私の紹介に感謝します。まず、私はオーガナイザーを感謝します。私はここにお話しすることを願っています。OK。まず、私の話題の紹介です。今日はカルティエトランスフォームのジェネラリズーを説明します。カルティエトランスフォームはシンプソンコレスポンデンスのポジティブキャラクタリスティックアナログを紹介します。オーガナイザーとボロドスキの紹介です。まず、私はオーガナイザーとボロドスキの紹介について説明します。私の話題は、PDノーツのフィクストプライムナンバーです。このスキムはカルティエトランスフォームのフィクストプライムナンバーです。このスキムはPDストラクタリスティックアナログを紹介します。このスキムはゼロIDを紹介します。そして、レッドXゼロIDのスラクタリスティックアナログです。レッドXはSスキムを紹介します。レッドXはSフローバルナンバーです。そして、リアタルフローバルナンバーです。そして、オーギーさんのボロゴドスキー2つの目標を比べます1つ目はOXモジュールE withインテグラブルコネクションナブラEとE-TENSAOXOmega-1XOmega-1X over SI omit to write2つ目はインテグラブルコネクションE-TENSAE-TENSAD-AXD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over SD0X over Sシンメテロッカーアルジブラの 単純とバンドルのX'OPSHIXモジュールはコミュタテブリングOXモジュールのコネクションは コミュタテブリングのコミュタテブリングオーグスボルドスキーを 使用していますアズマイアルジブラストラクチャーC4Dファレンシャルオペレーター オーバーHIXセンター実際 D0 コンテインズDシメトリカアルジブラHIXセンターでPカーバーD0 コンテインズ アズマイアルジブラセンター シメトリカアルジブラランク P2ZRP2RD0 コンテインズ アズマイアルジブラX オーバーSもう一度今 トボロジカルスペース X'だから アズマイアルジブラSシンボル プシャルドこれは無くて アズマイアルジブラX'D0 コンテインズ アズマイアルジブラD0 コンテインズ アズマイアルジブラアルジブラを使って、このオブジェクトを使って、それをモリタイクイバネスに使うことができます。問題は、スプリッティングモジュールをアルジブラに使うことができます。サフィッシェントコンディションを一つのサフィッシェントコンディションをスプリッティングモジュールに使うことができます。では、 let X be the anthropological spaceand let E be the azimile algebra.C4 of the azimile algebra over a commutative ring R on X.Then, assume we find a scalar extension of R.This is the ring and this is the commutative ring.and locally free azimile algebra of rank R to the square.and we find locally free S module of rank Rwith an action of E tensor R S on M.So, we have a map of algebra from S has nothing to do with S.S has nothing to do with S.I have a prime space on R.S has nothing to do with S.Sorry for my abuse of notation.It's okay?Okay, okay.Okay, okay, R prime.Okay, okay.Is that okay?So here, I'm sorry for my bad notation.So, if you also find an action of E tensor R prime on M, say alpha, then alpha must be isomorphism.Alpha is an isomorphism.So, we can conclude M is the splitting module of E tensor R prime.Actually, August Borodowski constructed two splitting modules.I would like to write here.So, this is a general situation.This is general situation.We apply this sufficient condition.In the case, E is the zero and R is the symmetric algebra.First, they constructed splitting module over the completion of...Now, we use Zariski topology, but we will use Etartopology later.First, they constructed splitting module of D0 over the completion of Pd algebra defined by tangent bundle of X prime.Here, the completion is taken by the augmentation ideal.In this case, the splitting module is K dual.This is the OX dual of K, but the definition is complicated.So, I don't explain that.They constructed a splitting module over this algebra under the assumption of an existence of a lifting of X prime to S module P to the square.This is the first splitting module.And the second splitting module is the splitting module over the completion of symmetric algebra.In this case, splitting module is Frobenius pullback over the completion of symmetric algebra.To construct this module, we need an assumption of a lifting of the relative Frobenius module P to the square.No flat lifting.Lifting means...OK.So, what is the M in the first example that...K?What is K?Sorry, the definition of K is complicated.So, I don't explain now.Sorry.What did you write about the K check?K dual.K check?Yes.So, I don't explain the definition of K.So, I'm sorry.So, what did you write about the lifting here?So, you leave it to a lift.Yes.Yes.Yes.No.I explain.Module P to the square is...This means the morphism of the schemes defined over P to the square are such that mod P reduction is nothing but X prime to S.And so, this is the partition.And we also assume X prime tilde and F tilde is flat over Z over P to the square.This is the definition of lifting.In the second case, that is F.So, it means you lift X, you lift X prime.And also, S tilde?Yes, yes, S tilde, definitely.OK.So, it follows that X prime morph to S morph is smooth.Yes.Yes, automatically.So, OK.I would like to give some remarks.Rata assumption is obviously stronger than Formas assumption.As a consequence, the Rata splitting module is smaller than the Formas splitting module in some sense.And a lifting of the relative Frobenius rarely exists globally on X.So, the Rata case is very local in some sense.So, in this sense, I call the Rata splitting module local version.And the Formas splitting module global version.And I also remarked that the splitting module must be locally free.And the local freeness of the splitting module comes from the local freeness of the relative Frobenius.This is the point.In some sense, yes.Actually, the Formas splitting module can be constructed by gruing argument.So, this means that the completion...This is like a formal scheme kind of...OK.It's a long zero section.Yes, completion by the argumentation ideal.So, this is like considering the formal scheme of associating the embedding of X by...Next, OK.This is August Borotsky's theory.And next, I would like to mention two generalizations of August Borotsky's theory.First generalization is a logarithmic version studied by Sheprach.He generalized August Borotsky's theory to the case of fine log schemes.And the second generalization is a higher-level version.This is studied by Michel, and Lucius and Quillus.This means generalization to the case of differential operators of higher-level defined by Beftelow.OK.Now, I can give you the aim of my talk.You can easily guess from the title of my talk.Today's aim is log and higher-level version of August Borotsky's theory.OK.So, plan of my talk is the following.The first section is introduction and the second section.My construction is also based on the azimilar structure of differential operator.So, first I would like to state azimilar property of our indexed version of C4 differential operators of higher-level.I will explain the definition of this guy.And the third section, this is final section of my talk.I will try to explain splitting module of this guy.This is plan of my talk.OK.OK.Let's start next section.From now on, we are mainly working on the category of fine log schemes.And I denote fine log scheme by a single letter, such as X.OK.Let X to S be a logarithms morphism of fine log scheme defined over a field of characteristic P.Let me introduce some notation.Let F be the M plus first relative problem of X over S.No.Let pi denote the natural projection from X prime to X.So, we have the usual Frobenius diagram in log case.We have relative Frobenius and projection.And the composition of these two guys are equal to…So, what is the X prime in terms of the pullback in the final scheme?No.OK.OK.We have X over S.And we take fiber product by the M plus first iteration of Frobenius of S.And we denote it by X double prime.But in the category of fine log schemes.Later, I will assume X to S is integral.But I don't assume of cartotype.Then, natural map from X to X double prime is uniquely factored as X to X prime to X double prime.Here, the first map is required to be purely inseparable.And the second map is log at R.I denote the composition X prime to X double prime to X by…I denote this guy by…OK.Next, let me introduce the divided power envelope.Let P…So, purely inseparable is in the sense of underlying scheme or also the log structure.I think also log structure.This means it's covered with only P.OK.OK.Let P be the divided power envelope.Log divided power envelope of level R.I forgot to introduce M.Level M over diagonal.Here, M denotes a positive integer.And let I bar be the defining ideal of this exact cross-immersion.Then, associated to I bar, we have filtration I bar,so-called N-p-diadic filtration.Sorry, I bar to the something bracket N.N-p-diadic filtration of I bar.And let script P be the structure shift of this PD envelope.And let P upper N be the quotient shift of P by I bar to the bracket N plus 1.Then, this forms OX module via the first projection.And these guys form the projective system.And we take OX dual of PN.This forms inductive system.And we take inductive limit.And we put it dMX.I see pretty denoted by d.This is the shift of log differential operators of level M.And as usual, this is OX algebra.This is my terminology.Next, let me mention difficulties in the study of the logarithmic version of Org's-Bordowski's theory.First difficulty is the fact.Relative problem is not finite flat in general.This is serious because the local freedom of splitting module comes from local freedom of relative problem.So, we cannot simply generalize to the case of logarithmic scheme.And the second difficulty is our shift d.This is not as my algebra in general.Actually, in logarithmic case, we can't describe its center by the symmetric algebra of tangent bundle.Okay, so these are roughly speaking difficulties in the study of logarithmic case.But we can overcome these difficulties by using indexed algebra associated to log structureIntroduced by Roaranson and Montagnon.First introduced by Roaranson.Let me recall the definition of indexed algebra.First, let's script y with a quotient shift of group envelope of log structureby the shift of invertible function on x.Okay, first I define a by quotient of mx group times ox over equivalence relation defined by axy is equivalent to xay for any a is in ox star.and x is in mx group and y is in ox.That a is not usual algebra, but that a has a structure of a kind of algebra called iindexed ox algebra.I don't recall the definition of iindexed ox algebra, but roughly speaking this is ox algebra graded by shift i.And it is known that shift of differential operator naturally acts on a which preserves index i.And this action satisfies certain ripenitz type formula.So if we consider scalar extension of d to a, then this has natural structure of iindexed ox algebra, iindexed ox algebra.I denote it by d tilde.This is appeared in the title of second section.We will study as my algebra structure of this guy instead of usual d.So a itself is viewed as indexed by trivially indexed or is it?So when you tens of two index things, so you the index of the tens of should be something like the sum of the two.Yes, this guy.This is, so we regard ox and d as c over trivial shift.And this tensize in the sense of indexed modules.Ok.Ok, next let me introduce another indexed algebra.It is written by bm plus 1x over s, but I simply denote it by b.This is internal homomorphism of the use dm plus 1 module f r plus star d0 x prime.By the theory of Froben's descent, dm plus 1 naturally acts on f r plus star d0.And we also use dm plus 1 module structure on a.Ok.And I define map from b to a by g maps to g of 1 tensor 1.And takes a composition with map.So by the composition of these maps, we regard b is c over i.And b has multiplication and addition induced from that of a.And we also end our b with ox prime action.We are the right multiplication on this guy.We form the i index to ox prime algebra.Let me state one theorem.Provide by Roiranson and Montagnan.Is that ok?It's better that you lower one because of the light.So the camera, it's better to lower one.白いところ?Ah,白いところきすと。Yes.Let me state theorem proved by Roiranson.In the case, m is equal to 0.And also started by Montagnan.The assertion is the following.a is an i indexed locally free b module of rank.p to the r times m plus1.Here r is ox rank of log cotangent bundle of x over s.So in some sense, this theorem overcome the first difficulty mentioned above.And I would like to first theorem, theorem 1.This theorem overcomes the second difficulty.The assertion is there exists a map data from symmetric algebra of tangent bundle of x prime to d.This is map of ox prime algebra, which is generalization of classical peak curvature to the case of higher level calledp to the n plus1 curvature.I don't explain the construction better such that the tilde.This is indexed version of d.Forms as my algebra over its center.And the center is isomorphic to scalar extension of symmetric algebra to binclusion from b to a tensor beta.This is first thing.Sorry, I forgot to write the rank of as my algebra.As my algebra of rank p to the 2 times r times m plus1.20 minutes.Next section, I will explain splitting module of this guy.Roughly speaking, main result is the following.We construct this splitting module of d tilde over corresponding to that table.B tensor the completion of pd algebra.Also B tensor the completion of symmetric algebra under certaindictability assumption module p to the square.So today I will explain about the latter case.So this is local version I mentioned in the first section.Let me first define one notion.Yes, yes, yes.I explain you more detail.Now I define a certain nice lifting of the relative problem.This is my method is natural generalization of that ofGro and Lussmann Quills.First, let f tilde be a lifting of the relative problem is module p to the square.So this means f tilde is mapped from x tilde to x prime tildeand module p reduction of this map is coincided with f.And we also assume underlying scheme of x tilde and x prime tilde is flatover z over p to the square.Then f tilde is we say f tilde is a log strong lifting for any m in this section of log structure of x with a lifting with a lift m tilde in mx tilde.There exists m prime tilde.This is a section of x prime tilde.This is a lift of pi up buster m and there exists g tilde.This is a section of ox tilde such that f tilde sends m prime tilde to m tildep to the m plus1 times1 plusp timesg tilde to the p to the m.This is a definition of log strong lifting.This is a definition of log strong lifting.Locally this is equation in stock.This is a definition of log strong lifting and we can always takeLocally on x.Actually if we have a lifting of the absolute flow of news of xthen we can construct log strong lifting in a natural way.Okay.Let me state second theorem.We need additional assumptions.Let x to s be we assume this is integral and log smooth.We also assume of finite type.Morphism of fine log schemes defined over a field of characteristic pand we also assume s is natalium and assume that we are givenlet's assume we are given log strong lifting f tilde.Then the assumption is we have an isomorphism of indexed algebrasfrom d tilde tensor.The statement is quite complicated.Skate like extension of d tilde to b tensor the completion ofsymmetric algebra to matrix ring of b tensor the completion ofsymmetric algebra.A tensor o x flow of news pullback of the completion ofsymmetric algebra.So a tensor flow of news pullback of the completion ofsymmetric algebra is a lifting of a splitting module of d tilde over b tensor the completion ofsymmetric algebra.This is second theorem.Okay.10 minutes.Finally, I will try to explain the sketch of proof.The sketch of proof.First point is this splitting module.Local freeness of this splitting module.This is locally free in the sense of indexed module over d tensorthe completion ofsymmetric algebra byGuaranson and Montagno and of rank p to the r times n plus 1.So this module is locally free and also has the suitable rank.And the next point is the action of thisskate like extension on this module.I'm sorry.Is that okay?Next point is to construct an action of this is not indexed version.d tensor completion ofsymmetric algebra on flow of news pullback of this guy.If we construct such an action and by takingskate like extension to athen we have a map appeared in the theorem.And by using as my algebra property of d tildeand the fact rank of a tensor f r pastathe completion ofsymmetric algebra is suitable.We can conclude such map is isomorphismby using the indexed variant of sufficient conditionstated in the first section.So next point is this.Finally I would like to explain how to construct such action.Today I have no time so maybe I can give youhow to construct an action of d on flow of news pullback of this guy.I explain this action.The first step is to construct themorphism of divided power algebra fromflow of news pullback of divided power algebra defined by long cotangent bundle ofx prime to p.p is the structure c of divided power algebra.This is the morphism of p d algebra.To construct this guy we use strong lifting.And the second step is to constructsecond step is to construct sorry.I will explain first step later to constructpsy.And the second step is to construct along stratification of the level m onvenus pullback of this guy by usingthis Psy say,long n.Then associated to this stratificationwe have d-action onfrpasta gamma.tx prime.And by taking ox dual,dual as d-module,ox dual offrpasta gamma.tx primeis nothing but flow of news pullback ofthe completion of symmetric algebra.So we have d-action.OK,finally let me explain step one.So next I will explain how to constructpsy.First let p tilde be the log divided power envelope of diagonal ofx tilde.Then we consider the following diagram.We have this p d envelope.And we have diagonal.And we composed withtake a composition by f tilde times f tilde.This is mapped from x tilde times x tildeto x prime tilde times x prime tilde.And we also have log strong liftingf tilde.There is diagonal of x prime tilde.OK,and by using ourfinite rate assumption of finite typeand netarian,we can check that therata horizontal map factor as loginfinitational neighborhood.This is loginfinitational neighborhood of order n for somelarge number n.I denote it by psy tilde sub n.And let J be the defining idealof OP tilde to OX tilde.Then by local calculation,the image of Junder psy tilde sub n is contained inideal generated by p.So this is psy tilde sub n.And multiplication byp exclamation induces isomorphism betweenp OP tilde to p OP tilde.Because we are working on themodule p to this square.And OP tilde is flat over p to this square by our assumption.And then this guy is mod p reduction ofstructure C of divided by envelope.So this is nothing but OP.OK,this is map of OX prime module.And because this map factor throughthe ideal generated by p,this isfactor as factor through omega 1X prime tilde and also factorit's mod p reduction.And the image of this guy iscontaining the underlying pdstructure of OP.So this map naturally extendsthe divided algebra.And by taking linearization.So p was just your block withoutbefore lifting.So do you mind us what was p?What was p?P is log divided by envelopeof diagonal of X.OK,and this map,as image of thismap is contained in the underlyingp structure of OP.So by universal mapping property ofdivided by algebra,we have map fromgamma.omega 1X prime to this guy.And by taking linearization,we havethis map psi.OK,I stopped here.1is psi weighted to p curvature.Yes.Could you really explain the case?Yes,we can see the p curvature.Ah.It's OK.This map is lift of p curvature.- エンディクアリティのコンディションは、マフィスを確認するために、フラットエッセクションを確認しますか?- 実際に、エンディクアリティのアサンプションは、OPチルダイズ、フラットオーバー、ログスムースエンディクアリティのフラットオーバー、OXチルダー、- エンディクアリティのアサンプションは、OPチルダイズ、フラットオーバー、OXチルダー、OXログスムースエンディクアリティのフラットオーバー、OXログスムースエンディクアリティのフラットオーバーを確認しますか?