98 Cuptalliueltodgeobnrutimandmotiuc

OIutrodup.ecall k perfect field of char p

W W k

公 smooth Injectivescheme Xn xqYn.cn lkoM.KoM theKoogx X respectively

⼼心 ⺕彐 Canonical mapKoMa K M a3 1 3k

Question let六 EK.CN underwhat condition onecan 1时 3 toQ

3GkoNk ie 3 EImlkoMQ kolx.IQ

p adicV koMQ kofXDQ3

liGldnanEg曮侧 ⼀一 𠺖 侧i

deRhamgmtlinecomparison

ㄒ凹 the following two conditions areE

a chB E f HiiMw canbe gedto 后的

然 ⼆二3 EKoMQ.s.t.ch131x Echl引

⼩小 引 is Hodge tie.ch 13 E区 李李下有箱1

maintnm.glloch EsnawltKerz_ undersome technicalcondition liepsdimlxnt6nnr.ie扣 Xofsmalldimthefollowing arei 3 can be t.gtedhmdly.iejEykoNnXst3k 3⼼心 了了 ⻔门 Hodge

BEKliterallyliftthe Kona class to anelementin幽⼼心们 notonlythechemchan

incrystallinecohomology see their Remarkiii in 7677

to this me authorsconstructed thefollowingdiagram withfirstl.ioexact

dilxb.is 蕊妒筑 x 加⼝口台蕊 CHiN.IQ_ ard i⼋八 ⼋八

crystallineHodgeobstructionmap

iii ofanf surjectivet

link刚 ⼀一 ⼩小从⻔门areduction

Where X inadicwupktimogXNCHintlxjwutinuouschowgp.li

N continuous Kon z

then 3 Hodge oh了了 如川引canbehgted to continuousChowgp

3 can be lgtedgwmdlytoyu.nl

Aimofthis talk define me first line ofthe diagram and show y a

nextmelting CouplettheConstructionof 1 1the theequivalencesflowsfondly

mtinuonchowgroupandtneohstruct.my

Recall Xf propersmooth isomorphismCH x E H Ki 不不 In

So continues Chow group is defined analogouslyusing me

continuesversionng Zxlr the continuum cohomology

pwjat.ve

Xh p adic smooth tr formalschemeoven W

X n X n n EN

Zxir 21 Ix.us EX.is concentrated in degree fr

log Zin ⼀一 折⽽而川的⼆二 ri rd W.bg的

choose an embedding sunthat

X Li Y INSmooth

I ⼆二 liftingofFrobenius F圷下 下

T.nlno one needsywmgectiuensieadlx.human Y

let D ⼆二 PD enuelopeyx.cn I compatible withme pdsnutuneoncpEWFGD.PEG the onespounding PD ided.io

Set for rap扣 贴 1 扣 ⼀一 丁

⼼心知知 如箲 ⼝口品 ___

f r Er REx Q.IN ēt

Tsr RG Jcr Rj不不 ⼝口⽩白 1

where f 瓦benius

lno.needmelgtingoghoben.ws FT

dredyri fundamentaltriangle

pmin soin W.Rxi.intthe connecting homomorphism W⼝口前的 ⼀一 in Rihis

w.Rii ni fnW.tk ÉP̌define the continuousmotivic

complex 不不 Ir by加⼝口矿

to⼗十⼆二们

Zx.ir hnelZxM Q.lrlsWRxi.gErJ

commutative diagram whoselines are exacttangles

plnRi zx.irsZxf

motivic fundamentaltriangle

H is加⼝口 x ⼤大 ⼼心

r 们 1cr W.Rx.bg ⼀一

些 tinnohowgsogx.CHhi ⼀一 嗞仙 不不吵

properties of continuous cohomology short exact sequence

on 暅 H lxi.hn C嵫侧 您 HT 2名以70

Exam e.FI

short exact sequence

as R些H x Zxdrj CHhnlXD yilx.ambut we have seen in ⼦子 proposition 7.2

2 12 三 Gmx们

So the queueabovebecomes

𣉖 HIM Gm们 CH钬们⼀一些 HKGun

but Xnlswotn.mn 恢 Qn MM Qmspawn withnilpotentkernel

ilxox 脈 hiR竺HIM Gm们⼆二0

CHhdx.li竺 脈 Gm们想 肶们that atht x E Pix if ㄨ ⼥女女 W的光阴ur

smooth

obstruction sequence

motivic fundamentaltriangle

pin Rip air 不不mobCH.milXD CHYXD ltflx.tn哎 1

exactness for 引ECHTXD 3 cmbeNd to CH筑 X

ok3Dso one needs to compare thelastandtim lie ok3 o

win the one that 3 is Hodge Me

Chai13 EH笖从 Q is containedincidlniedekhamcmpa.mn

In FrHail觅个噬侧 ⼀一 瞓觅so one needs to relate obstructionmapwith thecrystallinecyclemap

gstallmicydedbg.RirtWRx.is

then the crystalline cycle map is given by

cicxj ilx.ki Hhw.Rxi.gliBlochformula

Def for 3 E CHT define a3 九bethe imageof 3 via themap

CHYXD sHYXN.Rx.gg 1恼 Wiig的

1Hh WRillilx.fmWRX

called the refniemptall.nu cycle class of 3

the crystallinecycle classof了了 denotedby Criss is theimageof a 3 Via

H K fcnWRD

silx.WRDDefinillt3ECHTXDNmes.gs

that Can13 Imp CBD is Hodgeif.GGEIml 慌 X⼝口砂 噬价 Rxrap CB E Im 1 1喩所⼝口叕 喩如啪 喈不不扣顺了了

2 onesay anB isHodgemoddotniGnB7EImlHlX.RiYQ 11筑仙⼝口们

题 in many cases themaps Inthedefinitionabove are mjatwè

i degenerationof Hodge to deRham Turd Sequencemodulo mime

the canonicalmaplie 特征0上对我湆序列列正⾏行行

𡂿 X ⼝口们 Q 䐋K⼝口们is Injective

ii Assume int x Ri tnimgraWmoduletdla.be

guardsequence

Eit Hii 㬩 知们ant

Eid degenerates at Eithegrantid d Ef_Efbis zero after 0 Q

but Eibtm.in批 Wmodule

d E以 ⼀一 Efl b is zero aswell

ik.Dii ilx.ba injective

凹 the map 煰 K 加 Rio 嵫K Ri QQI is an isomorphism

1uainthe.tnlet XN.be asmotnmojativepadichmdscheme.lt 3GCHThThen

i its refined crystalline cycle class CB EH筑 x qmwRijnHodge iff 了了 canbe lifted to CHandX via the surjectivemap

Clint X CHIN

ii its crystallineclass Can仿 EH筑 xw.fi is Hodgemodulotorsion if 了了 QEImlCHcoilx.lfcilx.D.iean integralmultipleof 3 Canbe𠺫 to CH品 X

如上 Motivic fundamental triangle

pcnni zx.ir Zxlr

orequivdentlyZxr Zxlr spcrnir

III W.hiii pn.si

⺕彐

Hwumwtat.vebecause

byaxioms Ce gThm6.1i gnW.ligtriangulated i tcategoriesRǐ i 加⼝口台

taking cohomology find

CHailx.i7CHY

0 lt㼔 加⼝口的

IiinecydehvdwosHK.fi瞓伽咧 ⼀一 㽧 x poi

its analogue uptotorsion

int x Q sailXDQ sHK.mn ⾏行行

lar.ae l H嵫 x.Rik Hlxinib HK.in

11 1 1嗞 lxnik HK.li ⼀一 HintK ⼝口们

withthese two diagrams the main theorem followsby diagram chasing