 Okay, so I wasn't in the process of speaking about the total homology of diamonds. So let's try to remind ourselves a little bit where we are. So we were always working with this test category, it's called PERF, which was the category of perfectoid spaces of characteristic P. And this had two topologies, it has a pro etal topology, which was generated by open covers and co-filtered inverse limits, surjective etal maps of phenolites. And then there was another topology called the V-topology, where we allowed many more covers, so again that's generated by open covers and all surjective maps of phenolites. Okay, and then let me bring this back on the board, we had the definition that a diamond is a quotient, is a pro etal sheaf on PERF, it's called y actually, such that y can be written as x mod r, where x is a perfectoid space, or the sheaf represented by a perfectoid space, and r in x times x is a pro etal treatment solution. And there was a theorem that all diamonds are V-sheafs, and in fact many things generalize to all, and maybe there's a saturated condition of being small V-sheafs. And so in particular x is a small V-sheaf and r, so lambda again will be a ring, a coefficient ring, such that, and last time I was making some comments about statements which are true more generally, but at least from today on everything that is new today it will really be necessary to assume that n lambda is zero for some n which is not divisible by p. So if x is such a small V-sheafs then we have the drives category of lambda modules on x, by which I mean you take the V-side of x, which is just all small V-sheafs mapping to x, I mean ignoring saturated issues, and I'm looking at lambda modules on z, it takes the drive category of z. And in there there was a certain subcategory that I called d etal x lambda, which is essentially characterized by the properties that whenever I pull back to a perfectoid space it comes from the etal drive, I mean, okay let me first write this and then say what's true. So the thing to know about these categories is that containment in the subcategories can be checked locally and if x is a perfectoid space respectively n totally strictly totally disconnected, so strictly totally disconnected guys were these things which were essentially just profaned sets of points, and we were using often that anything can be covered by profaned set of points, then this d plus et was just the d plus of the etal side of x which can then be defined lambda, and if you want the statement to be true for the unbounded category you need this extra assumption. You can get in slightly more general settings if there is some bound on the chromological dimension. And in the strictly totally disconnected there isn't bound on the chromological dimension or not? Yes, it's bound on the chromological dimension zero, like for anything which is, any separated etal for the compact maps say to a strictly totally disconnected guys against strictly totally disconnected, and if you have something strictly totally disconnected then there is no higher chromology at all. Right, but there is a basis for the topology. So these are the objects we care about and actually the one we care most about is this guy. Okay, so let's talk about the six operations. So let me start with some simple ones. So first of all there is some tensor product, well that's a priori maybe defined dx lambda, just the usual thing we always have, and it respects this d et until you get it there. Let me say what I do for the r-home. If you take this, then this does not respect this guy. But you can somehow reflect it back, so this inclusion has an adjoint, which in some cases if you're a perfectoid space and line the d plus or so, you can sometimes more explicitly say what this really means. So this is actually a little bit tricky to understand what this means. Is it a left or right adjoint? It is a right adjoint. So this inclusion preserves all co-limits and then while you can upgrade the whole situation to infinity categories and then it follows from Lewis infinity categorical joint function theorem that there must be an adjoint. So if f from y to x is a map of small b-shefs, then the pullback preserves the d-ed. It commutes with tensor products. And so you also want to push forward. If f is q c q s, then the r-f lower star from dy lambda to dx lambda, it respects the d plus eta. It induces such a map. And under an assumption of bounded homological dimension, also on the unbounded derived category. I don't really find such an operation r-f lower star for q c q s morphisms, but I wanted this such an operation more generally. So for a general f, I can define the operation on this unbounded italic derived category as a right adjoint of f upper star. The homological dimension is for the r-f lower star or for both x and y? No for r-f lower star. So I think it's literally enough to assume that whenever you have an italic sheaf on y, then if I apply this r-f lower star, then r-f lower star is equal to 0 for sufficiently large i. So then there is this caution that this is not compatible. So equivalently, you can also embed the derived category for y into this four derived category of these sheaves. Then there is an r-f, let's call it b lower star, defined on the level of sites here. But this may not end in this d-ad, and then you can just reflect back. In f with qcqs, I recover it on the d plus eta and unbounded homological dimension also on the d8 in the form. So in the case of overlap, it's the same thing, but in more general cases, that is the operation I want to consider. So this takes care of four of the operations. And then there are two more, r-f lower star and r-f upper star. And at the end of the last lecture, I discussed a little bit the push forward with compact supports. And so let's remind ourselves about this. So for this, I need to talk about compactifications again. So let's remind ourselves how this went. So there was this funny thing that for edict spaces, there is this canonical compactifications. It's canonical compactification, which is somewhat slightly weird, which only adds a few higher rank points. If f from y to x is a separated map, then I can do the compactification of f over x. This goes from a space called y bar over x to x, which is given by the functor, which takes some spar r plus. And in some sense, forgets the information of cr plus, or at least it forgets all the information of r plus as regards y, but keeps it for x, as we still want something which lives over x. So it's the thing which maps this to the y of r, r-circ, where your points, sorry, sorry. So for example, if you want to map a, but I need to assume that this is totally disconnected. If you want to map some point, k, k plus to it, then you only need to map the rank one point, the maximal generalization to y, and maybe the full point, the higher rank point to x. So these kinds of single parametrize here, they are actually some of the kind of diagrams you would draw on the evaluative criterion of properties. And so we have this thing, and then f is compactifiable, f is separated. And so there's always a total, a natural map from J, an injective map into this compactification. And you ask that this is an open version. So let's just briefly discuss one example again. So if you have y, which is the edict spectrum of this guy mapping to x, which is spar k, okay. So this is a ball, a closed unit disk, but it's not quite closed as an edict space. It's not yet proper. And so you include one higher rank point. So then y bar over here is this thing, and then I take all k plus the maximal ideal. On the other hand, if y is the a1 over k as an edict space mapping to x, which is the first example, second example, if you have this guy, then actually y bar over x is just equal to y. And so this sounds strange, because you wouldn't think that the a1 should be proper. You would want to compactify it into a p1 maybe, but that's not what happens. And the reason is that actually if you consider the a1 as an edict space, then it's this increasing union of larger and larger disks, it's the union of increasing disks. And the kind of universal competition of each disk sits inside the next disk. And so it's what's called partially proper. Let's see the compact supported commodity of a1 you want it to be. Yes, that's why there will be a quasi-competitive condition in a second, okay. So you can, let's see how we define competitive support commodity. Partially proper is the same as this compactification, can we add anything? Right, partially proper is actually, well there's a more geometric definition for edict spaces, but one way to say it in our set up is that this map j is just an isomorphism. So it's a clue to saying that you have some kind of variative criterion. That whenever you have map from spa rr, cirk to y, and one, you draw such a diagram and then there's a unique map here. It's related to this one statement, this one variative criterion of properness that I stated last time. Okay, so this brings us to the following definition slash zero. So let's assume that f from y to x is a quasi-compact. So this, this allows this a1 example here for the moment. Because this is an increasing union of discs, quasi-compact morphism of v-shefs, small v-shefs, sorry, compactive phi bar, I need to add. Okay, so then we are in the situation where we have y, which embeds y and j into this compactification, which sits over x, this is f. And so then this map is actually open by assumption. And this map is proper, according to the definition of properness I gave last time. And so then if I look at rf lower shriek, which I defined to be the usual thing, you push forward after you extend by zero. So this is a map from dy lambda to dx lambda. This preserves the d plus eta, this induces. And also a map on the full derived categories. If rf lower shriek has bounded homological dimension. And it's compatible with base composition and base change. So a composite of compactifiable morphisms is again compactifiable as is a base change. Put them how the formation of y bar is smashed x is how strictly compacted with composition. Okay, so let me explain what happens. So if I z to y to x, then this has a compactification over x. This has a compactification over y. Yeah, and then this also has a compactification over x. And this little diagram here is a Cartesian diagram. So it induces me just by restriction without any adjoint. Right, right, right, right, I mean. So it really, if you apply this to something that's d plus eta, it really maps, it applies this function, it really lies in the subcategory. One funny thing about the situation is that you could already ask this compatibility with composition without passing to the d eta. Let me actually write this down and then I believe it's not true. So there are very two remarks about this, which are related. So, well, no, maybe this one, so we could ask. So let's consider the situation z to y to x, f and g, f, g as above. So quasi-compact, compactifiable. Then you could ask the question, is if I first apply g and the f, is it the same thing as this thing on the full, guys? Okay, and I should one day sit down and actually find a counter example to this, but I believe it's not true. And so you actually need to restrict to this d eta to make this true. So there is a natural map in one direction, which I'm not able to figure out right now. And well, you can show that this is an equivalent. If you plug in such a guy, such a d eta guy in this eta thing, but it might fail in general. And so for this reason, it's actually net, that's the main reason that it's necessary to really work with this d eta and not something more general. There is a map in one direction. There's a map in one direction, yes. And on d plus also it should fail. Yeah, even on d plus, yeah. And I think even on this, there was this other guy that I didn't recall this time. So yeah, I think even on this kind of direction. Okay, so this way we get this R of law streak. Okay, then there is another unrelated remark. So there's another thing if this map is not quasi-compact. So then we can do the following. So if f is compactifiable, maybe not quasi-compact. But it is representable in locally spatial diamonds, meaning that all x prime f into x, where x prime is a locally spatial diamond, y times x, x prime is a locally spatial diamond. Then one can define R of law streak as a direct limit. And that may allow to be a little bit work here. Of all u and y, which are quasi-compact open, quasi-compact over x, say if the base is locally spatial, in general by descent. Okay, so the logic is a little intricate here. I just want to say you can extend it. And the essential thing is that you just defined as a direct limit of the lower streak for open subsets, quasi-compact open subsets, and then take a direct limit. This only works if you have enough quasi-compact open subsets. This is this condition of being locally spatial essentially. And so if everything inside y and x were such locally spatial diamonds, and this formula I would have written would be okay. In general, it's enough to assume that the map is somehow locally spatial. In which case, you just reduce to the previous case by descent. It's not the most critical thing. So this gives us five of the six functors and we need to find the last one. There's also a projection formula, let's say it's a quasi-compact compactifiable. A small v-sheafs then, R of law streak of. Yeah, so once you have the projection formula, you can essentially form it in space change terms. You can essentially formally get a knitted formula. Okay, so of course the last factor, R of upper streak wants to be defined as the right adjoint to R of lower streak. But to make this work, you really need to be in the setup where R of lower streak is defined on the full derived category. And so we have this assumption there of bounded homological dimension. And so I actually want to talk a little bit about dimensions for this reason. So this is holding on the what, or which A and B? Sorry, A, B are in the D, so A is in D et al. Y lambda B is in D et al, X lambda, sorry. And I might have to assume some bounded homological dimension again. Where is this? Okay, so it holds for this under if something happens that I didn't define yet. If some, sorry, if some dimension of S is finite, which brings me to dimensions. It's some kind of transcendence degree dimension, geometric trend. So this G stands for geometric, this trit stands for transcendence degree, and this dims stands for the mentioned. We will come to it. I'm sorry, maybe I was assuming for this one other little thing. Yeah, representable. Well, I can remove this and say it's representable in locally spatial diamonds. And the tricky thing with the projection formula is that this R of lower streak is usually better defined on the D plus, whereas this derived transfer product goes very unbounded to the left and you have to be a little careful a bit. I need to talk about dimensions. So there are the following notions of dimensions. So the first thing we need is a cruel dimension. If X is a locally spectral space, then it's a cruel dimension. It's a supremum of all n such that there exists a chain of specializations. X naught, X1. Whereas specialization means that one is contained in the closure of the other. Put this here. If f from Y to X is a map, is a spectral map of locally spectral spaces. I guess in principle one could, well, no, let's assume that locally spectral spaces. Then I define the dimension of f to be the supremum of the dimensions. Now, by the way, why does this live? This is either a non-negative integer. It could be infinite, but it could also be minus infinite. That's if f X is empty. The dimension of f is the supremum over all X and X of the dimension. And again, this is either a non-negative integer or infinite or minus infinite. But this only happens if y is empty. Are there notions of dimension functions? Are there notions of dimension functions? Well, so far we're here in the completely abstract setup. One could ask this for diamonds. Not really, I'm afraid. So there is this issue that the local, the usual closed immersion that one would want to look at, like a point mapping into a closed unit disk. They're actually not so well behaved in some ways in this world of hectic spaces because the complement of this map is a non-quasi-compact open subset, which means that this map has a non-constructible image, which means, say, that the skyscraper sheath supported at this point is not constructible. It also related to the fact that there is no specialization of a point of the complement into this closed point, as you know for schemes. And so actually, if you want to define the core dimension of this point, you can't really define it directly on the spectral space because the spectral space actually doesn't see that there is any, that they are connected in any way, really. It's a bit tricky. All right, so anyway, I mean, these notions exist. And then there is another thing that I need. Let's assume we have K contained in L, an extension of complete non-accommodian fields. Then I define the kind of continuous transcendence degree. There might be a C. So these names are essentially due to Huber of the Stimtre of L over K to be the minimum of the set of all N such that exists some L prime and L, a dense subfield containing K such that the transcendence degree of L over K, L prime over K is equal to N. It's difficult to get your other coefficients because you get only one. Right, so this is actually a slightly tricky definition. So there's the following very basic questions that I don't know the answer to. So I mean, what is true is that, okay, let me make a remark. So if K is contained in L, is contained in M, then there is a sub-editivity. This transcendence degree for M over K is at most the sum of the transcendence degree here plus a transcendence degree, this continuous transcendence degree. Sorry, I didn't want to call this, sorry. I wanted to call this this continuous transcendence degree, sorry. And then when I go to morphisms, I want to call it the dimension of the morphism, but sorry. Okay, so this is not so hard to see. But, additivity may fail, equality may fail, and equality may fail for good reasons. Say if K is something like this guy, so some complete algebraic closed field of characteristic P. And M is algebraically closed, and of transcendence of this transcendence degree equal to 1. Then there exist some, so you would think that I mean algebraic geometry, or for usual fields if you have two algebraically closed fields, one is of transcendence degree one over the other. Then there are no algebraically closed fields in between them. But this fails in this setting, so then there exists some K contained in L contained in M. Both proper, where L is again algebraically closed and transcendence degree equal to 1. Well, over K, but then I thought theory also for M over L. I mean if this was less than one, namely zero, then this would mean that there is a dense subfield which is algebraic over the base. But if this is already algebraically closed, there can't be further algebraic elements. Thus, this would have to be dense. But if these are both complete, then this would have to be in equality. But it's not in equality, and so this has to be actually equal to 1. Can it be also in characteristic zero? I think this can't happen in characteristic zero. So there's a paper of Matignon and Reversa, which discusses this problem. One funny thing to note about this paper from Matignon and Reversa, which is, I think, from the 80s, early 80s, is that they already find the notion of a perfectoid field in this paper. They called it a hyper-perfect field, hm? Science has progressed. So, additivity may fail. You could at least ask whether there's some monotonicity. And even this I don't know, yeah? Question, if you have the situation k contained in L contained in M, is it true that this continuous transcendence degree for L over k is at least bounded by the 1? Let me just note that the similar question for M over L is actually true. It follows immediately from the definition. But for this thing, it's not so clear because here you would have some dense subfield of M, which has some small transcendence degree. But this dense subfield has no reason to intersect L in any way. So it's not clear how to find a dense subfield of L. So, even in the case where M is the algebraic closure of L, I don't know this. So, in the absence of knowledge of this, of the answer, I define a slightly different notion. It's called geometric, to be what you get for the corresponding extension of algebraic closed-complete fields. And you can eliminate for k this operation? You can eliminate for k, I mean, it's the same as, and it's always less or equal to the 1 for L. That's by sub-additivity. It's less or equal to this plus the corresponding thing for the algebraic closure over L, but it is transcendence degree 0. All right, this is all stupid, but... I mean, I think it's an interesting question just in general to try to establish some good series of transcendence degree for these non-accompanient fields and see what properties what can establish and which fail. So, for example, this counter example is also interesting. Is the problem with this monotonicity? No, even then I can't answer monotonicity. But the problem I have is that if I have a diamond, then it doesn't have a val-defined residue field because it has a val-defined algebraic closure of its residue field. And so, I need a notion of this transcendence degree which only depends on the algebraic closure. So, here's a basic proposition. Let's say that. Let's assume you have a quasi-separated diamond, such that the underlying topological space is a point. So, what are the diamonds with a small quasi-separation? Whose topological space is just a point? We can understand those. Namely, then x is of the form spa COC, modularity action of G, for some unique but up to non-unique isomorphism as usual, complete algebraically closed field C, with a continuous and faithful action of a profiled group G. So, you would like to think of C as the algebraically closed residue field, the completion of the algebraic closure of the residue field. And as G is the Galois group of this point, but G may not be the Galois group of any actual field, it's just something which acts. Conversely, whenever you have such data of such a complete algebraically closed field C of characteristic P, with such a continuous and faithful action of a profiled group G, you can define such a quasi-separated diamond here, whose space is a point. So, you can understand the structure of points of attic spaces. And so I can make the following definition. So, in general, the field of invariance can be very small? The field of invariance might just be Fp or something like this. So, you might have the case that if you take spa Qp and then take its diamond, then what you get is somehow this field moduloos the Galois group of Qp, but if you take the invariance in here, you just get Fp. And so there's not any actual field in the business here. Anyway, so before the break, let me finish by giving this kind of transcendence degree dimension. So, let's say that f from y to x, let's say it's a map of locally special diamonds, this G stands for geometric, this T stands for transcendence, dim stands for dimension, of course, is the supremum overall y mapping to x of this kind of geometric transcendence degree. Well, which is actually the same thing as this continuous guy because I'm doing algebraic closed skies here. So, let's say that this is the sum of c of y over c of x. And I can add some maximal points where, well, I mean, it's equal to the true because we're only doing geometric guys. I just want to stress that even in the case where these work perfected spaces, I'm actually doing something different by passing to this algebraic closed skies where, say, if I, this is the localization of x at x and similarly for the definition of c of y. And c of x is defined up to isomorphism, but I mean, I'm only computing numbers and these numbers are in the end independent. The map of c of x to c of y is still well defined up to actions of some groups. Right. They don't change the, okay. Okay, let's have a break now for 15 minutes. Well, the localization of a spectral, in general, if you have a spectral space and you have a point in it, then there's the notion of the localization of the spectral space at this point which is just a set of all generalizations of this point. But if you have a maximal point, then there are no further generalizations. Well, I mean, if you would want to define such a complete residue field as a non-maximal point, they would be the same as at the maximal generalization. And so it's not really, not much is lost by assuming this is a maximal, just not with. Okay, so another basic proposition is that, so before the break, I defined this notion of this transcendence dimension for morphism. And the basic fact is that the dimension of f considered just as a map of spectral spaces is bounded by this geometric transcendence dimension. Now, let me just sketch it. So if x is a maximal point, if you look at the closure of x, then there's always embeds into the terrisky Riemann space for... What did you define as dmf? Ah, the dm of the fibers. What? I'm trying to concentrate here, okay? Always in terrisky Riemann space where a copper of x is, you take the residue field of C of x, it's completed residue field, and then actually there you can take the GX invariance and get a reasonable field because there the action is actually... Continuous for the discrete topology and then you're in a better shape. Sorry, if x is a maximal point and say, let's assume for the moment that x is separated. So there is a relation between the specializations of points you see here and the terrisky Riemann space of some fields. And so this bounds essentially the dimension, in this case maybe of x by the dimension of these residue fields. And so that's how you get this. So this bounds the cold dimension in terms of the transcendence degree. And this is some kind of relative version of this observation. Okay, so we have this. And then another proposition is the following. Let's say that x is a spatial diamond and f is a sheaf on its et al. side. Then the et al. homology of f is zero for any i which is larger than the dimension of x but the supreme overall x and x maximal of the homological dimension. Oh, sorry. Let's assume it's an L torsion sheaf. L not equal to p. Well, I think it's not even necessary. The L homological dimension of gx. Okay, so you can balance the homological dimension of the et al. side in terms of just the cruel dimension and the somehow galvanic homological dimension of these points. Let me sketch this. Let's look at the map from the et al. side to the just the side of the underlying topological space. Let's call this lambda. Then it comes down to two things. The first checks that if you push forward f under this map then this is zero for i bigger than the supremum over the L homological dimensions of these points. For this you just observe that the fibers are given by gx homology at least at maximal points, at non-maximal points you need a small extra argument but it works. On the other hand, the homological dimension of x as a topological space is bounded by the dimension of x. This is actually a theorem of Scheiderer from 1992. This holds for any spectral space. Of course, everybody knows the theorem of Grotendieg that if you have such an Assyrian topological space then the homological dimension is bounded by the cruel dimension. But actually, there's an unnecessary Assyrian hypothesis so this is always true. These are the ingredients. This tells us why the cruel dimension is relevant to the picture but now we need to unbalance the homological dimensions of these residue fields here. For this, business of transcendence degrees will be important. Let's say we're in the following situation. It's a spatial diamond over some sparse COC. By f, I want to denote this map from x to sparse COC. If I look at the supremum over all x and x of the homological dimension of this profile group gx then it's bounded by this geometric transcendence degree dimension if L is not equal to p. This relates, let me say what this comes down to. The sketch we'd use is to c' over c as some other algebraic closed complete field and g is a profile group acting continuously and facefully on c' but over c. This trivial action on c. Then we need to bound the homological dimension of g which wants to be this Gaoua group by the transcendence degree of c' over c. Of course such a bound is well known if g was an actual Gaoua group of a field over c but this is just some profile group which apparently has nothing to do with any field but you can still make enough of Gaoua theory work. It was well known Gaoua theory if g were a Gaoua group. In this case you wouldn't need an assumption on air. It might be true that this is also true if L is equal to p but I don't know what to prove this. What turns out to be true is that up to some pro-p group you can essentially control what g can be. You can still define some p and i and then g. There is some kind of inertia where this is pro-p. This quotient i and p are bound by Gaoua groups. C' and c have non-acquaintance fields so they have Gaoua groups in the real numbers. This one for c' might be bigger than for c and this is bounded in terms of the difference you see on value groups. This is bounded by the transcendence degree of residue fields. There is a general theorem in Bob Ike that relates the transcendence degree of residue fields as the value groups and the transcendence degrees of the fields themselves. You finish using this inequality. I don't see any way of controlling what this Waldian inertia here looks like. It might be really wild. It's wrong for L equal to p because it will look like Delphan-Fuchs-Gormulge of the algebra vector fields. This is a zero-critistic analogy. This can act on such a non-acquaintance field. It should be more of the same. Maybe we can discuss this afterwards. Finally, I must state what all of this discussion is leading to. Let's say f from y to x is a compactifiable map of small v-shefs, which is representable on locally special diamonds. By the way, this discussion also tells you why I'm making two assumptions throughout. One is that I assume that my diamonds are spatial, so that the underlying topological space is spectral, so that I can apply this theorem of Scheiderer bounding the conmodical dimension. The other is that I assume that they are diamonds, which was this condition about points. This condition about points comes up somehow through this proposition that if you have these maps from algebraically closed fields into them, which are pro-Ital, then you can control what these points look like. That's why this notion of a locally special diamond is really the setup I want to work in. Alright, so then, if a is then d less or equal to n, while lambda are a floor strip of a, i is then d less or equal to n plus three times. You would expect two there. It might be true with two. My proof doesn't give it. Let me tell you why, what's the technical issue that makes the sphere appear. So if you have y to the x in the map of spatial diamonds, the compactifiable map of spatial diamonds, I do not know whether the compactification y bar over x is again spatial. So it is a diamond. But I don't know how to get enough quasi-compact open subsets of this guy. So it's not true that if I take a quasi-compact open subset of y and then compactify it, it will be a quasi-compact open subset of this compactification just because again the closure is automatically closed because you are taking some compactification. So there is no obvious way of producing any quasi-compact open subsets in here. And so this means that I cannot actually apply this kind of proposition there which applied to spatial diamonds to this compactification and then you need to work around this and then there is a workaround but I don't want to go into this. But in any case it shows that if your system trig of f is finite then R of Lore-Schrieg has finite common logical dimension. And so definition, let f be such a guy, so let f from y to x be compactifiable, representable in locally spatial diamonds. Such that this thing is finite. Then finally we can define R of upper Schrieg from d x to y is a right adjoint of Lore-Schrieg which in this case is defined on the full derived category. And in some senses it's a theorem because implicitly claimed there is such a right adjoint. So why is there a right adjoint? You use that R of Lore-Schrieg commutes with all direct sums in particular infinite direct sums and the infinity categorical adjoint function theorem. For this you have to briefly update this function to a function of infinity category and then apply it. Can we use the classical construction to go to more resolutions? Can we use the classical construction to go to more resolutions? I never learned to go to more resolutions. Sorry. I would believe that if these things are actually locally spatial diamonds then this should be okay. I mean, somewhere you need to be in the setup where this guy I call d ad is actually the derived category of etal sheeps on something. I think to apply this to go to more things. No? I don't know. And so then I would need to make very strong assumptions on x not just relatively but somewhere absolutely and I don't want to put absolute restrictions on x. Okay, I don't know what you could do there. Replace x by a simplistic system of special... Yeah. Well, I think once you pass through the simplest thing, I mean, you're not so far from this thing. Anyway, it exists. Okay. And so, okay, that's purely abstract right now and so we want to make this function explicit in some cases and so in particular there should be some kind of smooth morphisms for which f r of upper shriek is essentially just given by pullback twisted by something and then you can wonder what's the right definition of smoothness in our case. But you can also not do it and just define comodical smoothness as being satisfies what you wanted to satisfy. So f, as in the definition, as in previous definition, is l comma not particularly smooth. As usual, l is not p. If the following happens and essentially you just want to say that this function is given by pullback twisted with some invertible thing but you need to say this not just for x but for x to be base change but actually it's slightly simplifying that you only have to verify this for strictly totally disconnected guys. So the condition is following. If y is x prime to x where x prime is strictly totally disconnected if I look at, let's call this map fx prime it's a map from y times x, x prime to x prime if shriek pullback under this map it's equivalent to some l tensor fl star is functions from d h r x prime so the condition is just that there isn't equivalence of functions where l is locally isomorphic to fl n which may depend on where you are locally. So that's like the minimal requirement that you would want to put on a smooth morphism so if you have a smooth morphism it should be any pullback should be smooth so in particular all these fx prime should be smooth and if they are smooth then it should be of this form it doesn't matter it's all the same for this question. So that's the minimal thing you would want and the good news is that if this minimal requirement is satisfied then a few good properties follow so if it's comatologically smooth so in this definition of comatological smoothness I only used fl coefficients but actually things improve from there so then for all l power torsion rings lambda it is true that rf upper shriek is equivalent to something I call df turns out or maybe say this more canonically there is always a natural map from this kind of gadget to this guy this function is from d at x lambda to d at y lambda so this natural map is then an equivalent note that this is also something that's even for if lambda is just fl it's not part of the definition sorry I only asked this after pullbacks to strictly totally disconnected bases but it's already true on the basin what else? and rf upper shriek lambda is italically isomorphic to lambda and degree n sum n and z and the formation of rf upper shriek can be used with any base change meaning if I have y to x here I have some x prime mapping to it and I have some y prime the fiber product let's call this f prime and I have some fg, some fg tilde here then there is a natural transformation of functors from g tilde upper star rf lower shriek rf upper shriek maps to rf prime upper shriek g upper star so in particular the dualizing complex can be used with any base change so let's call this guy here df in particular the dualizing complex for its map f prime is canonically isomorphic to the pullback of the dualizing complex for f note that this is also something that wasn't asked in the definition that this sheaf L which appears there in this local equivalence is actually compatible with any base change it follows from does it suffice to prove to check this on one subjective? yes so if the base is already strictly totally disconnected you might ask yourself whether it's enough to check this not just on x itself not after any base change and this is something I don't know so you still need to check it for all further base changes to bigger algebraic closed fields somehow can this n be made explosive? what is this n that appears there? good question I don't know so this n somehow gives you another weight so you would expect that for example n is even but I don't know this for the affine space it's just what you expect okay I will come to some examples in a second there is still some time left what was the question? what I wanted to say what is this n? so this n somehow gives you another way to define the dimension of the smooth morphism so it's the degree shift in the dualizing complex and I don't know how to relate this in general to the these other dimensions I was discussing so there is some fuzziness about dimensions right now in the theory that needs to be resolved but anyway so if you have such a comodically smooth morphism then the theorem essentially tells you that everything you would expect to be true in the usual formalism is true but this doesn't tell you that there are any examples so maybe let me just note this I mean so color is that you you get a smooth space change in punk radiality but this is completely abstract so we need examples well what's good about this notion of comodologically smooth morphism is that it has a lot of good stability properties so let me first note these stability properties so once you have some example you get a lot of other examples and then in the end I will tell you one example one base case so composites of comodologically smooth are there is a two out of three properties so if you have g from z to y f from y to x this is surjective and if g and the composite are all comodologically smooth then f is all comodologically smooth so you can check whether you are all comodologically smooth you can check this all comodologically locally then this comodologically smoothness is v local on the target it's the first point composites are and then there is a funny operation so assume we are in the following situation f from y to x is all comodologically smooth and g is some pro p group pro prime to l but let's say pro p acting freely on y over x with trivial action on x then if I take the quotient by k from y mod k to x then this is still comodologically smooth so we call that a lot of the examples were constructed by taking something more explicit and then quotienting out by some action of what's often a pro p group and so this takes care of these quotients by pro p groups if you take something smooth and quotient by pro p group it's still smooth and then there is this one example that gets everything running it's that if b is a v sheaf which takes which is some of the closed unit ball which takes r plus to r plus then the projection from b to the point is alcohol in other words if I take spalf I can base change this to some non-accommodient field and then it's just this usual closed disc and the end is two and so rf upper shriek of lambda isomorphic to lambda t twisted by one shifted by two okay so sketch well the hardest part the one where most input comes into is of course the last part and this uses pretty much everything from Hooper's book so it uses extensively so the essential thing you need is you need a complete series of prompter radiality where smooth curves over algebraically closed but this is this is done by Hooper one thing to note is that here it's really important how I phrase the definitions that I only need to check something over these strictly totally disconnected bases and then by some reduction arguments I can actually get rid of this profiled set and just actually work over a point and so in the end everything is just reduced to the classical setup of the ball over a point and then again directly to side Hooper's book great and so now starting with this one example and using all the stability properties you can now verify there's lots and lots of things that are logically smooth so for example you could take the a1 over cp it takes the associated diamond so then this lives of a spar cp diamond which is just the edge of the spectrum of the tilt how does one see that this is alcohol logically smooth so this can be checked locally so and this is an increasing union of balls so reduced to well you could say it's an increasing union of balls but you could also do something slightly different and you can essentially reduce to the case of a torus the first reduce to the ball but then this ball is covered by two copies of a torus t is the set of all the points are the set of all x which are absolutely really t of cp is the space but then you actually t is the spar of cp this guy and so as in the first lecture you have this t infinity where you join all p-par roots of it and then t infinity flat it's the same thing for the tilt and so this is actually open in the ball it's just the subset of points where the absolute value of t is equal to 1 and so this implies that ah maybe they should have also been a part 0 that etal implies alcohol logically smooth so this is alcohol logically smooth this is open in it and so t infinity flat mapping to spar cp flat alcohol logically smooth but then you can recover t as a quotient let me finish this sword but then if I look at this torus of a cp and it passes the diamond then this is t infinity at some point I lost the subscript cp the tilt modulates the action of cp everything here over and so now I use this funny statement 4 so you use this and then you conclude that this guy is alcohol logically smooth let's do another example this shows that in general smooth attic spaces are yeah so in general you can use this to show that smooth attic spaces are alcohol logically smooth because well because composites are and well the products are also alcohol logically smooth so you get that f i in spaces are alcohol logically smooth in general something smooth over an f i in space and then and then it's always twice that I mentioned and right so I wanted to do another example so at one point I had this guy beat around plus mod fill 2 which had this extension which was an extension between diamond a1 over cp so this is an extension of group diamonds I guess these examples are called pannachal mess spaces over sparse cp claim is it smooth sorry I mean maybe I should say more precisely what I mean so I have this beat around plus mod fill 2 well it is a diamond and this maps to sparse cp diamond and I claim that this map here and as this is really an example which doesn't come from usual geometry it doesn't it's not a smooth attic space or anything and so it's really non-trivial to get your hands on this guy is any group object alcohol logically smooth well all these pannachal mess spaces are alcohol logically smooth well any group object I'm not sure I can talk about any group object I don't know what they are no no that's it's true for any I just want to consider the simplest example to explain how the arguments go so this maps to the a1 over cp diamond and we know this is smooth by the previous example and so it's enough to prove that this map here is smooth it's enough that f is smooth but now we use that this can be checked locally we locally and we locally this has a splitting and so once you fix the splitting and so if you have a map space x mapping so ejectively onto here such that there's a splitting then this relatively this just decomposes as a product of x with this a1 which is comodically smooth by base change and so let me just finish by saying that for those who know what they are is that all Banach commas spaces are comodically smooth I mean all positive yeah I could be underline not okay so all connected Banach commas spaces effective as I called so of course you would really like to have a more geometric definition of what it means to be smooth but the problem is that what we're working with is in some sense something like a topological space and it's not a manifold so if you had a map of many folds you would know what it means to be smooth because you can look at tangent spaces but if you just have a topological space what does it mean for a map of topological spaces to be smooth and so I'm unable to give a direct geometric definition of this and so but so far one can work around this just by working with this very abstract notion of comodical smoothness that can then be verified in the example to care about and the pro etal map is that the pro etal map is not comodical smooth so ah so another property of comodical smooth maps is that there is some finiteness properties for their comology so they automatically if lower streak preserves constructable sheaves for some notion of constructability but in particular if you had a pro etal map say over a geometric point it might be something like a pro finite set but the comodgy of a pro finite set is of course very big so this is not allowed the question was whether pro etal maps are smooth comodical smooth and they are not and the essential reason is that ah you don't have finiteness of comodgy for them so something like a pro finite set mapping to a point is pro etal but the comodgy of the pro finite set is of course very big because say the h0 is already very big ok so there is no higher comodgy but the h0 is really big ah and so that's not allowed ah did I want to say anything I don't know question how about localisoclasticity localisoclasticity come tomorrow I will talk about localisoclasticity ok is there some general reason for saying that something is compactified is there some general reason for saying that something is compactifiable no but again like in these procedures here why you were having some examples and you were like the same kind of reduction procedures that you were doing here they also worked for the compactifiable notion so in practices it can be checked