どうもありがとうございました。初めまして、インフィテーションのオーナーをご覧いただきありがとうございます。とても素晴らしいオーナーです。私のトークをお祝いします。お祝いするプロフェサーオファーガーバーの6日のお祝いです。私のトークのタイトルはリラティブ・ログ・ドラム・ビット・コンプレックスです。トークは、リラティブ・ログ・ドラム・ビット・コンプレックスの One-sing Of Theınt is introduced by performer student of Mind Matsui.コーモロジーとリラティブログのコーモロジーを説明しますコーモロジーのリラティブログとリラティブログのコーモロジーは最も知らない結果があるイルゼ・ランガジンク・ヨードカとエンドマッツウイルですこのコーモロジーはカズキ・ヒラヤマとしてフォーマー・スチューデント・ブーマインですこのコーモロジーは小さなプライムナンブルを作りますまずはフォーマー・スチューデント・ブーマインを作ります最初にターミナロジーを作りますアログリングはRとRGPとアルフを組み合わせますRはコンピュータリブリングとしてRGPはファイン・モノイドを組み合わせますモノイドはコンピュータリブリングとファインを組み合わせますアルフはファイン・モノイドを組み合わせますアルフはコンピュータリブリングとファインを組み合わせます普通にアルフとアルフを組み合わせますアルフとアルフを組み合わせます次にアルフとアルフを組み合わせます答えがこの後RWM R RGPFGPはLugglingBet Landesand BetaBetA is a composition from pアルファとアルファとタイヒミュアとWMRを組み合わせますではモルフィズムのログリングをR-PとS-Qのモルフィズムのログリングを組み合わせますこのモルフィズムのログリングはMと共有するとログフィズムのログフィズムを組み合わせますこのモルフィズムのプロジェクティブシステムはM.DM.MNはナンバーグレーターではなく、1はプロジェクティブのシステムで、ログ、デファレンシャル、グレーディドコミュタティブ、コミュタティブはハイパーコミュタティブ、アルジェブラー、アルジェブラーは、WMSQ、WMRPのシステムで、3のコミュタティブのシステムで、WMRPで、WMSQのマップスタイルで、デファレンシャル、グレーディドコミュタティブのシステムで、デファレンシャル、グレーディドコミュタティブのシステムで、マップスタイルのマップスタイルで、Fのマップスタイルで、EMのマップスタイルで、Vのマップスタイルで、EMのマップスタイルで、プロジェクティブシステムで、グレーディドコミュタティブのシステムで、3のコミュタティブのシステムで、WMSQとEM0のマップスタイルで、FとVのマップスタイルで、Fのマップスタイルで、グレーディドアジェブロスのマップスタイルで、Fのマップスタイルで、FとVのマップスタイルで、FDVで、Vのマップスタイルで、FDVで、はい、忘れちゃいましたでも3つの問題はFDタイヒミューFDタイヒミューはb-1-dタイヒミューと同じですそしてプロジェクションフォーミューラはVωF'I'm sorryωF'is equal to the Vω times'そして最後にFDωQQ is an element in large QDLOGD is equal to DLOGD.DLOGD is not a DLOGD, which is what you mean by differential.You assume there is a DLOGD map.Yes. A DLOGD map is a map Q to the E, E, M, 1.DLOGD is equal to 0 and DLOGD is a log derivation.PARDLOGD log M and D0M is a derivation and DM1 composed with DLOGM is equal to 0.DLOGD is a log derivation in the sense of log geometry.Yes, this is the definition.It is proven, of course, based on the work of UZ and Langer Zinc.This form is due to Matsue, that there exists the initial object.FG is probably, I should say, a log FB complex on something over something on SQ over RP.So RP to SQ is fixed.Then we have the category of log FB for complexes on SQ over RP.Then there exists the initial object in the category of...R is obviously something that is...Just for the existence, I don't think we need it.Ah, yes, I'm sorry.R is the bracket for algebra.Thank you very much.There exists the initial object in the category of log FB for complexeson SQ over RP.This is denoted by WM omega dot SQ over RP and DLOG.We omit to write MN dot here.And is called the relative load-landed complex on SQ over RP.And as a next step, the people there have shown that it satisfies UZ, Langer Zinc and Matsue.The people there have shown that if we are given F, the map from XM to YN,this is a morphism of fine log schemes on which P is need-potent.There exists canonically in some...canonically in the natural sense that the chief of complex is on the etal side of X,which is equal to the etal side of the B2 schemes,such that...Etal locally, it is written as the previous one.If XM to YN is written as induced by the morphism of log rings like this.So this is quasi-coherent?Ah, I'm sorry.Quasi-coherent.Yes, on B2 schemes.And yes, this is the definition.And next, I will explain briefly the comparison map.So let F as before, as in the proposition definition.And yes, OM, let OM is the structure crystal on log crystal insideof XM over B2M over YN and UM.Then we have the map...Sorry, quiz.Quiz in topos.And then UM, we have...We denote by UM the canonical projectionfrom the crystalline side to the etal sidebecause I'm in France.We like to read this.Etal.And then, yes.R log, yeah.Probably I stress.No, this is classical etal.Classical etal.And yes.And then we have canonically,we can define canonically the mapfrom the RUM star of M to the WM,our active log-drampt complex,which I call the comparison map in this talk.I will not recall the definition in general case,but if F is log smoothand there exists a following diagram,diagram which I write here.XM is over YN and this is map Fand this is in the bit log schemeand...P is nilpotent here.P is nilpotent.Yes, as before,means that includes the assumptionthat P is nilpotent.I'm sorry that...I should write...I'm sorry, yes.The crystal inside is a kind of logdivided power-thickeningcompact with speed,something like this.Divided power-thickeningshould be exact close image.And yes,with respect to the idea of defining this,this has a canonical log structureand PD structureand we respect this.On the base.And yes,with respect to classical etatopology.And if F is assumed to be log smooth,we can take at a locallythe log smooth liftand if local enoughwe have this diagram,this is over here.And then,C is the following map.C is from RUM star,OM,but this isbecause this is written aslog-dram complex ofXMM overVMYN.And this restrictslog-dram complex ofVM overVMXM overVMYN.So I forgot to saythis thing,but ourreactive-log-dram bit complexis a quotient of this.Actually,the quotient of the PD version of this,but yeah, anyway.And so this is a map.YMM is equal to,yeah,I forgot to say this thing also,but if M is equal to one,this isall the identityequal.YM is equal toDM is equal to X and this isthe function of the usual program.Yes,yes,yes,yes,yes,yes,yes,yes,yes,yes,yes,yes,yes,yes,yes,yes,yes,yes,yes,これがメニューサービスだとでもサーマーサービスだとプロジェクトはグローバリーですか?はいシンプルシャープテクニックメニューサービスを作りますヒガネマを作りますF from XM to YNもろフィズムのF-SログスキムログスヌースサチュレイティーズもろフィズムのF-Sログスキムどちらに必要があるのかこれが良いのかな?これがフォローリングアサンプトのフォローリングアサンプトを作りますサチュレイティーズもろフィズムのフォローリングアサンプトでこちらのコンディションはYNのコンディションでここでこの小さい小さい小さい大きい小さいYNQF-SモノイドエディアルJラディカルエディアルラディカルはルートJはRコンディションRもろフィズムYNのRQモノイドリングエディアルJでエンデイドQスムースモロフィズムスムースモロフィズムスムースモロフィズムスムースモロフィズムスムースモロフィズムチャートQI-NグッドアトワイグッドアトワイはQイスモロフィズムQNDivided by OY-ClosedSo we have to assumethe local structure of YNAhYesThen say that wasイスモロフィズムSo the really kind of very strange question herethat you always can locallysomething goodYou mean the general case andthis somethingOh thisI don't know the best change questionsof thereactive lockdown bit complexSo this is one of the problemsEven in the live sensorI don't still knowAh yes, the following things are not sufficiently flatYeahYeah, this kind of problemAh ok, so probably you formulatedis it sufficiently the live sensorAh maybeYeah maybeBut yeahI have not consideredin the live version yetSo I don't knowSo yeahMaybeDon't you think it's probably true without that assumption?Sorry?You probably think it's true without the star assumption?Not completely clearCompletely clearBut yeahI hope so, but I have no ideaSo ok, so this isAh, thenC is acquisitionAnd I willExplain previous worksOf course this kind of theoremis the first of proven byUZIn this case, there was no log structureyet, so N at trivialAnd Y was a perfect schemeof characteristic PAndYes, this is generalized byRanger-ZinkRanger-Zink isTrivialAnd Y isIn generalTrivial log caseOf course this assumption is automatically satisfiedSo star is nothingSo yeahAnd inLog-drum bit complexRanger-Zink isA work of YodokatoThey treat the caseWhere Y is a spec KA perfect fieldOf characteristic PP, P, P is always fixedSo P, PAndButAny log structure, I thinkAnd in this caseLog-drum morphism of character typeYeahAnd yesYesWhich I'd like to explain nowWith another definitionOfdrum bitDenoted byWm or small omega dotButActually we can proveThat they coincideIn this assumptionAnd comparison is compatibleWith this morphismOur resultSoWhen m andMn fs-log structuresOur resultContainsYodokatoBut in YodokatoThey prove the theoremWhen m and n areJust fine log structuresNot necessarily fs-log structureSo strictly speakingOur result does notContains the result of YodokatoAnd I alsoMentionedResult of MatsueIn the paperHe defines the relativeLog-drum bitMatsue treated the case thatTwo casesM is trivialAnd X over Y is smoothAnd M isM isAssociated to aN-closing deviceReactive normal-closing deviceOn X over YThis is one caseHe proved the comparisonAnd the second caseIs that N isLog structureAssociated to the zero mapFrom N to YAnd fThe morphism f isNormal-closing log varietySomething likeAnd in these two casesThey proved the comparison theoremAnd actually in these casesThis condition is satisfiedSo our resultGenerizes that of MatsueAnd this isomorphismWe can prove this isomorphismBy checking that thisYodokato definitionIs satisfied thisLog-fb complexAnd then we have a mapFrom here to hereAnd we know the kernelFrom the log-dram-bitLog-dram-complex toYodokato-dram-bitSo we canProve thatBy some calculationThat the kernel are equalSo in the case of the characterizationOf the runwayComplex is primary methodYou can also do it without the FYou can have a universal problemJust with VFB definitionI think you use the VOnly VSo this FB definitionIs invented by aLanguageAnd that the universal problemIs only driven by the logOf the V definitionIn this settingOkayWhat about all theStructure theoremI can only come to the questionExactlyNot yetSo you have a lot of different thingsYeahBut I don't use this filtrationIn the proofI will not use the filtrationIn the proof of this theoremOf course one should developThe such kind of theoryBut anyway it uses theLocal calculation in terms ofComplex of integral fourThe size structureOf theYou need toAnywayOkayYeahOf course we need toCalculate the structureBut yeahSo I will sketch the proofSo we may workAt a localAnd first we reduce to theSimple caseWe may assume thatF isThe monophism like thisSpec of rpDivided by jrpWith logStructure pToThe specr of qDivided by jr of qWith logStructure qDivided to the map from qTo pThis is an injectiveSaturatedMorphismMapof fs monoAnd that is our q-groupAnd p-group And the nrp-groupOf q-groupThese are Toshin freeToshin free z moduleG is as before, J is a radical ideal.You also assume it's the exact though?Yes, saturated in the integral.You could assume this because it's localizedbecause it's Modp to the n.ISR is a ring, actually after reducing to this casewe can work over z bracket P ringでも、このAの漸減を確認する必要はありません。それは、このAの漸減を確認する必要はあります。- では、このAの漸減を確認する必要はありません。- 絶対です。実際、このAの漸減を確認する必要はあります。はい。この場合は、WMω.SQ over Rpを解釈すると、WMω.SQ over Rpを解釈すると、まず、Jは開けます。これは、基本的に、この場合、この場合、WMrqを解釈すると、WMω.SQ over Rpを解釈すると、WMω.SQ over Rpを解釈すると、WMω.SQ over Rqを解釈すると、WMω.SQ over Rqを解釈すると、WMω.SQ over Rpを解釈すると、WMω.Rpを解釈すると、WMω.Rqを解釈すると、WMω.P over Qを解釈すると、WMω.P over Qを解釈すると、はい、必要があります。Pは、カロニカルマップを解釈すると、Rpを解釈すると、対比を解釈すると、RpのBit Timeを解釈すると、ここから、DMGx、小さめxを解釈すると、Rpを解釈させられなかったかもしれないですが、きでも、これは粗めないかもしれないです。また、DMGw。P over Rqを解釈すると、Fهاを解釈すると、Qグループはトーシャンフリースです。Iフィックスです。IススタンダードコーリネットのイメージをDログとラジエクスアイのコーリネットです。小さなXはP、P1のパーティーです。Pはトーシャンフリースです。このエレメントはYのPグループのGをG1のPに乗ることです。GはUの自然の数です。PはUのYのRGPに乗ることです。このセットはP1のパーティーです。XはP1のパーティーです。UXはUの自然の数に乗ることです。UはUの自然の数に乗ることです。PはUのXに乗ることです。Xのパーティーは、実際にサブコンプレックスがフォローのエレメントを作ることです。UのXはUのXに乗ることです。EのUXのパーティーは、Xに乗ること enemies of Playing X柄の Xに乗ることは states oi dwxixはフィックスタインですy is an element in the bit of m-uxri is a subset of the possibly empty rfrom 1 to r, r is here, there.p-group is an isomorphic to r.x part is a sum of…What's the v?Why the v is up?this follows the right way of writing of langa zingand I…it's awful?it sounds awful, yeah?maybe, okay.what does it mean?d-d-dyeah, so, yeah.we can write like this.yes, yes, yes.just I follow the way of writing of langa zingbut if you don't like it, so we can write like this.yeah.but then it would be a disaster to change our paper.so, okay.and then by a…with some calculation,we can prove that wm.p over qis equal to a sum of its x parts.x1 through p1 over p over the x parts.but I wish…we would like to prove that this is a direct sum.yes.to do so, one sees a following.so, we can consider the same drum bit complexwith p and q replaced by p-group and q-group.and if we write precisely,this is a relative log-drum bit complex like this.but it is canonically equal to the relative drum bit complexwithout log structure.and this is treated by langa and zing.and this ring is a relatively lowland polynomial ringover this ring.and then langa and zingintroduced a notion of basic bit differential.and proved that any element is written in a unique way as a sum of basic bit differential.and strictly speaking,langa and zing is a case of polynomial algebra.but we can modify…we can modify it to prove the similar descriptionin the case of lowland polynomial algebra.and I think it is also written in the paper of Bartmell Schultz.and yeah.and so, for this, we know better.and then, so we compare them.then one sees the following.so we have a wm omega dot p over qand this maps to the bifanctoriality to this one.and this was a sum of x, its x-parts.and this also has a sum of x-parts.but x is now run through the…p-group 1 over p.but here, we can check that the relationbetween the x-parts and the basic bit differentialand we can prove that this is the direction.and then next, we have a map…this map respects the x-parts.so we have this map for each xin the large p, p1 over p.and we can prove that this is an isomorphism.each x-parts.of course, x should be in p1 over p.is an isomorphism.the map between…and this is a…this is a…first, we consider the case q is trivial.and in this case,so we use the basic bit differential.so I said that the element is writtenin a unique way as a sum of basic bit differentialand we construct the mapin the opposite directionby defining the image of eachbasic bit differentialand check that this gives the converse.if I have time,maybe I don't have time,but I'm sorry, I don't have time.so if I had time,I would explain some calculation,concrete calculation,and thenthis shows that this is injectivein particular.and thenin the special case,q is trivial and p isconcrete character.so,yeah.the sum of the bottomwith the sum of all xor is it some partitions?I mean,this map,what's that sum?direct sum of all elementsin p group one of them.don't need some partitions?yeah,yeah,yeah,yeah,no.I'm sorry.no,no,no,no,I don't think so.it is writtenin the image of this formand this vanishes.do you go to the iand you need some modelwith some partitions?all the partsitions,yes,yes,yeah.so this isthe weight is fixedand the partitions aremoves.so thisx part is the sumof basic bit differential of fixedweight.so the partitions movesand sum is with respect to weights.yes.and then next wecompare some map.compare some map.yeah,againwe comparewith langaging case.before you didn't explainso you planned it.yes.you do itcompare some of the to show the direct sumyou have to comparethe terms of the compositionand you saidthat somehow you do ityeah,first weconstruct a map when q is trivialand we prove that this is a mapfor general q in this settingwith j empty.yeah,this isso we have something likeexact sequence ofwhich is valid forlogif we haveexact sequence of drun complexif we divide oversomething and we have a similar formulafor relative load and it.and yeah,this isproven by maths way.and yes,and we have this diagramand this is injective.this is that map.yes,and this isinjective.also this isessentially by langaging.and we can prove the injectivity here.here we should usecrucially in that this is saturated.andyes,and thenwe see that image c isinjective.c isinjective and image c isactually a directwe can check this mapconcretely and we see thatit is a mapimage isdirect summand.where p prime isq1 over pplus p.and thenimage is a direct summand.we should check that the other partthe remaining parts are cyclic.but in this casebecause we have shown already thatthis p ofour x part isisomorphic to the x partin langaging's senseand so this isa cyclicforx in thep1 over pis not contained in p primeby thewe know the effect of applyingd to the basic bit differentialso this isif we uselangaging calculation wefinish the proofand this is the case wherej is emptyand maybein the remainingfour minutesI will explain very brieflycase generaland in the case generalactually our map c isthe quotient ofsome quotientmap between some quotient ofthe previousthe map in the previous caseradical yes yesand we need toso itwe need to know what I and Iarewe introducefollowing terminologyx an element in pis called j minimalifone cannot writex as a sum of y and zwhen y isj and z is in pand this is the definitionofan element in large pbut we need todefine the jminimalityin p1 over pan element in p1 over pis called jminimaliffor anyn in zsuch that p to znx is in pthis element is jminimaland actually we can proveusing the radicalityradicalnessj is radicalthat for this for anynis equivalencefor some nand so we see thatthese two definitions arecompatible for an elementx in pand then weprove that the iprime dot is a direct sumofx partssuch that x is notjminimalby that is mappedby c isomorphicallyto the setof x in pprimepprime is a q1 over pplus1q1 over pplus psuch that x is not jminimalj is an ideal inqinj is an ideal worldqqqqjj isradical ideal in qq is in pby dividing outwe see that this is a direct sumof x parts in which indexis jminimal and also for thisand so this is the imageof a direct factor and the otherdirect factor is acyclicby the same reasons beforeso this finish the proofand if we drop the assumptionyeahcondition starit seems thatby constructing this questionto the non-trivialthis sum andcan bedivided in non-trivialway so we needmore calculation to provebeyond this casethank you very muchwe have an examplethat thethis is notquestionis notdirect sumthis isdivided non-trivialbut if both hands sizedivided non-trivialthis is possiblethat it remains to be quasiisomorphicso I don't knowthe situationso I don't have an examplefor the compilersumptionssuppose you take the projective limitof the wmis it p-torsion 3ahprobably yesprobably yesprobably yes maybebut is it known andso is it reduced to the langazineis it locked structureis it in this caseso you mean the langazine caseohreducein my caseit may not be reducedbutr is not reducedand I guess I can just be sureahokso if r is notreducedthen p-torsion can happenreducedahyeah yeahyeah I knowahsorryr isthis limit is p-torsion 3no I don't think sobecause I don't thinkyou know the kernelof d on wmyeahyou don't have theyou have some part of the local structurebut not muchso you want toyou want to knowthat it's p-torsion 3ahso youyou want to seeit's injectiveyeah yeahI have not yetexploredthis I think is not knownby langazineI don't think soyeahthank youI should explore