 Okay, so the course continues and I want to talk today really about the Tarkov module of diamonds. So let me try to briefly summarize a little bit of the last lecture by drawing a big diagram of various objects that occurred last time. So he had this category... so one notion that was introduced last time was categorically totally disconnected perfectoid spaces. These were ones which are essentially profinite sets of geometric points. Then you had totally disconnected effectoid spaces. These were in some sense profinite sets of points. Those were all examples of your phenoid perfectoid spaces, which of course are examples of perfectoid spaces. And then we enlarged the category of... I should put this here... perfectoid spaces into this category of diamonds, which were these quotients of... Yeah, yeah, sorry. I mean, let's say that everything in this diagram is of characteristic p. Quotients of perfectoid space of characteristic p by a pro-ital equivalence relation. And then I introduced a strange in this topological condition of spatial diamonds. And you can slightly weaken this and talk about locally spatial diamonds. So these are ones which have an open sub-cover by spatial. Well, just... I mean, this is a quite a compact and quite a separateness condition. This somehow... it's like locally spectral. So these have a spectral underlying topological space and these have a locally spectral underlying topological space. And if you have such a guy whose underlying topological space is spectral again, then you are spatial. And then you could further generalize the whole situation to include all V-shefs. You could also define a locally spatial V-shefs. These are... by definition I want all of these guys to be small, if I say that they are locally spatial. And these are contained in all possible V-shefs. There is another inclusion here and there is another inclusion. Okay? Right, so that's another thing I should draw. They said there's also a functor actually here from analytic edict spaces over Zp. This is functor x maps 2. So last time I stated that if you have a QCQS analytic edict space and the diamond is spatial and it's a more general statement, this is that if you have any analytic edict space then at least you're locally spatial. Because you have this open sub-cover by these QCQS guys. Maybe another thing I should have said is that if you're trying to understand containment here is just a condition about points. This was a final theorem I stated last time. I stated it for spatial guys but it extends immediately to the locally spatial case. All right. And so to other definitions that I want to briefly recall, also it will be important today. Amorphism of V-shefs et al. If for all the factored spaces X or the map X to Y, the fiber product is representable. I mean meaning the total space and the quasi pro et al. If for all strictly totally disconnected X, leaving to Y, the fiber product is representable. Well in the separated cases according to like the weakest possible statement so when it was separated. So in the non-separated cases it's slightly tricky in ocean still. So you can stay weakened if I bring it. Yeah okay. So in practice one can often replace Y prime with something some separated space and then if you map a separate space to anything the map is also separated. So usually there's not much of an issue. All right. So the main... So what is our separated V-shef? Maybe I can give these definitions right now. Let me give some other definitions. Let me say it's a closed immersion if for all totally disconnected times Y, X to X is representable and a closed immersion. And now I didn't say what this means. In this case it just means that this is equal to well as I said a totally disconnected space is essentially just a profiled set of points and so you just single out some close subset of this profiled set of points. Of course you have high rank valuations there. Yes but because everything it needs to be well it's supposed to be a closed immersion so all the specializations must be in there and all maps of analytic edict spaces are generalizing so all the generalizations must be in there too and so you don't really have any choice except to take the whole connected component. It's separated if the diagonal is closed immersion and while we're at it let me also define properness right now. Sorry yes proper if it's quasi- compact separated and universally closed. Where are universally closed? Universally closed means that okay quasi-compacity is some kind of relative smallest condition so it means that okay let me just say this. So for all let's say perfectoid space is X. I was a map into Y. The fiber product will be a small v-sheaf and so it has an unaligned topological space and you ask that the map to X is closed. Well I mean it's this usual thing which you have in any topos so whenever you pull back to some object the fiber product is quasi-compact and this means that whenever you have any cover of it there's a finite sub cover. Yes and so remark is that so you may wonder that I'm always asking slightly different conditions on the axis that I'm pulling back to and the reason is that I want all my notions to be v-local so that you can check some v-locally on the base and to arrange this you need to ensure that these notions satisfy appropriate descent theorems and these descent theorems are always true and like with the base is one of these sorts and so that's why you need these kinds of assumptions. So three, four and five and one and two in the separated case can be checked v-locally on the target meaning that if you have another sheaf y tilde mapping to Y which is a surjective map of v-sheafs and the pullback is S's property then already the guy itself has this property so he uses some descent results and these precise phrasing of this is the sense result explains the precise wordings of the different descent or separated atoms. Right but I had descent for separated atoms last time and also while we're at this let me explain that they are a variative criteria for separateness and problem is let's be a more efficient of v-sheafs and then f is separated if and only if f is quasi separated so again this is a notion which you have in any topos meaning well whenever you pull back to something say it's an infinite perfectoid space the pullback is quasi-separated and quasi-separated means that the fire product of any two quasi-compact things mapping to it is quasi-compact so this and for all perfectoid fields k with some open bounded variation suffering k plus and k if you look at diagram as follows so so whenever we have a non-archimedian field I denote by ok string of integers meaning the suffering of power bounded elements so it's what's sometimes denoted by case work for general ring and this k plus will always lie in there so this is thing which corresponds to the rank one valuation but you might possibly have a higher rank this k plus correspond could correspond to a higher rank valuation so the spark k ok is really just a point but it sits inside this slightly bigger space spark k k plus might which might be this totally ordered chain of points so this is what corresponds to the fraction field and the variation ring in the world of schemes and so you have this mapping to y prime you have this mapping to y and you wonder about lifts here and the condition is that there's at most one lift and then there's a similar criterion for proper maps this case you need to ask is further compact and quasi-separated and then you ask the same condition and the same diagrams there exists a unique error I mean there exists exactly one is it enough to demand existence of an error after an extension of k yes you can own the usual business that you have for schemes it works here as well you can you could also assume that case algebraic eclosed this kind of business works so let's say something from this page so now we want to next I want to talk about the task sites and for a general diamond it is not clear that there are any time maps into X on trivial because the only thing which we are given for a diamond is it says a quasi-pro etal map into it but it's not clear that there's certainly probably no etal maps from perfectoid spaces into here because then we could more easily write this diamond as a quotient of perfectoid space by etalic reverence relation so you can only expect that there are some other diamonds somewhere with an etal map but it's not clear how to produce such guys but the issue goes away under this topological condition of being spatial or just locally spatial but there are such maps x is and then also in the locally spatial case and that's because of the following theorem the x is a spatial diamond someone can find co-filtered in this system of etal maps x I till there I think to x I can assume that they are quasi compact and separated say I could even assume more such that the inverse limit the inverse limit taken of course in the category of pro etal or V sheeps on the category of perfectoid spaces of characteristic P such that this guy is representable and in fact is such a strictly totally disconnected perfected space so that's something that I said orally for a phenoid perfected spaces and in this case he would just take some kind of inverse limit of all possible etal maps but in fact this works for any such spatial diamond so just by using the etal maps you can build such a pro system which will completely disentangle the whole space and we can also arrange which is a technical point but a technically very important point that this map from x infinity tilde to x is universally open as yes otherwise this completely useless such it and it's automatically quasi pro etal so in particular this will give you one way to write x as a quotient of perfectoid space by by pro etal equivalence relation because this map here is quite all right for example one corollary of this is that is that the category of spatial diamonds is closed under fiber products I don't want to explain how this follows but it has to do with this points the topology underlying the definition of spatial guys and this fact that this guy's universally open no product there's an issue if you take two quasi compact objects their absolute product is not quasi compact again because if you take fp long series t or its perfection in the expected would you let me give the example in a second it's also closed under co-founded in the case of double errors and so all connected limits so yeah maybe I should make this general warning and should have done much earlier is that the product of quasi compact perfectoid spaces x1 x2 is not quasi compact and I think all cases except they were maybe one of them is empty so for example if you take the attic spectrum of this guy and takes the absolute product was itself then well one way to think about this is that this is some kind of you should would like to think of this as a punctured formal open unit disk like fp power series would be like a formal disk and then you puncture it by going to the by inverting t and you would expect that if you base change this to any non-acumenial field for example this guy you get like over this field now you haven't you have a physical punctured open unit disk and that's actually what you get so you get some d star so if you think of this as a rigid space it's a set of all x whose absolute value lies strictly between 0 and 1 but such a guy is of course not quasi compact so an equivalent way of saying this is that the final object is not quasi separated in the category of these chief say this V-shef which sends everything just to a point it's not quasi-separated and well this is slightly something funny sometimes something funny because but the product of quasi-separated is still quasi-separated that's okay now I think it's an algebraic talk but one consequence of this for example is that an object being quasi-compact different from asking is that's a map to the final object it's quasi-compact so this object here just one of these affinity perfectoid spaces is quasi-compact but the map to the final object is not because if you take the product with this guy you get something which is not quasi-compact sometimes you have to be careful about what you say well it's just the chief which sends everything to a point it's the problem is that it's not an object in our category because it's some of the adex spectrum of FP if you want but FP is not an analytic object okay so let me now introduce different sites so there's a definition of an etal site but I want to restrict this definition to the locally spatial case because otherwise I don't know in general if there are enough etal maps and then to not run into this potential problem I keep it out from the beginning so x etal the site is supposed to be all the y to x which are etal maps and unfortunately I don't know whether this really implies it why is the locally spatial diamond automatically so I just ask it again so this follows if the map is an addition quasi-separated but in the non-quasi-separated case I don't know I mean it's automatically a diamond the question is whether it's locally so this part is always okay this is automatic if x is just a diamond I can define this quasi-pro etal site to be our quasi-pro etal maps well let me say it automatically why is the diamond and then finally you can also go all the way to the v topology so if x is just any small v-shef I define x v to be all maps from y to x where y is small v-shef and in all cases covers are given by families of surjective maps covers are given that's just some satiric thing right it means that there is a surjection from some perfect that's based on to it I mean module you set theoretic issues you can always find such a surjection from a perfectoid space by just taking over all possible perfectoid spaces and all maps into why it takes a disjoint union of all of them and then you map it to it this would be surjective all maps are automatically Italian and all maps here automatically quasi-pro etal let me make this remark in a second are given by families of jointly surjective families of I mean surjective here as she's automatically and so maybe one one is really interested in our is a taco module so you want to a tall side but on the other hand in many cases you like diamonds are defined by just quasi-pro etal equivalence relations and so you need to make a lot of at least quasi-pro etal descent and sometimes even easier to the V descent and so you need to know how she's on these various different sites are related and it turns out that the relation is actually as simple as possible okay let me from now on also fix a coefficient ring before stating the serum some coefficient ring lambda such that n times lambda 0 for some n prime to P so of course there are maps of sites and then also these are very nice sites so we get all associated maps of topoi so you can pull back she's from the tall side to the quasi-pro etal side and then further back to the V side and these pullback functions turn out to be fully faithful so so by a tilde I denote the category of sheaves so this is a topos I mean in all cases where it makes sense in the first case I can ask that this true of X is a locally special diamond and the second case if you can ask this effects now percent of the tops of the entire site cannot define what should be the topos for non-locally special things or yes by descent yes by descent somehow so maybe I state this right now so containment in these subcategories can be defined be locally can be checked be locally let me just give one example what I mean by this X prime to X is a surjective map of locally special diamonds and F is a V sheave on X then if the pullback lies in say tall topos which sits inside the V topos then F actually lies so this means that for a general V say small V sheave you can define a category which would be some of the tall topos as a category of all V sheaves which has the properties of an ever you pull back to a locally special diamond they lie in the subcategory and this is something which then becomes a V stack and similarly for quasi-proletar sheaves right similarly for quasi-proletar sheaves so for a general small V sheave can define full subcategories these are just set where the sheaves stand I mean yes so far the lambda didn't appear sorry there will be a part 3 and 4 of this where lambda will appear so we will need a similar result for derived categories of lambda modules and that is also true I mean yeah checking this fully faithfulness of course because you have this adjunction it's equivalent to saying that some adjunction maybe some of you okay and so you also have full inclusions so pullback again induces full faithful embeddings from the D plus on the ATAR side of X so this is a derived category of lambda modules on the ATAR side this is bounded below comodically derived category of lambda modules so this is for any locally special diamond contained in the and I could provide a plus here totally logically but for the next statement I actually remove the plus and then if X the diamond it's even true that the next passage works on the unbounded thing and this biggest guy I actually want to abbreviate it just as the DX comma lambda and so then there are conditions that somehow comes is some quasi-proletar guy and there's a further condition that's an ATAR guy sometimes he actually gets this on the unbounded guy so sometimes you actually get full inclusion of DX etal lambda into here but that only works if X is strictly totally disconnected I mean this is related to some left completeness of the etal cat of this DX etal so if the X etal has some unbounded comodically dimension there are some convergence issues here with posting of towers okay you can also do non-nibirian yeah let's let's stick to the set or it'll be in group case and so the last part is the analog of two that containment and these subcategories can be checked real locally I let's see for this statement right now I can't try to look it up in the manuscript but so some parts work without being primed to pee I'm not sure if this already needs it let me see what I'm writing here here I don't seem to have any assumption I think this covers yeah so I think this statement is still true without primed to pee yes so this works actually for anywhere lambda okay so containment and also part for so this means that for any small V-shef X we can define so inside this full derived category which makes sense even just for small V-shef you can define for sub categories and let me switch as the position of the subscript to the left because now the X quasi pro etal doesn't make sense but this D quasi pro etal doesn't make sense okay if I would have put a tilde somewhere instead but I'm not actually sure if that's the same thing right right I don't know that enough of them but so they might this might not be the derived category of anything itself and this maybe contains the D plus etal X lambda what's the following property such that this is equal to this X quasi pro etal lambda if X is a diamond this D at plus X lambda is equal to the D plus X etal lambda X is locally spatial diamond and for the other guy I need to assume it's strictly totally disconnected otherwise it's a left completion of this guy a perfect space so for those things lying in the subcategory is it enough to just look at the homology sheeps and use the condition yes that's cool and so you can if you want to check whether some object here lies in one of these guys it's enough to check it on the homology sheeps okay and it is true that for the for the ones for you might unbounded the right categories what you say that right so this category derived categories are left complete everywhere I don't put a plus they are actually left complete which is what allows me to do anything okay so basically what I'm trying to do is to get all the basic definitions to have some available in the most general setting possible so and this kind of works okay well I want to say I want to say how to actually check in practice what does it actually mean to line these subcategories okay how to check containment so let's say X is a small V sheeps and K is in the derived category of lambda modules on X then the first thing I want to say is that being such a quasi-proletar sheaf is saying that the sheaf is invariant invariant and a change of algebraically closed base field in the following sense so I want to say the following if f from X 2 to X 1 over X is a map of strictly totally disconnected such that f is a the underlying topological spaces it's a homomorphism then the value on K on X 1 and X 2 agree so why do I say this isn't variance on the change of algebraically closed space field but it's strictly totally disconnected guys they essentially are just profanite sets of geometric points so let's forget about this profanite set so then both of these would be just geometric points so this would be automatic somehow sorry yes yes yes X then well these would be both points so it's a homomorphism and then well it's just a map of geometric points and so you're just enlarging some algebraically closed field here and well to get the correct statement you somehow need to do this with a profanite set in place okay so I didn't yet finish the statement one has that if you evaluate this complex on X 1 it maps isomorphically via some up a star to the value of this one n prime to p do you really want to know for every statement whether I need n prime to p I mean I can look it up I would like the enough technicalities you already so I think this could still be true in general I mean I think this is still for any lambda including entorgen I think so but if it's not true I don't take responsibility I will check during the break okay I think this is also true for any lambda so now I want to check whether it's in the tall guy so so assume it's already passed the first test it lies in this quadruple it's on then K lies in the etal X lambda and only if you have some kind of commutation I mean some kind of some kind of finite presentation thing that it takes the commodity of an inverse limit is a direct limit of the commodities so for if and only if for all co-filtered inverse systems spatial diamonds it's called a yi mapping to X and limit y infinity which is against spatial the value on why infinity of K it's just a direct limit so for spectral spaces you can have no spectral maps and here doesn't occur that is right any map between analytic or edict spaces it's always locally spectral so it's a spectral map so this is actually interesting in both directions it's also interesting to know that if you have something which lies on the etal site then you can compute homology on an inverse limit but just taking the direct limits of the commodities but on the other hand it also characterizes and it's probably true that you don't need to check this for all guys but you can somehow make it further assumptions if you want yes I think you can just check this for inverse limit of strictly totally disconnected guys yeah well if I say it's actually totally disconnected I always implicitly mean it's a perfect place all right so I guess it's time for a break so that's a break of 15 minutes okay let's continue so now we have these kinds of derived categories and we know what it means to be quasi pro etal and oh yes I checked it works for any lambda so far okay but now it becomes important so so now we want to do the simplest kind of operation or we can do pullback so that maybe let me just first do pullback just have it done so I mean that's essentially it's also a lot tautology if this is an any map of the pullback from D y lambda D y prime lambda preserves for subcategories D for the pro etal and and then it agrees with I mean so this F upper star some of the pullback for the induced function of these sides of veto point and it agrees with the respectively but you also have such guy pullback function is just on the etal or quasi pro etal sites if these guys line by primer diamonds respectively locally spatial diamonds sorry right I mean I can only say this for the D plus yeah F plus on D plus etal respectively so if I am by prime for example I diamonds then I just previously said that this quasi pro etal derived categories the same thing is the derived category on the quasi pro etal site and so you would have a different functor a priori relating these draft categories on the quasi pro etal site given by pullback on the induced map there but it's the same thing and this is still for any okay so that's the simplest operation but now let's go one step further let's try to push forward and so now it's really important that and lambda is equal to 0 by NP minus prime to P so again we're in the situation that f from sometimes I call them x let's say again y prime to y is a map of small issues I mean it's basically the same statement which is the RFP law star preserves the full subcategories and I have to be a little bit careful the quasi pro etal and also the D etal plus but not in general the D etal sorry D etal plus so this in general it preserves the D etal plus of fsqcqs and so again defined by primer diamonds then it agrees with RF pro etal law star and defined by primer locally spatial diamonds D quasi pro etal which in this case is the D of the length of fsqcqs then it agrees with the RF etal law star D plus etal which in this case is the same as the D plus of blank etal okay so in some sense is the first statement which actually has some input let me say what goes into this for the quasi pro etal case we need invariance of etal cohomology under change further back to a speed base field we just write cohomology it reduces to etal cohomology of some reasonable guys that's because checking that maps something in the D quasi pro etal into itself you somehow have to check that the image satisfies this condition here which means that some of the cohomology of y prime with these coefficients is invariant under this change of algebraic flow space field and so this so this really needs that n times lambda is zero and how to prove this you reduce the two reduced to a statement of Hoover for usual some kind of rigid spaces so this involves some elaborate limit procedures using such limits like this to eventually reduce some stuff which is a finite type and then what Huba does is that in turn reduces the statement about some rigid space of finite type to schemes by using nearby cycles and that nearby cycles invariant in a change of algebraic closed space field and the usual algebraic statement and so all this formalism works well enough really to means maybe kind of surprising that you can get statements about all of these issues by some elaborate reduction to the new Syrian case works and for etal well for etal you need to check this commutation that it takes an inverse limit of spaces to a direct limit of cohomologies but this is actually essentially just use the characterization so the hard part is the invariance on the change of algebraic closed space field okay I claim that a formal consequence of the serum is a base change result and so because it's a color to this serum it of course again needs these assumptions so this is a quasi-compact quasi-separated base change it says the following if say this is a Cartesian diagram of locally spatial diamonds and F is QCQS then base change holds true then the natural base change map from you see if I have names in the same way have my notes and yeah I always mean the etal operation so assume I would only be interested some are in locally maybe I am I'm only interested in locally spatial diamonds and their etal homology then I would be looking at this kind of map which is a map from the d plus etal of y to the d plus etal of x sorry I mean I can also write this really as a d plus of the etal side then this isn't right so Cooper has a similar serum for analytic edict spaces under some new Syrian hypothesis and this generalizes super statement at least for analytic edict spaces of a zp because well maybe I should say this again so if you have an analytic edict space of a zp then the etal side of the associated diamond is the same thing as the etal side of x itself so if you care about analytic edict spaces of a zp then you can now prove theorems about them by regarding them as diamonds and then using these general statements about diamonds so it implies same result basis of a zp as for such x something works to that I already stated last time that the etal side of x to come to the etal side of the diamond and so this is due to Hoover under the Syrian conditions why is this a collaring to the serum well the serum tells us that we may replace this operation on the etal side by the operation on the v side everywhere and all the operations are compatible so but what does it mean to push forward on the v side it automatically means that you're already she-fi-fi on all possible things mapping to x so actually well summer base base change to a slice in the top of this always a formality and so base change in the v topology is actually this autology so x prime v is the summer the slice of x v over x prime is a slice the v topology and base change to slices is formal slice well it's just all the objects mapping to some object on your side this localization okay so that seems like a pretty general base change result in particular it seems better than proper base change because proper mess around particular required to be QCQS unfortunately things work slightly differently with edict spaces and with schemes and so this summer doesn't give you the full analog of proper base change for some reason so the next thing I want to do is I want to actually define the RF lower shriek functor and then I need to show that this is well behaved and then I hope I will come to this I will say at one point where the usual proper base change theorem for schemes enters a proof and so I only at this point we really have used the full analog of proper base change I mean also see that's not so far we haven't used anything like proper base change in the world of schemes you haven't put this into the theory so yes an open immersion can be QCQS there is no complementary closed immersion for QCQS of new merchants the problem is that the subtleties was hiring points so whenever you have an edict space the map will be generalizing but this complementary closed subset will not be generalizing and so that that's exactly related to this warning there actually so now we have as a functor floor star I mean you can also easily define a tensor product there's also no issue was defining an arm except you and maybe you have to be a bit careful about this change of topology but I mean these operations in principle they are just formal tensoring and taking an arm so what we still really need our function is our floor shriek and our upper shriek no I mean if you map something into something else and usually this is only like this competition was direct limits for example in order to be tall you need that the thing you map out of is compact and things like this so okay so we want to define a functor so one very nice feature of edict space is that they have canonical compactifications if they have any but they don't look like what you would expect a compactification to look like so let me first do an example because it's really important to first understand this one example so let's say that k is some non-accommodian fields complete non-accommodian field and let's say B is a closed unit disc concretely it's the edict spectrum of K adjoint T close to the disc then what is the compactification so how to compactify B bar or usually what maybe probably embeds is why an open immersion into something like P1 but you might just embed this into many other proper smooth curves over your field if you want and so this doesn't this wouldn't be canonical but actually what happens is that you can let B bar be the closure of BMP1 and uproises maybe just topological space but actually it has a natural structure as an affinate edict space namely it is the edict space for the same ring except that you mess around a little bit with this suffering here we look at okay plus the maximum ideal so these are all the functions which mod over the versatility field would be constant this I'm another and so let me try to draw a picture so maybe you have the P1 here and okay so there's the usually draw pictures by getting your intuition from the complex numbers and so then the picture would be like so that somewhere you have here's a ball which is one same atmosphere and then well but then the picture you know it looks different because it actually has this Gauss point sitting somewhere near the boundary which is given by the supremum norm so recall that points on your edict space where some variations on norms and one such norm is given by the valuation as a premium or looking at the largest coefficient of such a power series this gives you a point which sits at the boundary and this point happens to have one rank two specialization so this is a rank one point and has a rank two specialization which lies outside the disk let's call this guy here x this little guy so this b bar will just be b union x and maybe gives a formula for what this variation is x is given by sending a sum a and t to the n that's the supremum of the absolute values of a to the n times gamma to the n where this is an element of the positive reals times gamma to the z where it makes this this is a totally ordered to be in group by asking that this gamma is infinitesimally larger than if I have any real number which is bigger than one then it's already bigger than this gamma so this point x thinks that t is infinitesimally bigger than one in absolute value so it doesn't lie in the disk where the absolute value of t is less or equal to one but it's very very close to it okay yeah sorry let's say this is not a zero function okay and so this b bar is not a finite type but it's a wonderful attic space and let me maybe make one remark and maybe one of the key points where I like the Huber's theory of attic space is better than the Berkowitz theory the difference between b bar and b makes the compact support commonology of b with coefficients and say f l b f l and degree minus 2 and so compact support commonology of this b bar was this some are already proper this would be f l so the usual commonology of these guys this would guys would just be f l and degree zero in both cases but the compact support commonology distinguishes very much between the two guys and I mean this is actually what you want to have to have punker a duality on b so whereas you get this answer in Berkowitz's theory so in Berkowitz's theory you also have a general notion of compact support commonology and if you evaluate this for the ball Berkowitz's theory somehow can't distinguish between these two guys and it will give you this answer that's f l which somehow means that for a space like the closed unit disk you don't have punker duality but you have it for b bar you also don't have it for b bar so punker duality only holds for spaces of finite type in the end but you need these spaces which are not a finite type for which this formalism so far works beautifully you need them to define compact support commonology in a canonical fashion commonology it's the same yeah it's both f l and degree zero so unusual commonology you don't really see these hiring points making a difference but for compact support commonology they are crucial okay all right so let's try to talk about canonical compactifications of issues so maybe I start with a proposition if f from y to x is proper map of these sheets and spa r plus some a phenoid guy then you have a more general kind of evaluative criteria namely you also always have the space r r-circ where someone takes the maximum possible r plus which is subspace of spa r plus and they always exist the unique map here in other words if you're only interested in proper spaces then for proper spaces this extra ingredient here this r plus doesn't make it play any role okay and so you can turn this kind of thing around and makes a following definition if y is separated or x is a separated issue you can also do this from maps let x bar it will be its compactification be the same which sends some spa r r plus which is an opportunity to restrict this to totally disconnected x guys maps this to x of r so if you want to map this guy into the compactification you only have to map the much smaller space bar a circ where you're somewhat essentially reduced to the rank one points you only have to map those into x and then you will automatically get a spread of map into x bar and similarly if f from y to x is a separated map of issues you can also define a compactification over x which takes some spa r plus maps to an r-circ point of y which is refined to an r plus point of x again for totally disconnected so if you want to define a v-shef it's enough to define it on the basis for the topology so it's enough to define it for those guys and the formula so if I would evaluate then this x bar the general thing it wouldn't have this formula so I need to write so that's the proposition so that x bar also this y bar over x or v-shefs if x is also separated sorry I mean the second case I don't assume it's separated but you can compute this as some kind of relative guy so it's a absolute compactification of y over the absolute compactification of x with x if x is separated so that this makes sense and if in fact f from y to x is quasi compact and separated then the kind of compactification of f over x this is always proper there's a more general notion of being partially proper so in general this guy f bar over x partially proper something like you forget quasi-compacity but it keeps a variety of criterion so yes no so there are natural maps from x injective maps that's by separatedness of x into x bar also from y into the relative compactification over x so in general you mean without quite a compact or without separated I mean separated is somewhat necessary condition so if f is separated if this map is not even separated I'm not even sure this is a v-shef again so it is separated and partially proper there so what does partially proper mean it means it's separated and satisfies this condition up there I mean for all these guys and there is a slightly different way of saying it where you only have to variety of criterion for fields but then the locally special case it's equivalent but then there was some other stuff well you need to assume it's tout also I don't want to discuss partially proper guys but they need not be open emergence well you can just take the difference but it's not an attic space again so what you're adding here somehow higher-ranked points so any rank one point will be an x but some higher-ranked points will be here so this means it's a boundary will never have the properties that it's stable on a generalizations and hence the boundary will never have the properties as itself an attic space or a diamond or anything like that topologically you can make sense maybe one example if I take the attic spectrum of some r r plus and I compactify it then it's this bar r and then you somehow put the minimal guy f plus plus I search search integral closure so this is a minimal possible so in this way taking the compactification some of what gets everything about the up so definition a separated map from x to y of small v she's compactifiable if if the maps and I make this definition even in the case where the map is not quite a compact and sorry f bar over x sorry no no sorry yeah absolutely right sorry it's just this map is it thank you right open emergence I should have to find means that whenever I map some perfected space onto it the fiber product is representable and given by an opening subspace of and yeah it's also enough to check it be locally should have been one of the definitions that's the beginning today all right time's almost up let me nonetheless finish with definition so let's assume I have such a map f from y to x which is a quasi-compact separated and compactifiable map of small issues so you have y y bar over x compactification over x and then I define our floor shriek to be well given by the usual formula you extend by zero upstairs and then you push forward yes compactifiable stable under composition I mean it's also compactifiable in the sense that if there is any other compactification somewhere so some open immersion to some proper space or partially proper space then also this open immersion to this kind of canonical guy will this map will always be an open immersion itself so if there is any compactification what I said here is if there is any then there is this canonical guy stable under composition and you can also check it be locally and has all sorts of good properties okay so we have this and the serum is affoing and then I stop and so here again for these things it's important that lambda is torsion prime to the characteristic so there's a notion that you are representable on spatial diamonds meaning whenever you map spatial diamond into it the fire project is special again then this takes the D plus it's our more a lambda into the D plus it's our X lambda and this competitor was composition and base change so there is one part of the argument which it doesn't need prime to the characteristic but I'm afraid I'm using something else where I do need it's prime to the characteristic because I'm defining the somehow implicitly with the beat apology to start with so in order to see that it's the same as a quasi-proletar thing I already need for invariance and algebraic closed so time is up so let me just comment by saying that so that maybe three things to prove here but let's discuss these final two that's compatible composition and base change usually what maybe expected I know that some things compared with composition should be easy and the base change should be hard but actually it's the other way around so this is actually easy using what we already have done what's hard is composition and for this part you actually need to use usual proper base change for schemes to prove a certain result about the rescue Riemann spaces so what happens here is that for example if I have spark k ok and compactified and then the underlying topological space is the same thing as a service key Riemann space of the residue field which also shows that the map from spark k ok into here must it's not an open immersion because this is just a generic point of the service key Riemann space mapping into here and this is not an open immersion and so these kind of canonical compactifications automatically makes the service key Riemann spaces appear which are these inverse limits of all proper things with this as a generic point and well to prove something about them some statement about the commod if service key Riemann spaces you need proper base change for schemes all right I need to stop here the easy part requires are you actually right yeah so this compatibility with composition this result is actually something which this result about service key Riemann spaces works even with p torsion coefficients it is what no no no sorry I mean under these assumptions under these assumptions words defined I mean these these are means these are even perfectoid spaces here if you know perfectoid space is a spark k ok and it's compactification but still as a map it's not an open immersion it just takes embeds a generic point into this huge service key Riemann space questions so what's going to be in the second half I need to finish this historical model discussion so so far we have some more proper base change but we still need something smooth space change and property and this kind of stuff may need another lecture and but then finally I want to get to bungee so I want to in particular I really want to do the classification of reflexive sheaves on bungee let's say at miscible representations which is a basic finance result that underlies as a whole you have credits for me I feel projection formula and then the usual things apply this one while some are compactifiable means that in this R plus like you have the minimal choice and then the condition that this bar plus mapping into here is some of the condition that there are only finally many inequalities imposed so some of this R plus must be generated by just finally many elements in a sense so it's what's Hubert calls plus weekly finite type the plus weekly part I mean you by this notion of morphism of plus weekly finite type and the plus refers to the condition that there is this plus suffering is finally generated in the sense it's related to this notion of people when you have this notion that yes but it's also equivalent in the sense so so that's how because I was had a symmetric worry of that is to say how common is they are quite common so for all the spaces that actually will appear in the geometric situation we interested in they will always be compactifiable so whenever you build something from usual which is based on something like this and even in some elaborate procedure you usually preserves this notion of compactifiability