 So, welcome back everyone. So today we're going to be doing a little bit of localization in a certain framework in the setting of solid analytic rings. But first I want to start with a recap of last time. So last time we defined the notion of analytic ring and that's a pair, r with funny triangle. And then the whole pair will be denoted just plain old r and it consists of an r with a funny triangle and what we would want to call the category of r modules where, so this triangle r is a condensed ring commutative and this mod r is a full subcategory of mod r triangle by which I mean condensed r triangle modules satisfying some axioms, some closure axioms. And the first one is that mod r is closed under all limits, co-limits and extensions in particular it should be an abelian subcategory but even more because I'm allowing infinite co-limits and limits here. And the second is a bit funny so if m is in mod r triangle and n is in mod r then we want to require that the internal x's in this category from m to n should also lie in the smaller category for all i. And this condition was necessary to get a good tensor product on this category and on the derived analog as well, I'll make a couple more remarks about that in a second. And then the third condition is that our triangle itself, the unit in this category mod r triangle should lie in mod r. What's about the existence of r joints with chunks instead of r? So there were a couple of technical points that came up last time that were not fully addressed that I'll take the time to address now. So there's a, and one of them was indeed about whether the existence of the left adjoint is automatic or not and there was a claim that it was and it's justified by this theorem of Ademek-Rozitski called reflection principle. So if c is a presentable category, which used to be called locally presentable, but if d is a full subcategory, so it's a version of an adjoint functor theorem, closed under all limits. Now that's the condition under which you might reasonably expect a left adjoint to the inclusion, but there are set theoretic technicalities that get in the way in general. But if you also, plus there exists a regular cardinal kappa such that d is closed under all kappa filtered co-limits. So if kappa is the first infinite cardinal, so I'll if not, then this is just kappa filtered co-limits is just filtered co-limits, but if you make kappa bigger it's a weaker condition that it be closed under kappa filtered co-limits. So but then d is presentable and there exists a left adjoint. Okay, so this is nice, the inclusion. Okay, let's think about it, I'm not about to derive the category. And the infinity category version is proved by Ragnarov-Schlank. Let me make sure I got the name completely correct. Yeah, Raghimov, I'm so sorry, Raghimov, I'm so sorry, Raghimov-Schlank. Yeah, but you have to, there was some circular thing, I've seen one of the discussions. I don't remember exactly that. If you want the presentation under limit, let's say infinity. Let me make another remark concerning a technical point that came up in the last talk. So recall that we were discussing the derived analog of this. And in this context of this definition, we made the definition that we defined the derived category of R as a full subcategory of the derived category of mod R triangle, let's say, let's just write derived category of R triangle. Consists of those M in here, such that on homology, you lie in this abelian category mod R for all I. And we wanted to show that this derived category satisfies the analogs of all of these conditions, in particular, these closure properties under limits and co-limits. And for proving limits, the thing to prove is to prove closure under products. And there was a sticking point that came up that we needed that if, say, M, alpha, alpha and A is a collection of elements of objects in mod R. We needed that if you take, if you view them now in the derived category, concentrated in degree zero and you take the product in the derived category, then it should satisfy this condition. And the subtlety is that infinite products, countable infinite products are exact in our setting of light condensed abelian groups, but arbitrary infinite products are not. So the product functor has right derived functors. And you need that this lies in mod R for all I. Okay, but this can be proved as follows from the axioms. But consider, but no to, so, but set to M to be the direct sum over alpha and A of M alpha. Then this guy is a retract of our I product alpha in AM. Because term wise, it's obviously, this system is a retract of this system here. But this is just the same thing as external x i from direct sum of copies of our triangle to M. Okay, and what about product of infinite complexes, unbounded below complexes? Yeah, the argument given in the last lecture, I mean, this was treated in the last lecture. Because they are, it's like, kind of, it's a positive conflict. So that you can reduce the countable, okay, and do this. Exactly, okay, so that's, and I think also that, I don't know, maybe another one. So if you have in general a groten-dix category and a groten-dix subcategories that closed under, closed under K, K, and filter under direct sum. But then in the derived category, when you consider the object with homology or homology, lies in the subcategories, and this is also presentable. Okay. This, I think, can be shown, but I don't know the reference for this. I don't know. Okay, but this is just an elementary thing. And complex, you can do it in the level of complexes. And I think this can be used to, to give a less, I mean, once you know this, which actually, one can formulate it in terms of every complex. It's kind of a limit of small ones with some bound. Yeah. So this, and then, then probably this can slide this, well. Yeah, I don't think, yeah. We'll talk to this, without using the- I don't think it's necessary to use this. I mean, actually, you can kind of explicitly construct the left adjoin on the derived level just by taking left drive functors of the, the left adjoint you have on the abelian level. Well, maybe it doesn't quite work like that. But you can, well, yeah, I, I, I'm willing to, I'm very much willing to believe that the infinity category version can be, can be avoided. Anyway, let's, Peter. You can question this, if you, you can question this, if you are actually presentable, it goes here, just out in the long, the HII, right, right, right, right, right, yeah, yeah, that's, that's a good argument. Yeah, cuz this is not necessarily the derived category of, of mod R, but what Peter said was that there's a general principle about like presentable categories being closed under limits and the category of categories. As long as you have functors that commute with co-limits and the homology functors commute with co-limits. So, then you can see that this thing has to be presentable. And then the infinity categorical adjoint functor theorem, the more naive version proved by Lurie, would give you the left adjoint. Okay, anyway, enough of that, right, let's move on to some real math. Okay, so. And when you did before without light, you did everything. So, is there a close relation between the notions of the light and the general set of that is sort of a condensed ring in the light? Gives you up in the general and the subcategory gives you subcategory and so on or maybe or it's more twisted. Well, though, the first statement is completely accurate. So light condensed rings embed fully faithfully into all condensed strings. But when it comes to the analytic ring structure, it's a little bit more, it's a little bit more subtle. Yeah, because if you have a light, an analytic ring in the light sense, you've a prior only decided what the free module should be on light condensed sets. Okay, so we also in the previous lecture, we also discussed some examples of analytic ring structures in the solid context. So let's say we have a solid ring. So I use this notation. So it's a condensed ring who's underlying a condensed abelian group happens to be solid. And then we had this subset of power bounded elements and then for any subset R plus contained in the power bounded elements and maybe I'll say that one lies in the subset. We get an analytic ring structure on R. Denoted like this, R triangle R plus solid. And well, as the notation indicates, the underlying condensed ring is just our ring R triangle. And then the category of modules such that if you look at this space of null sequences in M and you have an endomorphism from this to itself given by 1 minus shift times F. This should be an isomorphism for all F in R plus. And let me make a remark, so the condition that 1 is in R plus ensures that every M in mod R is actually solid, is in solid Z. That was actually, so if you plug in F equals 1, that was our definition of solid Z. And then this condition that R plus is contained in R zero. That's what gives that the ring itself is complete, so to speak. So I recall that the interpretation of this second coordinate in the analytic ring structure was that it's specifying some notion of complete modules adapted to whatever situation and you kind of want the ring itself to be complete, otherwise you would complete it. And that's the condition on power bounded elements by definition just translates to saying that the underline ring itself should be complete, which was one of our required axioms. And another remark is that there's no loss of generality. That R plus is an inner really closed subring with containing all the topologically nilpotent elements. Because we saw that if you have a solid analytic ring structure on a solid ring, so meaning an analytic ring structure such that every module over it is solid as an underlying abelian group, then the collection of F for which this is an isomorphism for all M is actually an integrally closed subring containing all the topologically nilpotent elements. So you could always just throw in this, these guys take the subring generated by them and take the integral closure and that will not change the theory. Okay, and then there was an example. So if this is a Huber pair, so recall that that means this is a Huber ring, or that, sorry, excuse me, sorry, I'm gonna rephrase, if, so it's a Huber ring. And those things can always be considered as solid condensed rings just by taking the associated condensed ring. So then the integrally closed subrings, R plus like this are the same thing as open integrally closed subrings like this. And these are exactly the R pluses in Huber's theory. So the general setup we have for an arbitrary solid ring of the possible choices of R plus. When you specialize to the case of a Huber ring, it recovers exactly the choices of R plus you have in Huber's theory. Yes, thank you, yes, let me say that from now on, whenever I say Huber ring, I'll mean complete Huber ring unless otherwise specified. So there is also sometimes people use so-called derived complete things. And if you have a derived complete thing, it also seems to give a condensed stack seeing and then it's also, is it possible to say that this is also a solid? Yes, yes, yes, actually I think I might, I was just considering talking about that. There's a fun story with that that maybe I'll talk about it later in the lecture. Yeah, yeah, but yes, the answer is yes, right. And moreover, in this case, the R plus is actually recovered from, recovered by the analytic ring. It's basically equivalent to the, yeah, so Huber's R R plus, or I should say R triangle R plus is exactly the same thing as this solid analytic ring structure. So in the general case, I didn't quite claim that if you start with an, I didn't claim and it's probably not true that if you start with an integrally close sub-ring satisfying these conditions and you form that theory, there might for some other reasons also be other things that F that satisfy this property for all your M possibly. But in the Huber case, you can show that no there aren't. No, we didn't do it or we didn't do it. Well, maybe we didn't do it last lecture, but it was done in a previous lecture. So the argument is in the analytic.pdf. I mean the second to last lecture, I think. Okay, questions? So now we're going to discuss localization. So let me make a remark, so localization. So I'll start with a remark, which might be a bit shocking at first glance, but it's actually trivial. So if R is a solid ring, so and then we have R plus satisfying these conditions. And again, you can feel free to assume it's an integrally close sub-ring containing the topologically no potent elements. Note that this condition defining the analytic ring structure is just a condition that you're imposing for all F in R plus. And R plus was by definition a subset of the underlying discrete ring R. So all the data that you're using to define the analytic ring structure actually already appears at the discrete level. So then get another pair just with the same R plus. And the power bounded elements in the discrete case are just all the elements. So certainly it's still going to be power bounded inside there. And this observation shows that if you're interested in solid modules over your original R, R plus solid. Well, you can take the ones over where you have a discrete ring. And then that already has all of the information about the analytic ring structure and all that remains is to observe that R will be a commutative algebra object in here. And you just kind of abstractly take R modules in this Abelian category. So it's important to note that since we're doing condensed modules, even when you have a discrete ring, you have a huge amount of new modules besides the discrete modules you can consider. And in particular, over this discrete condensed ring, you have R, the honest condensed ring. And the theory for an arbitrary R is actually base changed from the discrete case in this completely naive way. Okay, so we're going to discuss localization. So how these categories glue, but it's actually going to be sufficient to treat the discrete case. Because if you understand how this category glues, then you can just put the R module structure on top of that. And you'll understand how this category glues. And I want to stress from the beginning that I'm talking just about one kind of example of gluing. I'm not claiming this is the most general, but it is nonetheless quite general. But it's just a certain framework for gluing, you can call it. So now let me make an analogy. So, well, so we're going to be in the world of discrete rings now. But so if R is a commutative ring, then we have its usual derived category of R modules. So this is, for emphasis, I'll put discrete R modules. This localizes on over, this is a risky spectrum of R. And I'll say more precisely what that means in a second, but just, and what is the spec R? Well, it's the, yeah. So spec R, the set of prime ideals. And there's a basis of quasi-compact opens closed under finite intersection. These are the so-called distinguished opens. Something called principal. Principal opens? Well, it depends on, I don't know, I've seen the principal in some references, but it is not, I'm not sure what is. What do you prefer? I don't know because actually myself I didn't remember. I remember sometimes I said basic open because I did not know, but then I saw in some textbooks. It was principal, but I don't know how standard it is. Are you OK with distinguished open? OK. Distinguished opens, so let's say UF. Those are set of prime ideals, P such that F is not in P, so F maps to something non-zero in the residue field. And there's a structure sheaf, and its value on this distinguished open is, so, this ring R1 over F, localized at F, and it's kind of important to note that neither UF nor R1 over F kind of determine F, right? But they do determine each other, and in fact this UF actually gets identified with spec of R1 over F, and this matches up distinguished opens. And then, so if this is an inclusion of distinguished opens, sorry, sorry, sorry, sorry. So if U is distinguished open, to this we can assign the derived category of the value of the structure sheaf on U, and if U is, if V is contained in U, then you get a base change functor, and then the theorem, completely classical I guess, is that this sheaf, or this pre-sheaf, is a sheaf. No, that's true, yeah. So it's a sheaf of infinity categories. So, well, in this setting, you also have a sheaf of abelian categories. I could have told the same story with the mod, the usual abelian category. Yeah, yeah, yeah, I mean, yeah. OK, but this isn't, but also there is a certainty of a sheaf of hyper-cover versus shea-cover. Yes, that's right. So this is proved in Luri somewhere, or what is it? I assume it's proved in Luri somewhere, yeah. But people sometimes don't know it's, OK, it's kind of. In this language, I'm sure it's Luri, yeah. So I'll explain the argument in a second. But, right, so let me, so in this case, we also have a sheaf on the level of abelian categories and even more as Gebra says, a hyper-sheaf. But never mind that. I just want to warn you that in the setting I'm about to discuss here, we will only get a sheaf on the level of derived categories. We won't get a sheaf on the level of abelian categories. And the basic problem is one that came up in my previous lecture. So in this situation, the localization maps are flat. So O of U going to O of V is a flat map. Well, it's just a localization. And so if you have a derived statement, then using the flatness of localization, it's not too difficult to deduce an abelian statement. But in the setting I'm about to discuss, these localization maps will not in general be flat. And I kind of, so one example of such a localization map is going to be this localization from the affine to the closed-unit disk, which was this T solidification, or ZT solidification. And I already mentioned that it's not exactly T-exact. There's a discrepancy by one. And that actually obstructs the sheaf condition on the abelian level. So this is called an electrically related statement. So our statement is that the derived category of the ring is the same as the quasi-coherent derived category of spec of the ring. I think in general, the southern boundary, this is for the primary stacks project. And then there is another statement about the sheaf property, which I also don't know the reference, but I think that people, that it is on this line of Lore is saying that if you have any ring topos, then you can associate to every object in the site or the topos, the derived category of this U, of U. And this should also be a hyper sheaf. But is it, what do you, is it, so this, this is in Uluri also? So, it's possible, what they said should be correct, I think, is it? I'm sorry, my brain is a little, not working very well right now. I actually zoned out while you were talking, my apologies. Any topos, any, you associate to any object U, the derived category of the topos restricted to U with values in the sheaf, then this is a hyper sheaf. Hyper sheaf, no, sheaf, yeah. Hyper, and also a hyper sheaf. Well, why would it automatically be a hyper sheaf? No. No, I think I was able to do it in some classical, more classical formulation, but it's, it's a, is it, so what do you know about this? Oh yeah, the usual derived category, maybe it is a hyper sheaf, yeah, okay. I don't know, I don't know, yeah. You don't know the reference, yeah. I mean, I don't know the statement. I guess now that I think about it, it sounds plausible, I mean the hyper sheaf, but there's a short, certainly Luri approves it's a sheaf, yeah. And hyper sheaf maybe, I don't think. Goes it in one of his books. Yeah, I'm sure, yeah. There is also something that I saw that you mentioned, I mean, in some texts that I found on the internet, instead of looking at the derived category, you can look at functors, infinity functors, your sheafs, with values in D of a billion, yeah. And you claim that this is the same as the derived category of the ring. Oh, hyper sheafs, hyper sheafs, yeah. If you do hyper sheafs with values in D of a billion groups, that's the same thing as the derived category of the category of sheaves of a billion groups, yeah. Hi, this is not the problem. I assume it's somewhere in Luri, yeah. Now, I'd like to move on, so maybe we have this discussion at another time. Okay, so that's one part of the analogy, kind of, and the second part is, so now we have R, R plus, a discrete Hooper pair. So that just means R is an ordinary commutative ring and R plus is an integrally closed subring. Well, then we've assigned to this D, R, R plus, solid, and the claim is that this localizes on something else on the valueative spectrum of this pair, R, R plus. Okay, so what is this? So that was the set of prime ideals and the kind of purpose of a prime ideal in this setting is to let you know where functions vanish or don't vanish. So kind of, you could think of it that way, so kind of a binary condition, whether you're zero or non-zero, and in the valueative spectrum, you are allowed some more refined information, not just information about whether a given function vanishes or doesn't, but given two functions, you can ask whether one is bigger than the other. And the way you can measure that is by means of evaluation, so it's a function from R to gamma union zero, so this is an Abelian group written multiplicativity, and then there are axioms, so multiplicativity, V F G equals V F V G, and there is the obvious rule about zero times anything equals zero. There is the non-Archimedean condition, V F plus G is less than or equal to maximum of V F. V G, ordered, thank you, thank you. Ordered, but the order is reversed when you pass on the multiplicative. Yes, yes, yes, yes, yes, but thank you, yes. V of zero equals zero and V of one equals one, and then we involve the subring R plus, so we ask that all of these kind of be integral with respect to the valuation, so maybe V F for all F in R plus, and then there's an equivalence relation on valuations because you could always, for example, enlarge the order to be in group in some arbitrary way without really changing what's going on, and one way to describe this equivalence relation is that, so V is equivalent to W if and only if for all F and G in R, we have V of F less than or equal to V of G if and only if W of F less than or equal to W of G, so in other words, that what's really important about evaluation is this binary relation on functions, which is testing whether one is bigger than the other. Is it the same, in this case, is the same as SPA in this case? Oh, it's also SPA, yeah, yeah, yeah. Yeah, because the continuity condition is vacuous when the ring is discreet, yeah. Okay, that's a bit of a mouthful, you've never seen it before. There's another perspective on these things that maybe explains exactly in what sense it's bigger than spec R is. This is also the same thing as just a pair of a prime ideal and then sort of evaluation subring, so evaluation domain in the residue field. So you have the information which records whether your function vanishes or not and then you have this extra information saying when a function or a fraction of functions with a denominator doesn't vanish, should be less than or equal to one, basically, and that's the same thing as this. Yeah, thank you, yes, yes, yes, yes, yes, yeah. Off, you can see where this is right off, okay. Yeah. Containing R plus. Thank you, yes, containing R plus. Reinterpreting R plus. Yeah, thank you, yes, R plus, right, okay. So the point here is that now we have a much bigger category and there's more flexibility for how to localize and it connects with this classical discussion of valuations. So if you've never seen this before, then you can look, for example, at the rational numbers or something, then maybe you know the classification of valuations. There's the trivial valuation, which I guess corresponds to equality here. For every prime ideal you have the trivial valuation where it's zero if your element lies in the prime ideal and one otherwise, but then also for every prime P you have a Patic valuation. So you have generic point of spec Z or the special points of spec Z but then you have these things in between, which kind of are nearby P but not equal to P, these Patic valuations. Oh sorry, I was talking about Z not Q. And then, but then the fact that you can classify those is kind of a little bit misleading because once you add an extra variable then all of a sudden things explode and there's many different kinds of valuations basically because in a surface you can have lots of different kinds of curves passing through a given point. You have valuations of so-called higher rank which introduce additional complications into the theory. So I'm not gonna go too much into this but yeah, so I'll stick to mostly formal aspects for now. Okay, so let's continue the table of analogies. So there we add spec R and we have this particularly nice basis for the topology, quasi-compact closed under finite intersections and each of them was also of the same form as the global guy just for a different input datum, so R one over F. And we have the same thing here so we have a basis of quasi-compact opens closed under finite intersection and these are called the rational opens in this case and they depend on a choice of some elements in your ring. So you choose arbitrarily F one, F n and G inside your ring then you can form this thing and what is it? It's the set of those valuations V satisfying all of these conditions such that moreover, so V of G is non-zero and V of F I is less than or equal to V of G for all I. So in some sense it lives inside the distinguished open, the Zariski open given by just deciding G should be non-zero and then we use this extra flexibility of we can also impose inequalities so we're shrinking this Zariski open a little bit using some inequalities and we still get an open subset. Okay, continuing. So there's a structure, there actually, there's a structure sheaf but actually there are structure sheaves. So on this F one, F n over G you have one thing which just takes the algebraic Zariski localization but then you also get a choice of integral elements and that you get by, it's gonna be a subring of here and you get it by taking the integral elements you had before or rather their image in there and then looking also at these elements F one over G, F n over G and then that might not be integrally closed so you take the integral closure. So basically you just look at all of the elements which the valuations in your open subset think should be less than or equal to one. So you've kind of already have it for this by fiat and you've forced it for these and then the collection of those things is an integrally closed subring so you have it for all of those guys. And then again you have this nice recursive property that U of F one, F n over G is just the same thing as the valued of spectrum of O U, O plus U and this matches up rational opens. And here is another place you can see the kind of necessity of including the data of this R plus in the general theory because if you, I mean you could have said, okay well I want a bigger space, I'll leave out this condition, why should I ask for it? But then you define these rational open subsets and they will no longer be of the form SPV R for some ring R because you've forced certain elements, these elements F I over G to be less than or equal to one and the abstract ring R one over G doesn't remember that. So if you want this condition that the rational opens themselves should be of the same form as the global object then it's absolutely necessary to include this extra data of R plus from the start. So of course this, okay, so basically so this implies that when rational open including a data you say that this what is the statement matches up rational opens? What do you mean there? I mean that the, so there is a continuous map from this to this and it induces a bijection, the pullback say induces a bijection between rational opens in here which are contained in this open subset and rational opens in here. Okay, I know this, rational open, rational open. Okay, I'm not good, okay. Okay, and then the theorem, well maybe I should say again, okay, so U a rational open, to that we can attach D of OU O plus U solid. And then if U is contained in V inclusion then we get the pullback map. Well there's a, in fact there's a, it comes from a map of analytic rings OU plus or plus U solid to OV O plus V solid. In the sense of the previous lecture so we have a map of condensed rings which is just in this case a map of discrete rings such that if you have a complete module here then when you restrict scalars it's also complete here. So that's the kind of forgetful functor and then that always has a left of joint which is this base change functor and explicitly you get it by taking your module here, abstractly tensoring up from this ring to this ring and then re-completing in this theory here as this base change functor. And then the theorem, well I've kind of run out of space but maybe I'll put this pre-sheaf, that one over there is a sheaf infinity categories and I'll put the warning that this is not true on a Boolean level in contrast to classical case. These pullback functors are not T exact in general because the pullback involves solidification, a T solidification which as I said is not a flat operation. Does it have bounded homological dimension or not? Yes it does, yeah. So it'll be bounded by N. The solidification is bounded by N. I mean the homology is zero up above N. Yeah, yeah. Okay, so I think we'll take a five minute break before I get to the proofs. Proof? Is it hyper-complete or not? Probably not in general. But there's an abstract result that if your space has finite curl dimension then hyper-completeness is automatic, this is sometimes useful. So if I start with a solid ring and then I can take it on the line inside and which I'd always discreet about and we mentioned it's the same as modulus of z. And is there cases it's actually identotent of this thing? I guess identotent algebra. And there, no I don't think so. So the things these tend to, I mean so for example like, I don't know, z. So we showed that z power series t is identotent over z polynomial t. But I mean this discreet ring is gonna be way too big. This is not gonna be identotent. There's no extra reason why this should be. I mean I didn't think about it carefully but I would assume the answer is no. What's that? Right, right, right. So let's, so I've stated the theorem and now I wanna explain the proof. But to motivate it I'll give a certain proof of this classical theorem here. And so there's many, actually in this classical case there's many different possible arguments for this. Especially because these localizations are flat there's lots of flexibility in how you set things up. But I wanna describe a particular argument for this claim here which will kind of translate over without too much difficulty to this case here. So, but maybe I erase some boards. I don't know. There was maybe one remark that one can make in both settings that I forgot to make. So I said, so I defined this pre-chief of infinity categories on this irrational opens. Okay, not every open subset is rational. They're just a basis for the topology but there's this general result. And when you have a basis for the topology closed under finite intersections that condition being actually necessary in the infinity context. Then a sheaf on the basis uniquely extends to a sheaf on the whole space and kind of the naive manner of have an arbitrary open. You take limits. Yeah, so we're only describing this sheaf of categories on the rational opens but after the fact you get also a category attached to an arbitrary open whether or not it's rational. Okay, so proof of classical theorem. So meaning descent, zariski descent, you could call it for D of R. So you can start with the, so we're interested in this, you could say this site of distinguished opens inside spec R. So we have this, the open cover topology. And there is, and so we kind of have an understanding of what it means to have a cover. The distinguished opens cover or a ring, if a cover spec of a ring, if and only if you're inverting some elements and those elements should generate the unit ideal. But you can do a series of reductions actually which will show you that you only need to check the sheaf condition for a very specific, a very specific example of such a situation. So the lemma is that this Groton-Dictopology is generated by covers of the following form. So you take U spec R, a distinguished open. You take an element of the structure sheaf on U and you form the cover which is U of F and U of one minus F covering U. So this is a very simple example of two elements which generate the unit ideal inside this ring. And the claim is if you wanna check something as a sheaf, you only need to check the sheaf condition in this one specific situation. Yeah, this was originally in Quilin's proof of the, okay, it doesn't, I knew it's easy, but it was, there was something of Quilin and it proved the circumjection was true. Okay. It reduces to, I mean, you want to prove that if you have some ventral bundle or a fine space over a ring which is driven over a local ring then it is extended from the ring and it did it by reducing to this and it was a bit tricky. Yeah, Quilin's a clever guy. So let's give the proof. So, well, as I said, you know, we know you can describe algebraically the covers. So if you, in general, the covers would be described like this. You take F1 in Fn in an O of U generating the unit ideal. So such that there exists X1, Xn in O of U with X1, F1 plus dot the dot plus Xn, Fn equals one. And the general cover is the U of FI. FI. Ooh, darn it. I can't believe I didn't think of that. We'll not be okay because you cannot generate the empty cover. The empty set's from the empty sets. Damn it. I can't believe I forgot to check those things. Yeah. Thank you. I should know better by now, but plus empty cover of empty set. Okay. Checking the sheaf condition there just means you check that the value of your sheaf on the empty set is the terminal object in the category that is the target of your sheaf. Okay. So that usually can be done without much difficulty. Okay. Any, is there a question or comment from Bonn? No, okay. But note that this cover here is, but this cover is refined by another cover where you take FI times XI. So this is a smaller distinguished open and those still generate the unit ideal because of the same expression. So then we can assume just that F1 plus dot the dot plus Fn is equal to one. And then you can do an induction on N, so you can then induct on N. So let's say for example you had F1 plus F2 plus F3 equals one and you wanna check the sheaf condition, then because you're assuming the sheaf condition for covers of this form, you can localize to U of F1 plus F2 and U of F3 and it's enough to check the sheaf condition when you do these localizations. But when you localize to here, then F1 plus F2 is equal to a unit and by dividing by the, or multiplying by the inverse of the unit, you reduce to that case there. And then when F3 is a unit, well, there's something similar, right? First there is a statement about generation Ah, sorry, wait. Learning to probably containing this and then there is another statement which is probably loaded under some conditions that it's enough to check the sheaf condition on generators or on covers to generate the quality and the best changes. Was assuming that seems like the intersections exist. Yes, exactly. But yeah, I chose these collection to be compatible under base change. So that, yeah. So, I mean this collection here is closed under base change. So that is an important technical point when you get into making this argument completely precise but I set it up so that it's true. Oh yeah, and sorry, when you restrict your cover to this, the cover is split. I mean, because this was one of your covering elements. So the sheaf condition is automatic here and the sheaf condition here follows by induction. So that's basically the argument, fairly simple. Okay. So, but what about this case here? Okay, so what are we trying to show? So in now, so checking the sheaf condition for, well, I can, it's a distinguished open in an element F but there's no, now there's no loss of generality in assuming U equals spec R. So, well, so what is the sheaf condition in this case? So in this case we have, well, we have just two elements and then we have their intersection. So the sheaf condition says that if you look at the derived category of R and then the derived category of R one over F, the derived category of R one over one minus F and then the derived category on their intersection which is you invert both. This should be a pullback of infinity categories. And now let me make a pause because I didn't spend much time discussing what this notion means like sheaf of infinity categories, limit of infinity categories and so on but I can make it completely, so to speak, elementary in this case of these pullbacks so you get a feel for what the claim is. So, yeah, so claiming that this is a pullback of infinity categories, what does it mean concretely? Well, you have a, it means the functor from D of R to the pullback category, one minus F should be an equivalence of categories. So, now how to think of this infinity category? Well, so you give yourself an object in the derived category here and an object in the derived category here and then you give yourself extra data of an isomorphism between them here. And that's, but it's not an isomorphism in the usual derived category, it's an isomorphism in some infinity version. So you could imagine for example if this is represented by a complex of projective objects, this is represented by a complex of projective objects then you'd actually wanna give a chain homotopy equivalence between their images there, let me say they're bounded above just for simplicity and then you make an infinity category out of that so you find some notion of chain homotopy there and so on. Right, so then what is essential surjectivity mean? It means you can glue in the derived category. If you have a chain complex here, a chain complex here and an explicit identification between them, so maybe you choose some quasi isomorphic models and make a chain homotopy equivalence between them, then that collection of data uniquely comes from an element here up to quasi isomorphism. So the point being that you actually have to specify the data of the chain homotopy equivalence here in order to get the well-defined object there. That's the essential surjectivity. The fully faithfulness says something else. It says that if you have two objects here and you wanna know the hams between them, so you can think of calculating X groups, for example, so the r-homs between two objects here, you can get it by you base change here and you take r-homs, you base change here and you take r-homs, here and you take r-homs and then you do a homotopy pullback of those complexes for r-homs you have there, which is the same thing as like a shift of a mapping cone of some direct sum of these two mapping to that. But is it equivalent to like the drag category of the diagrams that you can say in a billion-level and a module-level diagram is just giving a module over every ring and maps and then, not necessarily without isomorphism, you take the drag category with something, posing some condition of homology. Maybe it sounds reasonable, but I'm not entirely sure. Yeah, yeah, that's certainly not how I think about it, but okay. Okay, so that's kind of how to think about this result. It's, it lets you glue objects that are defined locally in a derived sense, but it also lets you do global, if you can do global X calculations by localizing. Okay. But how do you now, how do you formally prove such a statement? So note that, so the proof, so note each base change functor has a right of joint which is just a forgetful functor. So from derived category of R1 over F to derived category of R. And then it, so that's for each of the individual maps in this diagram, but then it actually follows formally that this functor also has a right of joint does too. And you can explicitly describe what this right of joint is. So it sends, if you have a pair, so MN alpha, so alpha is an isomorphism, so M is a module here, N is a module here, and alpha is an isomorphism between their common base changes. Then you just take, you apply the right of joints to each of these objects, and then you take a limit. So you just take M cross over N with M1 over F, which is the same thing as N, one over one minus F. Yeah, you have to open up Luri and find the precise version and, but it works. Yeah. Okay. So the thing to be used to cover by two opponents, is just to have the easiest diagram. Yes, yes, yes, yes, yes. In principle, you could also do this argument without doing a reduction. That's correct, but it's certainly easier to talk about. Okay, because there are finitely many intersecting, just you intersect the particular, like alternate exactly, and it's fine like maybe. Yeah, I think in the, once you get to this statement we're trying to prove with the value of the spectrum, then you really probably don't wanna, well I don't know, maybe, you could probably organize it cleverly, but I think doing the reductions makes it much easier. Okay, right, oh, so, and this is good news. I mean, this is the great thing about proving something as a sheaf of categories. It's like, you have an automatic candidate for the inverse, it's some right of joint. So you have a function you wanna prove is an equivalence, you have a right of joint, that means you have a unit and a co-unit, you need to check or isomorphisms. So then you need to check. So one of them will be a map in this category and one of them will be a map in this category. So, for example, for the unit you need that if M is in D of R, you need that M, M one over F, M one over one minus F, M one over F one minus F. This is a pullback. So that would be the claim that the unit of the adjunction is an isomorphism. And so this follows from the statement that if you have any element, let me call it N in D of R such that N one over F equals N one over one minus F equals zero, then N equals zero. So you wanna test something as a homotopy pullback, you can measure the difference between this and the homotopy pullback by a mapping cone. And that object you can check with this, will satisfy this condition and then you wanna conclude that the object itself is zero. And then, well, this is, well, this is, for example, now you could reduce to the Abelian level because an object is zero if and only if its homology is zero and these are localizations that are flat. So it reduces to the same claim on the Abelian level which is kind of very easy to check. And then that was for the unit and the co-unit is a, it actually reduces to the exact same claim again. But so that's kind of the proof. But I wanna point out what's formally used. Okay, so we're using localization by F somehow and this right adjoint commute that is. I'm gonna say it, I'm gonna say it over. So each base change is a localization. So the right adjoint is fully faithful. The localizations commute with each other. Just a sec, I'll say what it means. It means that if you take something which is local with respect to say this operation, so something on which one minus FX invertibly and then you invert F on that, it will one minus F will still act invertibly. These are things which are all just obvious in this context but I wanna point out exactly what's being used in this argument. And the third condition was that yeah, if M maps to zero on each element of cover then M equals zero. So if you have a pullback diagram satisfying these three properties, then, I mean, yeah. Then you're gonna automatically get this, I mean a square like this satisfying these three properties then you're automatically gonna get a pullback. Okay, so then the proof of the solid analog. So implicitly the F you use the DR when you invert F one minus F is some of the intersection of the two. Right, right. Yes, yes, yes. Yeah, you wanna know that when you, yeah, exactly. You wanna know that when you pass the right adjoints this thing is just the intersection of those two things as well. Yeah, maybe I should have added that to the list. Yeah, thanks. Yeah. Well. Well, we have to understand it in a suitable way. I guess you're right. Yeah, and of course we have to know the language of lawry to make it precise. Yes. So it's not balanced well as it may be. Well, I mean, the right adjoints are fully faithful so it really is just kind of an object wise condition you could say just, yeah. Okay, so what's the analog? So again, we have the site of rational opens, U in this value of spectrum and the Grondick topology open covers. And then we have a lemma that topology is generated by. So the empty cover of the empty set and for all rational opens and all F in OR. Now we have to take care of two different kinds of covers. So we have UF1 and U1F cover. Cover you? Read the geometry because you have those two, yeah. Right, so the way you read this as well. You read this F less than or equal to one. So F should be integral and the way you read this is that F is non-zero but even more it's F is bigger than or equal to one so it's well away from zero. It's a compliment of the open unit disk. And U, so then yeah, one over F U and U1 over one minus F. So this is again, so again this is where F is non-zero and you require F to be bigger than or equal to one and similarly here. So this is actually a refinement of the Zariski cover we had previously where you just inverted F and one minus F and that was a cover. This is a smaller refinement of that which still covers. And the last thing I think one, it's enough to do it when F is in R plus. Okay, I don't think that will be helpful but that's nice to know. Yeah, it's like in read the geometry, Tate's original word, very good acyclicity theorem. Yeah. It reduces to, it didn't have rational domains but he has these two types of covers for which you can prove acyclicity and turns out that it generalizes to the other case. Yeah, I mean I don't know if maybe in the Tate setting that one can arrange that F is. No, that was there F is in unit because you have the, you call them Weierstrass and Lorentz, Weierstrass, no, not him, maybe some other people did it. Anyway, then the Weierstrass is just F in A zero. In that case, A zero is A plus. We didn't know about plus initially but I think that in Hubert that, okay, then I also wrote some letters to other people sometimes to explain something. Okay. So I will not give the argument for this, it's, it uses, well, it's just, it's a bit more complicated because, well, the value of spectrum is more complicated than spec but the idea is basically, well, the idea is somewhat similar you could say but it's actually a somewhat complicated argument, so. But it's, There is maybe a statement that it's enough to have a rational, that is you have a rush F1, FN generating the unit ideal and then you, it's enough to check for those. Yes, yes, yes, yes. You can do some little bit of work to reduce to this. Exactly, exactly, exactly. Yes, yes, so Hubert, so maybe I'll sketch so based on what Gabber said, so Hubert shows every cover is refined by one of the following form. So take F1, FN generating the unit ideal, so. And then you look at U, F1 up to FN but then you leave out FI and you put it on the bottom instead. So in the collection of these, I and I. So again, it's a refinement of the usual Zariski cover you get when F1 through FN generate the unit ideal but you can check just on valuations that it still covers the value of spectrum. And then you do some kind of, you would do something similar to what we did previously. You have with this here, you can, using these covers which refine all Zariski covers, you can assume one of the elements is equal to one. Anyway, you keep playing and playing and playing and eventually you get the desired thing. Okay, right, so now what do we reduce to? Analogous to there, so if R, R plus discrete Hubert pair and we take F in R, then we need that, right, so there's two different kinds of covers. There is the F over one and one over F and then, yeah, so I'll do that one first. Sorry, R and then, okay, so solid, solid, solid. We need that and we need the other one so we need that as well. Okay, so we're gonna basically just verify that all of these conditions hold. So what is this? So there's only really two types of covers here if you look at it. There's the cover U F over one and then there's the cover U one over F. This is of the form U one over F where F is just one minus F, so if F in R, so what is this D, R, R plus goes to D, R one over, sorry, let's say R, R plus to join F, integrally closed. This is just solidification, T solidification for Z, T goes to R, T goes to F. So by definition, so these are both analytic ring structures on the same ring, right? And the only difference is here we've enforced extra conditions of, it's just by definition, so here we have for all, everything in R plus, your T solid with respect to that variable. And here, and if you enforce this, then you also have that condition, not just for R plus but for F, but then we said you automatically then get it for the integrally closed subring generated by those, so then you exactly get this condition here. So the category that you're getting here is exactly this condition here and the left adjoint is just the T solidification. And recall that this was given by some R-hom. Okay, that was the first type. At least you need to invert F at first. Sorry? Do you at least invert F and then you pass out? No, I'm not doing this, sorry, I'm doing this one here. Yeah, so sorry, thank you. So this is U of F over one and then there's the one where U of one over F. So that gives DRR plus goes to DRR one over F and then R plus one over F and we're closed. So what is this? This is, so first invert T, our first invert F, then T solidify for ZT, for ZT, T goes to one over F, R one over F. So the modules here are full subcategory of R one over F modules, which is a full subcategory of R modules and the condition is FX invertibly and you have this extra thing that's supposed to be solid. But now let me make a remark about that second situation. So I claim that that whole process inverting F and then solidifying with respect to one over F, so that base change there is also described by just an R-home. So now I'm gonna take ZT mapping to R with T going to F, not one over F, and I take R-home over ZT, so Z power series T modulo ZT shifted by minus one there. So IE, so based on the formalism from two lectures ago, this is the localization which kills the idempotent algebra, idempotent object, Z power series T in mod, in DZT. Z solid. So we had ZT modules and solid Z modules. We said that this was idempotent. And I also said that when you have an idempotent algebra, then when you can kill it by just taking the mapping cone, or the homotopy fiber of the unit map hitting that thing and doing this R-hum formula. DZTZ, this is what you call D. This was, yeah, this would be the same thing as D of modules over a Z bracket T in solid Z. That's kind of, this is kind of the new notation fitting it in the general framework. Because before for Uber, you wrote D, A, D-R, R. Plus, you did not write solid. No, no, you had an integrally closed R, okay, you wrote D. So the D. So let me explain what the point is here. So this localization is supposed to be given by first inverting F and then doing the solidification. But again, this solidification, yeah, but the first claim is that this functor already inverts F. So if you have something F torsion, it's gonna be killed by this. And the reason is you're killing this whole guy, and therefore in particular, you're killing any module over this guy. But everything T torsion is a module over Z power series T. So this automatically inverts F because anything T torsion is a Z power series T module. So if you have a solid abelian group, which is a filtered co-limit of things killed by powers of T, then it is a ZT module. That's just a condition. You can, to check it, it's a condition closed under limits and co-limits. So you can reduce to checking for something which is uniformly killed by some power of T, but then it's obviously a Z power series T module because it's a module over the truncated power series ring. So then it would be the same thing to write this formula where you invert T, but then if you, that's just after a change of variables, it's exactly the same thing as T solidification as described by the, or T inverse solidification as described by the previous formula. So inverting F and then solidifying one over F is just the same thing as doing this here, okay? So basically all you need to check now, if you look at those conditions, most of them we already know. So it's a localization, kind of by construction. The localizations commute, that's because they're both given by arhoming out of some object. And any two functors are arhoming out of an object commute with each other just because the tensor product by a junction in the tensor product being commutative. And then what does this translate to in terms of these item potent algebras which determine these localization functors? So three translates to, well it's a different condition in, so four. It's just a, it translates to a simple condition on these item potent algebras here. So there's that one and then there's that one. It's that if you take Z power series T and tensor it in solid Z modules over ZT with Z Laurent series T inverse, you get zero. So if you have something that dies on arhome out of this and dies on arhome out of that, then by messing around you will, using this condition conclude that it just has to be zero. And this is a, you can use the geometric series to see that this is zero. Yeah, so what's the interpretation here by the way? Remember like, so this, you can think of this as localized away from the open unit disk. And this was localized to the closed unit disk or away from the open unit disk centered at infinity. And the reason those two cover intuitively is because if you take the open unit disk and the closed unit disk, then, or sorry, if you take the, sorry, if you take the closed unit disk centered at infinity and the closed unit disk centered at zero, then those union is the whole space. But in terms of the complements, that's saying if you take the open unit disk and the open unit disk at infinity, then they don't intersect. And that's exactly a, this is the algebraic translation of that fact. And then similarly, for the second kind of cover, you need the Z power series T tensor over ZT, Z power series one minus T. You need this to be zero. But again, this is just Z power series T U and then one minus U plus T. And that's also zero for the same reason. And again, the interpretation is that the open unit disk centered at zero intersect the open unit disk centered at one. They don't intersect. And that sounds strange until you remember what we're doing on our comedian geometry and then it sounds reasonable again. Okay. All right. So I wanna finish with just a couple of remarks. So, so a remark. So there's the corollary. So that if R is any solid ring and then R plus any one, no, no, no, sure. So the, so now you have to be a little bit careful. So you go, I could say that the DR R plus solid localizes on the value of spectrum of this discrete ring. But be careful. Namely, so there is something that is formal which is what I said that, so on the global sections, that this thing is just R modules in the category we assigned to this discrete Hooper pair. But if you then wanna get a sheaf, so then you send you a rational open, you have to send that to modules, R modules in this D of U discrete. Maybe I wanna say, so, oh, for the discrete ring. So this then recovered topology in the demo, non-discreet, okay. Yeah. So, so the, and in particular, so what is the unit object in this category? So you take R and then you invert G and then you derive solidify with respect to F I, all the F I over Gs. So necessarily when you do this object for a completely general solid ring, you're gonna end up with some derived phenomena here. And we'll, I was hoping to get to it today, but we'll probably discuss exactly how that happens later. And I wanna make, but I wanna also make another remark, which is that if, so in fact, this sheaf, so DR plus actually lives over, so it localizes on this big topological space for the discrete ring, but it actually lies over a much smaller subset. So the closed subset, so let's say spoof R star R plus R. If you remember the set of topologically nilpotent elements, then here you add the condition that if F here is topologically nilpotent, then you want the valuation to be strictly less than one. So for an individual F, that's a closed subset and then it's a big intersection of such things. That's a closed subset of this topological space. And my claim is just that if you take this sheaf of categories and you restrict it to the open complement, you just get zero. So it's really living over this closed subset here. On the other hand, so Huber studies, Huber considers spa R R plus, which is the continuous valuations, which are less than or equal to one on R plus. And that's the same thing as, so subset of this, it's a potentially smaller subset, generally smaller subset, such that satisfying a stronger condition, such that if you're topologically nilpotent, then for all gamma in gamma, there exists an N in N, such that the valuation of F to the N is less than gamma. Okay, so there's some subtlety here that the space that Huber localizes over is actually smaller than the space that we localize over. But, yeah, so the distinction only occurs for higher rank valuations, but I shouldn't say only because in higher dimensions, higher rank valuations are just everywhere. So there's actually a huge difference. But Huber shows, and I should say that this, yeah, continuing the remarks of this DR R plus does not live over the subset, spa R R plus, R R plus. In the same, I mean, I claimed here that the sheaf here becomes zero when you restrict to the complement of this subset. That's not what's happening here, but there does exist a map of a traction. Huh, what's up? Peter? Oh, what? Oh, thank you, thank you. Yeah, does not live over the subset, but there is a retraction like this, which is actually a quotient. Oh, sorry, R circles, yeah. Which is a realizing. So by definition, it was a subspace, but you can actually realize it as a quotient. And then you do get a sheaf of categories on spa. And that's the correct way to, so to speak, get a sheaf of categories on Huber's topological space. And it's the retraction that's kind of the good map in the sense that this is the quasi-compact map. So yeah, we'll probably, so in general, you get more flexibility for localization using this picture than with Huber's picture. And the kinds of extra things you get are something that we already, for example, the things we already discussed, something like this, so-called functions on the closed unit disk, will arise from the structure sheaf in this general setting, but doesn't arise from the structure sheaf in Huber's setting. And you can analyze these things, but I think I've now said enough. Thank you for your attention. So as far as I remember in this retraction context, so actually both are spectral spaces, but the inclusion is not spectral. In general, the retraction is spectral. Moreover, it is such that when you have a sheaf on the bigger thing, then the direct image by the retraction is the same as the restriction to the subspace. For an arbitrary sheaf? No. No. I just further remember the- Ah, yes, yeah, so let me make the- Oh, so it is a nice situation where in particular the higher direct images for the retraction are zero because of this, and now you are modeling with sheaves in this infinity state, so probably the same thing we learned, but not- Yeah, so, yeah. So there's a basis of rational opens here. You can actually parametrize it by similar data, but with an extra condition that these things generate an open ideal. And then if you pull those rational opens back here, you get exactly the corresponding rational opens as expected, but if you take a general rational open here, not satisfying that condition, and then restrict it, I don't know if it does satisfy that condition and you restrict it, you get the correct thing, but if it doesn't satisfy that condition and you restrict it, you get something new, which is not necessarily even quasi-compact and not a rational open, so you have to write it as a union of rational opens, and this is kind of like taking the open unit disk and writing it as a union of closed unit disks as a typical example of that phenomenon. So you said something about getting a structural sheet last time, can you comment on this or will it come later? Yeah, I was hoping to get to it again, but I didn't. So it is the structure sheet would just be you take R, which is living globally, and you apply the localization functor to get something living in here instead, and that's this. It gives you this object here, and one can analyze it, and so on and so forth. Yeah. And it's also some center category. Is there a way to think of it as a center of a category? Center? I don't know yet. Oh, no, but these are symmetric monoidal categories, so it's just the unit. I mean... The unit of the symmetric monoidal derived the category of in this sense. Yeah. So this, so you claim that when you take mod R of this, this is actually a good thing. So it is associated to the, in good cases, it is associated to the root of R, associated to the rational domain. Yes. Sometimes you have to do the right. Yes. So this is, I mean this, so this will also correspond to an analytic ring, but in the derived sense. So you have to, so the notion of analytic ring that we've discussed so far, you had an ordinary condenstering in a full subcategory. Here you need to not just remember that ordinary deriving, but you need to remember some derived enhancement of it as well. But then it is enough to just remember the ordinary abelian category of modules over the ordinary thing. And besides the derived stuff, there's also a quasi-separated issue where the value of the structure sheaf might be different from Hooper's, even if it lives in degree zero, it might, the quotient might not be by a closed ideal and so it might still differ from Hooper's, but again in practical cases that doesn't show up. And yeah, I guess even inverting G can introduce non-causi-separated behavior in general. Yeah, well I think we'll discuss some of this in coming lectures, all of these. So concerning those two spaces in Hooper's theory, where you have a reflection, which I think he maybe uses a slight differentiation, but anyway, with SPVA algorithm. So you have this subspace, you're living in a slightly bigger thing with a retraction, which is spectral. So you have sheaves, you can consider sheaves on both things. What I said, I think is correct that the restriction to the subspace is like the direct image. Then you have a sheaf, like in this case a sheaf of categories in some higher sense on the full thing and then you take the direct image to the subspace where you like restriction in some cases, I thought. But the question, it can also be asked about, so you have particular sheaves on the full thing which are direct images by the inclusion of sheaves on the subspace. So the question is like in this context, so you have your, let us say you have a rational open in the figure thing, and you consider its intersection with the smaller size, but you said that you can write as a union of six. So you can evaluate your sheaf by, in this way by, in this limit of those, is it equivalent to the sheaf, to the value on the original thing in the big, in SPV? So, like whether the sheaf of categories is the direct image of its restriction to the subspace. Which she, so. Which sheaf. You have on SPV are plus or zero, zero. You have, let us say, some stock, let us say, of categories. You can take its direct image to, which I think is the same as the restriction to that. And then it is a canonical junction map to R-Royal star to the, let's say, I is the inclusion, it has a canonical junction to I lost. Yeah, it's not the same, because for example on global sections you can see something like, well, no, it's not the same, because for the category, you have a rational subset corresponding to the closed unit disk, giving you this. And in this, in the category of modules over this guys, say the unit is compact, but when you go through that procedure again. An analytic function is not this thing. The global analytic function is just certain before you don't invert the fixed power of the objective convergence on each piece. So you get another ring of function and this is something else. This is another theory. So it is not, so the answer is not. No, there's really more data in the structure chief on this guy. And of what one can do in the usual way. Exactly, exactly. Which is okay. Okay, so yeah, maybe if there are more questions I can handle them on a personal basis. Let's release the people from the room.