 Let's give all stuck kids that person can't present pop so I switched to English. I'm sorry So I met I saw offer the first time First of all before I start I'd like to say that offer has been a very important participant in the seminar by interaction in Paris So many years and extremely grateful to him for that I met him the first time in Angers in 1979 Where there was an intensive discussion between probably these two characters over the distinction between the Chateau Moreau's 1947 and 1945 there were a couple of bottles on the table and offer offer us all the right question Is there a difference between the yellow one and the red one? And then after that well, there was a kind of remember one point when Susslin was lecturing and some of the term she's been mentioned namely Fugititi serum for algebraic and closed field where there was a cover so Susslin was lecturing there was a cover in this was 1981 1982 And so the curve was smooth and projected so at some point offer a study saying well Projecting not as history it's most not as it's like connected I have for connecting points by Okay. Yeah. Yeah remember this So in 83 He lent he gave us a great help to some six within me when we're right Studying pit ocean and in a child group and co-dimension to using the ramvid comma g and the crystalline comma g So he helped us an awful lot, but he doesn't remember. No, but I think your paper was in 82 You started about the paper The preprint I remember that I told I met in Sonsuk in his office around 83 the paper was published in 83 the paper in history so I can't tell a story about this or I have one minute So there was a diagram which is supposed to commute or anti commutes So one morning some sequence saying it would come up a lot of spectral sequences So I would come up saying it commutes and the next day would say it anti commutes at some point We decided that enough is enough it commutes and then after the paper was published one month later We got a letter again from gabbers. It was a map from CHI to HH to I It minus one to the bar I commutes Let's see and then I remember another story which is typical and which is generic of many things which all of us have Experienced it in 1997. I was giving a talk on somebody else's work after five fifty fifty minus fifty five minutes of lecturing There was a And then there was a flow of consciousness as they say in English And so of our told us many ways to fix the gap about the salt one To understand what he was saying, but he was of course he was right and they are of course you have several ways to fix it Okay, and then and then we come nearly to the topic of the of the conference because of his talk because in 2002 He showed us how to he showed me how to To show that a certain technology class which was non-ramified was in fact non-trivial by spreading it out to some some place where it is actually a ramified Yeah, but it published in 2002, but it was it was I remember I discussed with you in tits conference in the tits conference college the It was maybe no no no, but the discussion with you was before It was in maybe two thousand two thousand one I don't think we stood the tits story is 2006 there's a story in 2006 where You really managed to read it so a paper was submitted and there was a spreading out Out-argument of field which you use to a function field of something of five type of a spec Z and it was a very complicated argument reducing to the paddx and find the field and then Offer me talk about this and from the other end of the room You said oh no no you replace this by a spreading up arguments You take the field and you embed it any field can be a characteristic zero at least is At your back a close in the field of comical dimension one and that was enough to make a much better paper, okay? Well, thank you. Okay. Okay. Thank you with the introductions. I'm not spreading up spring up for families of rigid and active varieties Okay, so I thank you for the introduction and organizers. So this is actually not a new Completely new thing. It is something that I discussed with Brian Conrad starting in 2014 in the spring after he heard talks of Schultz No, maybe yeah In the MSRI workshop that didn't attend where it was based on something actually mentioned to show that some some ideas that I had about Spreading out and it was important for some arguments in P. I. The coach theory and then The I sent him some a scan and then anyway, it was I lectured about it here in the Event for the laboratory Alexander Grotendijk and also in Oberwollfach in 2016 in August so this is essentially so the abstract is actually just a copy for convenience for what I did in Oberwollfach but the content of the talk will not exactly Be the same because maybe I don't remember exactly and I did not There are some notational Anyway, so I will I will I will So the So now there is some of this so this work is actually quoted in the Paper of Bartmore Schultz on integral periodic Hodge theory. In fact one needs less than this. So they want to prove certain things on the work over Algebraically closed Nonarchimedean field Okay, so you haven't Okay Well, this is the razor or so there is a Nonarchimedean so complete for a rank one Valuation so it is a so we are working as usual. We are in the mixed characteristic zero P case and We have some proper Smooth Is a scheme or rigid analytic Space now the case of scheme is easier than the case of rigid analytic spaces because for scheme There is a there are many results on etalc homology for rigid analytic spaces it's there are equivalent definitions like either using berkovic or Hubert's approach with addyx spaces or The rescue Riemann approach in any case it's because there are comparison. This is the cohomology of schemes Now the the in padi coach theory wasn't one is interested in in several Homology theories like there is the etalc homology the deramc homology possibly the crystalline homology of the Special fiber if there is good reduction so And for the deramc homology there is a well-known spectral sequence that Converges to so the hodge to deramm spectral sequence which degenerates when we are dealing with proper smooth schemes or even Keller varieties or synced by rational together with certain class of So over the complex numbers using hodge theory and actually it does not always degenerate For some bad comp smooths I believe for some although I did not have time but I believe it's you need some assumption And in the periodic world one does not need any assumptions. So Schultz approved over so in So in the case of discreetly valued fields way with perfect residue fields one can Okay so So the question is to know whether this degenerates with particular it gives equality of dimension So originally fountains in his work on padi coach theory, but of course he has the case of I mean It comes out from the from the theories. It's coming off. There is a kind of padi approach to this and Also in particular there are comparison here and show you the dimensions of etal homology is the same as the dimension of the deram homology and the question is to do it in In most somewhat more generally for so social did it over discreet So in the general using the theory of perfect with spaces so but this was in In the more classical case of perfect residue field and discrete valuation which actually helps So actually there is a finiteness theorem in general which Also, which also gives some kind of whole state spectral sequence Which involves H Hp x omega q q Manus q yes, okay, so classic. So this is a originally originates from tates works on Pd visible groups and then using the gallo action one can see Tates uses the gallo action to get some results now So of course when the field is algebraically closed one doesn't have any gallo action So here in a in any case we are interested in the question just let us say for dimensions the dimension of the etal homology which is Known to be finite in fact over also over z p but shows as well but it's the question is to know that it's the same as the dimension of the Deram homology and also the sound dimension the hp omega q so for this It's enough To know So there was also a theory of Kind of crystalline homology of x over something that they call be the run plus in This paper of BMS and then this was a Kind of a complex of be the our plus modules Some of this specializes to the run homology But you have to invert some psi to get to be the run where actually then there is some comparison with With etal homology and so the question was whether this is a free be the R plus module and So the the result was that if X over C spreads out And also there was some other argument or there is a good reduction then there was also another argument so spreads out means Then all the questions one can answer all the questions well Then one knows all the facts that I mentioned like the dimensions are Are right so the spreads out is like this that you have the field qp inside see so what you want is a family of Rigid analytic spaces that is a proper smooth family proper smooth Where y is a rigid analytic? well That is a classical rigid analytic space of course you can always Then pass to other spaces if you want so then you you want a C point Of why so that the fiber is the space that you start with and So under the spreading out assumption The idea is that one can use for example for the old stood around the generation one is this family with one is a the kind of the spectral sequence Gives so you have current shifts and then you can specialize it some generic sufficient general point Okay, so now so actually in my work I Prove that this kind of spreading out exists, but in fact For the application in question to that is for the facts that are in this BMS Work one needs slightly less it is enough to have some So enough To spread Out over some Discretely valued field F Containing C Because then you could also make the residue feel perfect and Then the same arguments will work. You don't need it to be QP and now this can be done using more classical The formation theory, but I will need it in the form Which I well, I did not know the references usually in classical things like Schlesinger criterion come as the residue field is the same as the the residue field of the Basering somehow, but it is not difficult to see that the same statements hold so So I want now to review classical the formation theory just for At least I will assert that certain things in the literature extend when you allow finite type Residual field extension, which is not Difficult to see but I don't know exactly if there is a reference for this so not So so there are statements in there So there so in particular there is Schlesinger so-called Schlesinger criterion and The classical reference for the case of Stax is Dachshund Riem in SGA 7 and then There are other people like they but I don't I know that like There is log art in ring, but in any case there so basically we work over Let us say R is a complete netarian local ring with residue field K and so let us fix a Finitely generated field extension and then we consider the category C of Artinian R algebras with residue field given with an isomorphism to K prime and Then I have a functor from this to sets and So there are several natural conditions on the functor Okay, so Okay, so So I have f of First a So if I have a fibered product of So of course unlike unlike unlike the case where you are over small K, it is not clear in general that this fibered product Exit I mean But in any case we are because I consider only fibered product One map is subjective and then there is no problem to see this is still in my category so this maps to F a1 and then the There is also F of So in any case There is one result about when F is representable of F is Pro representable So this is star by Complete netarian local ring over so something in this With radiofil K prime if and only if Local artinian yes just So Okay, so In fact, you can always think of K prime is the residue field of some point in some affine space Over K and then you can pass to the case of trivial residue extension by Okay So it is presented with some condition. Okay. That star is an isoam and then you need Some finite dimensionality of the dual number thing and then For then there is a notion of a Versailles the formation so So a Versailles the formation is something like that The functor representable by some a Some be going to F is Surjective and satisfy the formal smooth some formal smoothness condition so some lifting and So in any case I will So anyway, there is a classical Criterion for which I claim I Mean it is isomorphic to home from completely in the terrain local ring to your so so that this the Schlesinger critique Well, you need this to be an isomorphism and the the tangent space to be I Mean you apply to do a number of a K prime you cannot endow it with vector space structure You want this to be finite dimensional that's and then you also want to know when there is a Versailles the formation space. So that's a slightly weaker condition. This is Well, this is a very classical Thing in any case you assume that this is an isoam and this is one the dual numbers and again the finite dimensionality This is the criterion And then I will actually so actually in algebraic geometry work like flat deformations of of scheme shifts and so on so we have categories so So the case of Stacks in group weights So you want Now that gen F is a pseudo functor and so this is the j7 case so it Is and then I want to have a smooth group or it so Some formal spectrum of some B and then something so I Have a completely and local ring with residue field K prime over R And then I want a kind of formally smooth equivalence relation And so then you have a get a necessary insufficient condition That star is an equivalence of categories plus finite dimensionality for Well, both the the isomorphism club well the set of isomorphism classes and For the automorphisms of the trivial thing and then you this is a quiz. Okay, so now in the case of So now let's let's look at the case of rigid and elitic spaces over C So we know that X over C comes from a formal scheme Topology proper formal scheme over OC so addick and flat So it corresponds to F Formal scheme of our doors which are flat or finite presentation will apply some pseudonythromizer for example P and then Yeah, so now the point is that in the cases that I I work with the ascertaining thing but Instead I Can work with notness some slightly bigger Rings because usually the infinitesimal deformations are So usually F commutes with direct limit of I mean This is the two limit of the categories. So and then the So we have the residue field So chi zero over Let us say the residue field of C descends to a finite type Subfield and then Okay, so Yeah, so there there is a finite type So there is a some Ring of finite type over the integers so that the the model of a mod P descends to R And then you can localize R And then you Yes, so the This are after localization clearly the maximum ideal we'll give you'll put it elements. We'll get Artinian Ring so it has some residue field, which is a finite type and now the There is elementary so the fact that this is an equivalence of categories for in algebraic geometry is is some Well, you some pinching arguments. So you don't have to show the flat scheme over a fiber product ring is the same as giving a collection of flat schemes is comparative with I mean a patching data for flat schemes of a 1 a 2 a 3 which is Not so difficult when the one kernel is new potent. So in any case I will replace OC by this so OC mod P I will replace by this R or R localize and then I will have OC mod P square and so on and then I will have the inverse images So those are not Artinian, but kind of limit. So anyway, then one can pass to the limit and find I have some Versailles deformation space. So I map to I can I can by versatility I can extend the Successively the maps of the Versailles deformation space so that this comes from the Versailles deformation Okay, so now So I have my Yes Yes Why do you need this finally generated extension? I mean, it's not the case of what you will some small power P to the epsilon of P the scheme Yeah, yeah, P to the epsilon somewhere constant. Yeah, the scheme. Yeah, it's true. So you can use Instead of Instead of observing the localization of this is Artinian you can one can do as you I mean as you said because You have the field certainly Lifts to to OC modulo Some principle ideal and then you can do the same thing with this you don't Certainly alright, so you you can yet just use this finally generated field instead of R in OC modulo something and then Do the I mean I don't have to write this P square or so so Can't you just literally use field K Because I mean you can't find this Yes, it's a special fiber over K and what it was some small power. It's already isomorphic to this Yeah, but I want to I want I said before that I am I It is not difficult to see that the classical Three results in the formations for that is a Representative Versa and for the case of stocks, which are usually stated was trivial resued field extension extend to the case of finitely generated Field extensions, but of course, maybe they're also more general field extension. The problem is that Somewhere it's implicit there that you are using things like that the differential the cotangent complex or finite degenerated for this field, I'm not sure that there is a Okay, so You want to read so you can say that the vid vectors over the fear as you feel map to the to the Ah, you can choose it in some way. Yes, and then there is will be no field extension at all because the you can choose Ah But this requires some up. Yes, I know that one can one can I mean Okay, I know that one can do it. Okay, but I did not Okay, so so in fact it is not Well, so let let us But in any case what I say works also for more general and not Archimedean fields. I don't need the to be algebraically close, so Okay, so So in any case I have some Some universal or Versa the formation over some Ring B But then I can this is a finite type So then I can pass to some DVR which is obtained by adding a transcendence basis of the Residue field and then this is finite and then I have this Complete, okay, then this is the kind of this is kind of a special form of skimms and of Berkowitz so it has a kind of General fiber and then you get a family of Proper smooth family over the general fiber of this which is the spreading out that you want So now I want to have a more general Spreading out result so I want So suppose now that X to Y is a Flat proper morphism of Rigid analytic spaces over some Non-Archimedean field and I want to approximate the ring of integers by some the Terian ring a So it's a Could be like So I have just some up and a 10 ring complete with specter ideal I and an attic map to okay now So I know okay, so locally You have a flattening theorem so you have After blowing so you have some formal models and you can after blowing up you can assume that this is a flat proper map or formal models, so Now I want to Yes, so I have the Yes, so I let us say that I have That Y is a finoid so I have a Like over speck or Y Modular power So I have a family of schemes proper flat over this So again the the this descends to some finite type Algebra over a mod I so the Okay, so now I want to prove that the following so let us say that I have Let us say in a fine finite type Morphism and I have a flat family so Properly, so I want to construct a compatible family of Versa and order the formations So the the power of the ideal so and so I want to embed D prime in something of Finite type is the ideal kill ends plus first power of the ideal is zero and extend the morphism to something flat over T prime and So So in particular and then this will give me a Versa kind of So the idea is that you can Define the notion of Versa loans and so the deformation in the sense that if you have any family over a and Some answer the formation it it is a pullback from the Versa line plus the and also in a stronger form that if you Give the the map on the close up scheme between t and t prime then you can extend it in and with extension of the isomorphism so then So the the main point is to construct a Versa first order the formation and for this One is to look at The I there's a I is an OT module Finite type and then I associate to this the category of X over T the X1 over T1 where This is So I want to have a Infinitesimal sickening with square zero ideal given by I and and the lifting of the morphism so So this varies with I so there is a notion. So in Grotten Dick's First or the original one of the first work on deformations was the Elections of Grotten Dick and additive cofibered categories. So an additive cofibered category is a cofibered category over Let's say you have an additive category a and I have some cofibered category over it and I assume so the assumption is that It commutes up to equivalence with finite products including this so zero is the one point and binary product And moreover if there is a notion of left exact when it is a nabillion category Which means that C of a? Kernel of a surjection is the equivalent to Okay, now the things coming from the formation theory are easily seen to be additive cofibered categories Dependence on I and now We have Okay, so For a complex They say in degree Zero and one we associate with the sake you of this complex so In additive Cofibered category, which is as follows you take home C prime X to home C X so this is a an Object, let us say in the additive. So this is a chain complex in of these two terms and then Giving this Gives you what is called a strict picard category where those are the objects and those are the morphisms between Well, the morphism between two objects is something here was that goes to the difference and then you get kind of true the fun goes to So you associate So the the category associated to ease is the one associated to X now so in the case of an a billion so category So the kind of the left exact Additive cofibered categories correspond to two term well always represented by two term complexes in The category policy let us say it is small to avoid so So this is a category with enough projectives and the complex in question is two terms where this is projective and This ensures the left exactness and In fact, it's slightly less and now There what we need some finiteness property. This is just We need some finiteness property in the case is coming from algebraic geometry. So the idea is that you calculate Using the formation theory like the cotangent complex theory of illusix that we need some slight some refinement so the We introduce a topos the top actually we have in general when there are no non constant function One is to look at this as a select the this is a diagram of Topo is there the total topos and so one is to look at the formations there So something classified by our home of the cotangent complex and and then pull back of I And so But also the usual tier the usual way it was written. It does gives you the set of isomorphism classes the Yeah, the isomorphism class the formations are classified by something and so Yeah, so the yeah, so there is So there in the in the course of this there is some use of current duality So if I have something like a proper flat Morphism, let us say and why as a dualizing complex is an Italian this is dualizing complex Let us say with the strong sense that is finite cool dimension as well So it's a like like an archon's book so it so the dual duality exchanges D minus and D plus then You can look at So you can construct some kind of at least partially defined Our joint to F upper star by kind of the and so So that and then we have there is also a generalization of this to to the case of the topos associated with finite diagram and So this this was also done by in some paper of Jack Hall several years ago but we need the case of a topos of this diagram is so and then if we want applies the Yes, if one applies the Only what is So that just the statement of the for a deformation theory in the sense of isomorphism classes One does not get that the additive cofiber category is given by a complex you only get that the the functor set of isomorphism classes is Is Isomorphic to what is given by this complex of course there should be a way to refine it and in fact one can object by object refine it by being very careful about the constructions of of Involving the cotangent complex, but then you have to var x there is some Hierocategorical coherence is seeing so the idea one idea is to look at all let us say all axis Let us say all x is some size restriction and somehow put them together in a topos I mean this and I mean they have a topos of diagram parameterized by this and walking distance and one can I think refine the the Information although Maybe now there are some techniques with higher categories that allow one to see that whenever Things look like it should be queer and they are queer and she's implicit in various talks that are now heard although the lecturers are not always willing to reveal the Well anyway, so Anyway, so Instead of doing this, or maybe I will I don't know if I mean can take a long time to So anyways, there is actually a trick here that you don't really know have to know it in the refined sense because you can Demonstrate that if you know that the additive microfiber category is representable In a weak sense just then it's actually there is a versatile object. So that's I want to explain so if an additive Cofiber Category C over a let us say a is a billion is such that Isomorphism classes of fx is isomorphic functorially to co-kernel home Qx to home px Then Where q is projective Okay, then q and the canonical map give a Versa, no Co-kernel P and the canonical map gives a versatile object. So the surjectivity is Clear but the problem is you have to know it in the refined form with a lifting So I have got let us say X prime I Have some class over X prime and it goes to And I can represent it suppose it's already represented by a Map from p to X prime and then I want to leave this map to a map that represents So I have some object actually a lifting So the point is you can pull back by this you can pull back and use the properties that It sends I mean that you get a pullback square of categories. So you can assume that it's it's a It's that X prime is pin. This is the identity map then the class alpha is represented by some up But then the composition must be the identity up to something that factorizes through q which is projective So you can correct because Q is projected you can leave then you can correct it So there is actually a section But then you have an isomorphism, but I don't see it. So I only know up to isomorphism So I know that my object here Comes from this in some way, but then I get an automorphism over this but since but now since I can lift automorphisms because The origins this comes from this so I can lift automorphisms and then I can fix it. So I get Okay, so now after one does some Well, there are some technical work with stop process and then some technical work on the finding successive ends order Deformations I mean Universal so the idea is that once you have the ends order one you want to pass the end first first order one But the end plus first order one is kind of a first order deformation of some ends order one so you you want to You have something which is where I mean the idea is to take Kind of the you some vessel first order the formation and cut it down so that the the ideal the finding that this well, some product of ideal should be zero and then you have to to Collapse something because after you take this model of the ends the end plus first power maybe you get slightly big something bigger than this you can retract it and And pinch something then you have to prove that you don't have to pinch father and it is still versatile. So that's some argument that I did around 2014 then So after one knows this One can generalize So the rigid and erythric description of so in the concerning the application to rigid geometry So if X over Y Let's say our topology this is topologically finite type of some ok I Can so I this I have my So there is some ideal of definition of of a it goes to an ideal of definition of ok and then I can See that this flat deformation comes from my vessel formal so This comes from Formal schemes Over SPFA except that they are not topologically of finite type They're more like special in the sense of work of it But then you can still make some blowing up reduced to topologically a finite type thing and then you get a map of rigid spaces like in the sense of Renault over over a so are or in the sense of the Book of Fujiwara Kato. So that's and then you get you so for example, I want to prove that The whole should around spectral sequence degenerates so So I want to prove that RPF l star omega q are locally free and The spectral sequence degenerates so So of course there is a question foundational question for which kinds of nice which kind of framework there is enough Theory to speak about this question for but at least I reduce So what one is to do some work with base change of query and current base change but the idea is to reduce to the nice neaterium case and Then you can actually use rings of finite type of a Z and then you have enough specializations enough Where are enough rig points which are Essentially coming from finite extension of qp So this is in the case where the general characteristic is zero, but notice that I can also have Residue characteristic zero, which is this cannot be handled by by just Approximating okay by DVR in okay, so one has to really Spray it over Z and use some characteristic P Residue characteristic P specialization so the the proper smooth so this is an analog of the result over C which is Proved in the paper of the lean that is that if you know for the fibers of the information of the generation it shows by a length argument that in 69 I think or in any case it shows that this is the So comparison is better come or G this paper series degenerate so here one is what? 60 Well, it is a paper on The generation spectral sequences. Okay. I don't okay Anyway, Grotten dick also mentioned it in recall to semi. That's really anyway. It's a Okay, don't so in any case that the idea here is that you reduce to The case of Artinian local rings finite dimensional over Qp and so some family of rigid analytic spaces over this and then the analog of Chris of Betty Comology would be to to develop some kind of Kind of crystalline Comology like what batmore and Schultz would do in this framework so you have to say that if you have a a Chi over SPR So I have also Chi the special fiber of Okay, so I want to see that the RAMc homology of chi doesn't depend on you You also have a constant deformation You want to show that the RAMc homology of chi doesn't is the same when you use a constant deformation So the idea is you can do it by hand without any you can just Define the analog of crystalline Comology by just using Small I mean like a fine cover or some cover where it lifts and then you use a check construction So you use a local local embedding in something smooth and the embedding of the intersection in the In the fiber product is so as known to the expert So you take the formal completion in the RAM complexes You make a complex of this and with hypercromology if needed if those things anyway, so you but So you somehow can compare any two choices by doing both of them You use some easy Poincaré-Lehmann argument in characteristic zero and then you see that this theory is well defined And it depends only on the special fiber and it gives the RAMc homology of any liftings in particular the lengths Is the same and this allows one to use the lens argument and then this gives us that At least in every context in which there are enough foundations for rigid geometry. So of course you need some base change and and Well, so it's it's a question. I mean Which at least in the classical rigid case and an Italian case should be okay But still one one is to use some some base change the argument. Okay, so this is the application which of This result and Then I also want to mention though the That they're spreading out. It doesn't work. You need the proper regionality space. It doesn't work for Non-proper ones and for non-proper ones. There are several ways to think about it So the in the manuscript there is an example in characteristic P based on inseparability So if it spreads out Then it is defined over a field extension of the series always a base DVR. Let's say it's a field extension of a Is a part of the completion of finite type field extension in particular the finite P rank in characteristic P and then The you can if it is not reduced you can make it You can after a finite extension it becomes generically smooth and this is not true for reduced regionality spaces in general because of infinitely many Coefficient so you can but you can also make an example with So for for affinuids are smooth like that can be algebra as well kicks So they're always the send to something good. So now I want to give an example of a non spread out So let us say that I have a Base So I want to find a large extension Such that this is an honor comedian field I want to find some pop some smooth Regionality space over it. It doesn't spread out to a flat family Over K So for this I will take let us in characteristic no, too. I will consider over a disk Cross P1 Kind of I have double covered like a family of hyper elliptic curves. So this is a parameter Z So I have something like y square is equal to X X minus one. Let's say a pseudo-uniformizer Pi then X minus Pi square one plus Z X minus Pi cube They say one plus Phi Z or Phi some infinite series Is coefficients in E. So the idea is that For such an E, which is it slides over a disc and the All of these the ring of global function the point if it descends to some non-alchymidian field and the ring of global function Descend of course it does not sell a disc but after some finite type extension and closure It becomes a disc and then you need some find the tall extent some fine extension to well You need cross-ratio between branch points when you can also have you can construct hyper elliptic Involution and the cross-ratio up to some finite extension. So then you use the fact that you can You can make the so-called topological transcendence degree to several the mission to infinite because you can introduce infinite Transcendence degree in the residue field here and you get a contradiction or you can also if the value group You can also introduce infinite Dimension of the value group this value group mod this value group in this coefficients and you can then see that you cannot Reasonable you cannot descend it to to a small you always need to any field to which it descends as as Residual transcendence degree which is infinite so it's or the value group extension Which is infinite? That question is infinite dimension so it cannot spread out so I guess Okay So in the first thing so if I have a Yeah, it depends on the precise problem which you so let us say I have a non-archimedean field E and the subfield K and I want to Yeah, but then I need always a So the first argument doesn't give the spread out spreading out to with a basis She's originally space over K. It's the best thing one can do is to Adjoin some transcendent else. You have some Gauss valuation over K And then you get it by the argument I said and what Schultz has said is actually in the case of algebraically closed Like Cp, but they know that you can embed you can lift the Residual field so the point is you can just lift the transcendence basis and then use Ansel and I mean you use and Taking peace routes and you you get Let me see I so I want to discuss before this anyway, it is a non-point I forgot now the the Yeah, it is not the classical successive lifting because you need you need to first lift it more the Let's okay, you have a you have FPA join some transcendence basis and separate closure and then you use just piece Route of sometimes Yeah, I think I see I see how it works in any case you can you can do it in non-unique way And then you actually can work over the bit vectors of this perfect fever. This is special in this situation. So But in any case there is a strongest thing that I said but anyway for the application to P. R. The coach theory in question is it's enough to use what he said When and when does a strange adjoint exist there What when does a strange adjoint exist there? What's your hypothesis on K? Okay, so there is the following also this is slightly so this adjoint this funny adjointness also works with twisting by Relatively F perfect when the more morphism is not flat is not flat but you have an F perfect complex and usually I work at least in the originally I just make the The K in D minus L in D plus E is bounded and F perfect then you can see that Tensoring this actually do this but anyway doing all of this still sends you to D plus and then the The assumption so let us say you work with Neterian scheme or finite cruel dimension admitting a dualism complex and then one can Make some argument which uses that the dualizing complex gives an equivalence between dick, I mean exchanges demons and deep class and I mean you need some unbounded complexes, but So the point is that this it looks like that It should so what I also after I know that it exists it So the system So I think it should exist without the assumption of dualizing complex and without the reality if you have enough approximation technique And also maybe you can relax of some burden assumptions but and I think that There is a part so I didn't discuss all of the formation problems and you can do the formation of coherent shapes Sometimes you have to use it for each one. So usually I get that this is bounded I can see that this is bounded when case in D less than or equal to zero This is in D less than or equal to zero in particular you get something for for just for home And of course the stable for home can be easily by approximation reduced to this case when it's a nice So then you can do more Not necessarily Neterian basis not Neterian Yeah, so I don't know the So the So there is a paper of Jack Hall around 2000s. Well, I saw a paper in 2013. I think maybe so Now what we need is the case of finite diagram is the same thing at least diagram we consider So there are one is to work a little and see the same similar effect hold but So But I think now when people have this viewpoint of infinity categories and some General criteria for existence of adjoint in some whatever say I mean you can see that this Must accept that they usually everything is This let me see so you you want that they call limits in K go to this It's an upper stock and it takes limits to limits You need Yeah, but of course here everything is is Coherent so I doesn't in this argument with duality everything is coherent with some boundless restrictions So one doesn't see those phenomena you are speaking about but if you want to do it very generally then First of all, this is in the quasi-coherent setup So I'm not sure about Yeah, so F uppers star commutes with No, no, but this means that At least if you restrict yourself to with some finance hypothesis you will get such a thing now So of course nowadays, maybe if you want to know what it means for Unbounded things, maybe you have to work In some of guess guru stuff where you take pro in all kind of things that I don't know about and then you Always get something in which certain things hold and then you can say that you What No, no, but I'm saying that there are people there are some talks where it's really important to understand the The subject to have to have this theory with With I mean, I would have I mean, I'm not saying that I but I already heard it several years ago and So I but I didn't study. I mean, so I'm still I'm still bounded in some sense, but Okay