Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz $(1,1)$ theorem

0.21 MB·

by  Christopher Lazda

CYCLE CLASSES IN OVERCONVERGENT RIGID COHOMOLOGY AND A SEMISTABLE
LEFSCHETZ (1,1) THEOREM
CHRISTOPHERLAZDAANDAMBRUSPA´L
ABSTRACT. In this article we prove a semistable version of the variational Tate conjecture for divisors in
7
crystalline cohomology, stating that arational (logarithmic) line bundle onthespecial fibreofasemistable
1
schemeoverkJtKliftstothetotalspaceifandonlyifitsfirstChernclassdoes. Theproofiselementary,using
0
2 standardpropertiesofthelogarithmicdeRham–Wittcomplex. Asacorollary,wededucesimilaralgebraicity
liftingresultsforcohomologyclassesonvarietiesoverglobalfunctionfields.Finally,wegiveacounterexample
n
a toshowthatthevariationalTateconjecturefordivisorscannotholdwithQp-coefficients.
J

8
1

] CONTENTS
G
A Introduction 1
. 1. Cycleclassmapsinoverconvergentrigidcohomology 3
h
t 2. PreliminariesonthedeRham–Wittcomplex 4
a
m 3. Morrow’svariationalTateconjecturefordivisors 8
[ 4. AsemistablevariationalTateconjecturefordivisors 11

5. Globalresults 15
1
v 6. Acounter-example 16
7
References 18
1
0
5
0
.
1
0 INTRODUCTION
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Manyofthedeepestconjecturesinarithmeticandalgebraicgeometryconcerntheexistenceofalgebraic
1
: cycles on varieties with certain properties. For example, the Hodge and Tate conjectures state, roughly
v
i speaking,thatonsmoothandprojectivevarietiesoverC (Hodge)orfinitely generatedfields(Tate) every
X
cohomologyclasswhich‘lookslike’theclassofacycleisindeedso.Onecanalsoposevariationalformsof
r
a theseconjectures,givingconditionsforextendingalgebraicclassesfromonefibreofasmooth,projective
morphism f :X→Stothewholespace.Fordivisors,theHodgeformsofboththeseconjectures(otherwise
knownastheLefschetz(1,1)theorem)arerelativelystraightforwardtoprove,usingtheexponentialmap,
butevenfordivisorstheTateconjectureremainswideopeningeneral.
Applying the principle that deformation problems in characteristic p should be studied using p-adic
cohomology,Morrowin[Mor14]formulatedacrystallinevariationalTateconjectureforsmoothandproper
families f :X →S ofvarietiesin characteristic p, andprovedtheconjecturefordivisors,atleast when f
isprojective. ThekeystepoftheproofisaversionofthisresultoverS=Spec(kJt ,...,t K),whichwhen
1 n
n=1isadirectequicharacteristicanalogueofBerthelotandOgus’theorem[BO83,Theorem3.8]onlifting
linebundlesfromcharacteristic0tocharacteristic p.
1
CycleclassesandLefschetz(1,1)
Morrow’sproofofthelocalstatementusessomefairlyheavymachineryfrommotivichomotopytheory,
in particulara ‘continuity’resultfor topologicalcyclic homology. In this article we providea new proof
ofthelocalcrystallinevariationalTateconjecturefordivisors,atleastoverthebaseS=Spec(kJtK),which
onlyusessomefairlybasicpropertiesofthedeRham–Wittcomplex. Thepointofgivingthisproofisthat
itadaptsessentiallyverbatimtothecaseofsemistablereduction,oncethecorrespondingbasicproperties
ofthelogarithmicdeRham–Wittcomplexareinplace.
SoletX beasemistable,projectiveschemeoverkJtK,withspecialfibreX andgenericfibreX. Then
0
thereisanisomorphism
H2 (X/R)(cid:209) =0∼=H2 (X×/K×)N=0
rig log-cris 0
betweenthe horizontalsectionsof the Robba ring-valuedrigid cohomologyof X and the partof the log-
crystallinecohomologyofX killedbythemonodromyoperator.TheformerisdefinedtobeH2 (X/E†)⊗
0 rig E†
R, where H2 (X/E†) is the bounded Robba ring-valued rigid cohomology of X constructed in [LP16].
rig
Thesegroupsare (j ,(cid:209) )-modulesoverR and E† respectively. Inparticular, if L is a line bundleon X ,
0
wecanviewitsfirstChernclassc (L)asanelementofH2 (X/R). Ourmainresultisthenthefollowing
1 rig
semistableversionofthelocalcrystallinevariationalTateconjecturefordivisors.
Theorem(4.5). L liftstoPic(X) ifandonlyifc (L)liesinH2 (X/E†)⊂H2 (X/R).
Q 1 rig rig
ThereisalsoaversionforlogarithmiclinebundlesonX .Thegeneralphilosophyofp-adiccohomology
0
overk((t))isthattheE†-structureHi (X/E†)⊂Hi (X/R)istheequicharactersiticanalogueoftheHogde
rig rig
filtrationon the p-adic cohomologyof varietiesovermixedcharacteristic localfields. With this in mind,
this is the direct analogue of Yamashita’s semistable Lefschetz (1,1) theorem [Yam11]. As a corollary,
we can deduce a global result on algebraicity of cohomology class as follow. Let F be a function field
of transcendence degree 1 over k, and X/F a smooth projective variety. Let v be a place of semistable
reductionforX,withreductionX . Thenthereisamap
v
sp :H2(X/K)(cid:209) =0→H2 (X×/K×)
v rig log-cris v v
fromthesecondcohomologyofX (see§5)tothelogcrystallinecohomologyofX .
v
Theorem(5.2). Aclassa ∈H2(X/K)(cid:209) =0 isintheimageofPic(X) undertheChernclassmapifand
rig Q
onlyifsp (a )isintheimageofPic(X ) .
v v Q
One might wonder whether the analogue of the crystalline variational Tate conjecture holds for line
bundles with Q -coefficients (in either the smooth or semistable case). Unfortunately, the answer is no.
p
Indeed,ifitweretrue, thenitfollowsrelativelyeasilythatthe analogueofTate’sisogenytheoremwould
holdoverk((t)),inotherwordsforanytwoabelianvarietiesA,Boverk((t)),themap
¥ ¥
Hom(A,B)⊗Q →Hom(A[p ],B[p ])⊗ Q
p Zp p
would be an isomorphism. Thatthis cannotbe true is well-known, and examplescan be easily provided
withbothAandBellipticcurves.
Let us now summarise the contents of this article. In §1 we show that the cycle class map in rigid
cohomology over k((t)) descends to the bounded Robba ring. In §2 we recall the relative logarithmic
deRham–Wittcomplex,andprovecertainbasicpropertiesofitthatwewillneedlateron.In§3wereprove
aspecialcaseofthekeystepinMorrow’sarticle[Mor14],showingthecrystallinevariationalTateconjec-
ture for smooth and projectiveschemes over kJtK. The argumentwe give is elementary. In §4 we prove
2
C.Lazda,A.Pa´l
the semistable versionof the crystalline variationalTate conjectureover kJtK, moreor less copyingword
for word the argumentin §3. In §5 we translate these results into algebraicitylifting results for varieties
overglobalfunctionfields. Finally,in§6we givea counter-exampletothe analogueoftheofcrystalline
variationalTateconjectureforlinebundleswithQ -coefficients.
p
Acknowledgements. A.Pa´lwaspartiallysupportedbytheEPSRCgrantP36794.C.Lazdawassupported
byaMarieCuriefellowshipoftheIstitutoNazionalediAltaMatematica“F.Severi”. Bothauthorswould
liketothankImperialCollegeLondonandtheUniversita` DegliStudidiPadovaforhospitalityduringthe
writingofthisarticle.
Notationsandconvenions. Throughoutwewillletk beaperfectfieldofcharacteristic p>0,W itsring
of Witt vectors and K =W[1/p]. In general we will let F =k((t)) be the field of Laurentseries over k,
and R=kJtK its ring of integers (although this will not be the case in §5). We will denote by E†,R,E
respectivelythe boundedRobbaring, the Robbaring, andthe Amice ringoverK, andwe willalso write
E+=WJtK⊗ K.ForanyoftheringsE+,E†,R,E wewilldenotebyMF (cid:209) thecorrespondingcategory
W (−)
of(j ,(cid:209) )-modules,i.e.finitefreemoduleswithconnectionandhorizontalFrobenius.Avarietyoveragiven
Noetherianbaseschemewillalwaysmeanaseparatedschemeoffinitetype. ForanyabeliangroupAand
anyringSwewillletA denoteA⊗ S.
S Z
1. CYCLE CLASS MAPS IN OVERCONVERGENTRIGID COHOMOLOGY
Recall that for varieties X/F over the field of Laurent series F =k((t)) the rigid cohomologygroups
Hi (X/E) are naturally (j ,(cid:209) )-modules over the Amice ring E. In the book [LP16] we showed how
rig
to canonically descend these cohomologygroupsto obtain ‘overconvergent’(j ,(cid:209) )-modulesHi (X/E†)
rig
over the bounded Robba ring E†, these groups satisfy all the expected properties of an ‘extended’ Weil
cohomologytheory.Inparticular,thereexistversionsHi (X/E),Hi (X/E†)withcompactsupport.
c,rig c,rig
Definition1.1. Definethe(overconvergent)rigidhomologyofavarietyX/F by
Hrig(X/E):=Hi (X/E)∨, Hrig(X/E†):=Hi (X/E†)∨
i rig i rig
andthe(overconvergent)Borel–Moorehomologyby
HBM,rig(X/E):=Hi (X/E)∨, HBM,rig(X/E†):=Hi (X/E†)∨.
i c,rig i c,rig
In[Pet03]the authorconstructscycleclassmapsinrigidcohomology,whichcanbeviewedashomo-
morphisms
A (X)→HBM,rig(X/E)
d 2d
fromthegroupofd-dimensionalcyclesmodulorationalequivalence.Ourgoalinthissectionisthefollow-
ingentirelystraightforwardresult.
Proposition1.2. Thecycleclassmapdescendstoahomomorphism
A (X)→HBM,rig(X/E†)(cid:209) =0,j =pd.
d 2d
Proof. NotethatsinceHBM,rig(X/E†)(cid:209) =0,j =pd ⊂HBM,rig(X/E)itsufficestoshowthatforeveryintegral
2d 2d
closedsubschemeZ⊂X ofdimensiond,thecycleclassh (Z)∈HBM,rig(X/E)actuallyliesinthesubspace
2d
HBM,rig(X/E†)(cid:209) =0,j =pd.
2d
3
CycleclassesandLefschetz(1,1)
Byconstruction,h (Z)istheimageofthefundamentalclassofZ(i.e.thetracemapTr :H2d (Z/E)→
Z c,rig
E(−d))underthemap
HBM,rig(Z/E)→HBM,rig(X/E)
2d 2d
induced by the natural map H2d (X/E)→H2d (Z/E) in compactly supported cohomology. Hence it
c,rig c,rig
sufficestosimplyobservethatboththismapandthetracemapdescendtohorizontal,Frobeniusequivariant
maps on the level of E†-valued cohomology. Alternatively, we could observe that both H2d (X/E)→
c,rig
H2d (Z/E)andTr arehorizontalandFrobeniusequivariantatthelevelofE-valuedcohomology,which
c,rig Z
gives
A (X)→HBM,rig(X/E)(cid:209) =0,j =pd,
d 2d
thenapplyingKedlaya’sfullfaithfulnesstheorem[Ked04,Theorem5.1]givesanisomorphism
HBM,rig(X/E)(cid:209) =0,j =pd ∼=HBM,rig(X/E†)(cid:209) =0,j =pd.
2d 2d
(cid:3)
2. PRELIMINARIES ON THEDERHAM–WITT COMPLEX
ThepurposeofthissectionistogathertogethersomeresultswewillneedonthevariousdeRham–Witt
complexesthat will be used throughoutthe article. These are all generalisations to the logarithmic case
ofwell-knownresultsfrom[Ill79],andshouldthereforepresentnosurprises. Thereaderwillnotlosetoo
muchbyskimmingthissectiononfirstreadingandreferringbacktotheresultsasneeded.
We will, as throughout, fix a perfect ground field k of characteristic p>0, all (log)-schemes will be
considered over k. Given a morphism (Y,N)→(S,L) of fine log schemes over k, Matsuue in [Mat16]
constructedarelativelogarithmicdeRham–WittcomplexW w ∗ ,denotedW L ∗ in[Mat16].
• (Y,N)/(S,L) • (Y,N)/(S,L)
Thisisane´talesheafonY equippedwithoperatorsF,V satisfyingalltheusualrelations(seeforexample
[Mat16,Definition3.4(v)])andwhichspecialisestovariouspreviousconstructionsinparticularcases.
(1) WhenS=Spec(k)andthelogstructuresLandNaretrivial,thenthisgivesthe(canonicalextension
ofthe)classicaldeRham–WittcomplexW W ∗ (toane´talesheafonY).
• Y
(2) Moregenerally,whenthemorphism(Y,N)→(S,L)isstrict,werecovertherelativedeRham–Witt
complexW W ∗ ofLangerandZink[LZ04].
• Y/S
(3) Whenthebase(S,L)istheschemeSpec(k)withthelogstructureofthepuncturedpoint,and(Y,N)
isofsemistable type(i.e. e´talelocallye´tale overk[x ,...,x ]/(x ···x )with thecanonicallog
1 d+1 1 c
structure)thenweobtainthelogarithmicdeRham–WittcomplexWw ∗ studiedin[HK94].
Y
(4) Ifwetake(Y,N) semistablebutinsteadequipSpec(k)withthetriviallogstructure,theresulting
complexisisomorphictotheonedenotedWw˜∗ in[HK94].
Y
Ifwearegivenamorphismoflogschemes(Y,N)→(S,L)overk,thenasin[Mat16,§2.2]wecanliftthe
logstructureN→O toalogstructureW N→WO ,wherebydefinitionWN=N⊕ker((WO )∗→O∗)
Y r r Y r r Y Y
and the map N →WO is the Techmu¨ller lift of N → O . Since Ww 1 is a quotient of the
r Y Y r (Y,N)/(S,L)
pd-log deRham complex w˘∗ (see [Mat16, §3.4]) there is a natural map dlog:W N →
(WrY,WrN)/(WrS,WrL) r
Ww 1 andhenceweobtainmaps
r (Y,N)/(S,L)
dlog:Ngp→W w 1
r (Y,N)/(S,L)
whicharecompatibleasrvaries. WeletWw 1 denotetheimage.
r (Y,N)/(S,L),log
4
C.Lazda,A.Pa´l
Whenbothlogstructuresaretrivial,andY →Spec(k)issmooth,then[Ill79,PropositionI.3.23.2]says
thatdloginducesanexactsequence
0→prO∗→O∗→WW 1 →0,
Y Y r Y,log
andourfirsttaskinthissectiontoobtainananalogueofthisresultforsemistablelogschemesoverk. In
fact, since we will really only be interested in the case whenY arises as the special fibre of a semistable
schemeoverkJtK,wewillonlytreatthisspecialcase.
WewillthereforeletX denoteasemistableschemeoverR=kJtK(notnecessarilyproper). Wewilllet
LdenotethelogstructuregivenbytheclosedpointofSpec(R),andwriteR×=(R,L). Wewilldenoteby
L theinverseimagelogstructureonR =kJtK/(tn+1),andwriteR×=(R ,L ). Wewillalsowritek×=
n n n n n
(k,L ). WewilldenotebyM thelogstructureonX givenbythespecialfibre,andwriteX×=(X,M).
0
Similarlywe havelogstructuresM on X =X ⊗ R , andwe will write X×=(X ,M ). Finally, when
n n R n n n n
consideringthelogarithmicdeRham–Wittcomplexrelativetok(withthetriviallogstructure)wewilldrop
kfromthenotation,e.g.wewillwriteWw ∗ insteadofW w ∗ .
r X× r X×/k
0 0
Proposition2.1. Thesequence
0→prMgp→Mgp−dl→ogWw 1 →0
0 0 r X×,log
0
isexact.
Proof. Thesurjectivityoftherighthandmapandtheinjectivityofthelefthandmaparebydefinition,and
since prWw 1 =0,thesequenceisclearlyacomplex. Thekeypointisthentoshowexactnessin the
r X×,log
0
middle. Sosupposethatwearegivenm∈Mgp issuchthatdlogm=0. Wewillshowthatm∈ prMgp by
0 0
inductiononr.
When r=1 we note that the claim is e´tale local, we may thereforeassume X× to be affine, e´tale and
0
strictoverSpec k[x1,...,xd] ,sayX =Spec(A). Wehave
(cid:16) (x1···xc) (cid:17) 0
c d
w (1A,Nc)∼=MA·dlogxi⊕ M A·dxi.
i=1 i=c+1
Nowsupposethatwearegivenalocalsectionn=u(cid:213) c xni ofNgp foru∈A∗andn ∈Z. Write
i=1 i i
c d
(cid:229) (cid:229)
dlogu= adlogx + adx
i i i i
i=1 i=c+1
witha ∈A,notethatsincedloguactuallycomesfromanelementofW 1 itfollowsthata ∈xAfor1≤i≤c.
i A i i
Inparticular,we haven =−xa for1≤i≤c, andpassingto A/xAitthereforefollowsthatn =0ink.
i i i i i
Henceeachn isdivisibleby p. Itfollowsthat(cid:213) c xni isin pNgp, anditsdlogvanishes. Bydividingby
i i=1 i
thiselementwemaythereforeassumethatn=u∈A∗. SincesemistableschemesareofCartiertype,we
mayapply[Kat89,Theorem4.12],whichtellsusthat(e´talelocally)u∈A(p)∗(sincedlogu=0⇒du=0).
Sincekisperfect,A(p)∗=(A∗)pandwemayconclude.
Whenr>1anddlogn=0∈W w 1 , theninparticulardlogn=0∈W w 1 ; hencebyapplying
r X×,log r−1 X×
0 0
theinductionhypothesiswe obtainn= pr−1n . Butnowthisimpliesthat pr−1dlogn =0∈Ww 1 , we
1 1 r X×
0
claim that in fact it follows that dlogn =0∈w 1 . Indeed, since w 1 is a locally free O -module, to
1 X× X× X0
0 0
provethata sectionvanishesitsufficestoshowthatitdoesso ona denseopensubscheme. Inparticular,
byrestrictingtothesmoothlocusofX wecanassumethatX issmoothandthelogstructureisgivenby
0 0
5
CycleclassesandLefschetz(1,1)
O∗ ⊕N,(u,m)7→u.0m. Wenowapply[Ill79,PropositionI.3.4]and[Mat16,Lemma7.4]toconcludethat
X0
dlogn =0asrequired.Thusapplyingthecaser=1finishestheproof. (cid:3)
1
Thefollowingisanalogousto[Ill79,CorollaireI.3.27].
Proposition2.2. Thesequencesofpro-sheaves
0→ W w 1 → Ww 1 1→−F Ww 1 →0,
n r X×,logor (cid:8) r X×(cid:9)r (cid:8) r X×(cid:9)r
0→ Ww 1 → Ww 1 1→−F Ww 1 →0
n r X0×/k×,logor n r X0×/k×or n r X0×/k×or
areexact.
Proof. Letusconsiderthefirst sequence. Using Ne´ron–Popescudesingularisation[Pop86, Theorem1.8]
andthe factthatthe logarithmicdeRham–Wittcomplexcommuteswith filtered colimits, we may reduce
to considering the analogous question for Y smooth over k with log structure N coming from a normal
crossingsdivisorD⊂Y. Theclaimise´talelocal,wemaythereforeassumethatY ise´taleoverk[x ,...,x ]
1 n
withDtheinverseimageof{x ···x =0}. Locally,N isgeneratedbyO∗ andx for1≤i≤c,soinorder
1 c Y i
tosee thatthesequenceisacomplex,orinotherwordsthat(1−F)(dlogn)=0,itsufficestocheckthat
(1−F)(dlogx)=0. Thisisa straightforwardcalculation. Forthesurjectivityof1−F weclaim infact
i
that
1−F:W w 1 →Ww 1
r+1 (Y,N) r (Y,N)
issurjective.Forthiswenotethatby[Mat16,§9]thereexistsanexactsequence
c
0→WW 1 →Ww 1 → WO ·dlogx →0
r Y r (Y,N) M r Di i
i=1
for all r, where D are the irreducible componentsof D. Denote the induced mapWw 1 →W O by
i r (Y,N) r Di
Res. Since(1−F)(dlogx)=0itfollowsthatwehavethecommutativediagram
i i
0 // W W 1 // W w 1 // c W O // 0
r+1 Y r+1 (Y,N) Li=1 r+1 Di
1−F 1−F 1−F
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // W W 1 // Ww 1 // c W O // 0
r Y r (Y,N) Li=1 r Di
where WW 1 is the usual (non-logarithmic) deRham–Witt complex of Y. It therefore suffices to apply
r Y
[Ill79, Propositions I.3.26, I.3.28], stating that the left and right vertical maps are surjective. Finally, to
showexactnessinthemiddle,supposethatwearegivenw ∈W w 1 suchthat(1−F)(w )=0. Then
r+1 (Y,N)
applying[Ill79,PropositionI.3.28]wecanseethat
Res(w )∈Z/pr+1Z+ker(W O →WO )
i r+1 Di r Di
foralli. Henceaftersubtractingoffanelementofdlog(Ngp)wemayassumethatinfact
w ∈W W 1+ker W w 1 →Ww 1 .
r+1 Y (cid:16) r+1 (Y,N) r (Y,N)(cid:17)
Nowapplying[Ill79,CorollaireI.3.27]tellsusthat
w ∈dlog(Ngp)+ker W w 1 →W w 1
(cid:16) r+1 (Y,N) r (Y,N)(cid:17)
andhencethegivensequenceofpro-sheavesisexactinthemiddle.
6
C.Lazda,A.Pa´l
For the second sequence, the surjectivity of 1−F follows from the corresponding claim for the first
sequence, since sections ofWw 1 can be lifted locally toW w 1 . We may also argue e´tale locally;
r X×/k× r X×
0
assumingthat X× is e´tale andstrict overSpec Nc→ k[x1,...,xd] . The factthat the claimed sequenceis a
0 (cid:16) (x1···xc) (cid:17)
complex follows again from observing that (1−F)(dlogx)=0 for 1≤i≤c. To see exactness in the
i
middleweusethefactthat(againworkinge´talelocally)wehaveanexactsequence
0→ WW 1 →Ww 1 → WO →0
M r Di r X0×/k× M r Dij
i ij
by[Mat16,Lemma8.4],whereD aretheirreduciblecomponentsofX×andD theirintersections.More-
i 0 ij
over,thisfitsintoadiagram
0 // O∗ // Mgp // Z // 0
Li Di 0 Lij Dij
dlog dlog
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // WW 1 // Ww 1 // WO // 0
Li r Di r X0×/k× Lij r Dij
1−F 1−F 1−F
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // W W 1 //W w 1 // W O // 0
Li r−1 Di r−1 X0×/k× Lij r−1 Dij
with exact rows. Exactness of the middle vertical sequence atWw 1 now follows from the classical
r X×/k×
result[Ill79,CorollaireI.3.27,PropositionI.3.28]andasimplediagram0 chase. (cid:3)
Next,wewillneedtounderstandthekernelofWw 1 →Ww 1 .
r X×,log r X×/k×,log
0 0
Lemma2.3. Forallr≥1thesequence
Z
0→ ∧dlogt→Ww 1 →W w 1 →0
prZ r X0×,log r X0×/k×,log
isexact.
Proof. Notethatby[Mat16,Lemma7.4]itsufficestoshowthat
Z
dlog(M )∩WO ∧dlogt= ∧dlogt
0 r X0 prZ
insideWw 1 ,thecontainment⊃isclear. Fortheother,supposethatwearegivenanelementoftheform
r X×
0
g∧dlogt∈Ww 1 whichisintheimageofdlog. Thenweknowthatg˜∧dlogt=dlogn+cinW w 1 ,
r X× r+1 X×
0 0
for some c∈ker W w 1 →Ww 1 and g˜∈W O lifting g. Arguingas in Proposition2.2 above
(cid:16) r+1 X× r X×(cid:17) r+1 X0
0 0
wecan see that(1−F)(dlogn)=0, andagainapplying[Mat16, Lemma7.4]we candeducethatinfact
g=F(g)inW O . Henceg∈Z/prZasclaimed. (cid:3)
r X0
Finally,wewillneedtoknowthatthelogarithmicdeRham–Wittcomplexcomputesthelogcrystalline
cohomologyofthesemistableschemeX.
Proposition2.4. Thereisanisomorphism
Hi (X ,Ww ∗ ) →∼ Hi (X×/K)
cont e´t X× Q log-cris
foralli≥0.
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CycleclassesandLefschetz(1,1)
Proof. ItsufficestoshowthatHi(Xe´t,Wrw X∗ ×)∼=Hliog-cris(X×/Wr)whereWr =Wr(k). Arguinglocally
onX wemayassumeinfactthatX isaffine,andinparticularadmitsaclosedembeddingX ֒→P into
someaffinespaceoverW JtK.
r
NowapplyingNe´ron–Popescudesingularisation[Pop86,Theorem1.8]toW →WJtK, wemayinfact
r r
writeX =lima Xa asalimitofsmoothk-schemes,suchthat:
• thatthereexistcompatiblenormalcrossingsdivisorsDa ⊂Xa whoseinverseimageinX ispre-
ciselythespecialfibreX ;
0
• there exist compatible closed embeddings Xa ֒→ Pa into smooth Wr-schemes such that P =
lima Pa .
SinceboththelogdeRham–Wittcomplexande´talecohomologycommutewithcofilteredlimitsofschemes,
itsufficestoshowthatthesameistrueoflog-crystallinecohomology,inotherwordsthatwehave
Hliog-cris(X×/Wr)=colima Hliog-cris(Xa×/Wr),
whereXa× denotestheschemeXa endowedwiththelogstructuregivenbyDa . By[Kat89,Theorem6.4],
Hliog-cris(Xa×/Wr)iscomputedasthedeRhamcohomologyofthelog-PDenvelopeofXa× insidePa . Since
log-PDenvelopescommutewithcofilteredlimitsofschemes(i.e. filteredcolimitsofrings), itsufficesto
showthatHi (X×/W)canbecomputedasthedeRhamcohomologyofthelog-PDenvelopeofX×
log-cris r
insideP.
In other words, what we require a logarithmic analogue of [Kat91, Theorem 1.7], or equivalently a
log-p-basisanalogueof[Kat89,Theorem6.4].ButthisfollowsfromProposition1.2.18of[CV15]. (cid:3)
3. MORROW’S VARIATIONAL TATECONJECTURE FORDIVISORS
Thegoalofthissectionistoofferasimplerproofofaspecialcaseof[Mor14,Theorem3.5]forsmooth
andproperschemesX overthepowerseriesringR=kJtK. Thisresultessentiallystatesthatalinebundle
onthespecialfibreofX liftsiffitsitsfirstChernclassinH2 does,andshouldbeviewedasanequichar-
cris
acteristicanalogueofBerthelotandOgus’stheorem[BO83,Theorem3.8]statingthatalinebundleonthe
special fibre of a smooth proper scheme over a DVR in mixed characteristic lifts iff its Chern class lies
in the firstpiece ofthe Hodgefiltration. We willalso givea slightly differentinterpretationof thisresult
thatemphasisesthephilosophythatinequicharacteristicthe‘correct’analogueofaHodgefiltrationisan
E†-structure. Ourproofissimplerinthatitdoesnotdependonanyresultsfromtopologicalcyclichomol-
ogy, but only on fairly standard propertiesof the deRham–Wittcomplex. As such, it is far morereadily
adaptabletothesemistablecase,whichweshalldoin§4below.
Throughoutthissection,X willbeasmoothandproperR=kJtK-scheme.LetR denotekJtK/(tn+1)and
n
setX =X ⊗ R . WriteX forthegenericfibreofX andXforitsformal(t-adic)completion. Sinceall
n R n
schemesinthissectionwillhavetriviallogstructure,wewillusethenotationW W ∗ forthedeRham–Witt
•
complexinsteadofW w ∗. Thekeytechnicalcalculationwewillmakeisthefollowing.
•
Lemma3.1. Fixn≥0,writen=pmn with(n ,p)=1,andletr=m+1.Thenthemap
0 0
dlog:1+tnO →WW 1
Xn r Xn,log
isinjective.
Proof. WemayassumethatX =Spec(A )isaffine,moreovere´taleoverR [x ,...,x ]. Inthiscasesince
n n n 1 d
deformationsofsmoothaffineschemesaretrivial,wehaveAn∼=A0⊗kRn. Hence1+tnAn=1+tnA0,and
8
C.Lazda,A.Pa´l
ourproblemthereforereducestoshowingthatifa∈A issuchthatdlog[1+atn]=0,theninfacta=0.
0
ButvanishingofamaybecheckedoverallclosedpointsofSpec(A ),sobyfunctorialityofthedlogmap
0
wemayinfactassumethatA isafiniteextensionofk;enlargingkwemaymoreoverassumethatA =k.
0 0
Inotherwordsweneedtoshowthatthemap
dlog:1+tnk→WW 1
r Rn
isinjective.Sincekisperfect,any1+atn∈1+tnkcanbewrittenuniquelyas(1+tn0b)pm forsomeb∈k,
hence dlog[1+atn]= pmdlog(1+tn0b). It follows that if dlog[1+atn]=0, then pmn0btn0−1dt =0 in
WrW 1Rn. Since any non-zerosuch b is invertible, the lemma will follow if we can show that pmtn0−1dt is
non-zeroinWW 1 . Thiscanbecheckedeasilyusingtheexactsequence
r Rn
Wr((tn+1)) →d W W 1 ⊗ WR →WW 1 →0
W((tn+1)2) r k[t] Wr(k[t]) r n r Rn
r
from[LZ05]. (cid:3)
Fromthiswededucethefollowing.
Proposition3.2. Forr≫0(dependingonn)thereisacommutativediagram
1 // 1+tO // O∗ // O∗ // 1
Xn Xn X0
dlog dlog
(cid:15)(cid:15) (cid:15)(cid:15)
1 // 1+tO dlog // WW 1 //W W 1 // 0
Xn r Xn,log r X0,log
withexactrows.
Proof. It is well-known that the top row is exact, and the diagram is clearly commutative, it therefore
sufficestoshowthatforallnthesequence
1→1+tO →WW 1 →W W 1 →0
Xn r Xn,log r X0,log
isexactforr≫0. FromthedefinitionofWW 1 andtheexactnessofthesequence
r Xn,log
1→1+tO →O∗ →O∗ →1
Xn Xn X0
it is immediate thatW W 1 →WW 1 is surjective and the composite1+tO →WW 1 is zero.
r Xn,log r X0,log Xn r X0,log
Given a ∈O∗ mapping to 0 in WW 1 , it follows from [Ill79, Proposition I.3.23.2] that there exists
Xn r X0,log
b ∈O∗ andg ∈1+tO suchthata =b pr+g ,andhencedloga =dlogg inWW 1 . Thesequence
Xn Xn r Xn,log
1+tO →WW 1 →WW 1 →0
Xn r Xn,log r X0,log
isthereforeexact,anditremainstoshowthat
dlog:1+tO →W W 1
Xn r Xn,log
isinjectiveforr≫0. ByinductiononnthisfollowsfromLemma3.1above. (cid:3)
Wenowset
WrW iX,log:=linmWrW iXn,log
assheavesonX anddefine
e´t
Hcjont(Xe´t,WW iX,log):=Hj(RlimRG (Xe´t,WrW iX,log)).
r
9
CycleclassesandLefschetz(1,1)
AsanessentiallyimmediatecorollaryofProposition3.2,wededucethekeystepofMorrow’sproofofthe
variationalTateconjectureinthiscase.
Corollary3.3. Let L ∈Pic(X ), with first Chern class c (L)∈H1 (X ,WW 1 ). Then L lifts to
0 1 cont 0,e´t X0,log
Pic(X)ifandonlyifc (L)liftstoH1 (X ,WW 1 )
1 cont e´t X,log
Proof. One directionis obvious. For the other direction, assume that the first Chern class c (L) lifts to
1
H1 (X ,WW 1 ), in particular it therefore lifts to H1 (X ,WW 1 ). Hence by Proposition 3.2 it
cont e´t X,log cont e´t X,log
follows that L lifts to Pic(X), and we may conclude using Grothendieck’s algebrisation theorem that it
liftstoPic(X). (cid:3)
Fromthistheform(crys-f )formofthevariationalTateconjecturefollowsasin[Mor14].
Corollary3.4. LetL ∈Pic(X ) ,withfirstChernclassc (L)∈H2 (X/K)j =p.ThenL liftstoPic(X)
0 Q 1 cris Q
ifandonlyifc (L)liftstoH2 (X/K)j =p.
1 cris
Proof. Let us first assume that k is algebraically closed. By [Mor14, Proposition 3.2] the inclusions
WW 1 [−1]→WW ∗ andWW 1 [−1]→WW ∗ induceanisomorphism
X,log X,log X0,log X0,log
H1 (X ,WW 1 ) →∼ H2 (X /K)j =p
cont 0,e´t X0,log Q cris 0
andasurjection
Hc1ont(Xe´t,WW 1X,log)Q։Hc2ris(X/K)j =p.
Theclaimfollows. Ingeneral,weargueasin[Mor14,Theorem1.4]: theclaimfork algebraicallyclosed
shows that L lifts to Pic(X) after making the base change kJtK→kJtK. Let kJtKsh denote the strict
Q
HenselisationofkJtKinsidekJtK,byNe´ron–PopescudesingularisationthereexistssomesmoothlocalkJtKsh-
algebraAsuchthatL liftstoPic(X) aftermakingthebasechangekJtK→A. ButthemapkJtKsh →A
Q
hasasection,fromwhichitfollowsthatinfactL liftstoPic(X) aftermakingsomefinitefieldextension
Q
k →k′. But now simply taking the pushforward via X ⊗ k′ →X and dividing by [k′ :k] gives the
k
result. (cid:3)
Tofinishoffthissection,wewishtogiveaslightlydifferentformulationofCorollary3.4. After[LP16]
we can consider the ‘overconvergent’ rigid cohomology Hi (X/E†) of the generic fibre X, which is a
rig
(j ,(cid:209) )-module over the bounded Robba ring E†. Set Hi (X/R):=Hi (X/E†)⊗ R. By combining
rig rig E†
Dwork’strickwithsmoothandproperbasechangeincrystallinecohomologywehaveanisomorphism
Hriig(X/R)(cid:209) =0∼=Hriig(X0/K)
forall i. In particular, forany L ∈Pic(X ) we can considerc (L) as an elementofHi (X/R)(cid:209) =0⊂
0 Q 1 rig
Hi (X/R). Oneofthe generalphilosophiesof p-adiccohomologyin equicharacteristicisthatwhile the
rig
cohomology groups Hi (X/R) in some sense only depend on the special fibre X , the ‘lift’ X of X is
rig 0 0
seenintheE†-latticeHi (X/E†)⊂Hi (X/R). ThecorrectequicharacteristicanalogueofaHodgefiltra-
rig rig
tion, therefore,is an E†-structure. With this in mind, then, a statementof the variationalTate conjecture
for divisors which is perhaps slightly more transparently analogous to that in mixed characteristic is the
following.
Theorem3.5. AssumethatX isprojectiveoverR. ThenalinebundleL ∈Pic(X ) liftstoPic(X) if
0 Q Q
andonlyifc (L)∈H2 (X/R)liesinH2 (X/E†).
1 rig rig
Proof. Thisissimplyanotherwayofstatingthecondition(flat)in[Mor14,Theorem3.5]. (cid:3)
10