.

[00:08:15]So if you allow yourself maybe not to localize for the usual like analytic topology but for much finer topology where you allow like inverse limits of ital maps then uh anything becomes like any rigid space become perfect and maybe the typical example is like if you take the data algebra in one variable over QP and then you can well first you you make the base field for example by going to the cyclomic extension And then you also extract like piece roots of the or maybe I want to take the rigid and taurus and so that's a typical example of a perfect cover it's even ga in this case with g or some direct product of the of a nice simple rigid space over and of course when you look at this example it looks like you have replaced this nice necessarily state algebra by some like complicated ring a non necessarily ring but the in which the way in which rings are simpler than like the rings you have you have to deal with in rigid geometry is because they can be tilted to characteristic P and when you go to characteristic P they like the condition of being perfect to it just becomes the condition of being perfect and this allows you that for benus is an isomorphism it allows you to uh typically ally you have some kind of vanishing of by the cyclicity you have some vanishing of commology up to some power of uniformizer but using that probenus is an isomorphism you can improve this to some almost vanishing so like will in higher degree will be killed by any power fractional power and then when you invert P again you get some some vanishing on the nose for com so perfect rings are are big But they are like they are very you have a lot of vanishing of commodity for them. And one typical way you can try to use this is uh on the portal site you have uh you can like you have some art sequence you can resolve uh constant shift p in terms of you have the structure shift you take the power bounded element and then you go more p and this is for minus identity and this gives you a way to like compute FPL in terms of this kind of commology. But say for example you want to prove in the proper smooth case you want to prove finness property for FPology.

[00:11:04]uh you could try to then by this you could try to analyze inside this kind of commology and uh in the complex case have a nice argument for proving finness of coherent commology of a compact Keller manifold by choosing cover by sign spaces which are like two covers two finite covers one being embedded in the strictly embedded in the other and then you just compare the chromology of both and they both compute the same the coherent commodity but you observe that the restriction map is compact and then some kind of functional analysis input will force the commodity to be fine dimensional.

[00:11:43]If you want to do similar argument in this setup then you need some kind of like topology for which locally things become a secret like replacement for sty spaces but for this kind of chief so and that's exactly what aphinoid perfect spaces do for you okay so that that kind of application you find in the work of falings or in the work of right and so if you push this idea further then you uh end up with a notion of a vstack or an arc stack. uh so maybe in the work of shops typically uh it's this notion of like proal or v topology that is mostly used but for technical reasons that will show up later I will prefer to work not in the setting of addict spaces but rather back spaces and so this means I will work with this uh notion of arc stack instead of the maybe more familiar notion of well depending on your background uh more familiar notion of of a vstack so uh you define the category of arids.

[00:12:49]That will be my notation. uh these a category of uh sheeps or like strictly speaking it's really hyper sheeps uh of animal on Perf which is my notation for category of perfect spaces in character Okay.

[00:13:22]Uh undo with the with the arc topology and here I like arc topology is a very fine topology. So morph is an A to B of perfect to rings defines an AR cover.

[00:13:50]If when you look at the back of it spectrum uh the associated map is subjective.

[00:14:06]And so uh typical example if if you have x rigid space then you can naturally associate to it a certain arc type which I would denote x called the diamond of x well it's it's it's even arc shape uh which sends a a perfect ring in characteristic p so here okay maybe I even want to restrict to aphinoid perfect spaces that's good enough to define this category uh by sending this to some antilt a sharp of a uh plus a map from plus an so some antitar plus some a valued point of And so yeah, if you care about things like ealcomology, Palcomology, this arc stack as X diamond associated to X is a very good replacement of X. So that's really if you push this idea of really localizing for the proal or even arc topology so that everything becomes perfectoid that's some really just implicit the idea of replacing this rigid space by its incarnation as an arc stack.

[00:15:35]Uh yeah and so you you can do quite a lot with this uh with this idea. uh but this series of arax has some uh limitations and uh I want to maybe list two of them uh to explain why we we want some more to to to go beyond this this geometric setup. Um so and this will be some going back to my two motivations.

[00:16:04]Uh the first uh maybe drawback is related to 0.1. Uh and so it's just a basic observation that there are no nonzero differential forms on perfect to space.

[00:16:28]So in this example here for example if you look at the differential of t like you see that when you pass to this when you replace the state algebra by this perfect cover it becomes like infinity divisible by p and so in the p completion it will die.

[00:16:45]So this means basically that theory of arite is very good at capturing like topological properties of X like for example it's it's tell you things about it sphereology but it's not so good at telling you how this compares to Dam or filter Damology for example and precisely because we want to understand this kind of comparison results would be nice to have class of wings which somehow sees also this kind of differential properties.

[00:17:20]You mean there are no non >> there are non zero differential forces?

[00:17:28]>> No.

[00:17:31]>> Thanks.

[00:17:33]Um so that's one one issue and maybe another one is uh so there is a funtor from category of topological spaces to arc sacks um which sends a topological space t to an arc sack which is usually denoted by t on the line uh which is again it's sending a perfect space s uh to uh just It's like continuous functions from like the it's underlying topological space to to t and so in particular if G is a topological group let's say even like say a periodically group well then you can just you can in particular realize it as an arc stack and you can look at the classifying stack of G underline where point just means the final object of this topos. So it's just like some something which is sometimes also denoted sp or maybe fp bar and you would like this is like she on this stack would be some category of representations but this only some remembers the G as a topological group. uh but if G is a periodically group there may be as we saw earlier today in le talk there are like maybe other categories of with periodic coefficients like various class of categories of representations we may want to to work with like continuous representations smooth representations locally analytic representations uh and maybe we don't want to just have one geometric object and then change the ship theory to to get these various categories of representations it would maybe not even be clear how to do um rather we would like that this periodically group is incarnated as as a group object in some geometric category in different ways depending on which kind of representations we want to work with and this is not something you can really do uh in the world of arax and that's not really I want to stress that that's not really due to the fact that I restrict myself to perfect space perfected rings instead of more general Tate Huber rings things uh like in theory of BOVIC or in theory of Huber when you take a say in the world of Huber for example when you take a Tate Huber ring sorry a Huber ring and you look at its associated addict space it only depends on the completion so typically you can't really distinguish between like rings of locally analytic functions or rings of continuous functions so you you you need a geometry a geometric framework where you can distinguish between these different rings.

[00:20:21]Uh okay. And so that's what we'll try to I want to explain now how to do that. Uh so maybe there first before I move to this are there any questions about that?

[00:20:40]Okay. Um Okay. And so uh just to come back one more time to this. Uh so here what was the idea? We have a class of rings which we like a class of perfect rings on which we would like to model our geometry and then we just undo this category of rings with a certain topology and look at shifts or stacks for this topology. Right? So it's very analogous to uh maybe it's a usual algebraic geometry. You just have the category of a fine schemes which is just the opposite of the category of commutative rings and then you pick any goendic topology you like ital fpppf and so on and you look at sheets or stats for this topology on a schemes. So that's a bit similar idea that we have here with a very different class of rings and a different uh kind of topology. Um and so we want to enlarge our category of test rings so that we can typically distinguish between these different incarnations of G. Uh so there is a very general way of and not that the rings we we will be working with they will have some natural topology on them. So there is a very general way of dealing with these kind of questions which is analytic geometry in the sense of cloud and cha and so maybe I first want to briefly recall uh very briefly recall how how this goes. Um so the idea will be a bit the same. We first define a certain class of topological rings we'll be working with.

[00:22:25]Pick a topology on this category and then look at like shapes for this topology. Um all right. So uh for the first step the key definition is uh the notion of an analytic ring which is a correct correct notion of like topological ring one has to work with. So, so it's it's not just a ring, a topological ring. It's actually a pair a triangle da.

[00:23:02]So where a triangle is a a condom string well and everything I'm doing is will be at least at some point always like derived so it's really animated from three and but yeah if you don't know yeah you can for most many things I will say you can also just assume that it's a usual like it's static not derived and if If you don't know too much about uh condensed mathematics, you can also maybe just in first approximation imagine that this just means topological.

[00:23:42]Um and then DA will be a certain full subcategory of just a category of all a derived category of like condensed a triangle modules uh which has certain properties. So uh and I won't list all of them maybe just want to point out two important properties. So it's stable by arbitrary limits and collimits.

[00:24:20]Uh and it also contains a triangle itself and uh yeah there are some other properties which I don't want to list.

[00:24:38]Um and here again if you think to this as like the category of all topological modules over this topological ring maybe a good approximation to this is to you should think that there is a certain like you specify a certain notion of being complete. So it's something like complete a modules except that okay here you are basically for example requiring that the ring itself is complete.

[00:25:03]What's maybe slightly counterintuitive with this analogy is like it's stable under all columns which maybe you would not expect with the usual notion of a complete module.

[00:25:15]Okay. And so uh uh maybe then as a notation uh then you just define exactly like a scheme you could define as just you take the category of analytic rings.

[00:25:32]So you have a natural notion of a morphism of analytic rings and you take just the opposite category. So you just declare that aic stacks are just like opposite category of analytic rings and the notation one uses is it's purely psychological it's just notation to such an analytic ring a the corresponding aphetic sack is denoted and spec.

[00:25:57]So that's almost step one of the construction and then you define the category of analytic sets as a category of sheav of like for a certain topology called the shake topology on Uh so it's shipped on animal on the category of a fine analytic sax or the sh topology and I don't want to define the shake topology again in terms of intuition you can just think that it's something like the finest topology for which you have descent of uh so of this this this category DA of complete a models but not just descent of this category but also like the six operations you can define it. So in particular as a consequence of the choice and I will give later an example where this can be made a bit more concrete but an important consequence of this is precisely because you chooses topology in this way for any analytic sax x you have uh a certain infinity category of u what I will call a category of quasa coherent ships.

[00:27:32]on X in such a way that if X is a fine UH then this category of queries is precisely this category of complete models which is part of your B when you define what an analytic Okay. Uh so very brief recollection on this general notion of analytic stack.

[00:28:08]And of course to make this interesting you have to supply examples of such uh and I won't make an overview of all the nice examples one has maybe just list like two relevant examples. Uh so first this is this category of this analytic ring structure that you have on QP called the solid analytic ring structure. Uh so what is this analytic ring? Well it's just QP I will just use the same notation for the ring topological ring QP and the corresponding condense ring. So it's just QP unode with its natural topology topology and then you have a certain category which in this case is just a derived category of a certain abil category uh called solid QP and this is a category which has it's an abil category which has one compact projective generator which is given by the product countable product of copies of ZP and then invert okay so it's an abilion category with one compact projective generator so you can just like then it's just really the category of modules over the endommorphism ring of this of this object. There are other ways to define this this category. uh yeah so that's that's that's an example of of an analytic link structure and uh maybe I just want to stress here that okay like this contains for example this category contains like QP uh like banner spaces but it also contains like it's table also under limits and colit and here I just want to stress that when you compute like say filtered co limits it's really like uncompleted right and the underlying ring it's it's really just the uncompleted filtered poly. So typically something which is complete is just an arbitrary direct sums of copies of QP.

[00:30:20]Uh this is also complete sorry and once you have one example you can make many more examples uh by a general construction. So if a is a solid QP algebra so it means that it's like a QP like condens QP algebra which is like the underlying QP module is solid. Then you define uh an analytic ring structure on a called the induced analytic ring structure by just taking as a condensed ring. you just take a uh and then this category of modules that you work with is just the category of all like um condensed a modules. Sorry my notation is not really compatible like yeah this is what I would have denoted before a triangle uh and you just look at all corners a module which are solid such that M is whereas the forgetful map is solid as a QP model.

[00:31:45]And in fact that's all I need for my for the this lectures because uh we will always just work with induced in this sense induced analytic thing structures and as a remark maybe uh uh if you have a morphism A to B of solid QP algebbras and with like induced.

[00:32:20]So then you you get a morphism of analytic rings when you undo them. Uh so sorry maybe I should say uh then uh the induced morphism of 183 is a sh cover if and only if the morphism a to b is this map a to b is descendable.

[00:32:50]meaning that uh a belong basically to the smallest subcategory of da which is generated under limits limits and retracts by by b. So it has some kind of fire resolution by by b motors. So that's a notion that has been introduced by Akil in slightly different context but which also makes sense in this setup and yeah so that's for one example of simplification that you get by restricting to induced analytic thing structure uh in terms of intuition again you may think to this assumption yes this restriction to a bit analogous to something I already said that I will not work in the setting of Huber addict spaces but I will work in the inovich setting that's a bit analogous to this kind of restrictions.

[00:33:42]>> Say again.

[00:33:45]>> Yeah. And is that something one can do?

[00:33:53]fun to use the uh >> well you you can still do it with uh sorry I'm not sure what the question like it's just that no amorphism between a fine kinetic stacks would be proper right >> so lower would be lower >> lower street would be lower star in this case but yeah that's that's that's fine it's just yeah it's okay maybe I come back to this a bit Um okay and so still uh so I want to do nonarchian like periodic geometry so certainly I want to I don't want to to work with general analytic rings I certainly want to restrict to the one which leaves over QP solid and even more as I said I want to reduce to just considering the one with induced analytic ring structure but even then this is still not restrictive enough because we really care about rigid geometry And again in this analogy with maybe Huber theory or theory in Huber theory in the category of addict spaces you not only have like rigid analytic spaces but you also have for example schemes and formal schemes uh and this is the kind of examples I want to discard. So I want to like an example of a ring which is solid QP algebra is something like the polomial like the underlying QP module is just a direct sense of copy of QP and that's that's certainly solid and that's the kind of example that I don't want to work with because like unspec of this would be something like the algebraic aine line and we want to really be closer to rigid analytic geometry where you would be considering something like the analytic aine 9 instead. So this means I want to impose some further restrictions on the category of solid QP algebra that we'll be working with and that's some of the key definition uh that I want to give now.

[00:36:01]So to to to make sense of this definition I have to like do it in several steps. So let me fix a solid QP algebra A.

[00:36:16]So the first observation is that if you have a say non- negative real or rational number you can make sense of one can define a certain condensed subset or subanigma because we work everything is implicitly derived. uh denoted a less or equal to r inside a uh which intuitively speaking will be the subsets of sub animal of elements of norm less or equal to r and I won't give a proper definition of this just some information maybe what happens on underlying ring so for like any condens ring so it's a shift of rings on light profile sets And in particular I can just evaluate on the point and then I get a honest ring and this is what I call the underlying ring. So what does that look like in this in this example? What is this subset? Uh so an element of f of the underlying ring of a so a evaluated on the point or it's just the same as a morphism from uh maybe I still denote it by f from poly one variable to a and I want to say that it is of norm less or equal to r. If this map extends if f extends to a map uh from uh what's my notation from I will denote it like this.

[00:38:10]Well no the notation is maybe not optimal. I change the position of this symbol. So now it's not a an exponent.

[00:38:19]It's it's not a subscript. And what is this is just algebra over convergent rigid analytic functions on the disk of on the aphinoid disc of radius R.

[00:38:39]Sorry, that's maybe a bit small. Okay, so just to repeat, I look at all like I have this aphino disc of radius r in one variable and I look at this uh just not functions on it but over convergent functions and I want yeah I think it's a bit rather natural to imagine that if these elements have n less than r basically I can form like arbitrary like if I evaluate if t goes to my element f then it makes sense that this kind of sum will converge in uh a being solid and okay this is like a really a property because it's this QP this algebra over the polomial algebra is is important so it's really a condition and in general you do you do something analogous so you are not just trying to define some some anima but some condensed anima so you have to to do it for arbitary light profile set So to really define so that that's would just define the underlying uh set or anima and to really define this in general you have to define something which is denoted QP wait what what's our notation QP and less equal to R some it algebra of algebra of over convergent functions on the s dimensional disc whatever that mean I mean if s is a point or maybe just a finite set that would really be some kind of overcon convergent state algebra infinite many variables in general it's a certain important algebra over the free solid QP algebra over S.

[00:40:43]>> Can I ask >> okay >> just why is the why do you have the over convergent condition?

[00:40:50]I think it's just a little bit uh better behaved uh if you use this overcon convergent version uh rather than just the usual like the aphinoid disc like this for example would be like comfortably presented uh but not not the apheninoid version and also the idea is that this is like really some defining some kind of map from unspec of a to uh the interval 0 infinity like more precisely the basis stack of this interval and then if you look at some interval 0 r it's really the limit of the interval of limit of the interval 0 s when s is bigger than r so whatever if you want to have such a map then you are forced some to to to look at the over conversion data not the dat theta gema Uh okay so maybe quick just an observation at this point is that this is a subring when I take r to be one I get a sub ring of a and uh you always have a natural map you say that from so now you say that a is bounded if the natural map from A lesser or equal to one with P inverted to A is an isomorphism and the terminology is justified by the intuition that uh yeah again uh you basically you are saying that any element of A is bounded any function on unspec a is bounded because basically you are asking that when you have such a map it will extend to the over convergent state algebra of some radius smaller than infinity.

[00:42:55]>> So if you take an element bounded by R and you multiply by the right power of P, do you always kind of shift it to be less equal to one?

[00:43:02]>> Yes, that's correct. Yes.

[00:43:07]>> Uh and finally you say that A is gon if not only A. So first of all a is bounded but also you want to uh require that the uniform completion of a which I will denote by this symbol a u which is defined to be so you take this what I will call power bonded elements of a you take the p p completion of this and then you invert p and this you require to be a banner algebra.

[00:44:03]So equivalently when you look at this power bonded element and the reduce mode P instead of being an arbitrary solid FP algebra it's discrete >> what >> discrete like just has a discrete topology okay so I I now give some examples and non-examples but maybe first one more remark but if a is bounded the reason for why this condition is useful uh is because this uh you can also plug in r being zero That's something that is also sometimes denoted nil dagger of this is a dagger nil dagger a and called the dagger nil radical uh then this is an ideal of a.

[00:45:10]So this you have to like this uh uh subset you subma you have to think to as like the so it's by definition the elements which have norm zero so it's like elements which are over convergently close to zero like infinitely like they live in the power bonded part but they are also whenever you divide them by a power of p they still remain power bonded uh and so in particular in the if a is bonded you can make sense of the quotient of a by its diagonal radical and we will make use of this later today.

[00:45:47]So just know some quick examples. So uh well banner QP algebbras are gant but also this category is stable under co limits but also also co limits of such.

[00:46:15]So all the rings that you see in rigid dynic geometry. So t algebra but also over convergent t algebra which are like fed cos of banner qp algebra. They are examples of gon rings.

[00:46:29]>> Yes.

[00:46:30]>> Is it correct that a to a less than or equal to one elements?

[00:46:38]>> Uh yes I yes that's I think correct.

[00:46:41]Yes. actually is is that because of this uh because it's overcon convergent functions that would it be different >> h wait no maybe uh you have to be a bit uh careful about that I'm not sure this is actually true maybe if you take strictly less than r for any value of r then this would be I think fine but uh yeah know I don't I don't think that's true that uh like if you take just this for example this this when I don't know for any like say r equals 1 you take this this to be your a you take the identity map and then it won't really factors through but if you replace this less or equal by by some kind of like open uh condition then I think it becomes true >> so that being bounded is closed under filter Is that right?

[00:47:42]>> Yes. Even like arbitrary colits. Yeah.

[00:47:45]So yeah, I forgot to say category of finalic rings as like small colits.

[00:47:51]This category bounded or gon rings is also stable under this colit. But again these colits filtered colits are really I stress it again are really uncompleted.

[00:48:04]uh so non-examples uh somehow then when you check the gang condition you kind of have this curl limit of banak thing I mean somehow the things a pro not banak but individually when they complete their banak then when you kind of take a cur limit and then you complete there it's like you complete them individually take some kind of co limit and complete again >> that's correct yes exactly where complete meaning uniform this kind of uniform completion yeah >> so a toy model will be like taking a direct sum of copies of QP and thenically completely. I think I >> Mhm. Yeah. Exactly.

[00:48:38]Um Yeah. So non-examples uh uh so QPT is is not gon because it's not even bounded.

[00:48:52]So the idea is that this element T is not a bounded function. Just if you stare at the definition you see that like you just take the identity map from QPT to QPT you cannot extend it to some some over conversion t algebra. So T is not bounded but there are also examples which are bounded but not gon like typically something like like this. So power series ring over ZP and with P inverted is not gine it is bounded uh but the power bounded element will be given by just this power series ring but it's not keon because for example if you go mod P then like power bonded elements are given by this then you reduce mod P you get a power series algebra over FP which has its natural tic topology and that's not discrete All right.

[00:49:54]Okay. Uh and so now we can that's a class of rings we want to work with. And then we can just redo this definition two here and define gelfon sax uh the category of gon tax as the category of sheeps on maybe I should have introduced this notation. Uh so gon sorry g ring will be the category of gon rings and then I just look at this category I take the opposite category uh so shifts for the uh because I always work with induced andic structure I can just say for the desendable topology Okay. So the relation between this cate so this is if you want a variant of the category of analytic stacks of a QP solid like there is a natural like colic preserving fun from gelfon stacks to arc stacks.

[00:51:23]uh so by the way as notation if again I need some notation so if a is gon like I will again I will denote by gspec a instead of unspec a like the the funtor which is co represented by a so that's the corresponding aine gon stack associated to a and so I was saying there's a natural fun from colit presuming font from gon stacks to all analytic stacks which sends gpspec a tospect Okay. But uh it's I don't know if it's it's probably not fully faithful in general. Uh but it's also not maybe so so important. We'll just stick to to Gon tax stacks from now on >> in general.

[00:52:09]>> Mhm.

[00:52:09]>> You said that everything is over QP. Is there an analog if you replace QP by something else?

[00:52:14]>> Uh yes. I think B as long as you have a pseudo uniformizer basically then you can make sense of this. So I think more precisely I think you can make sense of all these constructions over the stack of nodes. I think that that may be discussed at the end of this lectures by Maximillian.

[00:52:32]Yeah. So for example you can work over something like like ZP double power series pi for some pi.

[00:52:42]Sorry, stupid question. But um so property of being gant is only on underlying condensed property of being >> say that again.

[00:52:54]>> I mean the property of being gant is only something defined on just the underlying condensed.

[00:53:01]>> Yeah, >> you're saying that gans. Okay.

[00:53:04]I mean it's implicit of course you mentioned earlier about >> sorry maybe I should have made this more clear like all like yeah this kelon rings is a is like certain subcategory of all solici algebraas and then I to see them as analytic rings I always undo them with induced analytic structure and yes the condition of being gund also only depends on the pi zero uh yeah so I was saying that there is a monitor from uh G fun stacks to analytic stacks which sends Gspec A to anspec A and it has a writer joint.

[00:53:48]Uh so I will never use that but I just want to maybe point out what this writer John does in these two examples. So we saw examples of solid QP ones which are not gon. So what happens when I apply this uh like gelfundification functor to them. Uh so for example if I take ansp spec of qpt so that's really the analytic stack version of the algebraic aine line and I take its gondification I take I get the analytic aine line over which is a nonfine analytic gon sack. So that's gon stack version. So it's really a colit over a union over all uh radius of like gstack of of uh of like uh autists.

[00:54:52]>> Why are they called gelfon stats? Uh yeah so for gon rings you have a a notion of a bid spect associated beovid spectrum uh just basically by taking the bid spectrum of the uniform completion and uh yeah I think at some point we we thought that yeah I think we made a mistake at some point and thought we can make sense of this spectrum construction for like all arbitrary bounded rings and then the property of being Gelfon would have been equivalent to saying that a certain kind of Gelfon transform you get is is like well defined as a as a map of condensed uh sets but so that's where the terminology is coming from I think it was from a discussion with testing clausen but yeah then we realized that there was a mistake in this construction so then I don't know we kept this terminology but uh with no no good motivation Uh all right so that's one basic one example. So yeah and maybe just for this other example if I take anspec of this and again I take its associated gon stack uh this will be just open unit disk.

[00:56:21]So that's my notation for the open unit.

[00:56:30]Yeah. So just to give two examples of this.

[00:56:41]Okay. uh of course for other purposes it's good to have like like this general flexibility given by the SE of analytic stack that you can also make sense of this kind of of analytic stacks and you want maybe to for some problems you may want to distinguish this from this open unit but at least for what we want to do uh that's uh good enough and uh other remark is that this category of gon stacks it contains fully faithfully the category of dagger or overcon convergent rigid spaces in the sense of gross a cloner.

[00:57:20]Uh in fact even better than this. So this itself sits inside a very general version of like a derived notion of beovich spaces and this also sits fully faithfully inside this category.

[00:57:40]So what you said earlier are you saying that if you take the so there's two definitions the spectrum of CP double brackets T invert P is the gon stack and you're saying that's the same as the union of the union of the closed disc of radius less than one >> that's correct yes that's defined as like yeah as union of I don't know that's maybe not very standard notation I'll try to Uh yeah so like this notion of derived backward spaces is basically like stack that you get by so it's just something that you take like gspec of some arbitrary gon ring and then you glue such for the analytic topology so you can define some version of the analytic topology where you just glue along rational localizations which are basically pull back from the back like betty stack of the back of it spectrum Uh and so there you get this general notion of a burovich space.

[00:58:41]And the reason I look at like over convergent rigid spaces and usual rigid spaces is again because of this kind of pay attention to induced analytic structure. So the fact that I work in the back of it rather than uber setting but so for example like you can look at partially proper rigid spaces. These are example of such Okay. So this also shows that this category of geometric objects that I have defined is like contains maybe the main examples we care about for which we want to to define some version of prismatic homology.

[00:59:17]Uh and so maybe I will make a small five minute break but maybe let me first uh end with before the break uh state a proposition.

[00:59:28]Uh so it's a key property of of gon rings.

[00:59:35]Uh so is the following statement.

[00:59:39]So let a be a gon ring and then there exist a desendable cover.

[01:00:03]A to B with B uh is what we call a near perfect ring.

[01:00:15]And so a ne perfect ring is a gon ring.

[01:00:18]Uh so B is you get found and it has the property that uh B when you model this diagonal radical of B h then you want this to be perfect to it or t when I say perfect to it I always mean like t perfect to so in this case it also happens naturally maps to the uniform complete this will also be the same as the uniform completion. So >> that's not always true for >> this property you mean? No, that's not always true. But that will be if I require that this like dagger reduction is perfect to it, then it implies this in general. You just have an injective map. It may not be subjective.

[01:01:08]Uh right. Okay. So that's just the geometrically this corresponds to some kind like of overcon convergent sickenings of perfect aphinoid perfect spaces.

[01:01:21]Uh and you can do even better you can even assume can even assume that b uh that is this dagger reduction is uh what is called strictly totally disconnected. So it's an even even stronger condition. So it's not only just pertoate but basically it means so maybe let me introduce a notation for this kind of question that's what I will call the dagger reduction and denote by bar so it means that the back of it spectrum of bar is is profite and residue fields are perfect to algebraically closed So it's just a bunch of profite bunch of uh back of it spectra of perfected algebraically close to field.

[01:02:27]So here I'm a bit cheating. I mean I've been cheating already several times in all the like like let me maybe point out that the definition of analytic stacks or gon stacks that I was giving was a lie. I mean the correct condition is a bit stronger than being just a sheath. Uh and here I'm also a bit cheating to to make this true. You have to impose some kind of countability and fine dimensionality assumption on the gon rings you work with and they will be implicit in everything I do from now on.

[01:03:00]uh and so as examples so maybe first this this is this subcategory of of n perfected rings is not stable under col limits but it's stable under push outs and or finite colits and so in particular it's really like you could have defined gon stacks just as stacks on this category of n perfectly rings instead and if you think about it this way you see that it's really close to the category of arcs STS I I started with except that I allow myself to replace like aphinoic perfected spaces by over convergent thickenings of such so it's just some of the minimal choice I can make for geometric theory where I which is close to the theory of arax but where where I can al also speak about something like the diagon radical in a non-trivial way and so just some examples of such so of course if you are perfectoid So perfect rings are not perfecto but less real examples are also if you take something I will denote uh by CPT so if a is cpt dagger so that's by definition the co limit over all positive real numbers Oh sorry natural numbers of state algebra like this. So it's like really the over conversion functions around zero in a1.

[01:04:37]So that's also need perfect and in fact the idea generated by t is exactly the diagonal radical. You see that in this col you made t of non zero.

[01:04:50]You made it divisible by arbit high powers of t. So in this case a bar is c which is a perfectly field and more like also more interesting class of examples is if you have any banner QP algebra and b is a quotient then b is near perfect.

[01:05:22]Uh sorry uh is a perfect to QP you take any quotient of of A then B is also is near perfect and in this case it's uniform completion will be given by the or it stagger reduction will be given by the generic fiber of the perfectization in the sense of of the perfectization of the power bonded part of a in the sense of of uh of what do I want to write? Uh uh so this something else hope I get that right. Um so you see that you can that's the example also suggests that you can think of um need perfected rings as a bit analogous in this context to semi-perfected rings in integral per code theory analogously like semi-perfected rings for a basis of say quasintomic topology and there as well like we have this desendable topology and we can uh we have this nice basis of near perfect rings Uh okay. So I don't know maybe it's a good time for a five minutes break. Uh I don't know if there are questions. Uh let me before moving to part two I just uh wanted to add one more general remark about gon tax stacks uh which is that it's actually important for later uh that uh for gon st there is a well behaved A category of perfect complexes on on X is a subcategory of like quazmanships on X satisfying descent for the top for for the for the shake or descendable topology which or maybe a bit more precisely what I'm saying here is that like you can even bound the the amplitude. So for any interval a integral interval AB you have some the fun which sends a gon ring a to perfect complexes on like just underlying ring uh and the claim I'm making is that this defines a shift for the desendable topology and so then in particular you have like a well definfined notion of a category of perfect complexes even with bounded amplitude for any any gon stack. Uh and the key the key input to this is the fact that gon rings are fred home terminology which has been introduced by clausen and shelter uh which basically means that all dualizable objects are just perfect perfect modules and the reason like why this enters is is basically that okay like a pory It's just on this galon rings, it would not be clear that if you take some such an A and you look at G-spec A and then you take an arbitrary trick or desable covers by other aine gelfon stacks, it would not be so clear that the something which is locally for this cover a perfect module is really a perfect module on a itself.

[01:09:21]But what you at least know is that it's it it will be like it will be realizable because this is a local condition and then this property of being fedom is exactly telling you that this this will be automatically a perfect complex. So we will use this fact several times later.

[01:09:44]Okay. And so uh now what I want to do is uh to discuss to some illustrate what this formalism is good for. I want to discuss the theory of the sax and I should also maybe have said that everything I'm discussing actually I real I sorry I should have said it at the very beginning uh what I'm discussing today is also joint with Bosco and it's actually written down so everything I have discussed so far is in a joint paper with an bosco over and sher which is archive uh but okay that won't be the case for the what I'm discussing in the next lectures but in the paper of maximian and analytics inification there is a very nice discussion of or summary of the theory of analytic prismatization that you can also uh look at and hopefully the paper will be available soon but okay so back to the ramstack so the maybe I want to start with a toy model for what we we will be doing. Um so it's it's an example in characteristic P. So maybe I want to write I don't know stack FP is a category of stacks on FP algebraas let's say I don't know for example for the desendable topology and then maybe I denote by stack FP pair the same category but now I only look that perfect FP algebraas which for belief is an isomorphism and again for the desendable topology and now you have the restriction functor uh or maybe I could also call it perfection uh which just okay you have a funtor on on all FP algebas you just stick to like perfect FP algebra has and this has a left and the right adjint.

[01:12:09]So the left adjint is basically just a funtor which sends if you have a R a perfect FP algebra uh well or the funtor co presented by R you just send it to to ST R and the right adjint is maybe more interesting uh because it's a version of the drumstack and more precisely. So the dam stack will be like I say a version because apparent R to some FP algebra to to so the DAM stack of some some some scheme in characteristically you would like to define it as a funtor which sends R to X of the reduction of R the quotient by the radical But then you that's a maybe only a pre-shift and here we work with a descendable topology. So you take the shiftification of this for the descendable topology.

[01:13:31]But uh a basis of the descendable topology is like formed by perfect FP algebra and oh sorry semi-perfect FP algebra and for semi-perfect rings the reduction is the same as the perfection R is semi-perfect and then that's just the same as the colit perfection of of R and on semi-perfect FP algebra this R goes to uh xrp is already a shift.

[01:14:08]>> So this the fact that you had these attitude by these formulas at the level of pre-shapes is it correct that at the level of pre-shapes this would be true if you said lower perf and upper perf on both sides that pre-shieves on perfect ranks with heads by pre-omposition with lower perf and upper perf.

[01:14:26]>> Yes, I think so. Yes.

[01:14:30]uh right so so that's if you want the reason I'm explaining this is because it shows that so it's maybe not quite the dam stack you'd like to consider in characteristic p there are no divided powers and so on uh but if you look at this version so you just take us reduction and you try for something like the descendable topology then basically what you get is a right adjint to this perfection functor and so this shows that yeah you can came to the Dam stack funtor as being adjoined to perfection and so that's exactly what we will mimic uh in the setting of gon stacks we will define the dam stack funtor as right adjin to some kind of perfectization fun so what is so analog for gon st I never managed to write this uh so you have again factor from category of G fun to the category of arax over QP. So over the arc stack of QP uh which is again basically just restriction to from arbitrary gon rings to to perfect rings over QP.

[01:15:56]And this is what I would call the perfectization functor or sometimes also maybe diamond funtor uh yeah so I will I will denote it with so x goes to x and in fact this like it's call it preserving and it sends if I have something like gspec r where a is a gon ring it sends it to the arc sack of a. So the fun co represented by a on perfect.

[01:16:43]So that's a notation the arc sack notation for what you would maybe call uh I don't know maybe something like spad of a if you use vstack that's maybe what you would the notation you would use that's notation produced by console Uh yeah. So that's what this fun does.

[01:17:15]Oops. And again this fun has a left and a right adjint.

[01:17:21]Uh so the left adjint is so it will be a functor. Uh so from R S to G S which I will denote X goes to X.

[01:17:40]So it's a left joint. So again it preserves co limits and it sends if you have R which is a perfect Q sorry it's R tax over QP it's a perfect to QP algebra and then you send M R of R to to G S of R.

[01:18:02]So that's a way to realize if you like arax inside gon tax stacks. Uh so the reason for this notation is that the when you look at the coherent commology of this stack uh it's basically the so x is an arc stack over qp because it's an arc stack over qp you have a structure shift in characteristic zero or hat and that's basically what it what it's computing that's more or less from the definition.

[01:18:37]So in general you write your arc stack as a co limit of aine aphinoid perctoids and then you just for them you just map this to to spec R that's a useful construction which we will also use later but for today I maybe mostly care about the right joint and that's no like my definition of what the drama funtor Um yeah. And so uh so if x is one more piece of notation is that if x is gon gon sack then I will slightly abuse notation and I will also use the notation x to run. So strictly speaking would not make sense because I need to input an onx but that would be defined to be like first I apply this perfectization functor and then I take the corresponding ar and uh if a is a perfect ring then and x is a gon sack then I have a function of points description for what x term of a is and this is the same as x of the uniform or dagger reaction.

[01:20:18]Okay, so that's very analogous to the the formula for the usual DAM stack except that I have replaced the usual reduction by this reduction.

[01:20:30]So that go.

[01:20:31]>> Yeah.

[01:20:32]>> So I guess maybe in some talks in the last few years there was some informal picture where in this analytic world sort of the armstack construction should be questioning out whether groupoid which is kind of the overcon convergent thickening of the diagonal >> right >> is that happening philosophically >> yeah I will I will state result where this yeah you're right if you have yeah in some class of examples this is really just this kind of question. So I will write it later. Yeah.

[01:20:58]>> Just in this FP picture, if X was A1, this is not what I would typically think of as the DM structure. This would be like GA mod GA hat.

[01:21:07]>> Yeah. Yeah. No divided powers. So it's Yeah, that's why I said a version of the D.

[01:21:12]>> So it's more like differential operators with like divided powers as opposed to crystal differential operators.

[01:21:18]>> Yes, exactly. But also I think usually maybe people do not shify for such a fine like Uh okay. Um yeah so that's a formula that is valid on like it's it's not true if I input an arbitrary gon ring but if I input some n perfectly ring I have this formula like without any need to like shify so that's really a funtor of points description of this the ramstack funtor on some nice basis of the of the shake topology and that's useful that you don't have to perform any further shification um okay So now let me state a proposition like a theorem which is in particular an answer to to Matt's question. Uh so the first part of the statement is that oh and maybe what's hidden in this claim that you have a left and right joint is statements of the following form that uh if you have a desendable cover of gon rings and you pass to uniform completion it induces an arc cover that's actually a statement for which the gon condition is useful but conversely also if you have an arc cover of uh say for example imperfected rings then uh this this is also a desendable cover. So this is the kind of statement that you need to to check this assumptions.

[01:22:53]Okay. Um and they also explain a bit some of this section that I put on my wings like countability or or fine dimensional.

[01:23:05]So the first uh part of the theorem is that this function commutes with limits and calling it.

[01:23:17]So in particular it preserves eporphisms.

[01:23:27]So the fact that it commutes with limits is just because it's a right it's defined as a right joint. But still it's interesting because uh note that limits in arc stacks are really like some kind of again like like co limits would really be something like at the level of rings it's some there is some completion involved but when I so typically if I look at something like uh I don't know like the the perfect taurus which I write as an inverse limit of of of rigid to uh and I I take the corresponding ar stack and then I apply the ram functor it will be a limit of the ram stacks of to which are nice rigid spaces but this limit is not computed in g so it's really some kind of uncompleted limit so that's this feature that the damac is kind of decomping things but that's in this definition that's sort of built in the definition because I define it as a right this is much less obvious and in particular This consequence you should think to it as telling you that this funtor x goes to x has very strong disarm properties. If you have an arc cover of arc stacks then sorry yeah if you have an epommorphism of arax you get an eporphism on the ram stacks. So that gives you a way to to access computing examples.

[01:24:58]>> Mhm. So if you say if you restrict to no perfectoids then you have a formula for extra and I mean at least at the level of pre-shape that with colts is there an issue of shapification and so forth.

[01:25:17]>> Uh can you sorry can you say that again?

[01:25:19]Are you saying that that formula if you have that at the level that that commutes with col >> uh limits in in xum mean?

[01:25:29]>> Yeah. Yeah.

[01:25:29]>> Yes.

[01:25:31]Um but I was I don't know I mean there's some cheap vacation so what that one has to do um sorry no I don't think there is Okay. Uh yeah, but I I'm claiming Okay, maybe I'm claiming commutation with colits in arax, right? And here I was like like for this formula to hold I was sorry I think sorry I'm saying something stupid now. Uh uh yeah, I don't think there was any need for this. Oh wait, maybe it was >> going to you saidorphism and maybe an is is it is it an issue of starting with like an an arc cover and lifting that to a cover?

[01:27:04]>> That won't necessarily be a shake cover, right?

[01:27:08]>> But I think this is precisely the issue, right? I mean the key thing that you have to prove is that if X2 Y is is a suggestion of astax then X round to Y around suggestion of GT and that's that involves lifting this guy >> yes okay I'm still confused by your remark no but uh >> my remark was the pre preacks but uh but maybe the issue is two different topologies Yeah, I think that you have different topologies.

[01:27:45]>> Okay. Okay. Yes. Now I get it. Yeah.

[01:27:47]Sorry.

[01:27:49]>> Um Okay. So the second statement is the answer to Matt's uh question. So if X is a Oh, sorry. Dit space.

[01:28:00]So eg partially proper created space or something like an over convergent disc then the dam sack of x which know I'm using this this this notation I see this as a gon stack as I explained before have this fully faceful fun from tiger rigid spaces to gon s and know this is just the quotient of x by the over convergent neighborhood of the diagonal.

[01:28:32]Uh um yeah, I'm not sure how how to denote this. So I have the diagonal embedding of X in X * X and I just look at the over conversion neighborhood of the diagonal and in particular okay so this you see that this is maybe another more familiar description of the D stack as some kind of portions. uh in general you always have a natural hand map from X to X the RAM just because you are applying like first some fun and then it's right joint so then you you do get a map like this it's just not true that in general that this is an epommorphism but under this assumption uh you can prove that and so it looks like the formula for the the ramstack of Simpson except that observe that you have replaced the like formal completion as an insurgent completion Um and this means that you are not really when I said to RAM stacks I should always maybe say analytic to stacks because really the category of sheets on these stacks you have to think not as like algebraic modules but as some kind of analytic models.

[01:29:44]So in the case of the aphine line you you could look at some algebraic ramstack of the analytic aine line and that would be some imposing no convergence condition in the cotangent direction but here you are also somehow imposing some convergence conditions >> parallel 2D models with the infinity differential operators of infinite order in usual usual magnetics >> I guess it's related I just don't know very What is this d infinity uh means?

[01:30:16]Uh yeah. So uh maybe let let me >> infinity is something usual usual differential operator areology you take square you take diagonal and consider chology with support functions exterior product forms something forms and in algebraic geometry and and this is if you do it in analytic geometry.

[01:30:41]>> Uhhuh. Okay. Yeah. Yeah, that doesn't sounds plausible. I I think in this periodic situation there is a like a precise statement you can make that u the category like maybe let's say this one like here I did not even assume it's smooth but for a smooth let's say partially proper rigid space then the category of quakan shifts on this stack should be the category of decap modules like aracov and watslay have defined some of pd models on rigid analytic varieties I think that's precisely what you get I mean maybe a condens version of But one consequence is that in particular this will be computing the damology of X.

[01:31:21]which somehat justifies the name for this uh construction.

[01:31:28]And uh yeah, you also have a a nice aspect of the theory of analytic stacks in particular of the theory of gon stacks which also will be a motivation for the theory of with analytic prismatization is that if you have defined something as an analytic or gon stack you have an a category of quazan chips on it and six operations.

[01:31:51]So you have a full six functor formalism. So uh this you can use for for the DAM stack and say if you have morphism uh which is smooth uh of relative dimension D. Then if you look at the induced morphism on the ram stacks and then this is commonologically smooth meaning that upper shake and upper star funtor are the same up to twist by some invertible object and this invertible object is upper shake of one and it's called the dualizing shift and this dualizing shift is a is just a structure shift on the on X direction with a shift by two times the dimension.

[01:32:52]>> Is it possible to see accurately what this applique is doing? say if you have more.

[01:33:01]Uh yes, I think for an open disc it would just be uh just be QP sitting in degree 2 cological degree two.

[01:33:14]>> Sorry I mean the lower shrink the lower >> the lower streak. Yes, I think that's >> I mean module >> ah so sorry I thought you were trying to just compute for the for the structure for for the units some sort of compactly supported complex >> yeah some kind of compactly supportedology uh I don't think I have a a much better answer than this uh yeah I also think okay I also have to say that some we did not really investigate. I I made this claim that this is related to the series of decap modules. This is not part of our paper. This is I think work in progress of Rodriguez Rasinto Rodriguez Kamago Rodriguez Rasinto. But so the exact relation with like six functors on usual D module or like decap modules this is not something I have thought about really.

[01:34:17]Okay. So let's give a few examples. Uh so if X is analytic F9 then it's the RAM stack that's just a particular case of 2. I mean this has a structure of a group. So you just get a description as a quotient of A1 analytic by something I will denote by GA dagger and GA dagger is notation for like some kind of over conversion disc around zero. So it's a spec of Gsp spec of of this t like this.

[01:34:57]So algebra of gems of of genetic functions around zero that I introduced before. I mean I I wrote it over CP but you can also of course define it over cubes.

[01:35:10]It's just a question for the natural action of this by translation.

[01:35:16]Uh another interesting example is if you take the aphinoid disc d so that's just dspec of qp t uh then xam is given by uh so in this case in fact the natural map from x to xam is not subjective. So uh really the correct description of the dam stack is you take the overcon convergent disc of radius one and again you modeled by the action of g dagger by translation.

[01:35:48]So this is uh by definition gspec of this algebra of over conversion functions of radius one. And so this means that in this example uh the homology of this stack will be computing like overcon convergent damology instead of just naive damology which is in some sense good because this is the one which has which has better properties.

[01:36:22]>> So the drums have sort of erases the difference between D and P.

[01:36:27]Is that is that right? It's supposed to be a dagger.

[01:36:36]>> Uh yeah, that's right. The uniform completion of this ring would just be this one. Yeah. Yeah. So the corresponding arc stack uh for both are the same, right? After the associated the perfectization only depends on the of of like a gon ring only depends on the uniform completion.

[01:36:56]Yeah. Yeah. So if I could apply 02 to the over conversion disk and get this description but what I'm saying is because they have un same uniform completion they have the same DAM stack.

[01:37:07]So in particular for this you can think to this as some kind of compactified disk of radius one and the claim is that it's the RAM stack like computed like over converge on the RAM.

[01:37:20]uh another example if if you take X to be just the arc stack associated to a topological spaceation I introduced before uh then the DAM stack of X is a construction called the BI stack so it's a construction due to claus and shter which is a way to realize um topological spaces or even condensed anima as analytic stacks and this also works in gon stacks So like quazon shifts on this bis stack are just the same as shifts on the underlying topological space. Sorry on the underlying topological space between >> so t underline is an arstack for me.

[01:38:07]>> Sorry that's the notation but maybe okay now I base chance to I see it over QP and now I yeah I take its bet st. So like the betty stack is living over Z.

[01:38:19]So maybe implicitly I'm doing a base change to QP solid.

[01:38:27]Uh okay. Uh and so I will use this to to make some kind of this one is not really an example but maybe more a definition.

[01:38:38]If you have an algebraic like a rigid analytic group of a QP. So for example the analytification of an algebraic group uh then you know that the dam stack of G okay so it's something it's like generalization of first example here it's like a quotient of G by something I denote by G dagger the over convergent neighborhood of the unit section uh and now you you have a natural map if you if you look at arc stacks you you if you take the QP points that's seen as a topology logical group. So G QP underline maps to the arc stack of of G.

[01:39:17]Uh and when you pass to the RAM stacks, so okay, that's by definition what I call G dam. Uh but now by this example, this is just the the Betty stack. So this is like locally profite. So that's also something I could write as the smooth incarnation of GQP. So just something which if G is compact this is this would really be just like spec of locally constant functions on GQP.

[01:39:49]So I get a map like this which is an immersion. And so now I you can define the locally analytic incarnation of G QP as a fiber product of GQP smooth with G.

[01:40:08]So I'm using this map here and there's a natural map from G to G.

[01:40:13]And so intuitive like geometrically it's some kind of like you are looking at by doing this fiber product you are taking GQP sitting inside the rigid dynic group G and you are doing like looking at over conversion neighborhood of GQP inside G.

[01:40:32]So that one is as I said more definition than really an example.

[01:40:41]Okay. So I don't have much time left. So I wanted to discuss two more things. Uh uh I want to discuss. Okay. First thing I want to discuss maybe I do it in the form of a remark. Uh so we just defined a stack which some geometries rigid damology of digger rigid spaces. uh but on the ramology we have the hot filtration so it's of course natural to also try to geometryize this piece of structure and well also from work of Simpson and more recently badeld and so on we know how to do that it's a construction called the filter the ramstack uh and I wanted to do it in a way which is maybe not exactly the way it's usually done it's let me do it in a slightly over complicated way because this allows me to introduce useful notation for future talks. Uh for this I need a definition.

[01:41:42]So if I have an arbiton stack I define a something what is called the cone construction define y cone.

[01:41:54]So it will be a stack over a1 gm and from now on when I write a1 and gm it always means analytic a1 analytic gm.

[01:42:03]I just don't want to write over QP as a funtor which sends so if you have a map a gon stack Z with a map to A1 mod GM it's the same as giving yourself a a line bundle so that's a map to BGM and a section like possibly zero uh so sending so in other words some kind of generalized cart divisor on Z sending uh the gelfon stack with a cart divisor D on Z.

[01:42:46]So I just write this as a notation but be aware that as I just said this is not really a C divisor in the usual sense like you have line bundle with a section which may not be injective uh and you send this to uh uh to y evaluated on okay so that's what I we call the cone construction and uh it has funny geometry. So, uh slightly strange.

[01:43:19]So, if your base change to A1 mod GM to uh GM mod GN, which is just a point, then this just becomes a point.

[01:43:34]That's basically because this is the locus where you wrote Y. Did you mean Y mean? Thanks for that.

[01:43:45]That's basically because this is a locus where this section is invertible.

[01:43:49]So this divisor will be just empty and so you just get uh just get spark up.

[01:43:57]Uh but over if so uh if you now look at what happens over bgm.

[01:44:04]So over zero the origin of a1.

[01:44:08]So now there is a natural map to y and uh if y admits a cotangent complex uh this will be just given by uh uh now I have to write this correctly.

[01:44:31]This identifies with uh this base change to uh I mean this base change of the current construction with uh some truncation of the tangent complex.

[01:44:44]So if I wrote it correctly, it's you take truncation. So you take the the tangent complex of y just defined as as linear of the cotangent complex and you truncate here and then you shift by one.

[01:45:00]So okay maybe most importantly if y is a smooth say for example a smooth rigid space this tell you that over this uh over bgm what you get here so maybe as notation I call this y hodge and the reason is that because of this if y is a smooth face this will really just be the classifying stack of the tangent complex of y which is yeah just dual component and I call this we call this cone construction I don't know if it's a standard terminology because there is also a relative construction so more generally if you have like a map y to x then you could define a relative cone construction by by taking a so you take the cone construction it's fromtorial and then you observe that there's always a map from X * A1 mod GM to X cone just because there is always a map from like D to Z. So at the level of like evaluating like taking the points this is gives a map from X to X cone and of course you also have everything leaves over I1 GM.

[01:46:21]So you can form this base change. And if I would be like in a situation where this would be if y is a regular closed immersion.

[01:46:36]uh then this would be some kind of deformation to the normal code like generically this is no for this constant generically this is no x so over gm or gm but if you look at the fiber over zero no you really get uh just the normal bundle of y / x and sorry here I forgot some kind of there is also a twist this twist means twist by the invertible like there's a universal line bundle on BGM and I pull it back uh and this gives me also so there is some twist involved.

[01:47:16]Sorry that's maybe a bit too quick but uh I just want to get to the definition uh so I define now the filter the RAM stack.

[01:47:31]So for for this definition it's no important that I start with a gal stack and not with not an not just the arc stack and you define yam plus called the filter dram as uh so you take uh the ram stack of y the dam stack of a1 mod gm And uh so as I said there is a map from Y * A1 mod GM to Y cone. I take the induced map on the RAM stacks and I just base change it to Y cone.

[01:48:21]Okay. So that's certainly a complicated way of writing what this uh filter ramp is doing. But maybe for time reason I leave as a as an exercise just as writing on nil perfect ring what this is doing and see that this maybe on the maybe you need some small condition on the gon stack and check that this if you are know already what the filter the RAM stack is that this will coincide with with the usual definition. uh but how I want you to think about the the right hand side is some kind of relative stack for this map y * a1 gm uh mapping to y con if you know what the relative dam construction is this is exactly this in this particular for this particular map and we will see later as in the analytic prismatization there is a similar behavior except that the source here will be replaced by something more complicated so that's the reason I wanted to present it in this way yeah and so basically yeah theology of this analytic damac is is related to as as you would expect to the much filter the ramology.

[01:49:55]Okay. And so that was maybe remark one and to finish another remark is uh about like what is called yato stacks.

[01:50:09]So there is something a bit surprising about this series of analytic dam stacks. It is that it makes sense analytic dramax by definition makes sense for any arc stack. If you think intuitively what the DAM stack is supposed to be, it's like supposed to be about DM modules which are defined in terms of differential operators and because of what I said at the beginning that there are no like I don't know like in general you would have imagined maybe that this is only reasonable when X is smooth or something like this and then you have to extend uh but really the fact that we work not with algebraic DAM stack but with analytic DAM stack makes it a very topological object so to say. So in the complex case you can also do a similar thing and then the by result of claus and so the analytic dram is just a petty stack which is maybe a case where you can really like yes make more precise this idea that this analytic dram is really a topological construction.

[01:51:09]Uh and so this is nice because this gives us a lot of flexibility. Now we can make sense of the dam stack of very strong objects. So uh an example of this is if you have any arc sack so no not necessarily over QP then you can define it to stack.

[01:51:31]So you take X you take so it leaves over the arc stack of FP which is a final object.

[01:51:39]Uh and then you just take this fiber product with arc stack of QP. So now it's an arc stack over QP and now you can take it stack h sorry and then if you want this yod stack I further so there is a natural forenius on x just because it's a funtor on perfected spaces in characteristic p and so I want to model out by forenus on x and identity on the other factor and then I take the dam so that's something you can know make sense of and as I will recall on Wednesday. Uh this is basically uh the like something called the far contain curve of X as just as a really as a diamond.

[01:52:26]Um and so this is a D of a relative far content curve over X. So shapes on this stack you imagine as some kind of analytic dem modules on this relative far content curves.

[01:52:41]uh yeah and so maybe I don't want to say much more about so the idea is that uh why is this notation so if x is just the arc stack associated to a scheme in character p theology of this stack is is basically computing rigidology that's a work in progress of kin um if no instead x is something like a rigid space say a smooth rigid space partially proper over QP. Uh then what you are computing is this yudoku like at least expectation for now is that this is computing some maybe over convergent version of the yolk of common.

[01:53:28]So they have defined some yudato commology for rigid spaces which maybe smooth rigid spaces which if you have a nice semi model it's just a usual use of special fiber but uh yeah so they have a definition which works for arbitrary smooth rigid spaces and the expectation is that this is a stack which geometricizes you um right and finally you can combine both remarks by defining some kind of filtered version of the yudoko sack. Uh and this will play a role in but I think this will be explained maybe by by Max in his lectures but yeah I wanted to give this examples because this this uh stack will play a role in my discussion of the prismatization of rigid spaces.

[01:54:20]Okay that's that's all for today.

[01:54:36]So you had a theorem that if X Y is smooth then used transmologically smooth. So what what which category was your X and Y living?

[01:54:48]>> Uh they can be arax.

[01:54:51]>> So there's a notion of smoothness. Uh yeah it's just something which locally is like compos like composition of like some ital map to some a relative a space.

[01:55:03]Yeah smooth.

[01:55:16]>> Yes. Well, uh, no, you have to be slightly careful with this just because we work in this buck setting. So, like this, uh, D that I wrote over there is not smooth anymore.

[01:55:33]Uh, yeah. So, it's proper, but it's it's not smooth. So, it's kind of a little bit switched with. So, that's why I always want to restrict to partially proper or work with dagger space.

[01:55:47]But >> so >> so rational localization is not as smooth >> a smooth out of digger things will be okay.

[01:55:55]>> Yeah.

[01:56:01]>> So like if you have some if you have some rigid space let's say buffer then the map given by rational localization is is not smooth.

[01:56:09]>> No it's not. Yeah.

[01:56:14]If you instead say that instead of saying that f is less than or equal to one, if you say that it's less than one, >> say that again.

[01:56:22]>> Mean if you say that f is one.

[01:56:26]>> Yeah. Then then that's good. Yeah.

[01:56:30]>> Like open this would be smooth for example.

[01:56:40]So I guess next talk will be on Wednesday with the announcement location.

[01:56:45]Let's let's just START CATWORTH.