A p-adic regulator map and finiteness results for arithmetic schemes

0.57 MB·English

by  Shuji Saito

A p-ADIC REGULATOR MAP AND FINITENESS RESULTS
FOR ARITHMETIC SCHEMES1
SHUJISAITOANDKANETOMOSATO
ABSTRACT. A main theme of the paper is a conjecture of Bloch-Kato on the image of p-
adic regulator maps for a proper smooth variety X over an algebraic number field k. The
9 conjectureforaregulatormapofparticulardegreeandweightisrelatedtofinitenessoftwo
0
arithmeticobjects: Oneisthep-primarytorsionpartoftheChowgroupincodimension2of
0
X. AnotherisanunramifiedcohomologygroupofX.Asanapplication,foraregularmodel
2
X ofXovertheintegerringofk,weshowaninjectivityresultontorsionofacycleclassmap
n fromtheChowgroupincodimension2ofX toanewp-adiccohomologyofX introduced
a
J bythesecondauthor,whichisacandidateoftheconjecturale´talemotiviccohomologywith
finitecoefficientsofBeilinson-Lichtenbaum.
2
2

]
G
A 1. INTRODUCTION
.
h
at Let k be an algebraic number field and let Gk be the absolute Galois group Gal(k/k),
m where k denotes a fixed algebraic closure of k. Let X be a projective smooth variety over k
[ and put X := X k k. Fix a prime p and integers r,m 1. A main themeof this paper is a
⊗ ≥
conjectureofBloch andKato concerning theimageofthep-adicregulatormap
5
v
regr,m : CHr(X,m) Q H1 (k,H2r−m−1(X,Q (r)))
1 ⊗ p −→ cont ´et p
8
0 fromBloch’shigherChowgrouptocontinuousGaloiscohomologyofG ([BK2],Conjecture
k
2 5.3). See 3 below for the definition of this map in the case (r,m) = (2,1). This conjecture
1 §
affirms that its image agrees with the subspace H1(k,H2r−m−1(X,Q (r))) defined in loc.
6 g ´et p
0 cit., and plays a crucial role in the so-called Tamagawa number conjecture on special values
/
h ofL-functionsattached toX. IntermsofGaloisrepresentations,theconjecturemeansthata
at 1-extensionofcontinuousp-adicrepresentationsofGk
m
0 H2r−m−1(X,Q (r)) E Q 0
v: −→ ´et p −→ −→ p −→
i arises froma 1-extensionofmotivesoverk
X
r 0 h2r−m−1(X)(r) M h(Spec(k)) 0,
a
−→ −→ −→ −→
if and only if E is a de Rham representation of G . There has been only very few known
k
results on the conjecture. In this paper we consider the following condition, which is the
Bloch-Kato conjectureinthespecialcase (r,m) = (2,1):
Date:January21,2009.
Keywordsandphrases. p-adicregulator,unramifiedcohomology,Chowgroups,p-adice´taleTatetwists.
2000MathematicsSubjectClassification:Primary14C25,14G40;Secondary14F30,19F25,11G45
(1)Theearlierversionwasentitled‘Torsioncycleclassmapsincodimensiontwoofarithmeticschemes’.
1
2 S.SAITOANDK.SATO
H1: Theimageof theregulatormap
reg := reg2,1 : CH2(X,1) Q H1 (k,H2(X,Q (2))).
⊗ p −→ cont ´et p
agrees with H1(k,H2(X,Q (2))).
g ´et p
Wealso consideravariant:
H1*: The imageof theregulatormapwith Q /Z -coefficients
p p
reg : CH2(X,1) Q /Z H1 (k,H2(X,Q /Z (2)))
Qp/Zp ⊗ p p −→ Gal ´et p p
agrees with H1(k,H2(X,Q /Z (2))) (see 2.1 forthedefinitionofthisgroup).
g ´et p p Div §
We will show that H1 always impliesH1*, which is not straight-forward. On the otherhand
theconverseholdsas wellundersomeassumptions. See Remark 3.2.4belowfordetails.
Fact1.1. TheconditionH1holdsin thefollowingcases:
(1) H2(X,O ) = 0 ([CTR1], [CTR2], [Sal]).
X
(2) X is the self-productof an ellipticcurve over k = Q with square-freeconductor and
withoutcomplexmultiplication,andp 5 ([Md], [Fl], [LS], [La1]).
≥
(3) X istheellipticmodularsurfaceof level4 over k = Q andp 5([La2]).
≥
(4) X isa Fermatquarticsurfaceover k = Qor Q(√ 1)and p 5 ([O]).
− ≥
A main result of this paper relates the condition H1* to finiteness of two arithmetic objects.
One is the p-primary torsion part of the Chow group CH2(X) of algebraic cycles of codi-
mension two on X modulo rational equivalence. Another is an unramified cohomology of
X, whichweare goingtointroducein whatfollows.
Let o be the integer ring of k, and put S := Spec(o ). We assume that there exists a
k k
regular scheme X which is proper flat of finite type over S and whose generic fiber is X.
Wealso assumethefollowing:
( ) X hasgood orsemistablereductionateach closedpointofS ofcharacteristicp.
∗
LetK = k(X)bethefunctionfieldofX. Foranintegerq 0,letX q bethesetofallpoints
x X ofcodimensionq. Fixanintegern 0. Roughlys≥peaking,theunramifiedcohomol-
∈ ≥
ogy group Hn+1(K,Q /Z (n)) is defined as the subgroup of Hn+1(Spec(K),Q /Z (n))
ur p p ´et p p
consisting of those elements that are “unramified” along all y X 1. For a precise defini-
tion, we need the p-adic e´tale Tate twist Tr(n) = Tr(n)X introd∈uced in [SH]. This object is
defined in Db(X ,Z/prZ), the derived category of bounded complexes of e´tale sheaves of
´et
Z/prZ-modules on X , and expected to coincide with Γ(2)X L Z/prZ. Here Γ(2)X de-
´et ⊗ ´et
notes the conjectural e´tale motivic complex of Beilinson-Lichtenbaum [Be], [Li1]. We note
that the restriction of T (n) to X [p−1] := X Z[p−1] is isomorphic to µ⊗n, where µ
r ⊗Z pr pr
denotes the e´tale sheaf of prth roots of unity. Then Hn+1(K,Q /Z (n)) is defined as the
ur p p
kernel oftheboundary mapofe´talecohomologygroups
H´ent+1(Spec(K),Qp/Zp(n)) −→ Hxn+2(Spec(OX,x),T∞(n)),
x∈X1
M
whereT (n) denoteslim T (n). Therearenatural isomorphisms
∞ r≥1 r
−→
H1 (K,Q /Z (0)) H1(X,Q /Z ) and H2 (K,Q /Z (1)) Br(X ) ,
ur p p ≃ ´et p p ur p p ≃ p-tors
p-ADICREGULATORANDFINITENESS 3
whereBr(X )denotestheGrothendieck-BrauergroupH2(X,G ),andforanabeliangroup
´et m
M, M denotes its p-primary torsion part. An intriguing question is as to whether the
p-tors
group Hn+1(K,Q /Z (n)) is finite, which is related to several significant theorems and
ur p p
conjectures in arithmetic geometry (see Remark 4.2.10 below). In this paper we are con-
cerned with the case n = 2. A crucial role will be played by the following subgroup of
H3 (K,Q /Z (2)):
ur p p
H3 (K,X;Q /Z (2))
ur p p
:= Im H3(X,Q /Z (2)) H3(Spec(K),Q /Z (2)) H3 (K,Q /Z (2)).
´et p p → ´et p p ∩ ur p p
(cid:16) (cid:17)
Ourfinitenessresultis thefollowing:
Theorem 1.2. Let X and X beasabove, andassumep 5. Then:
≥
(1) H1*impliesthatCH2(X) and H3 (K,X;Q /Z (2))arefinite.
p-tors ur p p
(2) Assumethatthereduced partofeveryclosedfiber ofX /S has simplenormalcross-
ings on X , and that the Tate conjecture holds in codimension 1 for the irreducible
components of those fibers. Then the finiteness of the groups CH2(X) and
p-tors
H3 (K,X;Q /Z (2))impliesH1*.
ur p p
Theassertion(2)is aconverseof(1) undertheassumptionoftheTateconjecture. Weobtain
thefollowingresult fromTheorem 1.2(1)(seealsotheproofofTheorem 1.5in 5.1below):
§
Corollary1.3. H3 (K,X;Q /Z (2))is finiteinthefourcasesin Fact1.1.
ur p p
WewillalsoprovevariantsofTheorem1.2overlocalintegerrings(seeTheorems3.1.1,5.1.1
and 7.1.1 below). As for the finiteness of H3 (K,Q /Z (2)) over local integer rings, Spiess
ur p p
proved that H3 (K,Q /Z (2)) = 0, assuming that o is an ℓ-adic local integer ring with
ur p p k
ℓ = pandthateitherH2(X,O ) = 0orX isaproductoftwosmoothellipticcurvesoverS
X
6
([Spi], 4). In[SSa],theauthorsextendedhisvanishingresulttoamoregeneralsituationthat
o is ℓ-§adic local with ℓ = p and that X has generalized semistable reduction. Finally we
k
6
have to remark that there exists a smooth projective surface X with p (X) = 0 over a local
g
6
field k forwhich theconditionH1* doesnotholdand such thatCH2(X) isinfinite[AS].
tors
We next explain an application of the above finiteness result to a cycle class map of arith-
metic schemes. Let us recall the following fact due to Colliot-The´le`ne, Sansuc, Soule´ and
Gros:
Fact1.4([CTSS],[Gr]). LetX beapropersmoothvarietyoverafinitefieldofcharacteristic
ℓ > 0. Let p be a prime number, which may be the same as ℓ. Then the cycle class map
restrictedto thep-primarytorsionpart
CH2(X) H4(X,Z/prZ(2))
p-tors −→ ´et
is injectivefor a sufficientlylarger > 0. If ℓ = p, then Z/prZ(2) denotesµ⊗2. If ℓ = p, then
6 pr
Z/prZ(2) denotes W Ω2 [ 2] with W Ω2 the e´tale subsheaf of the logarithmic part of
r X,log − r X,log
theHodge-WittsheafW Ω2 ([Bl1], [Il]).
r X
Inthispaper,westudyan arithmeticvariantofthisfact. Weexpectthatasimilarresultholds
for proper regular arithmetic schemes, i.e., regular schemes which are proper flat of finite
type over the integer ring of a number field or a local field. To be more precise, let k, o , X
k
4 S.SAITOANDK.SATO
and X beas inTheorem 1.2. Thep-adice´taleTatetwistTr(2) = Tr(2)X mentionedbefore
replaces Z/prZ(2)in Fact 1.4, and thereisacycleclass map
̺2 : CH2(X )/pr H4(X,T (2)).
r −→ ´et r
Weare concerned withtheinducedmap
̺2 : CH2(X ) H4(X,T (2)).
p-tors,r p-tors −→ ´et r
It is shown in [SH] that the group on the right hand side is finite. So the injectivity of this
map is closely related with the finiteness of CH2(X ) . The second main result of this
p-tors
paperconcerns theinjectivityofthismap:
Theorem 1.5 ( 5). AssumethatH2(X,O ) = 0. Then CH2(X ) is finiteand ̺2 is
§ X p-tors p-tors,r
injectivefor asufficientlylarger > 0.
The finiteness of CH2(X ) in this theorem is originally due to Salberger [Sal], Colliot-
p-tors
The´le`ne and Raskind [CTR1], [CTR2]. Note that there exists a projectivesmooth surface V
overanumberfield withH2(V,O ) = 0 whosetorsioncycleclass map
V
CH2(V) H4(V,µ⊗2)
p-tors −→ ´et pr
isnotinjectiveforsomebadprimepandanyr 1[Su](cf.[PS]). Ourresultsuggeststhatwe
≥
are ableto recovertheinjectivityoftorsioncycleclass maps by consideringaproperregular
model of V over the ring of integers in k. The fundamental ideas of Theorem 1.5 are the
following. AcrucialpointoftheproofofFact1.4in[CTSS]and[Gr]isDeligne’sproofofthe
Weil conjecture [De2]. In thearithmeticsituation,theroleof theWeil conjecture is replaced
bytheconditionH1,whichimpliesthefinitenessofCH2(X) andH3 (K,X;Q /Z (2))
p-tors ur p p
by Theorem 1.2(1). The injectivity result in Theorem 1.5 is derived from the finiteness of
thoseobjects.
This paper is organized as follows. In 2, we will review some fundamental facts on Ga-
§
lois cohomology groups and Selmer groups which will be used frequently in this paper. In
3, we will prove the finiteness of CH2(X) in Theorem 1.2(1). In 4, we will review
p-tors
§ §
p-adic e´tale Tate twists briefly and then provide some fundamental lemmas on cycle class
maps and unramified cohomology groups. In 5, we will first reduce Theorem 1.5 to The-
§
orem 1.2(1), and then reduce the finiteness of H3 (K,X;Q /Z (2)) in Theorem 1.2(1) to
ur p p
Key Lemma 5.4.1. In 6, we will prove that key lemma, which will complete the proof of
§
Theorem 1.2(1). 7 will be devoted to the proof of Theorem 1.2(2). In the appendix A,
§
we will include an observation that the finiteness of H3 (K,Q /Z (2)) is deduced from the
ur p p
Beilinson–Lichtenbaumconjectures on motiviccomplexes.
Acknowledgements. The research for this article was partially supported by JSPS Postdoc-
toral Fellowship for Research Abroad and EPSRC grant. The second author expresses his
gratitude to University of Southern California and The University of Nottingham for their
great hospitality. The authors also express their gratitude to Professors Wayne Raskind,
ThomasGeisserandIvan Fesenko forvaluablecommentsand discussions.
p-ADICREGULATORANDFINITENESS 5
NOTATION
1.6. Foran abelian group M and a positiveintegern, let M and M/n bethe kernel and the
n
×n
cokernel of the map M M, respectively. See 2.3 below for other notation for abelian
−→ §
groups. For a field k, let k be a fixed separable closure, and let G be the absolute Galois
k
group Gal(k/k). For a discrete G -module M, let H∗(k,M) be the Galois cohomology
k
groups H∗ (G ,M), which is the same as the e´tale cohomology groups of Spec(k) with
Gal k
coefficients inthee´talesheafassociatedwithM.
1.7. Unless indicated otherwise, all cohomology groups of schemes are taken over the e´tale
topology. For a scheme X, an e´tale sheaf F on X (or more generally an object in the
derived category of sheaves on X ) and a point x X, we often write H∗(X,F) for
H∗(Spec(O ),F). For a pure-d´eitmensional scheme∈X and a non-negative ixnteger q, let
x X,x
Xq betheset of all pointson X of codimensionq. Forapointx X, let κ(x) beitsresidue
∈
field. Foranintegern 0andanoetherianexcellentschemeX,CH (X)denotestheChow
n
≥
group of algebraic cycles on X of dimension n modulo rational equivalence. If X is pure-
dimensional and regular, we will often write CHdim(X)−n(X) for this group. For an integral
scheme X of finite type over Spec(Q), Spec(Z) or Spec(Z ), we define CH2(X,1) as the
ℓ
cohomologygroup,at themiddle,oftheGersten complexofMilnorK-groups
KM(L) κ(y)× Z,
2 −→ −→
y∈X1 x∈X2
M M
whereLdenotesthefunctionfieldofX. Asiswell-known,thisgroupcoincideswithahigher
Chowgroup([Bl3],[Le2])bythelocalizationtheory([Bl4],[Le1])andtheNesterenko-Suslin
theorem[NS](cf. [To]).
1.8. In 4–7, we will work under the following setting. Let k be an algebraic number field
orits co§m§pletionat a finiteplace. Let o be theintegerring of k and put S := Spec(o ). Let
k k
p be a prime number, and let X be a regular scheme which is proper flat of finite type over
S and satisfiesthefollowingcondition:
Ifp isnotinvertibleonX ,thenX hasgoodorsemistablereductionateach
closed pointof S of characteristicp.
Let K be the function field of X . We define H3 (K,Q /Z (2)) and H3 (K,X;Q /Z (2))
ur p p ur p p
inthesameway as intheintroduction:
H3 (K,Q /Z (2)) := Ker H3(K,Q /Z (2)) H4(X,T (2)) ,
ur p p p p −→ y∈X1 y ∞
H3 (K,X;Q /Z (2)) (cid:16) M (cid:17)
ur p p
:= Im H3(X,Q /Z (2)) H3(K,Q /Z (2)) H3 (K,Q /Z (2)),
p p → p p ∩ ur p p
(cid:16) (cid:17)
whereT (n) denoteslim T (n). If k isan algebraicnumberfield, then thissettingisthe
∞ r≥1 r
sameas thatintheintr−o→duction.
1.9. Let k be an algebraic number field, and let X S = Spec(o ) be as in 1.8. In this
k
situation,we willoftenusethefollowingnotation. Fo→raclosed pointv S, leto (resp. k )
v v
∈
6 S.SAITOANDK.SATO
bethecompletionofo (resp. k)at v, andlet F betheresiduefield ofk . Weput
k v v
X := X o , X := X k , Y := X F
v ⊗ok v v ⊗ok v v ⊗ok v
andwritej : X ֒ X (resp.i : Y ֒ X )forthenaturalopen(resp.closed)immersion.
v v v v v v
→ →
Weput Y := Y F , and writeΣ fortheset ofallclosed pointon S ofcharacteristicp.
v v ×Fv v
1.10. Letk bean ℓ-adiclocal fieldwithℓaprimenumber,andletX S = Spec(o ) beas
k
→
in1.8. Inthissituation,wewilloftenusethefollowingnotation. LetFbetheresiduefieldof
k and put
X := X k, Y := X F.
o o
⊗ k ⊗ k
Wewritej : X ֒ X (resp. i : Y ֒ X )forthenatural open(resp. closed)immersion. Let
kur bethemaxim→alunramified exte→nsionofk, and letour beits integerring. Weput
X ur := X our, Xur := X kur, Y := Y F.
o o F
⊗ k ⊗ k ×
p-ADICREGULATORANDFINITENESS 7
2. PRELIMINARIES ON GALOIS COHOMOLOGY
Inthissection,weprovidesomepreliminarylemmaswhichwillbefrequentlyusedinthis
paper. Let k be an algebraic number field (global field) or its completion at a finite place
(localfield). Leto betheintegerringofk,andputS := Spec(o ). Letpbeaprimenumber.
k k
Ifk is global,weoftenwriteΣ forthesetoftheclosedpointsonS ofcharacteristicp.
2.1. Selmer group. Let X beapropersmoothvarietyoverSpec(k), andputX := X k.
k
⊗
If k is global, we fix a non-empty open subset U S Σ for which there exists a proper
0
smooth morphism X U with X k X⊂. Fo\r v S1, let k and F be as in the
U0 → 0 U0 ×U0 ≃ ∈ v v
notation1.9. In thissectionweareconcerned withG -modules
k
V := Hi(X,Q (n)) and A := Hi(X,Q /Z (n)).
p p p
For M = V or A and a non-empty open subset U U , let H∗(U,M) denote the e´tale
0
⊂
cohomologygroupswithcoefficients inthesmoothsheafon U associatedto M.
´et
Definition 2.1.1. (1) Assumethatk islocal. LetH1(k,V)andH1(k,V)beasdefinedin
f g
[BK2], (3.7). For f,g ,we define
∗ ∈ { }
H1(k,A) := Im H1(k,V) H1(k,A) .
∗ ∗ −→
(2) Assume that k is global. For M V,A and a non-empty open subset U S, we
(cid:0) (cid:1)
∈ { } ⊂
definethesubgroupH1 (k,M) H1 (k,M) as thekernel ofthenaturalmap
f,U ⊂ cont
H1 (k,M) H1 (k ,M)/H1(k ,M) H1 (k ,M)/H1(k ,M).
cont −→ cont v f v × cont v g v
v∈U1 v∈S\U
Y Y
If U U , we have
0
⊂
H1 (k,M) = Ker H1(U,M) H1 (k ,M)/H1(k ,M) .
f,U −→ v∈S\U cont v g v
WedefinethegroupH(cid:16)1(k,M) and H1Y(k,M) as (cid:17)
g ind
H1(k,M) := lim H1 (k,M), H1 (k,M) := lim H1(U,M),
g f,U ind
U−⊂→U0 U−⊂→U0
where U runs through all non-empty open subsets of U . These groups are indepen-
0
dent ofthechoiceof U andX (cf.[EGA4], 8.8.2.5).
0 U0
(3) If k islocal,we defineH1 (k,M) tobeH1 (k,M) forM V,A .
ind cont ∈ { }
NotethatH1 (k,A) = H1(k,A).
ind
2.2. p-adic point of motives. We provide a key lemma from p-adic Hodge theory which
play crucial roles in this paper (see Theorem 2.2.1 below). Assume that k is a p-adic local
field, and that there exists a regular scheme X which is proper flat of finite type over S =
Spec(o )withX k X andwhichhassemistablereduction. Letiandnbenon-negative
k ⊗ok ≃
integers. Put
Vi := Hi+1(X,Q ), Vi(n) := Vi Q (n),
p ⊗Qp p
and
Hi+1(X,τ Rj Q (n)) := lim Hi+1(X,τ Rj µ⊗n) Q ,
≤n ∗ p r≥1 ≤n ∗ pr ⊗Zp p
wherej denotesthenatural openimmersi←o−nX ֒ X . There isanatural pull-backmap
(cid:8) → (cid:9)
α : Hi+1(X,τ Rj Q (n)) Hi+1(X,Q (n)).
≤n ∗ p p
−→
8 S.SAITOANDK.SATO
Let Hi+1(X,τ Rj Q (n))0 bethekernel ofthecompositemap
≤r ∗ p
α′ : Hi+1(X,τ Rj Q (n)) α Hi+1(X,Q (n)) Vi+1(n) Gk.
≤n ∗ p p
−→ −→
Forthisgroup,thereisa compositemap (cid:0) (cid:1)
α : Hi+1(X,τ Rj Q (n))0 F1Hi+1(X,Q (n)) H1 (k,Vi(n)),
≤n ∗ p −→ p −→ cont
whosefirstarrowisinducedbyα. ThesecondarrowisanedgehomomorphismaHochschild-
Serre spectral sequence
Eu,v := Hu (k,Vv(n)) = Hu+v(X,Q (n))( Hu+v(X,Q (n))).
2 cont ⇒ cont p ≃ p
and F• denotesthefiltration onHi+1(X,Q (n)) resultingfrom thisspectral sequence. Con-
p
cerning theimageofα, weshowthefollowing:
Theorem 2.2.1. Assumethatp n+2. Then Im(α) = H1(k,Vi(n)).
≥ g
Proof. We use the following comparison theorem of log syntomic complexes and p-adic
vanishing cycles due to Kato, Kurihara and Tsuji ([Ka1], [Ku], [Ka2], [Ts2]). Let Y be the
closed fiberofX S andlet ι : Y ֒ X bethenatural closedimmersion.
→ →
Theorem 2.2.2 (Kato/Kurihara/Tsuji). For integers n,r with 0 n p 2 and r 1,
≤ ≤ − ≥
thereisa canonicalisomorphism
η : slog(n) ∼ ι ι∗(τ Rj µ⊗n) in Db(X ,Z/prZ),
r −→ ∗ ≤n ∗ pr ´et
where slog(n) = slog(n)X isthelogsyntomiccomplexdefined byKato[Ka1](cf. [Ts1]).
r r
Put
H∗(X,slog(n)) := lim H∗(X,slog(n)) Q ,
Qp r≥1 r ⊗Zp p
←−
and defineHi+1(X,slog(n))0 as theke(cid:8)rnel ofthecompositema(cid:9)p
Q
p
Hi+1(X,slog(n)) ∼ Hi+1(X,τ Rj Q (n)) α′ Vi+1(n) Gk,
Qp −η→ ≤n ∗ p −→
wherewehaveusedtheproperness ofX overS. Thereis anindu(cid:0)ced map (cid:1)
η : Hi+1(X,slog(n))0 ∼ Hi+1(X,τ Rj Q (n))0 α H1 (k,Vi(n)).
Qp −η→ ≤n ∗ p −→ cont
On theotherhand,we havethefollowingfact ([La3], [Ne2], Theorem 3.1):
Theorem 2.2.3(Langer/Nekova´rˇ). Im(η) agrees withH1(k,Vi(n)).
g
By thesefacts, weobtainTheorem2.2.1. (cid:3)
Remark 2.2.4. (1) Theorem2.2.3isanextensionofthep-adicpointconjectureraisedby
Schneider in the good reduction case [Sch]. This conjecture was proved by Langer-
Saito[LS]in a specialcaseandbyNekova´rˇ [Ne1]inthegeneralcase.
(2) Theorem 2.2.3 holds unconditionally on p, if we define the space Hi+1(X,slog(n))
Q
p
usingTsuji’sversionof logsyntomiccomplexes S (n)(r 1)in[Ts1], 2.
r
≥ §
e
p-ADICREGULATORANDFINITENESS 9
2.3. Elementary facts on Z -modules. For an abelian group M, let M be its maximal
p Div
divisible subgroup. For a torsion abelian group M, let Cotor(M) be the cotorsion part
M/M . We say that a Z -module M is cofinitely generated over Z (or simply, cofinitely
Div p p
generated),ifitsPontryagindualHom (M,Q /Z ) isa finitelygenerated Z -module.
Zp p p p
Lemma 2.3.1. Let 0 L M N 0bea shortexact sequenceofZ -modules.
p
→ → → →
(1) AssumethatL, M andN arecofinitelygenerated. Then thereis apositiveinteger r
0
suchthatforanyr r wehavean exact sequenceof finiteabelianp-groups
0
≥
0 L M N Cotor(L) Cotor(M) Cotor(N) 0.
pr pr pr
→ → → → → → →
Consequently,takingtheprojectivelimitofthisexactsequencewithrespecttor r
0
≥
thereisan exact sequenceof finitelygeneratedZ -modules
p
0 T (L) T (M) T (N) Cotor(L) Cotor(M) Cotor(N) 0,
p p p
→ → → → → → →
where foran abeliangroupA, T (A)denotes itsp-adicTatemodule.
p
(2) Assume that L is cofinitely generated up to a group of finite exponent, i.e., L is
Div
cofinitely generated and Cotor(L) has a finite exponent. Assume further that M
is divisible, and that N is cofinitely generated and divisible. Then L and M are
cofinitelygenerated.
(3) AssumethatLiscofinitelygenerateduptoagroupoffiniteexponent. Thenforadivis-
iblesubgroupD N anditsinverseimageD′ M, theinducedmap(D′) D
Div
⊂ ⊂ →
is surjective. In particular,thenaturalmap M N issurjective.
Div Div
→
(4) If L = N = 0,then wehaveM = 0.
Div Div Div
Proof. (1)Thereisa commutativediagramwithexact rows
0 // L // M // N // 0
×pr ×pr ×pr
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // L // M // N // 0.
One obtains the assertion by applying the snake lemma to this diagram, noting Cotor(A)
≃
A/pr foracofinitelygenerated Z -moduleA andasufficientlylarger 1.
p
≥
(2)OurtaskistoshowthatCotor(L)isfinite. Byasimilarargumentasfor(1),thereisan
exact sequenceforasufficientlylarger 1
≥
0 L M N Cotor(L) 0,
pr pr pr
−→ −→ −→ −→ −→
where wehaveused theassumptionson L and M. Hencethefiniteness ofCotor(L) follows
from theassumptionthatN iscofinitelygenerated.
(3)Wehaveonly toshowthecase D = N . ForaZ -moduleA, wehave
Div p
A = Im Hom (Q ,A) A
Div Zp p →
by [J1], Lemma (4.3.a). Since Ext1Z (Q(cid:0)p,L) = 0 by the ass(cid:1)umption on L, the following
p
natural mapissurjective:
Hom (Q ,M) Hom (Q ,N).
Zp p −→ Zp p
By thesefacts, thenatural mapM N issurjective.
Div Div
→
(4)ForaZ -moduleA, wehave
p
A = 0 Hom (Q ,A) = 0
Div ⇐⇒ Zp p
10 S.SAITOANDK.SATO
by [J1], Remark (4.7). Theassertionfollowsfromthisfact and theexactsequence
0 Hom (Q ,L) Hom (Q ,M) Hom (Q ,N)
−→ Zp p −→ Zp p −→ Zp p
Thiscompletestheproofofthelemma. (cid:3)
2.4. DivisiblepartofH1(k,A). Letthenotationbeasin 2.1. Weproveherethefollowing
§
general lemma,which willbeused frequentlyin 3–7:
§§
Lemma 2.4.1. Under thenotationin Definition2.1.1we have
Im H1 (k,V) H1(k,A) = H1(k,A) ,
ind → Div
Im H1(k,V) H1(k,A) = H1(k,A) .
(cid:0) g → (cid:1) g Div
Proof. Theassertion isclear(cid:0)ifk is local. Assumeth(cid:1)at k isglobal. Withoutloss ofgenerality
we may assumethat A is divisible. We proveonly the second equality and omitthe first one
(seeRemark 2.4.9(2)below). LetU S beas in 2.1. We have
0
⊂ §
(2.4.2) Im H1 (k,V) H1(U,A) = H1 (k,A)
f,U → f,U Div
fornon-emptyopenU U . Thisfollowsfromacommutativediagram withexact rows
(cid:0) 0 (cid:1)
⊂
0 // H1 (k,V) // H1(U,V) // H1 (k ,V)/H1(k ,V)
f,U v∈S\U cont v g v
α Q β
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // H1 (k,A) // H1(U,A) // H1(k ,A)/H1(k ,A)
f,U v∈S\U v g v
andthefactsthatCoker(α)isfiniteandthatKer(β)QisfinitelygeneratedoverZ . By (2.4.2),
p
thesecondequalityofthelemmaisreduced to thefollowingassertion:
(2.4.3) lim H1 (k,A) = lim H1 (k,A)) .
f,U Div f,U Div
U−⊂→U0 U−⊂→U0 !
(cid:0) (cid:1)
To showthisequality,wewillprovethefollowingsublemma:
Sublemma 2.4.4. Foran open subsetU U ,put
0
⊂
C := Coker H1 (k,A) H1 (k,A) .
U f,U0 → f,U
Thenthereexistsanon-emptyopensubsetU U suchthatthequotientC /C isdivisible
(cid:0) 1 ⊂ 0 (cid:1) U U1
foranyopen subsetU U .
1
⊂
We first finish our proof of(2.4.3) admittingthis sublemma. Let U U be a non-empty
1 0
⊂
opensubsetas inSublemma2.4.4. NotingthatH1 (k,A)iscofinitelygenerated, thereisan
f,U
exact sequenceoffinitegroups
Cotor H1 (k,A) Cotor H1 (k,A) Cotor(C /C ) 0
f,U1 −→ f,U −→ U U1 −→
for open U U by Lemma 2.3.1(1). By this exact sequence and Sublemma 2.4.4, the
(cid:0) 1 (cid:1) (cid:0) (cid:1)
⊂
natural map Cotor(H1 (k,A)) Cotor(H1 (k,A)) is surjective for any open U U ,
f,U1 → f,U ⊂ 1
whichimpliesthattheinductivelimit
lim Cotor(H1 (k,A))
f,U
U−⊂→U0
isa finitegroup. Theequality(2.4.3)followseasilyfrom this.