 and I thank the organizers for the invitation. So I guess that it must have been really hard work to organize such a conference under this different situation. So before this pandemic, I met Ruki Rizio regularly. So it's very rare that I don't see him physically for such a long time. So before this pandemic, so we visited the Beijing together in September. So then there I was preparing to give a lecture. So at that time, so as maybe as usual, so I was studying finite extension, finite Galois extension of local field. So I was thinking about the problem, the whole new problem. So look at this reactive omega one and take tau or module with the residue field. So it is a residue field. So this can be the natural action of Galois group. So I expect to have this action un-lifed. So it's a inertia group must act trivially on this tau group. So at that time, so we are together in Beijing. So I met Ruki every day. So under such situation, it was very natural to look at the cotangent complex. So there is injection from this group to the H1 of the cotangent complex. So it is stands for cotangent complex. Then so this is, so if you have such injection, then on this group, the Galois action is un-lifed. So it's obvious that this action is un-lifed once you get to such injection. So it was two years ago and the last year, so shortly before we started, we stopped moving. So it was in January. So I went to Paris, more precisely, I went to the Duke. So I gave a talk on this subject there. Then Gaba was there and he suggested another construction very similar to this one. So Gaba was there and he has a question. No, no, is it the first page? Yes, it's okay. Then it's okay. Also, I don't see because of the vapor from the mask. So I don't know if I can take. Okay, so this guy is really the subject of today's talk. And after this discussion, so I came back to Tokyo and Luke suggested that I should give a name on this module. So this is suggested by Luke. So my choice is Provinous, bit different. I'm sorry. Probably the problem is only in Paris, but the last line, we don't see it because of some software problems already. Up to here, it's okay or it's already too long? You know, Luke, we can see. Okay, I can go to the next page. Give a name, but you will barely see it. Okay, so you see this bottom line? Okay. Yes, good, you can go. Okay. We don't see the last thing. I will stop here. Now I chose a name. So it will be called Provinous, bit different. So this will be the subject of my talk today. So this is just one example, but this example already shows that how much I owe to Luke in such a study. So thank you very much. Thank you, Luke, for your influence on me. Okay, so let me continue. Okay, so today I want to talk on, sorry, singular support and character cycle in mixed character situation. So we know that we have a nice theory for singular support and character cycle in equal geometric case. So now we are interested in the arithmetic case, and so then there are several opposed to actions. So let me like some of them. So first, this singular support or character cycle, so they live on cotangent bundle. So first we have to find the right place to find the singular support. So the first question is how we define the cotangent bundle. And the next one is how to define singular support. And the next one will be how to prove the existence of singular support. Then after this, we will work on definition of character cycle. So there are at least four steps of section. So up to now, I can tell you what I can do on this first two. But on this last two, I don't know how to answer such question. So today I will focus on this first two points. So now on this cotangent bundle. So we want to have, so let me say X will be some regular scheme. Then we want to say dimension is N. Then we want that this cotangent bundle has also rank N. But this is something hard to find. So for example, if X is just ZP, then, or spec of ZP, then we have dimension one. But omega one is just zero. So we don't have a nice object here. But cut, so as we have seen yesterday, he likes log geometry. So he can define some logarithmic stuff which has dimension one. But in the story of singular support and character cycle in geometric case, we will not work with log. So we don't take this approach. So we have to find some another construction to define cotangent bundle. So this is the first part we are going to talk today. So this is the first thing we have to understand. And, okay. So you said something about log stack. I mean, probably you mean log differential. Yeah, yeah, just so you take. Yes, okay. You put log four. Okay. Yeah, this is just, yeah, this is what I mean. Okay. So before going. So for difficulty. For the difficulty in this third part. So in the third part variation find that we can use a lot on transform. So this is a very powerful tool in geometric case. But so of course we can try not on transform, but it is not strong enough to mix our characters. And for this first last part. So we use a million of them. But so of course we can imitate this approach, but again, I don't understand what you can do. In this case. Okay, so today, let me go more on detail. So today. So on this first part. So I want to introduce some of the news for the back. So but this I have to put in quotation mark of the restriction of the cotangent bundle on the closed fiber. So we want to define this cotangent bundle. But I can define it only on the closed fiber. And it's further. It's not the cotangent bundle itself, but it's rather look like it's full of this pullback. So I will tell you what this means later. And on the second part. So this is a part of where part of the definition of the senior support. So here. So we work with some sheaf on et al. So in geometric case. So let me recall what happens in geometric case. So in this case. So we have we work with a closed conical subset on the cotangent bundle. So G star breaks with the cotangent bundle. And this C is a closed conical subset. Yeah, this clinical means that so we work with the vector bundle. So vector bundle has natural action with multiplication, scalar multiplication. So clinical means this is a closed random scalar multiplication. Can you see this sentence? Is that okay? Yes, actually we can see everything now actually. Okay, thank you. Okay. So to define senior support we have to study the relation between sheaf, et al. sheaf and such subset on the cotangent bundle. So this step is divided into two sub steps. So first one is to introduce the notion of micro supported. So we say, we introduce a notion that such sheaf f is micro supported. So I have to explain what this means. And once this step is done then the senior support will be the smallest one such C on which f is micro supported. So once you get the definition of this micro support then you can define the senior support as the smallest one. But of course the next question is the existence. So I don't know the answer in general but I will give an example at the end of the talk. So this is the plant leaf. So are there any questions so far? Okay, so let me continue. So do you also want it to be of certain dimension? So this is a tricky point because as I said this cotangent bundle is defined only on the close of fiber. So this means that my senior support is also defined only on the close of fiber. So I have to cut the generic fiber. So this makes the, sometimes this makes the dimension one less. So the problem is in the case that the singular support of the restriction to the general fiber Samo is related to the singular support that you have on the on the special fiber, some of specialization or something like this. So this is not so clear because on the close of fiber, the length of the bundle is dimension of X, but on the j-x fiber the length of the bundle is dimension one less. So it's not straight forward to connect to the j-x fiber and the close of fiber. So I don't know how to see this precisely. Okay, so let me continue. So the point in this second step is the notion of this micro support. So micro support is a, so this is a relation between this F. So F is again on the entire site of X and the C is living on the cotangent bundle. So still we are in the geometric case. So in this case, the definition of this notion so we use more systems from X to somewhere else. But it may not be defined on the whole of X. So this is only defined at all locally. So let's say we have a map from X to somewhere else. So X and Y are smooth. So we are in the geometric case. So we can fix some field and X and Y are smooth. And F is some morphs between them. And in this case, we introduce the notion which is called C acyclic. So I will not record the definition here, but anyway, we know if such a map is C acyclic or not resisted to such closed conical subset. So this used to be called C transversal. But there are two notions C transversalities. So to avoid confusion, I chose another name. So this is now I call this C acyclic. And micro support condition means that if F has this property with F to this C, then this should imply that F is locally acyclic relatively to the C. So such relation gives the relation between our C, F and closed conical subset. So C acyclicity means local acyclicity of F. So this definition works well in geometric case, but not so well in isometric in mixed characteristic situation. So this works well because in geometric case then there are many maps from X to Y. For example, if Y is just A1 then to take a map up to A1 just means that you take a function on X. So you can find many functions. So you can choose many such F to check that this is a good property. But in the case, in mixed characteristic case for example, in mixed characteristic case for example, if you take X as a spec of ZP then there is only one map from spec ZP to spec ZP. So it's not enough. So we have to make another definition. So instead of using the moves from X so we use morphism into X. So this time we use map going to X. So this time, so I will introduce C-transversality. So this I will define later and this should imply this H is F-transversal. So these two notions we will introduce later. So in this way, so we replace this definition which worked well in geometric case by such definition. And in geometric case they are equivalent in fact. In geometric case these two definitions are equivalent. So we will use this definition to consider mixed characteristic case. Okay. F-transversal means F-local characteristic. So I will define this property later. Thank you. So the content today I already have taken half of my time the content today so first I will introduce a little bit of definition and then I need to explain this C-transversality and F-transversality and I don't know if I have time but if I have time I will give some example of single asset. Okay, so this will be the plan today. So let me start with Robin's V2 definition. So let me fix some prime number and then we can define polynomial. So this is so you this polynomial using the definition of V2 vectors is different. Then we know that this is divisible by P. So we can define polynomial with Z question. Okay. Now let me let me take any link and m will be a module. Then we say a mapping W. So this so this is some some modification of the derivation is Robin's V2 definition derivation. So this is a modification of derivation and Robin's and V2 appears in this derivation modification. So first for any A and B so derivation must be additive. So so first we modify this additivity. So to modify this additivity I use this polynomial. By the way, if you want to see my note you can find it on my web page. So yes. So I sent a link. So if you want to see other pages, you can click on this link to see more information. Okay. So let me modify this additivity. So of course we have these two terms but I now use this polynomial to multiply double of p. So if this double p is there, then we just additivity one more but we have this modification. So this is the first modification. So this is the reason for this. And the second one so for derivation we have a ripening tool but here we use full of news to modify this ripening tool. So we take this power to modify the ripening tool. So here we have full of news. So this is the definition of full of news with derivation. So now I can define this I have to say something. Yes. So this is related to p derivation by dm or deltaling appearing in the definition of prism. Now we can define the sorry the universal object. So if omega one Ae is universal so they are universal pair of A modules and full of news grid derivation. So we can define such a universal pair and we call this the module of full of news bit differential. So in fact such construction is done by four authors the cats Nobinov and Juleik so this is for four persons. So this is one person. But they make essentially the same construction. And if if A is so in our application we are interested in the case where A is over Zp then localization then this full of news bit differential killed by P so this is in fact A divided by P a module but this is this is a nice object which works well for us. So now if x is a scheme so we can see this construction then we get kazikofen OX module or OX EOX module F OMEGA 1 EX and we under some some so really we most interested in the case where X is the finite type of K typically Zp so this will be dvr is perfect which is your field and and over so this is the field of character if it is a field K and the character K is P and sorry X is I will assume that X is regular then this F OMEGA 1 is so this is locally free OX divided by P OX module of rank equal to dimension of X so we have to work on the closed fiber but we have the locally free shift with correct rank so this is important point for us so in particular if X is smooth this is smooth then this shift fits in exactly so F OMEGA 1 X so I left it to the closed fiber then this fits in so first you take the full-of-news pullback of the conormal shift so this is conormal and this is full-of-news pullback and we get again full-of-news pullback and the differentials of the closed fiber so we have such an extension in the case where X is smooth so we have some almost important shift but we need to put full-of-news here and such an extension is constructed by the linear lazy and this extension is in fact studied by these four authors and slightly different so they call this shift as a total differential so they like this in this notation so this is the case when X is smooth now I go back to the general case where I just assume X is regular then we can define this full-of-news cotangent bundle but because of this property I distribute the closed fiber so this is a vector bundle on the closed fiber associated to this local receive so on this closed fiber we have a nice local receive so we can take associated vector bundle so if n is the dimension of X then this space has dimension n-1 but this vector bundle has rank n it is such a geometric object okay so this is the first part of the construction so for any question is that if and only if what is the if and only if regularity under some some finite condition in fact this regularity and the local space is equivalent under some condition thank you that's important thank you okay now let me continue to this second part the subject of the second part is comparison of if C transversality okay so let me start with this C transversality so now C will stand for closed conical subset of this vector bundle I just defined so C will be closed conical subset so again this conical means that it's stable under multiplication okay now transversality is a notion for map going to X so we look at most of schemes like this so W is also a regular scheme of finite type of okay now then we have the following diagram so we start from a patented bundle this full of news one makes then by this map we can take its product so this is just a fiber product then because of the functionality of this full of news with the differentials we have a conical map to full of this cotangent bundle on the W so they are living on this WK so this is just full of news product so we can make such diagram now our closed conical subset is living here so I can just take the pullback so I get some closed conical subset here and this is a linear map so we can consider the intersection with the kernel so we can take the intersection of this pullback with the kernel of this map of vector bundles kernel of a star of this full of news with cotangent bundle okay then so we say H is C transversal if this intersection is subset of the data section so if we have such inclusion then we say H is transversal so let me give you a few examples so if C is the subset of zero section or if H is smooth then H is C transversal so this is more or less a trivial example more interesting example is the following case so suppose we have a regular closed subset scheme of C then we have a kernel map from the full of news pullback of the conical bundle listed to the closed fiber so this is the conical bundle and listed to the closed fiber and then you take the full news pullback then there is a natural map so this is in fact injection so we can take this guy as a closed conical subset then such H is C transversal if and only if this pullback so this is a closed subset of W these are the regular and codiment joins the same but on the neighborhood of the closed fiber because let me remind you that everything is leaving only on the closed fiber so we get some condition only on the neighborhood of the closed fiber so this is such so in this case the condition means that this immersion is transversal to this map so this is the reason for the name of this transversality so this transversality let me repeat that this is the notion on the neighborhood of the closed fiber so this is the definition of C transversality so maybe in the remaining let me explain F transversality but any questions so far for C transversality so let me continue to F transversality so here F is constructive at our site of X of lambda modules maybe I should add constructable where lambda is finite extension of FL it is different from P so we have such contrast C from X and again this transversality is a condition for map into X so again we consider motion from W to X so here we don't need to assume any regretting condition so now I define canonical motion from this tensor product so first you just take the usual pullback and we want to compare this usual pullback with this extra ordinary pullback and to compare them I need to put the extra ordinary pullback for the constant C then I can define canonical motion like this so let me recall the definition so this is defined as the so by the assumption so this such motion correspond to this motion so if you have such motion then by assumption we get this motion and to define this motion I can apply projection formula so by projection formula so this is communically identified with this tensor product so now there is an assumption map from this to lambda so I can just take one tensor adjunction so I define this as a composition and this canonical map is defined as the adjoint of this motion ok now we have this canonical motion so now we can make the definition so we say so we say this h is f transpose if this Cf is so let me give you an example so this example correspond to the previous one so for the transversality p the C subset of adjunction correspond to f is locally constant or h is smooth then such h is f transpose also so for this locally constant this is obvious but in this smooth case this transversality is another example is this cross-imagine so suppose we have regular cross-imagine regular cross-imagine scheme C then so in this case if I take f to be the direct term of this constant c on z then f transpose also means that this h is this C transpose also so C as before sorry f transpose also let me write it here so f is h is f transpose also on the neighborhood on the neighborhood of the cross-imagine so this is a consequence of purity ok so we have so this is an example yes the table is regular here ah yes yes yes yes yes so here here w is regular in this example thank you so the purity implies one direction only one, only one thank you, thank you, yes and then the other one you have to work as opposed to do you have the other direction no no no it was my mistake, thank you only one direction yes, thank you ok now I can, so I have explained what is c transpose also and what is f transpose also so using these two properties I can define micro support definition so f is constant c from the other side and c is closed conical subset of this cotangent bundle then we say f is micro supported c if so there are two conditions so first condition is on the support so support of f intersection is the closed fiber is the subset of the base of c so this base is the intersection of c with the zero section so this vector bundle is living on the closed fiber so this base is in fact a subset of closed fiber so we have to take so this is the reason why I have to take so this is the first condition so on the support and the second condition is the essential one so for any most from w to x so here w is regular then then the c transversality implies f transversality but so everything is on the neighborhood of the closed fiber so once this definition is made then I can define let me give you one example easy example so f is a locally constant if f is a locally constant on the neighborhood of closed fiber you find only if f is micro supported on the zero on the zero section and I can define the single support so single support is the smallest one such that f is c such that f is micro supported c for example for the constant shift is just the zero section but as I said before I don't know so I can make a conjecture but I don't have time to explain the example so let me stop here so thank you thank you very much for your attention okay thank you very much are there any questions if there are questions from the audience you can type using the chat or Q&A can I ask okay so the example where f is locally constant in the neighborhood if it only is micro supported on the zero section this you can pull so you explain one direction is trivial if it is locally constant then it's always micro supported on any c on any c because but it does a direction you still have to pull if it's not locally constant you should get something which violates transversality yeah yeah yeah yes yeah so this so so I use so I said there are two possibilities of definition so in the beginning so one is one is the original one is mac from X and the definition I explained today is using map 2x so there is a relation between this property and that property and there so we have this local acyclicity so using my definition I can prove this implication so I get this local acyclicity and using this property I prove this the other implication so another small question maybe so if you have if you have a discrete valuation ring with not perfect residue fee but let's say with finite P rank then of course you can define similar things with just the vector value of slightly larger rank depending on the P rank so then do you have the same it should be having the same I mean yeah yeah yeah it should work also yeah yeah thank you and the existence means that the intersection that if it has singular support in C1 and C2 then it has singular support in the intersection and that we say italocalization and so this is what existence means yeah yeah yes so the property I don't know that if F is supported on 2 closed conical surfaces C1 and C2 then I don't know if it is micro supported on the intersection so if we knew such property then we get this existence but I don't know this property and of course F could be a complex in the right category this is what F is in DBC F is in DBC always okay yes yeah so it seems that this with SHIF is related to H-1 of the continental complex of special materials to the base so on the other hand in the geometric case then the main object for studying is omega-1 so omega-1 is a zero continental complex so it seems that maybe I'd like to bounce back on the first question if you want to somehow mix the two things both the quick-actors and also the mixed one so in the relative situations somehow it might be good to have an object a geometric object which encompasses both the omega-1 and the H-1 so this is a quick-actors context the problem is that for a locally complicated section this object is a perfect complex of perfect amplitude in minus one zero and then there is no immediate geometric object but I think that there is a stack associated to that which we use to the object we considered before and probably I'm too old now but it seems that it would be a nice way to unify the two approaches and for example to discuss what happens when you pass on generic to special father so the geometry of the stack so this is some kind of a pica stack something like that and then I think some people use this thing, this approach so it should be the complex not the H-1 H-1 so this is just a great intuition so I remember that I was very happy that the potential complex was able to put together the omega-1 and the H-1 somehow he had discovered H-1 much before studying in several extensions of fields and then he was very happy with that and so again it's appearing here so now the thing I don't see is that the Fabinius is very important in your case how it appears I'm not so sure it slightly appears in VI that you mentioned at the beginning that directly mentioned that the complex is some kind of truncation of potential complex but I'm not so sure but anyway this is just some hope that maybe someday there will be some geometry and compressing the two approaches what do you think thank you very much I know that I have to work more thank you are there other questions let me ask a very naive question so you mentioned something about the radon transform not working explain a little bit yeah yeah so in your geometry case you have a radon transform and you can take the inverse radon transform and then you almost get the original one but there is some side difference but this side difference is just constant one so this is very important property but in mixed character case so this is precisely speaking this is not constant so this is geometrically constant so this is coming from the pullback of something on the base so in mixed character case you already have a wide dimension on the base so this gives us some problem is this also your question yes thank you there are other questions now let's thank our speaker again thank you very much