Spreading-out for families of rigid analytic spaces
Gabber, Ofer
Introduction by Jean-Louis Colliot-Thélène (Université Paris-Sud) Let K be a complete rank 1 valued field with ring of integers OK, A an adic nœtherian ring and φ: A → OK an adic morphism. We show that if g: X → Y is a proper flat morphism between rigid analytic spaces over K then locally on Y a flat formal model of g is the pullback of a proper flat morphism between formal schemes topologically of finite type over A. For this, if S is an affine nœtherian scheme, T0→S affine of finite type and X0→T0 proper flat, we construct a compatible system of versal n-th order deformations of X0→T0 over S. As an application, one can prove that for a proper smooth g and K of characteristic 0, the Hodge to de Rham spectral sequence for g degenerates and the R q g∗ΩpX/Y are locally free. This is reduced to the case where K is a finite extension of Qp and Y is a nilpotent thickening of Sp K, where the result over K was proved by Scholze and follows for Y by imitating the proof of Deligne over C using a construction of crystalline cohomology in this case.
 Let's give all stuck kids. 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 Auffaire has been a very important participant in the Seminar Varia Interation in Paris over many years. I'm 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 Auffaire asked the right question. Is there a difference between the yellow one and the red one? In the meantime, yes. And then after that, well, I remember one point when Susslin was lecturing at a ceremony which has been mentioned, namely Fugititi Serum for Agile Break at Klosfield, where there was a cove. So Susslin was lecturing. There was a cove in which this was 1981. 82. 1982. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. And so the cove was smooth and projected. So at some point, Auffaire started saying, well, projected not Susslin. Smooth not Susslin. Connected. Ah, yeah, for connecting points by, OK, yeah, yeah. Remember this. And then he left. So in 83, he lent, he gave us a great help to Sonsu Xulainmi when we were studying pitotian in the Chow Group and Co-Demption 2 using the Ramvid Comal G and the crystalline Comal G. So he helped us an awful lot, but he doesn't remember. No, no, but I think your paper was in 82. You started the preprint. The preprint, I remember, I told, I met in Sonsu in his office around 83. The paper was published in 83. The paper in these three, so I can tell the story about this. I have one minute. So there was a diagram which was supposed to commute or anti-commute. So one morning, Sonsu Xulainmi would come up, there were lots of spectral sequences, Sonsu Xulainmi would come up saying, it commutes. Then the next day, we'd 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 from Gabbers. It was a math from CHI to H2I. Minus one to the bar I commutes. And then I remember another story, which is typical and which is generic of many things which all of us have experimented. In 1997, I was giving a talk on somebody else's work. After 15 minutes of lecturing, there was a growl. And then there was a flow of consciousness, as they say in English. And so Alfer told us many ways to fix the gap. About the salt one. Salt money. So we took up all the lunch to understand what he was saying. But of course, he was right. And of course, he had several ways to fix it. And then we come nearly to the topic of the conference. Because of his talk, because in 2002, he showed us how to show that a certain biology class, which was non-remifed, was in fact non-trivial by spreading it out to some place where it is actually a remifed. Yeah, you're not going to hear that one. Is it in the same with the pen? Yeah, but it published in 2002. But it was there. I remember I discussed with you in the TITS conference college. It was maybe. No, no, no. But the discussion with you was before. It was in maybe 2000, 2001. No, the thing with the TITS is 2006. There's a story in 2006 where he managed to read it. So a paper was submitted. And there was a spreading out argument of field, which you use to function field of something of phi type of a spec z. And it was a complicated argument reducing to the p at x and finding field. And then Hoffer asked me, talking about this. And from the other end of the room, he said, oh, no, no, you replace this by a spreading of arguments. You take the field and you embed it. Any field can be, characterizing zero at least, is algebraic closed in the field of Complicical Dungeon 1. And that was enough to make a much better paper. OK, well, thank you very much. OK, thank you for the introduction. I'm not spreading out. Spraying out for families of rigid and active varieties, joint work with Brian Pudder. OK, so I thank you for the introduction and organizers. So this is actually not a 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, it didn't attend, where it was based on something I actually mentioned to Schultz some ideas that I had about spreading out. And it was important for some arguments in the periodic coach theory. And then I sent him some scan. And then anyway, I lectured about it here in the event for the Laboratory Alexander Grotenweg 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 now some of this work is actually quoted in the paper of Bartmauer Schultz on the Integral Piadic Coach theory. In fact, one needs less than this. So they want to prove certain things on the work of an algebraically closed non-archimedean field. So could you write? OK. So you haven't. OK. Well, this is the eraser. So there is an onarchimedean, so complete for rank one valuation. So it is a, so we are, as usual, we are in the mixed characteristic 0p 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 space. It's because for scheme, there are many results on etalc homology for rigid analytic space. It's there are equivalent definitions, like either using Berkovich or Huber's approach with adic spaces or the risk-e-rimman approach. In any case, it's because there are comparisons with the homology of schemes. Now, the in Piadic Coach theory wasn't, one is interested in several homology theories, like there is the etalc homology, the deram homology, possibly the crystalline homology of the special fiber, if there is good reduction. And for the deram homology, there is a well-known spectral sequence that converges to, so the Hodge-to-deram spectral sequence, which degenerates when we are dealing with proper smooth schemes or even Keller varieties or seems barrational to Keller. There was a certain class of, so over the complex numbers using Hodge theory. And actually, it does not always degenerate for some bad, smooth, I believe, or some, although I did not have time, but I believe it's, you need some assumption. And in the Piadic world, one does not need any assumption. So Schultz approved over, so in the case of discreetly valued fields with perfect residue fields, one can, OK. So the question is to know whether this degenerates, when particular it gives a quality of dimension. So originally, Falking's in his work on Piadic Hodge theory, but of course, he has the case of, I mean, it comes out from the theories, it's coming of the reason of the Piadic approach to this. And also, in particular, there are comparison theories on the dimensions of etalc homology. It's the same as the dimension of the deram homology. And the question is to do it in, more generally for, so Schultz did it over discreet, so in the general, using the theory of perfected spaces. So but this was in the more classical case of perfect a residue field, when discrete valuation, which actually helps. So actually, there is a finiteness theorem in general, which also gives some kind of Hodge-Tate spectral sequence, which involves Hpx omega q, q, something, minus q, yes. So this is originally originates from Tate's works on the visible groups. And then using the Galois action, one can see, Tate uses the Galois action to get some results now. So of course, when the field is algebraically closed, one doesn't have any Galois action. So here in any case, we are interested in the question, just let us say for dimensions, the dimension of the etalc homology, which is known to be finite. In fact, over also, over Zipi, but Schultz has worked. But the question is to know that it's the same as the dimension of the deram homology, and also the sum of dimensions of Hp omega q. So for this, it's enough to know, oops, like this. So there was also a theory of a kind of crystalline homology of x over something that they call Bderam plus in this paper of BMS. And then this was kind of a complex of BDR plus modules. Some of this specializes to the deram homology. But you have to invert some psi to get to Bderam, where actually then there is some comparison with etalc homology. And so the question was whether this is a free BDR plus module. And so 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 right. So the spreads out is like this, that you have the field qp inside c. So what you want is a family of rigid analytic spaces, let us say proper smooth family, proper smooth, where y is a rigid analytic, well, let us say classical rigid analytic space. Of course, you can always then pass to other spaces. If you want, so then you want a c point of y, 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 student, the round generation, one has this family, one has the kind of the spectral sequence. So you have coherent shifts, and then you can specialize it, some generic sufficiently general point. So now, so actually in my work, I prove that this kind of spreading out exists. But in fact, for the application in question, 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 discreetly valued field, f containing c, because then you could also make the residue field 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 deformation theory, but I will need it in a form which I, well, I did not, I don't know the references, usually in classical things like Schlesinger criterion, some of the residue field is the same as the residue field of the basering somehow. But it is not difficult to see that the same statements hold. So I want now to review classical deformation theory just for, at least I will assert that certain things in the literature extend when you allow finite type residue field extension, which is not difficult to see, but I don't know exactly if there is a reference for this. So the statements in the, so in particular, there is Schlesinger, so-called Schlesinger criterion, and the classical reference for the case of Stax is Dockstein-Greem in SGA 7, and then there are other people like the, but I don't, I know that like there is log-art in ring, but in any case, 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-algebra 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. OK. So OK. So I have f of first a. So if I have a fiber product of, so of course, unlike the case where you are over small k, it is not clear in general that this fiber product, I mean, but in any case, I consider only fiber product per, one map is subjective, and then there is no problem to see that it is still in my category. So this maps to FA1. And then there is also f of, so in any case, there is one result about when f is representable, is pro-representable. So this is star by a complete netarian local ring over. So something in this with residue field k prime, if and only if. Is the object a local ring? Local artinian, yes. Just so. OK. So in fact, you can always think of k prime as 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. OK. So it is pro-representable with some condition. OK. That star is an isome, and then you need some finite dimensionality of the dual number thing. And then there is a notion of a Versailles deformation. So a Versailles deformation is something like that the factor representable by some b going to f is a surjective and satisfy the formal smoothness condition. So some lifting. And so in any case, I will. So anyway, there is a classical criterion for which I claim. You are reading it over, because I'm the programming that has the loop. I mean, it is isomorphic to home from completely an Italian local ring to your. So the Schlesinger criterion, well, you need this to be an isomorphism and the tangent space to be, I mean, you apply to dual number over k prime. You can endow it with vector space structure. You want this to be finite dimensional. And then you also want to know when there is a Versailles deformation 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 isom, and this is one of the dual numbers. And again, the finite dimensionality. This is the criterion. And then I will actually, so actually in algebraic geometry, we work like flat deformations of scheme shifts and so on. So we have categories. So the case of stacks and group weights. So you want, now gen F is a pseudo-factor. And so this is the J7 case. So it is, and then I want to have a smooth group weight. So some formal spectrum of some b and then something. So I have a completely 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 necessary and sufficient condition that star is an equivalence of categories plus finite dimensionality for, well, both the isomorphism class. Well, the set of isomorphism classes. And for the automorphisms of the trivial thing. And then you, this is a, OK. So now in the case of, so now 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, topologic proper formal scheme over O C. So addic and flat. In fact, so it corresponds to a formal scheme over those, which are flat or finite presentation where pi is some pseudo-uniformizer, 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 not 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 0 over, let us say, the residue field of C descends to a finite type subfield. And then, okay, so, yeah, so there is a finite type, so there is some, let us say, ring of finite type over the integers so that the model of mod P descends to R, and then you can localize R. And then you, just a moment. Yeah, so the, this R, after localization, clearly the maximal ideal, we give nilpotet elements, we get Artinian ring, so it has some residue field which is a finite type, and now there is elementary, so the fact that this is an equivalence of categories in a Gebraic geometry is some, well, some pinching argument, so you have to show that the flat scheme of a revival product ring is the same as giving a collection of flat schemes is compared with, I mean, a patching data for flat schemes of A1, A2, A3, which is not so difficult when one kernel is nilpotent, 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 squared 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, by Versailles, I can extend the, successively the maps of the Versailles deformation space so that this comes from the Versailles deformation. Okay, so now I have my, yes, yes, why do you need this finally generated extension, I mean, it's not the case of what you will sum small power P to the epsilon of P, the scheme, what will P to the epsilon sum of constant? Yeah, the scheme, yeah, it's true, so you can use, instead of, yeah, 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, it's lift to, to Oc modulo sum principle ideal, and then you can do the same thing with this, you don't, certainly you are right, so you, you can just use this finally generated field, instead of R in Oc modulo sum thing, and then do the, I mean, you don't have to write this P square also, so, Can you just literally use field K, because, I mean, you can find this, you get some special fiber over K, and modulo sum small power is already isomorphic to this constant extension. Yeah, but I want, I want, I said before that I, I, I, it is not difficult to see that the classical three results in the formations for the, for a representative Versa, and for the case of Starks, which are usually stated was trivial residue, field extension extend to the case of finitely generated field extensions, but of course maybe there are, there are also more general field extensions, the problem is that some of it's implicit there, that you are using things like the differential, the cotangent complex of finite degenerated for this field, I'm not sure that there is I want to reduce the, there's no field exchange at all. Ah, okay, so, ah, you want to, so you can say that the V vectors over the field, residue field map to the, to the, ah, you can choose it in some way, yes, and then there is, there will be no field extension at all, because the, you can choose, ah, but this requires some, 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, ah, so in fact it is not, well, so let, let us, ah, but in any case what I say works also for more general non-archimedean fields, I don't need the, the two to be algebraically closed, so, okay, so, ah, so in any case I have some, ah, some universal or, ah, a universal deformation over some, some ring B, but then I can, this is a finite type, ah, 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 has a kind of, this is kind of a special form of skew and send of vercovich, so it has a kind of general fiber, and then you get a family of, ah, proper smooth family over the general fiber of this, which is the spreading out that you want, ah, so now, ah, I want to have a, a more general, ah, this, spreading out result, so, ah, I want, ah, so suppose now that X to Y is a, a flat, proper morphism of rigid analytic spaces over some non-alchimedian field, and I want to approximate the ring of integers by some, the terian ring A, so it's, ah, could be like, ah, so I have just some map, and the terian ring complete with respect to the ideal I, and add the map to okay, now, ah, so I know, okay, so locally, ah, you have a flattening theorem, so you have, ah, after blowing, so you have some formal models, and you can, after blowing up, you can assume that this is a flat, proper map of formal models, so, ah, now, ah, I want to, ah, yes, so I, I have the, ah, yes, so I, let us say that I have, ah, that Y is a finoid, so I have, ah, like over spec OI, modulo power, so I have a family, a family of schemes, proper flat over this, ah, so again, the, the, this descends to some finite type, ah, algebra over A mod I, so the, ah, okay, so now, I, ah, want to prove that, ah, the following, so let us say that I have, ah, ah, let us say in a fine, a finite type, ah, morphism, and I have a flat, ah, family, so, ah, proper letters, so I want to construct a compatible family of, ah, versal N's order deformations, so the, the power of the ideal, so, ah, so I want to, ah, embed D prime in something of a finite type, so the ideal, the N's plus first power of the ideal is zero and extend the morphism to something flat over T prime, and, so the, so in particular, ah, and then this will give me a versal, ah, kind of, ah, so the idea is that you can, ah, define the notion of versal N's order deformation in the sense that if you have, ah, any, ah, family over, ah, and some N's order deformation, it, ah, it is a pullback from the versal one plus the, and also in a stronger form that if you give the, the map on the closed subskim between T and T prime, you can extend it in, in this extension of the isomorphism, so, ah, then, ah, so the, the main point is to construct a, a versal first order deformation, and for this, ah, one is to, ah, look at, ah, the I, there's the I's and R T module finite type, and then I associate to this the category of X over T, say X1 over T1, where, ah, this is, ah, ah, so I want to have, ah, ah, infinitesimal seconding with square zero ideal given by I and the, and the lifting of the morphism, so, ah, so this varies with I, so there is a notion, so in Grotten-Dick's, ah, first order, origin, one of the first work on deformations was the, ah, elections of Grotten-Dick and additive cofibre categories, so an additive cofibre category is a cofibre, and I have a category over, ah, let's say I have an additive category, A, and I have some cofibre category over it, and I assume, so the assumption is that, ah, 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 an abelian category, which means that C of a kernel of a surjection is the equivalent to OK. Now the things coming from the formation theory are easily seen to be additive cofibre categories, dependence on I, and now, ah, we have, ah, OK, so for a complex, let's say in degree, ah, zero and one, we associate, let's say Q of this complex, so an additive cofibre category, which is as follows, take home C prime X to home CX, so this is an object, let us say, in the additive, so this is a chain complex with 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 that goes to the difference, and then you get kind of two different those two, ah, so you associate, so the category associated to this is the one associated to X now, so in the case of an abelian category, ah, so the kind of the left exact, ah, additive cofibre categories correspond to two term, well, always represented by two term complexes in, ah, in the category Pro C, let us say it's small to avoid, so, ah, 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, you need slightly less, and now, ah, what we need some finiteness property, this is just, we need some finiteness property in the case coming from algebraic geometry, so the idea is that you calculate, ah, using the formation theory, like the cotangent complex theory of illusics, and we need some slight, some refinement, so the, we introduce a topos, the topo actually we have in general, well, there are no, no, no constant function, one is to look at this as a, so like the, this is a diagram of topo, it's a total topos, and so one is to look at the formations there, ah, so something classified by our home of the cotangent complex and, and then pull back of I, and so, but also the usual, the usual way it was written, it just gives you the set of isomorphism classes, the, the isomorphism class, the formations are classified by something, and so, ah, yeah, so the, yeah, so there is, ah, so in the course of this, there is some use of quaren duality, and so, if I have something like a proper flat morphism, let us say, and why as a dualizing complex, is in the terriand, it's a dualizing complex, so the same is the strong sense that is finite cold dimension as well, so it's a, like, like in Archon's book, so it, so the duality exchanges D minus and D plus, then, ah, you can look at, ah, so you can construct some kind of at least partially defined, ah, joint to F upper star by, ah, kind of the, and so, ah, 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, ah, so this, this was also done by some paper of Jack Hall several years ago, but we need the case of a topos of this diagram as a, so, ah, and then, if we want to apply the, ah, if one applies the, only what is, ah, so just the statement of the, the formation theory in the sense of isomorphism classes, one does not get that the additive confibled category is given by a complex, you only get that the, the fact or set of isomorphism classes is, ah, 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 higher categorical coherency thing, so, the idea, one idea is to look at all, let us say, all x's, let us say, all x's is some size restriction and somehow put them together in a topos, I mean, with, and, I mean, the topos of diagram parameterized by this and working this and then one can, I think, refine the, the, the, the information, although maybe now there are some techniques with higher categories that allow one to see that whenever things look like they should be coherent, they are coherent, which is 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, it 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 confable 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 cofibre category C over A, let's say A is a billion, is such that the isomorphism classes of fx is isomorphic functorially to kernel, home, qx, to home, px, then, where q is projective, okay, then q and the canonical map give a versatile, no, a co-kernel, p and the canonical map gives a versatile object. So the surjectivity is clear, but the problem is that you have to know it in 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 lift this map to a map that represents, so I have some object actually 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 pulvic square of categories, so you can assume that it's, it's, it's, that x prime is p and this is the identity map. Then the class alpha is represented by some map, 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 projective, you can lift and you can correct it, so there is actually a section. But then you have an isomorphism, but I don't see it, I only know opto-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 object, 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, there are some technical work with stop process and then some technical work on defining successive N's order deformations, I mean universal. So the idea is that once you have the N's order one, you want to pass the N first order one, but the N plus first order one is kind of a first order deformation of some N's order one, so you, you want to, you have something which is, I mean the idea is to take kind of the, some verse of first order deformation and cut it down so that the ideal defining 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 N's, the N 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 further 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 erratic description of, so in the concerning application to rigid geometry, so if x over y, let's say this topologically finite type of some okay, I can, so I, let's say I have my, so there is some ideal of definition of, of, of A that goes to an ideal of definition of okay, and then I can see that this flat deformation comes from my verse, so this comes from formal schemes over SPFA, except that they are not topologically finite type, they are more like special in the sense of Verkovich, but then you can still make some blowing up, reduce to topologically finite type thing, and then you get a map of rigid spaces like in the sense of Renault over, over A, so R, or in the sense of the book of Fujiwara Akato, so that's, and then you get, you, so for example, I want to prove that the, hold should the ram spectral sequence degenerates, so, so I want to prove that RPF star omega q are locally free, and the spectral sequence degenerates, so of course there is a question, foundational question for which kind of nice, which kind of framework there is enough theory to speak about this question for, but at least I reduce, so one is to do some work with base change of quaren, quaren base change, but the idea is to reduce to the nice Neterian case, and then you can actually use rings of finite type over 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 rescue characteristics zero, which is, this cannot be handled by, by just approximating okay by DVR, so one is to really spray it over Z and use some characteristic P, rescue characteristic P specialization, so the proper smooth, so this is an analog of the result over C, which is proved in the paper of Deline, that is that if you know for the fibers, the information of the generation, it shows by a length's argument that in 69 I think, or in any case, it shows that this is the, so comparison with Betik homology, the spectral sequence degenerates, so here one is, what, 60, well it is the paper on the generation of spectral sequences, okay, I don't, okay, anyway Grottenbeek also mentioned it in the call to semi, it's really, anyway, it's, okay, don't, so in any case the idea here is that you reduce to the case of a Artinian local rings, finite dimensional over qp, and so some family of rigid analytic spaces over this, and then the analog of Betik homology would be to develop some kind of crystalline homology, like what Bartmoulin-Scholze do in this framework, so you have to say that if you have a chi over SpR, so I have also chi, the special fiber of, okay, so I want to see that the Dirac homology of chi doesn't depend, you also have a constant deformation, you want to show that the Dirac 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 homology by just using a small, I mean, like a fine cover, you have some cover where it lifts, and then you use a check construction, so you use local embedding in something smooth, and the embedding of the intersection in the fiber product is known to the experts, so you take the formal completion and the Ram complexes, you make a complex of this, and reciprocal homology if needed, if so, 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 Ram homology of any liftings, in particular the length is the same, it allows one to use the Lienz 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, so it's a question, I mean, which, at least in the classical rigid case and in the utilian case should be okay, but still one is to use some base change argument, okay, so this is the application which, I think, is the result of this result, and then I also want to mention though that the spreading out, it doesn't work, you need the proper rigid analytic space, it doesn't work for non-proper ones, there are several ways to think about it, so in the manuscript there is an example in characteristic p, based on separability, so if it spreads out, then it is defined over a field extension of the, so it's always a base DVR, let's say, it's a field extension of, it's a part of the completion of finite type field extension, in particular the finite p rank in characteristic p, and then you can, if it is not reduced, you can make it, after a finite extension it becomes generically smooth, and this is not true for reduced regionality spaces in general because of infinitely many coefficients, so you can, but you can also make an example with, so for affinity there can be algebraic by a kick, so they always descend to something good, so now I want to give an example of 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-chimedian field, I want to find some smooth regionality space over E that doesn't spread out to a flat family over K, so for this I will take, let us say in characteristic no 2, I will consider over a disk cross p1, kind of a double cover, like a family of hyper elliptic curves, so this is a parameter z, so I have something like y squared is equal to x, let us say absolute uniformizer of pi, then x minus pi squared 1 plus z, x minus pi cubed, let's say 1 plus phi z, phi is some infinite series, this coefficients in E, so the idea is that for such an E which is, it slides over a disk and all of these are in a global function, the point is if it descends to a synonyms of the field and the ring of global function descends, of course it is not necessarily a disk but after some finite type extension and closure it becomes a disk and then you need some finite extension to, well, you need cross ratio between branch points, you can also construct hyper elliptic evolution and the cross ratio up to some finite extension, so then you use the fact that you can make the so called topological transcendence degree, this is 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, this value group in these coefficients and you can see that you cannot descend it to a small, you always need the any field to which it descends as residue transcendence degree which is infinite, so it's the value group extension which is infinite, the quotient is infinite dimension, so it cannot spread out, so I guess, that's the way. So in the first thing, so if I have, yeah, it depends on the precise problem which you, so let's say I have a nonarchimedean field E and a subfield K and I want to, yeah, but then I will need always say, so the first argument doesn't give the spread out, spreading out to, with the basis is regionality space of okay, so the best thing one can do is to adjoin some transcendentals, you have some Gauss valuation of okay and then you get it by the argument they said and what Schultz said is actually in the case of algebraically closed, like CP, they know that you can, you can lift the residue field, so the point is you can just lift the transcendence basis and then use Enzel and, I mean, you use, and taking peace routes and you get, let me see, I saw, 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 to first lift it more the, okay, you have a, you have FPA join some transcendence basis and separate closure and then you use just peace route of sometimes, yeah, I think, I see, I see how it works, so in any case you can, you can do it in a nonunique way and then you actually can work over the width vectors with these perfect people, this is special in this situation, so, but in any case there is a stronger thing that I said, but anyway for the application to piatic college theory in question it's enough to use what he said, so. How do you put, when does a strange adjoint exist there? What? When does a strange adjoint exist there, what's the hypothesis on K? Okay, so there is the following also this is slightly, so this adjoint, this final adjointness also works with twisting by a relatively F perfect when the 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 in F perfect, then you can see that tensoring this actually, but anyway doing all of this still sends you to D plus and then the, the assumption, so let us say you work with a netarian scheme of finite cool dimension admitting a dualizing complex and then one can make some argument which uses the dualizing complex gives an equivalence between D, I mean, exchanges, D minus and D plus 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 it exists, no, what? So I think it should exist without the assumption of dualizing complex and without netarianity if you have enough approximation technique and also maybe you can relax some bounding assumptions but, and I think that there, there is a part, so I didn't discuss all of the formation problems then you can do differential covariance shifts or sometimes you have to use it for H1, so usually I get that this is bounded, I can see that this is bounded when K is 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 nice, so then you can do more, not necessarily netarian basis, not netarian, yeah so I don't know the, so the, so there is a paper of Jack Hall around two thousand, well I saw it probably in 2013 I think maybe, so, now what we need is the case of finite diagram is the same thing, or at least diagram we consider, so there are, there are, one has to work a little and see the similar effect hold, but, I think now when people have this viewpoint of infinity categories and some general criteria for existence of adjoint in some whatever, I mean you can see that this must accept that usually everything is in, up a star usually doesn't commute with provokes because, I mean there was limits because there's some tendering involved, what, this, let me see, so you want that the core limits in K go to this, you need, up a star takes limits to limits, you need, yeah but of course here everything is, is, is coherent so it doesn't, in this argument with duality everything is coherent with some boundless restriction 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 upper star commutes with, commutes with cold limits, with, with cold limits, with our, no, no, but this means that at least if you restrict yourself to, with some finalist hypothesis you will get such a signal, 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, but I didn't study, I mean, so I'm still, I'm still bounded in some sense, but, okay.
 Let's give all stuck kids.  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 Auffaire  has been a very important participant  in the Seminar Varia Interation in Paris  over many years.  I'm 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 Auffaire asked the right question.  Is there a difference between the yellow one and the red one?  In the meantime, yes.  And then after that, well, I remember one point  when Susslin was lecturing at a ceremony which has been mentioned,  namely Fugititi Serum for Agile Break at Klosfield,  where there was a cove.  So Susslin was lecturing.  There was a cove in which this was 1981.  82.  1982.  82.  82.  82.  82.  82.  82.  82.  82.  82.  82.  82.  82.  And so the cove was smooth and projected.  So at some point, Auffaire started saying,  well, projected not Susslin.  Smooth not Susslin.  Connected.  Ah, yeah, for connecting points by, OK, yeah, yeah.  Remember this.  And then he left.  So in 83, he lent, he gave us a great help  to Sonsu Xulainmi when we were studying  pitotian in the Chow Group and Co-Demption 2  using the Ramvid Comal G and the crystalline Comal G.  So he helped us an awful lot, but he doesn't remember.  No, no, but I think your paper was in 82.  You started the preprint.  The preprint, I remember, I told,  I met in Sonsu in his office around 83.  The paper was published in 83.  The paper in these three, so I can tell the story about this.  I have one minute.  So there was a diagram which was supposed  to commute or anti-commute.  So one morning, Sonsu Xulainmi would come up,  there were lots of spectral sequences,  Sonsu Xulainmi would come up saying, it commutes.  Then the next day, we'd 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 from Gabbers.  It was a math from CHI to H2I.  Minus one to the bar I commutes.  And then I remember another story,  which is typical and which is generic of many things  which all of us have experimented.  In 1997, I was giving a talk on somebody else's work.  After 15 minutes of lecturing, there  was a growl.  And then there was a flow of consciousness,  as they say in English.  And so Alfer told us many ways to fix the gap.  About the salt one.  Salt money.  So we took up all the lunch to understand what he was saying.  But of course, he was right.  And of course, he had several ways to fix it.  And then we come nearly to the topic of the conference.  Because of his talk, because in 2002, he showed us how to show  that a certain biology class, which was non-remifed,  was in fact non-trivial by spreading it out  to some place where it is actually a remifed.  Yeah, you're not going to hear that one.  Is it in the same with the pen?  Yeah, but it published in 2002.  But it was there.  I remember I discussed with you in the TITS conference  college.  It was maybe.  No, no, no.  But the discussion with you was before.  It was in maybe 2000, 2001.  No, the thing with the TITS is 2006.  There's a story in 2006 where he managed to read it.  So a paper was submitted.  And there was a spreading out argument of field,  which you use to function field of something of phi type  of a spec z.  And it was a complicated argument reducing  to the p at x and finding field.  And then Hoffer asked me, talking about this.  And from the other end of the room,  he said, oh, no, no, you replace this  by a spreading of arguments.  You take the field and you embed it.  Any field can be, characterizing zero at least,  is algebraic closed in the field of Complicical Dungeon 1.  And that was enough to make a much better paper.  OK, well, thank you very much.  OK, thank you for the introduction.  I'm not spreading out.  Spraying out for families of rigid and active varieties,  joint work with Brian Pudder.  OK, so I thank you for the introduction and organizers.  So this is actually not a 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,  it didn't attend, where it was based on something  I actually mentioned to Schultz some ideas that I had  about spreading out.  And it was important for some arguments in the periodic coach  theory.  And then I sent him some scan.  And then anyway, I lectured about it here  in the event for the Laboratory Alexander Grotenweg  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 now some of this work is actually  quoted in the paper of Bartmauer Schultz  on the Integral Piadic Coach theory.  In fact, one needs less than this.  So they want to prove certain things  on the work of an algebraically closed non-archimedean field.  So could you write?  OK.  So you haven't.  OK.  Well, this is the eraser.  So there is an onarchimedean, so complete for rank one  valuation.  So it is a, so we are, as usual, we  are in the mixed characteristic 0p 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 space.  It's because for scheme, there are  many results on etalc homology for rigid analytic space.  It's there are equivalent definitions,  like either using Berkovich or Huber's approach  with adic spaces or the risk-e-rimman approach.  In any case, it's because there are comparisons  with the homology of schemes.  Now, the in Piadic Coach theory wasn't,  one is interested in several homology theories,  like there is the etalc homology, the deram homology,  possibly the crystalline homology of the special fiber,  if there is good reduction.  And for the deram homology, there  is a well-known spectral sequence that converges to,  so the Hodge-to-deram spectral sequence,  which degenerates when we are dealing  with proper smooth schemes or even  Keller varieties or seems barrational to Keller.  There was a certain class of, so over the complex numbers  using Hodge theory.  And actually, it does not always degenerate  for some bad, smooth, I believe, or some,  although I did not have time, but I believe it's,  you need some assumption.  And in the Piadic world, one does not need any assumption.  So Schultz approved over, so in the case of discreetly valued  fields with perfect residue fields, one can, OK.  So the question is to know whether this degenerates,  when particular it gives a quality of dimension.  So originally, Falking's in his work on Piadic Hodge theory,  but of course, he has the case of, I mean,  it comes out from the theories, it's  coming of the reason of the Piadic approach to this.  And also, in particular, there are comparison theories  on the dimensions of etalc homology.  It's the same as the dimension of the deram homology.  And the question is to do it in, more generally for,  so Schultz did it over discreet, so in the general,  using the theory of perfected spaces.  So but this was in the more classical case of perfect  a residue field, when discrete valuation, which actually helps.  So actually, there is a finiteness theorem in general,  which also gives some kind of Hodge-Tate spectral sequence,  which involves Hpx omega q, q, something, minus q, yes.  So this is originally originates from Tate's works  on the visible groups.  And then using the Galois action, one can see,  Tate uses the Galois action to get some results now.  So of course, when the field is algebraically closed,  one doesn't have any Galois action.  So here in any case, we are interested in the question,  just let us say for dimensions, the dimension of the etalc  homology, which is known to be finite.  In fact, over also, over Zipi, but Schultz has worked.  But the question is to know that it's  the same as the dimension of the deram homology,  and also the sum of dimensions of Hp omega q.  So for this, it's enough to know, oops, like this.  So there was also a theory of a kind of crystalline  homology of x over something that they call Bderam plus  in this paper of BMS.  And then this was kind of a complex of BDR plus modules.  Some of this specializes to the deram homology.  But you have to invert some psi to get to Bderam,  where actually then there is some comparison  with etalc homology.  And so the question was whether this  is a free BDR plus module.  And so 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 right.  So the spreads out is like this, that you have the field qp  inside c.  So what you want is a family of rigid analytic spaces,  let us say proper smooth family, proper smooth,  where y is a rigid analytic, well,  let us say classical rigid analytic space.  Of course, you can always then pass to other spaces.  If you want, so then you want a c point of y,  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 student, the round generation,  one has this family, one has the kind of the spectral sequence.  So you have coherent shifts, and then you  can specialize it, some generic sufficiently general point.  So now, so actually in my work, I  prove that this kind of spreading out exists.  But in fact, for the application in question,  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 discreetly valued field, f containing c,  because then you could also make the residue field 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 deformation  theory, but I will need it in a form which I, well,  I did not, I don't know the references,  usually in classical things like Schlesinger criterion,  some of the residue field is the same as the residue field  of the basering somehow.  But it is not difficult to see that the same statements hold.  So I want now to review classical deformation theory  just for, at least I will assert that certain things  in the literature extend when you allow finite type residue  field extension, which is not difficult to see,  but I don't know exactly if there is a reference for this.  So the statements in the, so in particular,  there is Schlesinger, so-called Schlesinger criterion,  and the classical reference for the case of Stax  is Dockstein-Greem in SGA 7, and then there  are other people like the, but I don't,  I know that like there is log-art in ring, but in any case,  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-algebra 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.  OK.  So OK.  So I have f of first a.  So if I have a fiber product of, so of course,  unlike the case where you are over small k,  it is not clear in general that this fiber product, I mean,  but in any case, I consider only fiber product per,  one map is subjective, and then there is no problem  to see that it is still in my category.  So this maps to FA1.  And then there is also f of, so in any case,  there is one result about when f is representable,  is pro-representable.  So this is star by a complete netarian local ring over.  So something in this with residue field k prime,  if and only if.  Is the object a local ring?  Local artinian, yes.  Just so.  OK.  So in fact, you can always think of k prime  as 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.  OK.  So it is pro-representable with some condition.  OK.  That star is an isome, and then you  need some finite dimensionality of the dual number thing.  And then there is a notion of a Versailles deformation.  So a Versailles deformation is something  like that the factor representable by some b going to f  is a surjective and satisfy the formal smoothness condition.  So some lifting.  And so in any case, I will.  So anyway, there is a classical criterion for which I claim.  You are reading it over, because I'm  the programming that has the loop.  I mean, it is isomorphic to home from completely  an Italian local ring to your.  So the Schlesinger criterion, well,  you need this to be an isomorphism and the tangent  space to be, I mean, you apply to dual number over k prime.  You can endow it with vector space structure.  You want this to be finite dimensional.  And then you also want to know when  there is a Versailles deformation 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 isom,  and this is one of the dual numbers.  And again, the finite dimensionality.  This is the criterion.  And then I will actually, so actually in algebraic geometry,  we work like flat deformations of scheme shifts and so on.  So we have categories.  So the case of stacks and group weights.  So you want, now gen F is a pseudo-factor.  And so this is the J7 case.  So it is, and then I want to have a smooth group weight.  So some formal spectrum of some b and then something.  So I have a completely 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 necessary and sufficient condition  that star is an equivalence of categories  plus finite dimensionality for, well, both the isomorphism class.  Well, the set of isomorphism classes.  And for the automorphisms of the trivial thing.  And then you, this is a, OK.  So now in the case of, so now 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, topologic proper formal scheme over O C.  So addic and flat.  In fact, so it corresponds to a formal scheme over those,  which are flat or finite presentation where pi is some  pseudo-uniformizer, 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  not 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 0 over, let us say, the residue field of C descends to a  finite type subfield.  And then, okay, so, yeah, so there is a finite type,  so there is some, let us say, ring of finite type over the  integers so that the model of mod P descends to R,  and then you can localize R.  And then you,  just a moment.  Yeah, so the, this R, after localization, clearly the  maximal ideal, we give nilpotet elements, we get  Artinian ring, so it has some residue field which is a finite  type, and now there is elementary, so the fact that this is an  equivalence of categories in a Gebraic geometry is some,  well, some pinching argument, so you have to show that the flat  scheme of a revival product ring is the same as giving a  collection of flat schemes is compared with, I mean,  a patching data for flat schemes of A1, A2, A3, which is not  so difficult when one kernel is nilpotent, 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 squared 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, by Versailles, I can extend the,  successively the maps of the Versailles deformation space so  that this comes from the Versailles deformation.  Okay, so now  I have my, yes, yes, why do you need this  finally generated extension, I mean, it's not the case of  what you will sum small power P to the epsilon of P, the scheme,  what will P to the epsilon sum of constant?  Yeah, the scheme, yeah, it's true, so you can use,  instead of, yeah, 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, it's lift to, to Oc  modulo sum principle ideal, and then you can do the same thing  with this, you don't, certainly you are right, so you, you can  just use this finally generated field, instead of R in Oc modulo  sum thing, and then do the, I mean, you don't have to write  this P square also, so,  Can you just literally use field K, because, I mean, you can  find this, you get some special fiber over K, and modulo sum  small power is already isomorphic to this constant extension.  Yeah, but I want, I want, I said before that I, I, I, it is  not difficult to see that the classical three results in the  formations for the, for a representative Versa, and for the  case of Starks, which are usually stated was trivial  residue, field extension extend to the case of finitely  generated field extensions, but of course maybe there are,  there are also more general field extensions, the problem is that  some of it's implicit there, that you are using things like  the differential, the cotangent complex of finite  degenerated for this field, I'm not sure that there is  I want to reduce the, there's no field exchange at all.  Ah, okay, so, ah, you want to, so you can say that the  V vectors over the field, residue field map to the, to the, ah,  you can choose it in some way, yes, and then there is, there  will be no field extension at all, because the, you can choose,  ah, but this requires some, 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, ah, so in fact it is not, well, so let, let us,  ah, but in any case what I say works also for more general  non-archimedean fields, I don't need the, the two  to be algebraically closed, so, okay, so, ah, so in any case I have  some, ah, some universal or, ah, a universal deformation over  some, some ring B, but then I can, this is a finite type, ah, 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 has a  kind of, this is kind of a special form of skew and send of  vercovich, so it has a kind of general fiber, and then you get  a family of, ah, proper smooth family over the general  fiber of this, which is the spreading out that you want,  ah, so now, ah, I want to have a, a more general, ah,  this, spreading out result, so, ah, I want, ah, so suppose now  that X to Y is a, a flat, proper morphism of rigid analytic  spaces over some non-alchimedian field, and I want to  approximate the ring of integers by some, the terian ring A,  so it's, ah, could be like, ah, so I have just some map,  and the terian ring complete with respect to the ideal I,  and add the map to okay, now, ah, so I know, okay, so locally, ah,  you have a flattening theorem, so you have, ah, after blowing,  so you have some formal models, and you can, after blowing up,  you can assume that this is a flat, proper map of formal models,  so, ah, now, ah, I want to, ah, yes, so I, I have the, ah,  yes, so I, let us say that I have, ah, that Y is a finoid,  so I have, ah, like over spec OI, modulo power, so I have a family,  a family of schemes, proper flat over this, ah, so again,  the, the, this descends to some finite type, ah, algebra over A mod I,  so the, ah,  okay, so now, I, ah, want to prove that, ah, the following,  so let us say that I have, ah, ah, let us say in a fine,  a finite type, ah, morphism, and I have a flat, ah, family, so, ah, proper letters,  so I want to construct a compatible family of, ah,  versal N's order deformations, so the, the power of the ideal, so, ah,  so I want to, ah, embed D prime in something of a finite type,  so the ideal, the N's plus first power of the ideal is zero and extend the morphism to something flat over T prime,  and, so the, so in particular, ah, and then this will give me a versal, ah, kind of, ah,  so the idea is that you can, ah, define the notion of versal N's order deformation in the sense that if you have, ah, any, ah, family over, ah,  and some N's order deformation, it, ah, it is a pullback from the versal one plus the,  and also in a stronger form that if you give the, the map on the closed subskim between T and T prime,  you can extend it in, in this extension of the isomorphism, so, ah, then, ah, so the, the main point is to construct a, a versal first order deformation,  and for this, ah, one is to, ah, look at, ah, the I, there's the I's and R T module finite type,  and then I associate to this the category of X over T, say X1 over T1, where, ah, this is, ah, ah, so I want to have, ah, ah, infinitesimal seconding  with square zero ideal given by I and the, and the lifting of the morphism, so, ah, so this varies with I, so there is a notion,  so in Grotten-Dick's, ah, first order, origin, one of the first work on deformations was the,  ah, elections of Grotten-Dick and additive cofibre categories, so an additive cofibre category is a cofibre,  and I have a category over, ah, let's say I have an additive category, A, and I have some cofibre category over it, and I assume, so the assumption is that, ah, 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 an abelian category, which means that C of a kernel of a surjection is the equivalent to OK.  Now the things coming from the formation theory are easily seen to be additive cofibre categories, dependence on I, and now, ah, we have, ah,  OK, so for a complex, let's say in degree, ah, zero and one, we associate, let's say Q of this complex, so an additive cofibre category, which is as follows,  take home C prime X to home CX, so this is an object, let us say, in the additive, so this is a chain complex with 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 that goes to the difference, and then you get kind of two different those two, ah, so you associate, so the category associated to this is the one associated to X now,  so in the case of an abelian category, ah, so the kind of the left exact, ah, additive cofibre categories correspond to two term, well, always represented by two term complexes in, ah, in the category Pro C,  let us say it's small to avoid, so, ah, 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, you need slightly less,  and now, ah, what we need some finiteness property, this is just, we need some finiteness property in the case coming from algebraic geometry, so the idea is that you calculate, ah, using the formation theory, like the cotangent complex theory of illusics,  and we need some slight, some refinement, so the, we introduce a topos, the topo actually we have in general, well, there are no, no, no constant function, one is to look at this as a, so like the, this is a diagram of topo,  it's a total topos, and so one is to look at the formations there, ah, so something classified by our home of the cotangent complex and, and then pull back of I,  and so, but also the usual, the usual way it was written, it just gives you the set of isomorphism classes, the, the isomorphism class, the formations are classified by something,  and so, ah, yeah, so the, yeah, so there is, ah, so in the course of this, there is some use of quaren duality,  and so, if I have something like a proper flat morphism, let us say, and why as a dualizing complex, is in the terriand, it's a dualizing complex,  so the same is the strong sense that is finite cold dimension as well, so it's a, like, like in Archon's book, so it, so the duality exchanges D minus and D plus, then, ah, you can look at, ah, so you can construct some kind of at least partially defined, ah,  joint to F upper star by, ah, kind of the, and so, ah, 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, ah, so this, this was also done by some paper of Jack Hall several years ago, but we need the case of a topos of this diagram as a, so, ah, and then, if we want to apply the, ah,  if one applies the, only what is, ah, so just the statement of the, the formation theory in the sense of isomorphism classes, one does not get that the additive confibled category is given by a complex,  you only get that the, the fact or set of isomorphism classes is, ah, 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 higher categorical coherency thing, so, the idea, one idea is to look at all, let us say, all x's, let us say, all x's is some size restriction and somehow put them together in a topos,  I mean, with, and, I mean, the topos of diagram parameterized by this and working this and then one can, I think, refine the, the, the, the information, although maybe now there are some techniques with higher categories that allow one to see that whenever  things look like they should be coherent, they are coherent, which is 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, it 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  confable 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 cofibre category C over A, let's say A is a billion, is such that the isomorphism classes of fx is isomorphic functorially to  kernel, home, qx, to home, px, then, where q is projective, okay, then q and the canonical map give a  versatile, no, a co-kernel, p and the canonical map gives a versatile object. So the surjectivity is clear, but the problem is that you have to know it in 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 lift this map to a map that represents,  so I have some object actually 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 pulvic square of categories, so you can assume that it's,  it's, it's, that x prime is p and this is the identity map. Then the class alpha is represented by some map, 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 projective, you can lift and you can correct it, so there is actually a section. But then you have an isomorphism, but I don't see it, I only know opto-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 object, 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,  there are some technical work with stop process and then some technical work on defining successive N's order deformations, I mean universal.  So the idea is that once you have the N's order one, you want to pass the N first order one, but the N plus first order one is kind of a first order deformation of some N's order one, so you, you want to,  you have something which is, I mean the idea is to take kind of the, some verse of first order deformation and cut it down so that the ideal defining 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 N's, the N 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 further 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 erratic description of, so in the concerning application to rigid geometry,  so if x over y, let's say this topologically finite type of some okay, I can,  so I, let's say I have my, so there is some ideal of definition of, of, of A that goes to an ideal of definition of okay, and then I can see that this flat deformation comes from my verse,  so this comes from formal schemes over SPFA, except that they are not topologically finite type, they are more like special in the sense of Verkovich, but then you can still make some blowing up, reduce to topologically finite type thing,  and then you get a map of rigid spaces like in the sense of Renault over, over A, so R, or in the sense of the book of Fujiwara Akato, so that's, and then you get, you, so for example, I want to prove that the,  hold should the ram spectral sequence degenerates, so, so I want to prove that RPF star omega q are locally free, and the spectral sequence degenerates,  so of course there is a question, foundational question for which kind of nice, which kind of framework there is enough theory to speak about this question for, but at least I reduce, so one is to do some work with base change of quaren, quaren base change,  but the idea is to reduce to the nice Neterian case, and then you can actually use rings of finite type over 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 rescue characteristics zero, which is, this cannot be handled by, by just approximating okay by DVR,  so one is to really spray it over Z and use some characteristic P, rescue characteristic P specialization, so the proper smooth, so this is an analog of the result over C, which is proved in the paper of Deline,  that is that if you know for the fibers, the information of the generation, it shows by a length's argument that in 69 I think, or in any case, it shows that this is the, so comparison with Betik homology, the spectral sequence degenerates,  so here one is, what, 60, well it is the paper on the generation of spectral sequences, okay, I don't, okay, anyway Grottenbeek also mentioned it in the call to semi, it's really, anyway, it's, okay, don't,  so in any case the idea here is that you reduce to the case of a Artinian local rings, finite dimensional over qp, and so some family of rigid analytic spaces over this,  and then the analog of Betik homology would be to develop some kind of crystalline homology, like what Bartmoulin-Scholze do in this framework, so you have to say that if you have a chi over SpR,  so I have also chi, the special fiber of, okay, so I want to see that the Dirac homology of chi doesn't depend, you also have a constant deformation,  you want to show that the Dirac 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 homology by just using a small, I mean, like a fine cover,  you have some cover where it lifts, and then you use a check construction, so you use local embedding in something smooth, and the embedding of the intersection in the fiber product is known to the experts, so you take the formal completion and the Ram complexes,  you make a complex of this, and reciprocal homology if needed, if so, 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 Ram homology of any liftings, in particular the length is the same,  it allows one to use the Lienz 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,  so it's a question, I mean, which, at least in the classical rigid case and in the utilian case should be okay, but still one is to use some base change argument, okay, so this is the application which,  I think, is the result of this result, and then I also want to mention though that the spreading out, it doesn't work, you need the proper rigid analytic space, it doesn't work for non-proper ones,  there are several ways to think about it, so in the manuscript there is an example in characteristic p, based on separability, so if it spreads out, then it is defined over a field extension of the, so it's always a base DVR,  let's say, it's a field extension of, it's a part of the completion of finite type field extension, in particular the finite p rank in characteristic p, and then you can, if it is not reduced, you can make it,  after a finite extension it becomes generically smooth, and this is not true for reduced regionality spaces in general because of infinitely many coefficients, so you can, but you can also make an example with,  so for affinity there can be algebraic by a kick, so they always descend to something good, so now I want to give an example of 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-chimedian field, I want to find some smooth regionality space over E that doesn't spread out to a flat family over K,  so for this I will take, let us say in characteristic no 2, I will consider over a disk cross p1, kind of a double cover, like a family of hyper elliptic curves, so this is a parameter z, so I have something like y squared is equal to x,  let us say absolute uniformizer of pi, then x minus pi squared 1 plus z, x minus pi cubed, let's say 1 plus phi z, phi is some infinite series,  this coefficients in E, so the idea is that for such an E which is, it slides over a disk and all of these are in a global function, the point is if it descends to a synonyms of the field and the ring of global function descends,  of course it is not necessarily a disk but after some finite type extension and closure it becomes a disk and then you need some finite extension to, well, you need cross ratio between branch points,  you can also construct hyper elliptic evolution and the cross ratio up to some finite extension, so then you use the fact that you can make the so called topological transcendence degree, this is 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, this value group in these coefficients and you can see that you cannot  descend it to a small, you always need the any field to which it descends as residue transcendence degree which is infinite, so it's the value group extension which is infinite, the quotient is infinite dimension, so it cannot spread out, so I guess,  that's the way.  So in the first thing, so if I have, yeah, it depends on the precise problem which you, so let's say I have a nonarchimedean field E and a subfield K and I want to, yeah, but then I will need always say,  so the first argument doesn't give the spread out, spreading out to, with the basis is regionality space of okay, so the best thing one can do is to adjoin some transcendentals, you have some Gauss valuation of okay and then you get it by the argument they said and what Schultz said is actually in the case of algebraically closed,  like CP, they know that you can, you can lift the residue field, so the point is you can just lift the transcendence basis and then use Enzel and, I mean, you use, and taking peace routes and you get,  let me see, I saw, 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 to first lift it more the,  okay, you have a, you have FPA join some transcendence basis and separate closure and then you use just peace route of sometimes, yeah, I think, I see, I see how it works, so in any case you can, you can do it in a nonunique way and then you actually can work over the width vectors with these perfect people,  this is special in this situation, so, but in any case there is a stronger thing that I said, but anyway for the application to piatic college theory in question it's enough to use what he said, so.  How do you put, when does a strange adjoint exist there? What? When does a strange adjoint exist there, what's the hypothesis on K? Okay, so there is the following also this is slightly, so this adjoint, this final adjointness also works with twisting by a relatively F perfect when the 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 in F perfect, then you can see that tensoring this actually, but anyway doing all of this still sends you to D plus and then the, the assumption, so let us say you work with  a netarian scheme of finite cool dimension admitting a dualizing complex and then one can make some argument which uses the dualizing complex gives an equivalence between D, I mean, exchanges, D minus and D plus 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 it exists, no, what?  So I think it should exist without the assumption of dualizing complex and without netarianity if you have enough approximation technique and also maybe you can relax some bounding assumptions but, and I think that there, there is a part, so I didn't discuss all of the formation problems  then you can do differential covariance shifts or sometimes you have to use it for H1, so usually I get that this is bounded, I can see that this is bounded when K is 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 nice, so then you can do more,  not necessarily netarian basis, not netarian, yeah so I don't know the, so the, so there is a paper of Jack Hall around two thousand, well I saw it probably in 2013 I think maybe, so,  now what we need is the case of finite diagram is the same thing, or at least diagram we consider, so there are, there are, one has to work a little and see the similar effect hold, but,  I think now when people have this viewpoint of infinity categories and some general criteria for existence of adjoint in some whatever, I mean you can see that this must accept that usually everything is in,  up a star usually doesn't commute with provokes because, I mean there was limits because there's some tendering involved, what, this, let me see, so you want that the core limits in K go to this,  you need, up a star takes limits to limits, you need, yeah but of course here everything is, is, is coherent so it doesn't, in this argument with duality everything is coherent with some boundless restriction 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 upper star commutes with,  commutes with cold limits, with, with cold limits, with our, no, no, but this means that at least if you restrict yourself to, with some finalist hypothesis you will get such a signal, 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,  but I didn't study, I mean, so I'm still, I'm still bounded in some sense, but, okay.
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Let's give all stuck kids. 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 Auffaire has been a very important participant in the Seminar Varia Interation in Paris over many years. I'm 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 Auffaire asked the right question. Is there a difference between the yellow one and the red one? In the meantime, yes. And then after that, well, I remember one point when Susslin was lecturing at a ceremony which has been mentioned, namely Fugititi Serum for Agile Break at Klosfield, where there was a cove. So Susslin was lecturing. There was a cove in which this was 1981. 82. 1982. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. And so the cove was smooth and projected. So at some point, Auffaire started saying, well, projected not Susslin. Smooth not Susslin. Connected. Ah, yeah, for connecting points by, OK, yeah, yeah. Remember this. And then he left. So in 83, he lent, he gave us a great help to Sonsu Xulainmi when we were studying pitotian in the Chow Group and Co-Demption 2 using the Ramvid Comal G and the crystalline Comal G. So he helped us an awful lot, but he doesn't remember. No, no, but I think your paper was in 82. You started the preprint. The preprint, I remember, I told, I met in Sonsu in his office around 83. The paper was published in 83. The paper in these three, so I can tell the story about this. I have one minute. So there was a diagram which was supposed to commute or anti-commute. So one morning, Sonsu Xulainmi would come up, there were lots of spectral sequences, Sonsu Xulainmi would come up saying, it commutes. Then the next day, we'd 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 from Gabbers. It was a math from CHI to H2I. Minus one to the bar I commutes. And then I remember another story, which is typical and which is generic of many things which all of us have experimented. In 1997, I was giving a talk on somebody else's work. After 15 minutes of lecturing, there was a growl. And then there was a flow of consciousness, as they say in English. And so Alfer told us many ways to fix the gap. About the salt one. Salt money. So we took up all the lunch to understand what he was saying. But of course, he was right. And of course, he had several ways to fix it. And then we come nearly to the topic of the conference. Because of his talk, because in 2002, he showed us how to show that a certain biology class, which was non-remifed, was in fact non-trivial by spreading it out to some place where it is actually a remifed. Yeah, you're not going to hear that one. Is it in the same with the pen? Yeah, but it published in 2002. But it was there. I remember I discussed with you in the TITS conference college. It was maybe. No, no, no. But the discussion with you was before. It was in maybe 2000, 2001. No, the thing with the TITS is 2006. There's a story in 2006 where he managed to read it. So a paper was submitted. And there was a spreading out argument of field, which you use to function field of something of phi type of a spec z. And it was a complicated argument reducing to the p at x and finding field. And then Hoffer asked me, talking about this. And from the other end of the room, he said, oh, no, no, you replace this by a spreading of arguments. You take the field and you embed it. Any field can be, characterizing zero at least, is algebraic closed in the field of Complicical Dungeon 1. And that was enough to make a much better paper. OK, well, thank you very much. OK, thank you for the introduction. I'm not spreading out. Spraying out for families of rigid and active varieties, joint work with Brian Pudder. OK, so I thank you for the introduction and organizers. So this is actually not a 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, it didn't attend, where it was based on something I actually mentioned to Schultz some ideas that I had about spreading out. And it was important for some arguments in the periodic coach theory. And then I sent him some scan. And then anyway, I lectured about it here in the event for the Laboratory Alexander Grotenweg 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 now some of this work is actually quoted in the paper of Bartmauer Schultz on the Integral Piadic Coach theory. In fact, one needs less than this. So they want to prove certain things on the work of an algebraically closed non-archimedean field. So could you write? OK. So you haven't. OK. Well, this is the eraser. So there is an onarchimedean, so complete for rank one valuation. So it is a, so we are, as usual, we are in the mixed characteristic 0p 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 space. It's because for scheme, there are many results on etalc homology for rigid analytic space. It's there are equivalent definitions, like either using Berkovich or Huber's approach with adic spaces or the risk-e-rimman approach. In any case, it's because there are comparisons with the homology of schemes. Now, the in Piadic Coach theory wasn't, one is interested in several homology theories, like there is the etalc homology, the deram homology, possibly the crystalline homology of the special fiber, if there is good reduction. And for the deram homology, there is a well-known spectral sequence that converges to, so the Hodge-to-deram spectral sequence, which degenerates when we are dealing with proper smooth schemes or even Keller varieties or seems barrational to Keller. There was a certain class of, so over the complex numbers using Hodge theory. And actually, it does not always degenerate for some bad, smooth, I believe, or some, although I did not have time, but I believe it's, you need some assumption. And in the Piadic world, one does not need any assumption. So Schultz approved over, so in the case of discreetly valued fields with perfect residue fields, one can, OK. So the question is to know whether this degenerates, when particular it gives a quality of dimension. So originally, Falking's in his work on Piadic Hodge theory, but of course, he has the case of, I mean, it comes out from the theories, it's coming of the reason of the Piadic approach to this. And also, in particular, there are comparison theories on the dimensions of etalc homology. It's the same as the dimension of the deram homology. And the question is to do it in, more generally for, so Schultz did it over discreet, so in the general, using the theory of perfected spaces. So but this was in the more classical case of perfect a residue field, when discrete valuation, which actually helps. So actually, there is a finiteness theorem in general, which also gives some kind of Hodge-Tate spectral sequence, which involves Hpx omega q, q, something, minus q, yes. So this is originally originates from Tate's works on the visible groups. And then using the Galois action, one can see, Tate uses the Galois action to get some results now. So of course, when the field is algebraically closed, one doesn't have any Galois action. So here in any case, we are interested in the question, just let us say for dimensions, the dimension of the etalc homology, which is known to be finite. In fact, over also, over Zipi, but Schultz has worked. But the question is to know that it's the same as the dimension of the deram homology, and also the sum of dimensions of Hp omega q. So for this, it's enough to know, oops, like this. So there was also a theory of a kind of crystalline homology of x over something that they call Bderam plus in this paper of BMS. And then this was kind of a complex of BDR plus modules. Some of this specializes to the deram homology. But you have to invert some psi to get to Bderam, where actually then there is some comparison with etalc homology. And so the question was whether this is a free BDR plus module. And so 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 right. So the spreads out is like this, that you have the field qp inside c. So what you want is a family of rigid analytic spaces, let us say proper smooth family, proper smooth, where y is a rigid analytic, well, let us say classical rigid analytic space. Of course, you can always then pass to other spaces. If you want, so then you want a c point of y, 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 student, the round generation, one has this family, one has the kind of the spectral sequence. So you have coherent shifts, and then you can specialize it, some generic sufficiently general point. So now, so actually in my work, I prove that this kind of spreading out exists. But in fact, for the application in question, 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 discreetly valued field, f containing c, because then you could also make the residue field 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 deformation theory, but I will need it in a form which I, well, I did not, I don't know the references, usually in classical things like Schlesinger criterion, some of the residue field is the same as the residue field of the basering somehow. But it is not difficult to see that the same statements hold. So I want now to review classical deformation theory just for, at least I will assert that certain things in the literature extend when you allow finite type residue field extension, which is not difficult to see, but I don't know exactly if there is a reference for this. So the statements in the, so in particular, there is Schlesinger, so-called Schlesinger criterion, and the classical reference for the case of Stax is Dockstein-Greem in SGA 7, and then there are other people like the, but I don't, I know that like there is log-art in ring, but in any case, 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-algebra 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. OK. So OK. So I have f of first a. So if I have a fiber product of, so of course, unlike the case where you are over small k, it is not clear in general that this fiber product, I mean, but in any case, I consider only fiber product per, one map is subjective, and then there is no problem to see that it is still in my category. So this maps to FA1. And then there is also f of, so in any case, there is one result about when f is representable, is pro-representable. So this is star by a complete netarian local ring over. So something in this with residue field k prime, if and only if. Is the object a local ring? Local artinian, yes. Just so. OK. So in fact, you can always think of k prime as 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. OK. So it is pro-representable with some condition. OK. That star is an isome, and then you need some finite dimensionality of the dual number thing. And then there is a notion of a Versailles deformation. So a Versailles deformation is something like that the factor representable by some b going to f is a surjective and satisfy the formal smoothness condition. So some lifting. And so in any case, I will. So anyway, there is a classical criterion for which I claim. You are reading it over, because I'm the programming that has the loop. I mean, it is isomorphic to home from completely an Italian local ring to your. So the Schlesinger criterion, well, you need this to be an isomorphism and the tangent space to be, I mean, you apply to dual number over k prime. You can endow it with vector space structure. You want this to be finite dimensional. And then you also want to know when there is a Versailles deformation 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 isom, and this is one of the dual numbers. And again, the finite dimensionality. This is the criterion. And then I will actually, so actually in algebraic geometry, we work like flat deformations of scheme shifts and so on. So we have categories. So the case of stacks and group weights. So you want, now gen F is a pseudo-factor. And so this is the J7 case. So it is, and then I want to have a smooth group weight. So some formal spectrum of some b and then something. So I have a completely 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 necessary and sufficient condition that star is an equivalence of categories plus finite dimensionality for, well, both the isomorphism class. Well, the set of isomorphism classes. And for the automorphisms of the trivial thing. And then you, this is a, OK. So now in the case of, so now 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, topologic proper formal scheme over O C. So addic and flat. In fact, so it corresponds to a formal scheme over those, which are flat or finite presentation where pi is some pseudo-uniformizer, 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 not 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 0 over, let us say, the residue field of C descends to a finite type subfield. And then, okay, so, yeah, so there is a finite type, so there is some, let us say, ring of finite type over the integers so that the model of mod P descends to R, and then you can localize R. And then you, just a moment. Yeah, so the, this R, after localization, clearly the maximal ideal, we give nilpotet elements, we get Artinian ring, so it has some residue field which is a finite type, and now there is elementary, so the fact that this is an equivalence of categories in a Gebraic geometry is some, well, some pinching argument, so you have to show that the flat scheme of a revival product ring is the same as giving a collection of flat schemes is compared with, I mean, a patching data for flat schemes of A1, A2, A3, which is not so difficult when one kernel is nilpotent, 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 squared 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, by Versailles, I can extend the, successively the maps of the Versailles deformation space so that this comes from the Versailles deformation. Okay, so now I have my, yes, yes, why do you need this finally generated extension, I mean, it's not the case of what you will sum small power P to the epsilon of P, the scheme, what will P to the epsilon sum of constant? Yeah, the scheme, yeah, it's true, so you can use, instead of, yeah, 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, it's lift to, to Oc modulo sum principle ideal, and then you can do the same thing with this, you don't, certainly you are right, so you, you can just use this finally generated field, instead of R in Oc modulo sum thing, and then do the, I mean, you don't have to write this P square also, so, Can you just literally use field K, because, I mean, you can find this, you get some special fiber over K, and modulo sum small power is already isomorphic to this constant extension. Yeah, but I want, I want, I said before that I, I, I, it is not difficult to see that the classical three results in the formations for the, for a representative Versa, and for the case of Starks, which are usually stated was trivial residue, field extension extend to the case of finitely generated field extensions, but of course maybe there are, there are also more general field extensions, the problem is that some of it's implicit there, that you are using things like the differential, the cotangent complex of finite degenerated for this field, I'm not sure that there is I want to reduce the, there's no field exchange at all. Ah, okay, so, ah, you want to, so you can say that the V vectors over the field, residue field map to the, to the, ah, you can choose it in some way, yes, and then there is, there will be no field extension at all, because the, you can choose, ah, but this requires some, 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, ah, so in fact it is not, well, so let, let us, ah, but in any case what I say works also for more general non-archimedean fields, I don't need the, the two to be algebraically closed, so, okay, so, ah, so in any case I have some, ah, some universal or, ah, a universal deformation over some, some ring B, but then I can, this is a finite type, ah, 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 has a kind of, this is kind of a special form of skew and send of vercovich, so it has a kind of general fiber, and then you get a family of, ah, proper smooth family over the general fiber of this, which is the spreading out that you want, ah, so now, ah, I want to have a, a more general, ah, this, spreading out result, so, ah, I want, ah, so suppose now that X to Y is a, a flat, proper morphism of rigid analytic spaces over some non-alchimedian field, and I want to approximate the ring of integers by some, the terian ring A, so it's, ah, could be like, ah, so I have just some map, and the terian ring complete with respect to the ideal I, and add the map to okay, now, ah, so I know, okay, so locally, ah, you have a flattening theorem, so you have, ah, after blowing, so you have some formal models, and you can, after blowing up, you can assume that this is a flat, proper map of formal models, so, ah, now, ah, I want to, ah, yes, so I, I have the, ah, yes, so I, let us say that I have, ah, that Y is a finoid, so I have, ah, like over spec OI, modulo power, so I have a family, a family of schemes, proper flat over this, ah, so again, the, the, this descends to some finite type, ah, algebra over A mod I, so the, ah, okay, so now, I, ah, want to prove that, ah, the following, so let us say that I have, ah, ah, let us say in a fine, a finite type, ah, morphism, and I have a flat, ah, family, so, ah, proper letters, so I want to construct a compatible family of, ah, versal N's order deformations, so the, the power of the ideal, so, ah, so I want to, ah, embed D prime in something of a finite type, so the ideal, the N's plus first power of the ideal is zero and extend the morphism to something flat over T prime, and, so the, so in particular, ah, and then this will give me a versal, ah, kind of, ah, so the idea is that you can, ah, define the notion of versal N's order deformation in the sense that if you have, ah, any, ah, family over, ah, and some N's order deformation, it, ah, it is a pullback from the versal one plus the, and also in a stronger form that if you give the, the map on the closed subskim between T and T prime, you can extend it in, in this extension of the isomorphism, so, ah, then, ah, so the, the main point is to construct a, a versal first order deformation, and for this, ah, one is to, ah, look at, ah, the I, there's the I's and R T module finite type, and then I associate to this the category of X over T, say X1 over T1, where, ah, this is, ah, ah, so I want to have, ah, ah, infinitesimal seconding with square zero ideal given by I and the, and the lifting of the morphism, so, ah, so this varies with I, so there is a notion, so in Grotten-Dick's, ah, first order, origin, one of the first work on deformations was the, ah, elections of Grotten-Dick and additive cofibre categories, so an additive cofibre category is a cofibre, and I have a category over, ah, let's say I have an additive category, A, and I have some cofibre category over it, and I assume, so the assumption is that, ah, 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 an abelian category, which means that C of a kernel of a surjection is the equivalent to OK. Now the things coming from the formation theory are easily seen to be additive cofibre categories, dependence on I, and now, ah, we have, ah, OK, so for a complex, let's say in degree, ah, zero and one, we associate, let's say Q of this complex, so an additive cofibre category, which is as follows, take home C prime X to home CX, so this is an object, let us say, in the additive, so this is a chain complex with 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 that goes to the difference, and then you get kind of two different those two, ah, so you associate, so the category associated to this is the one associated to X now, so in the case of an abelian category, ah, so the kind of the left exact, ah, additive cofibre categories correspond to two term, well, always represented by two term complexes in, ah, in the category Pro C, let us say it's small to avoid, so, ah, 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, you need slightly less, and now, ah, what we need some finiteness property, this is just, we need some finiteness property in the case coming from algebraic geometry, so the idea is that you calculate, ah, using the formation theory, like the cotangent complex theory of illusics, and we need some slight, some refinement, so the, we introduce a topos, the topo actually we have in general, well, there are no, no, no constant function, one is to look at this as a, so like the, this is a diagram of topo, it's a total topos, and so one is to look at the formations there, ah, so something classified by our home of the cotangent complex and, and then pull back of I, and so, but also the usual, the usual way it was written, it just gives you the set of isomorphism classes, the, the isomorphism class, the formations are classified by something, and so, ah, yeah, so the, yeah, so there is, ah, so in the course of this, there is some use of quaren duality, and so, if I have something like a proper flat morphism, let us say, and why as a dualizing complex, is in the terriand, it's a dualizing complex, so the same is the strong sense that is finite cold dimension as well, so it's a, like, like in Archon's book, so it, so the duality exchanges D minus and D plus, then, ah, you can look at, ah, so you can construct some kind of at least partially defined, ah, joint to F upper star by, ah, kind of the, and so, ah, 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, ah, so this, this was also done by some paper of Jack Hall several years ago, but we need the case of a topos of this diagram as a, so, ah, and then, if we want to apply the, ah, if one applies the, only what is, ah, so just the statement of the, the formation theory in the sense of isomorphism classes, one does not get that the additive confibled category is given by a complex, you only get that the, the fact or set of isomorphism classes is, ah, 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 higher categorical coherency thing, so, the idea, one idea is to look at all, let us say, all x's, let us say, all x's is some size restriction and somehow put them together in a topos, I mean, with, and, I mean, the topos of diagram parameterized by this and working this and then one can, I think, refine the, the, the, the information, although maybe now there are some techniques with higher categories that allow one to see that whenever things look like they should be coherent, they are coherent, which is 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, it 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 confable 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 cofibre category C over A, let's say A is a billion, is such that the isomorphism classes of fx is isomorphic functorially to kernel, home, qx, to home, px, then, where q is projective, okay, then q and the canonical map give a versatile, no, a co-kernel, p and the canonical map gives a versatile object. So the surjectivity is clear, but the problem is that you have to know it in 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 lift this map to a map that represents, so I have some object actually 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 pulvic square of categories, so you can assume that it's, it's, it's, that x prime is p and this is the identity map. Then the class alpha is represented by some map, 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 projective, you can lift and you can correct it, so there is actually a section. But then you have an isomorphism, but I don't see it, I only know opto-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 object, 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, there are some technical work with stop process and then some technical work on defining successive N's order deformations, I mean universal. So the idea is that once you have the N's order one, you want to pass the N first order one, but the N plus first order one is kind of a first order deformation of some N's order one, so you, you want to, you have something which is, I mean the idea is to take kind of the, some verse of first order deformation and cut it down so that the ideal defining 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 N's, the N 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 further 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 erratic description of, so in the concerning application to rigid geometry, so if x over y, let's say this topologically finite type of some okay, I can, so I, let's say I have my, so there is some ideal of definition of, of, of A that goes to an ideal of definition of okay, and then I can see that this flat deformation comes from my verse, so this comes from formal schemes over SPFA, except that they are not topologically finite type, they are more like special in the sense of Verkovich, but then you can still make some blowing up, reduce to topologically finite type thing, and then you get a map of rigid spaces like in the sense of Renault over, over A, so R, or in the sense of the book of Fujiwara Akato, so that's, and then you get, you, so for example, I want to prove that the, hold should the ram spectral sequence degenerates, so, so I want to prove that RPF star omega q are locally free, and the spectral sequence degenerates, so of course there is a question, foundational question for which kind of nice, which kind of framework there is enough theory to speak about this question for, but at least I reduce, so one is to do some work with base change of quaren, quaren base change, but the idea is to reduce to the nice Neterian case, and then you can actually use rings of finite type over 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 rescue characteristics zero, which is, this cannot be handled by, by just approximating okay by DVR, so one is to really spray it over Z and use some characteristic P, rescue characteristic P specialization, so the proper smooth, so this is an analog of the result over C, which is proved in the paper of Deline, that is that if you know for the fibers, the information of the generation, it shows by a length's argument that in 69 I think, or in any case, it shows that this is the, so comparison with Betik homology, the spectral sequence degenerates, so here one is, what, 60, well it is the paper on the generation of spectral sequences, okay, I don't, okay, anyway Grottenbeek also mentioned it in the call to semi, it's really, anyway, it's, okay, don't, so in any case the idea here is that you reduce to the case of a Artinian local rings, finite dimensional over qp, and so some family of rigid analytic spaces over this, and then the analog of Betik homology would be to develop some kind of crystalline homology, like what Bartmoulin-Scholze do in this framework, so you have to say that if you have a chi over SpR, so I have also chi, the special fiber of, okay, so I want to see that the Dirac homology of chi doesn't depend, you also have a constant deformation, you want to show that the Dirac 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 homology by just using a small, I mean, like a fine cover, you have some cover where it lifts, and then you use a check construction, so you use local embedding in something smooth, and the embedding of the intersection in the fiber product is known to the experts, so you take the formal completion and the Ram complexes, you make a complex of this, and reciprocal homology if needed, if so, 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 Ram homology of any liftings, in particular the length is the same, it allows one to use the Lienz 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, so it's a question, I mean, which, at least in the classical rigid case and in the utilian case should be okay, but still one is to use some base change argument, okay, so this is the application which, I think, is the result of this result, and then I also want to mention though that the spreading out, it doesn't work, you need the proper rigid analytic space, it doesn't work for non-proper ones, there are several ways to think about it, so in the manuscript there is an example in characteristic p, based on separability, so if it spreads out, then it is defined over a field extension of the, so it's always a base DVR, let's say, it's a field extension of, it's a part of the completion of finite type field extension, in particular the finite p rank in characteristic p, and then you can, if it is not reduced, you can make it, after a finite extension it becomes generically smooth, and this is not true for reduced regionality spaces in general because of infinitely many coefficients, so you can, but you can also make an example with, so for affinity there can be algebraic by a kick, so they always descend to something good, so now I want to give an example of 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-chimedian field, I want to find some smooth regionality space over E that doesn't spread out to a flat family over K, so for this I will take, let us say in characteristic no 2, I will consider over a disk cross p1, kind of a double cover, like a family of hyper elliptic curves, so this is a parameter z, so I have something like y squared is equal to x, let us say absolute uniformizer of pi, then x minus pi squared 1 plus z, x minus pi cubed, let's say 1 plus phi z, phi is some infinite series, this coefficients in E, so the idea is that for such an E which is, it slides over a disk and all of these are in a global function, the point is if it descends to a synonyms of the field and the ring of global function descends, of course it is not necessarily a disk but after some finite type extension and closure it becomes a disk and then you need some finite extension to, well, you need cross ratio between branch points, you can also construct hyper elliptic evolution and the cross ratio up to some finite extension, so then you use the fact that you can make the so called topological transcendence degree, this is 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, this value group in these coefficients and you can see that you cannot descend it to a small, you always need the any field to which it descends as residue transcendence degree which is infinite, so it's the value group extension which is infinite, the quotient is infinite dimension, so it cannot spread out, so I guess, that's the way. So in the first thing, so if I have, yeah, it depends on the precise problem which you, so let's say I have a nonarchimedean field E and a subfield K and I want to, yeah, but then I will need always say, so the first argument doesn't give the spread out, spreading out to, with the basis is regionality space of okay, so the best thing one can do is to adjoin some transcendentals, you have some Gauss valuation of okay and then you get it by the argument they said and what Schultz said is actually in the case of algebraically closed, like CP, they know that you can, you can lift the residue field, so the point is you can just lift the transcendence basis and then use Enzel and, I mean, you use, and taking peace routes and you get, let me see, I saw, 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 to first lift it more the, okay, you have a, you have FPA join some transcendence basis and separate closure and then you use just peace route of sometimes, yeah, I think, I see, I see how it works, so in any case you can, you can do it in a nonunique way and then you actually can work over the width vectors with these perfect people, this is special in this situation, so, but in any case there is a stronger thing that I said, but anyway for the application to piatic college theory in question it's enough to use what he said, so. How do you put, when does a strange adjoint exist there? What? When does a strange adjoint exist there, what's the hypothesis on K? Okay, so there is the following also this is slightly, so this adjoint, this final adjointness also works with twisting by a relatively F perfect when the 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 in F perfect, then you can see that tensoring this actually, but anyway doing all of this still sends you to D plus and then the, the assumption, so let us say you work with a netarian scheme of finite cool dimension admitting a dualizing complex and then one can make some argument which uses the dualizing complex gives an equivalence between D, I mean, exchanges, D minus and D plus 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 it exists, no, what? So I think it should exist without the assumption of dualizing complex and without netarianity if you have enough approximation technique and also maybe you can relax some bounding assumptions but, and I think that there, there is a part, so I didn't discuss all of the formation problems then you can do differential covariance shifts or sometimes you have to use it for H1, so usually I get that this is bounded, I can see that this is bounded when K is 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 nice, so then you can do more, not necessarily netarian basis, not netarian, yeah so I don't know the, so the, so there is a paper of Jack Hall around two thousand, well I saw it probably in 2013 I think maybe, so, now what we need is the case of finite diagram is the same thing, or at least diagram we consider, so there are, there are, one has to work a little and see the similar effect hold, but, I think now when people have this viewpoint of infinity categories and some general criteria for existence of adjoint in some whatever, I mean you can see that this must accept that usually everything is in, up a star usually doesn't commute with provokes because, I mean there was limits because there's some tendering involved, what, this, let me see, so you want that the core limits in K go to this, you need, up a star takes limits to limits, you need, yeah but of course here everything is, is, is coherent so it doesn't, in this argument with duality everything is coherent with some boundless restriction 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 upper star commutes with, commutes with cold limits, with, with cold limits, with our, no, no, but this means that at least if you restrict yourself to, with some finalist hypothesis you will get such a signal, 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, but I didn't study, I mean, so I'm still, I'm still bounded in some sense, but, okay.
us say, R is a complete netarian local ring with flat deformations of scheme shifts and so on. So the, so we have the residue field, so chi 0 over, us say, ring of finite type over the integers so which is a finite type, and now there is have to show that the flat scheme of a revival limit, so anyway, then one can pass to the limit ah, so suppose now that X to Y is a, a flat, proper morphism of rigid analytic spaces over family, a family of schemes, proper flat over proper letters, so I want to construct a compatible family of, ah, versal N's order have, ah, OK, so for a complex, let's say in So if an additive cofibre category C over A, degenerates, so, so I want to prove that RPF star omega q are locally free, and the spectral extension of qp, so this is in the case where the proper smooth, so this is an analog of the result with Betik homology, the spectral sequence analog of crystalline homology by just using a this is the application which, I think, is the finite p rank in characteristic p, and then you approximation reduced to this case when it's