A refined Bloch group and the third homology of SL_2 of a field

0.28 MB·English

by  Kevin Hutchinson

A REFINED BLOCH GROUP AND THE THIRD HOMOLOGY OF SL OF A
2
FIELD
KEVINHUTCHINSON
1 Abstract. WeusethepropertiesoftherefinedBlochgrouptoprovethatH3 ofSL2 ofaglobal
1 field is never finitely-generated, and to calculate H of SL of local fields with odd residue
3 2
0 characteristicuptosome2-torsion. Wealsogivelowerboundsforthe3-rankofthehomology
2 groupsH (SL (O ),Z).
3 2 S
v
o
N
Contents

1
2 1. Introduction 1

] 2. ReviewofBlochGroups 4
T
3. SomealgebrainRP(F) 7
K
. 4. ValuationsandSpecialization 16
h
t 5. Someapplications 23
a
m
6. ThethirdhomologyofSL oflocalfields 25
2
[
References 34

4
v
9
7
2 1. Introduction
3
.
1 In [10], Chih-Han Sah quotes S. Lichtenbaum who mentions our lack of knowledge of the
0 precisestructureofH (SL (Q),Z)asanexampleoftheunsatisfactorystateofourunderstanding
1 3 2
1 of the homology of linear groups. This was nearly twenty five years ago, and to the author’s
: knowledge the precise structure of this group is still unknown. We will return to this question
v
i below. (Observe, however, that H (SL (Q),Z) (cid:27) Kind(Q) (cid:27) Z/24 for all n ≥ 3 - by the results
X 3 n 3
of[5],forexample.)
r
a
In this article, we study the structure of the third homology of SL of fields by using the prop-
2
erties of the refined Bloch group of the field, which was introduced in [4]. We are particularly
interested in understanding H (SL (F),Z) as a functor of F, and its possible relation to other
3 2
functorsin K-theoryandalgebraicgeometry.
What are now referred to as Bloch groups of fields first appeared in the work of S. Bloch in
the late 1970s (see [1] for the lecture notes) as a way of constructing explicit maps - and, in
particular, regulators - on K of fields. In the 1980s, they were studied by Dupont and Sah
3
(under the name scissors congruence group) because of their connection with 3-dimensional
hyperbolic geometry ([2], [10]). This connection is still actively studied today: Bloch groups
of number fields are targets for Bloch invariants of certain finite-volume oriented hyperbolic
Date:November22,2011.
1991MathematicsSubjectClassification. 19G99,20G10.
Keywordsandphrases. K-theory,GroupHomology.
1
2 KEVINHUTCHINSON
3-manifolds ([8], [9], [3]). These invariants are amenable to explicit calculation and are related
in a known way to the Chern-Simons invariant. There are also intriguing connections between
Blochgroups,conformalfieldtheoriesandevenmodularformtheory([12],[7]).
The precise relationship between the Bloch group and K-theory was established via their mu-
tualconnectiontothehomologyoflineargroups. Theseconnectionsweregreatlyclarifiedand
exploited in the work of Suslin ([11]): For an infinite field F, the Bloch group, B(F), arises
naturally as a quotient of H (GL (F),Z). Of course, K (F) admits a Hurewicz homomorphism
3 2 3
toH (GL(F),Z) = H (GL (F),Z). Suslinprovedthatthereisanaturalshortexactsequence
3 3 3
(cid:93)
0 → TorZ(µ ,µ ) → Kind(F) → B(F) → 0
1 F F 3
(cid:93)
whereTorZ(µ ,µ )istheuniquenontrivialextensionofTorZ(µ ,µ )byZ/2and
1 F F 1 F F
Kind(F) = Coker(KM(F) → K (F)).
3 3 3
TheseresultsofSuslinwereextendedtofinitefields(withatleast4elements)in[4].
InalettertoSah,SuslinaskedthequestionwhetherthecompositeH (SL (F),Z) → H (GL (F),Z) →
3 2 3 2
Kind(F)inducesanisomorphism
3
H (SL (F),Z) (cid:27) Kind(F).
3 2 F× 3
The current state of knowledge on this question is that the map is surjective ([5]) and that the
inducedmap
H (SL (F),Z[1]) → Kind(F)[1]
3 2 2 F× 3 2
isanisomorphism,where A[1]denotes A⊗Z[1]foranyabeliangroup A([6]).
2 2
ThehomologygroupsofthespeciallineargroupsSL (F)arenaturallymodulesoverthegroup
n
ringZ[F×]viatheshortexactsequences
1 (cid:47)(cid:47) SL (F) (cid:47)(cid:47) GL (F) det (cid:47)(cid:47) F× (cid:47)(cid:47) 1.
n n
Since the scalar matrices a · I are central and have determinant an, it, of course, follows that
n
(F×)n acts trivially on H (SL (F),Z). In particular, the groups H (SL (F),Z) are modules for
k n k 2
thegroupringR := Z[F×/(F×)2].
F
Whenn > k (orn ≥ k whenk isodd),weareintherangeofstability(see[10],[5]and[?])and
thismodulestructureisnecessarilytrivial. Butbelowtherangeofstability,themodulestructure
appears to be nontrivial and interesting. For example, the unstable groups H (SL (F),Z) are
2n 2n
modules over the Grothendieck-Witt ring of the field (which is a quotient of R ) and surject
F
ontotheMilnor-Witt K-theorygroups KMW(F)([?]).
2n
In [4] the author defined a refined Bloch group, RB(F), of a field F which was shown to have
thefollowingproperties:
(1) The group RB(F) is an R -module and there is a natural surjective homomorphism of
F
R -modules
F
H (SL (F),Z) (cid:47)(cid:47) (cid:47)(cid:47) RB(F)
3 2
(2) Thisinducesacommutativediagram(ofR [1]-modules)withexactrows
F 2
0 (cid:47)(cid:47) TorZ(µ ,µ )[1] (cid:47)(cid:47) H (SL (F),Z[1]) (cid:47)(cid:47) RB(F)[1] (cid:47)(cid:47) 0
1 F F 2 3 2 2 2
=
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 (cid:47)(cid:47) TorZ(µ ,µ )[1] (cid:47)(cid:47) Kind(F)[1] (cid:47)(cid:47) B(F)[1] (cid:47)(cid:47) 0
1 F F 2 3 2 2
HereR actstriviallyonthebottomrowandfurthermore
F
RefinedBlochgroupandH ofSL 3
3 2
(3) On taking F×-coinvariants, the top row becomes isomorphic to the bottom row. In
particular,RB(F)[1] (cid:27) B(F)[1]and
2 F× 2
I RB(F)[1] (cid:27) I H (SL (F),Z[1]) = Ker(H (SL (F),Z[1]) → Kind(F)[1])
F 2 F 3 2 2 3 2 2 3 2
whereI denotestheaugmentationidealofthegroupringR .
F F
(ThegroupI RB(F)[1]isalsothekernelofthestabilizationhomomorphism
F 2
H (SL (F),Z[1]) → H (SL (F),Z[1]).
3 2 2 3 3 2
ThecokernelofthismapisthethirdMilnor K-group KM(F)[1]andtheimageisisomorphicto
3 2
Kind(F)[1]. )
3 2
The main result of the current article (Theorem 4.4 and its corollaries) tells us that given a
valuation v on the field F with residue field k, there are surjective reduction or specialization
homomorphisms
RB(F) (cid:47)(cid:47) (cid:47)(cid:47) R(cid:91)P(k)
(cid:91)
whereRP(k)isacertainquotientoftherefinedpre-Blochgroup,RP(k),ofthefieldk.
In particular, if a ∈ F× and v(a) is not a multiple of 2, there is a specialization homomorphism
whichinducesasurjection
e−RB(F)[1] (cid:47)(cid:47) (cid:47)(cid:47) P(cid:100)(k)[1]
a 2 2
where
1−(cid:104)a(cid:105)
e− = ∈ I [1]
a 2 F 2
and (cid:104)a(cid:105) denotes the square class of a in R and P(cid:100)(k) is a certain quotient of the classical pre-
F
Blochgroup,P(k),oftheresiduefield.
Using these results, we can prove that I RB(F)[1] is large if F is a field with many valua-
F 2
tions. In particular, it follows that H (SL (F),Z) is not finitely generated for any global field
3 2
F. (Bycontrast,if F isaglobalfieldthenH (SL (F),Z) = Kind(F)iswell-knowntobefinitely
3 3 3
generated.)
For example, (see Theorem 5.1) if F is a global field whose class group has odd order then
thereisanaturalsurjection
(cid:76)
I RB(F)[1] (cid:47)(cid:47) (cid:47)(cid:47) B(k )[1]
F 2 v v 2
where v runs through all the finite places of F and k is the residue field at v. (By the results
v
of [4], if F is the finite field with q elements, then B(F ) is cyclic of order (q + 1)/2 or q + 1
q q
accordingasqisoddoreven.)
Asanotherapplication,wealsousethetechniquesdevelopedtoconstructexplicitlynon-trivial
F -vectorspaces of known dimension inside groups of the form H (SL (O ),Z) where O is a
3 3 2 S S
ring of S-integers, and thus to give lower bounds on the 3-ranks of such groups. We hope that
thesetechniqueswillbeusefulinprovingmoregeneralresultsofthistypeinthefuture.
Finally, we use these specialization maps, together with the basic algebraic properties of the
refined Bloch groups, developed in section 3 below, to give a calculation of H (SL (F),Z[1])
3 2 2
for local fields F with finite residue field of odd order (Theorem 6.14). In particular, for such
fieldswehave
H (SL (F),Z[1]) (cid:27) Kind(F)[1]⊕B(k)[1]
3 2 2 3 2 2
where k is the residue field. Here the right-hand side is an R module with F× acting trivially
F
onthefirstfactorwhileanyuniformizeractsas−1onthesecondfactor.
4 KEVINHUTCHINSON
To return to our opening remarks, it follows from the results here that there is a natural surjec-
tion
(cid:16)(cid:76) (cid:17)
H (SL (Q),Z)⊗Z[1] = H (SL (Q),Z[1]) (cid:47)(cid:47) (cid:47)(cid:47) Kind(Q)[1]⊕ B(F )[1] .
3 2 2 3 2 2 3 2 p p 2
It is natural to ask whether this is an isomorphism and, furthermore, what adjustments, if any,
needtobemadetoobtainacorrespondingstatementwithintegralcoefficients.
2. ReviewofBlochGroups
2.1. Preliminaries and Notation. For a field F, we let G denote the multiplicative group,
F
F×/(F×)2, of nonzero square classes of the field. For x ∈ F×, we will let (cid:104)x(cid:105) ∈ G denote the
F
corresponding square class. Let R denote the integral group ring Z[G ] of the groupG . We
F F F
willusethenotation(cid:104)(cid:104)x(cid:105)(cid:105)forthebasiselements,(cid:104)x(cid:105)−1,oftheaugmentationidealI ofR .
F F
For any a ∈ F×, we will let p+ and p− denote respectively the elements 1+(cid:104)a(cid:105) and 1−(cid:104)a(cid:105) in
a a
R .
F
ForanyabeliangroupAwewillletA[1]denoteA⊗Z[1]. Foranintegern,wewillletn(cid:48) denote
2 2
theoddpartofn. ThusifAisafiniteabeliangroupofordern,thenA[1]isafiniteabeliangroup
2
ofordern(cid:48).
Welete+ ande− denoterespectivelythemutuallyorthogonalidempotents
a a
p+ 1+(cid:104)a(cid:105) p− 1−(cid:104)a(cid:105)
e+ := a = , e− := a = ∈ R [1].
a 2 2 a 2 2 F 2
(Ofcourse,theseoperatorsdependonlyontheclassofainG .)
F
2.2. The classical Bloch group. For a field F, with at least 4 elements, the pre-Bloch group,
P(F),isthegroupgeneratedbytheelements[x], x ∈ F× \{1},subjecttotherelations
R : [x]−(cid:2)y(cid:3)+(cid:2)y/x(cid:3)−(cid:104)(1− x−1)/(1−y−1)(cid:105)+(cid:2)(1− x)/(1−y)(cid:3) x (cid:44) y.
x,y
LetS2(F×)denotethegroup
Z
F× ⊗ F×
Z
< x⊗y+y⊗ x|x,y ∈ F× >
anddenoteby x◦ytheimageof x⊗yinS2(F×).
Z
Themap
λ : P(F) → S2(F×), [x] (cid:55)→ (1− x)◦ x
Z
iswell-defined,andtheBlochgroupof F,B(F) ⊂ P(F),isdefinedtobethekernelofλ.
2.3. TherefinedBlochgroup. Therefinedpre-Blochgroup,RP(F),ofafield F whichhasat
least 4 elements, is the R -module with generators [x], x ∈ F× subject to the relations [1] = 0
F
and
S : 0 = [x]−[y]+(cid:104)x(cid:105)(cid:2)y/x(cid:3)−(cid:68)x−1 −1(cid:69)(cid:104)(1− x−1)/(1−y−1)(cid:105)+(cid:104)1− x(cid:105)(cid:2)(1− x)/(1−y)(cid:3), x,y (cid:44) 1
x,y
Ofcourse,fromthedefinitionitfollowsimmediatelythatP(F) = (RP(F)) = H (F×,RP(F)).
F× 0
Foranyfield F wedefinetheR -module
F
RS2(F×) := I2 × S2(F×) ⊂ I2 ⊕S2(F×)
Z F Sym2F (GF) Z F Z
2
whereS2(F×)hasthetrivialR -modulestructure.
Z F
RefinedBlochgroupandH ofSL 5
3 2
AsanR -module,RS2(F×)isgeneratedbytheelements
F Z
[a,b] := ((cid:104)(cid:104)a(cid:105)(cid:105)(cid:104)(cid:104)b(cid:105)(cid:105),a◦b) ∈ RS2(F×).
Z
TherefinedBloch-WignerhomomorphismΛtobetheR -modulehomomorphism
F
Λ : RP(F) → RS2(F), [x] (cid:55)→ [1− x,x]
Z
(whichcanbeshowntobewell-defined).
In view of the definition of RS2(F×), we can express Λ = (λ ,λ ) where λ : RP(F) → I2 is
Z 1 2 1 F
themap[x] (cid:55)→ (cid:104)(cid:104)1− x(cid:105)(cid:105)(cid:104)(cid:104)x(cid:105)(cid:105),andλ isthecomposite
2
RP(F) (cid:47)(cid:47) (cid:47)(cid:47) P(F) λ (cid:47)(cid:47) S2(F×).
Z
Finally,wecandefinetherefinedBlochgroupofthefield F (withatleast4elements)tobethe
R -module
F
RB(F) := Ker(Λ : RP(F) → RS2(F×)).
Z
2.4. The fields F and F . Throughout this paper it will be convenient for us to have (refined
2 3
andclassical)pre-BlochandBlochgroupsforthefieldswith2and3elements. Forthisreason,
weintroducethefollowingsomewhatadhocdefinitions.
P(F ) = RP(F ) = RB(F ) = B(F ) is simply an additive group of order 3 with distinguished
2 2 2 2
generator,denotedC .
F
2
RP(F )isthecyclicR -modulegeneratedbythesymbol[−1]andsubjecttotheonerelation
3 F3
0 = 2·([−1]+(cid:104)−1(cid:105)[−1]).
Thehomomorphism
Λ : RP(F ) → RS2(F×) = I2 = 2·Z(cid:104)(cid:104)−1(cid:105)(cid:105)
3 Z 3 F3
istheR -homomorphismsending[−1]to(cid:104)(cid:104)−1(cid:105)(cid:105)2 = −2(cid:104)(cid:104)−1(cid:105)(cid:105).
F
3
ThenRB(F ) = Ker(Λ)isthesubmoduleoforder2generatedby[−1]+(cid:104)−1(cid:105)[−1].
3
Furthermore, we let P(F ) = RP(F ) . This is a cyclic Z-module of order 4 with generator
3 3 F×
3
[−1]. Letλ : P(F ) → I2 bethemap[−1] (cid:55)→ −2(cid:104)(cid:104)−1(cid:105)(cid:105). ThenB(F ) := Ker(λ) = RB(F ).
3 F 3 3
3
2.5. TherefinedBlochGroupandH (SL (F),Z). Werecallsomeresultsfrom[4]: Themain
3 2
resultthereis
Theorem2.1. Let F beafieldwithatleast4elements.
If F isinfinite,thereisanaturalcomplex
0 → TorZ(µ ,µ ) → H (SL (F),Z) → RB(F) → 0.
1 F F 3 2
which is exact everywhere except possibly at the middle term. The middle homology is annihi-
latedby4.
Inparticular,foranyinfinitefieldthereisanaturalshortexactsequence
0 → TorZ(µ ,µ )[1] → H (SL (F),Z[1]) → RB(F)[1] → 0.
1 F F 2 3 2 2 2
ThefollowingresultisCorollary5.1in[4]:
Lemma 2.2. Let F be an infinite field. Then the natural map RB(F) → B(F) is surjective and
theinducedmapRB(F) → B(F)hasa2-primarytorsionkernel.
F×
6 KEVINHUTCHINSON
Nowforanyfield F,let
H (SL (F),Z) := Ker(H (SL (F),Z) → Kind(F))
3 2 0 3 2 3
and
RB(F) := Ker(RB(F) → B(F))
0
ThefollowingisLemma5.2in[4].
Lemma2.3. Let F beaninfinitefield. Then
(1) H (SL (F),Z[1]) = RB(F)[1]
3 2 2 0 2 0
(2) H (SL (F),Z[1]) = I H (SL (F),Z[1])andRB(F)[1] = I RB(F)[1].
3 2 2 0 F 3 2 2 2 0 F 2
(3)
H (SL (F),Z[1]) = Ker(H (SL (F),Z[1]) → H (SL (F),Z[1]))
3 2 2 0 3 2 2 3 3 2
= Ker(H (SL (F),Z[1]) → H (GL (F),Z[1]))
3 2 2 3 2 2
On the other hand, the corresponding results for finite fields are as follows (the results in [4]
applytofieldswithatleast4elements,butitisstraightforwardtoverifythattheyextendtothe
fieldsF andF withthedefinitionssuppliedabove):
2 3
Lemma 2.4. For a finite field k the natural map RP(k) → P(k) induces an isomorphism
RB(k) (cid:27) B(k).
(ThisisLemma7.1in[4].)
ForafieldF,weletTorZ(µ(cid:93),µ )denotetheuniquenontrivialextensionofTorZ(µ ,µ )byZ/2
1 F F 1 F F
Z
ifthecharacteristicof F isnot2,andTor (µ ,µ )incharacteristic2.
1 F F
Theorem2.5. Thereisanaturalshortexactsequence
0 → TorZ(µ(cid:93),µ ) → H (SL (F ),Z[1/p]) → B(F ) → 0
1 Fq Fq 3 2 q q
foranyfinitefieldF oforderq = pf.
q
Furthermore,thereisanaturalisomorphism
H (SL (F ),Z[1/p]) (cid:27) Kind(F ).
3 2 q 3 q
(See[4],Corollary7.5. )
Furthermore,thecalculationsin[4],sections5and7,showthat
Lemma2.6.
(cid:40)
Z/(q+1)/2, qodd
B(F ) (cid:27)
q Z/(q+1), qeven
andif K ⊂ SL (F )isacyclicsubgroupoforder(q+1)(cid:48) thenthecompositemap
2 q
Z/(q+1)(cid:48) (cid:27) H (K,Z[1]) → H (SL (F ),Z[1]) → B(F )[1]
3 2 3 2 q 2 q 2
isanisomorphism.
RefinedBlochgroupandH ofSL 7
3 2
2.6. The map H (G,Z) → RB(F). In section 6 of [4] a recipe is given for calculating the
3
homomorphism H (G,Z) → RB(F) for subgroups G of SL (F). We recall this calculation in
3 2
thecasethatG isafinitecyclicsubgroup.
First,given4distinctpointsx ,x ,x ,x ∈ P1(F)wedefinetherefinedcrossratiocr(x ,x ,x ,x ) ∈
0 1 2 3 0 1 2 3
RP(F)by
 (cid:68) (cid:69)(cid:104) (cid:105)
cr(x ,x ,x ,x ) =  (cid:104)(cid:104)xx(x12−−−xx02xx)−(2xx10(cid:105)(cid:105)−(cid:104)(cid:104)xxxx1113)−−−xxx302(((cid:105)(cid:105)xx22,,−−xx10))((xx33−−xx01)) , iiifff xxxi0(cid:44)==∞∞∞
0 1 2 3  (cid:104)(cid:68)x(x202−−x0x)(01x0(cid:105)−(cid:104)xxxx1233)−−−(cid:69)xxx(cid:104)010x(cid:105)2,−x1(cid:105), iiff xx12 == ∞∞
x2−x1 x2−x0 3
NowsupposethatGisafinitecyclicsubgroupofSL (F)oforderrwithgeneratort. Wechoose
2
x ∈ P1(F)withtrivialstabilizerG = 1,andchoosey ∈ P1(F)\G· x.
x
Lemma2.7. Thecomposite
Z/n (cid:27) H (G,Z) → RP(F)
3
isgivenbytheformula
(cid:88)r−1
1 (cid:55)→ cr(βx,y(1,t,ti+1,ti+2)).
3
i=0
where
βx,y(1,t,t,t2) = 0
3
βx,y(1,t,ti+1,ti+2) = (x,t(x),ti+1(x),ti+2(x))for1 ≤ i ≤ r−3
3
βx,y(1,t,tr−1,1) = (y,t(x),t−1(x),x)−(y,x,t(x),t−1(x))
3
βx,y(1,t,tr,tr+1) = βx,y(1,t,1,t)
3 3
(cid:40)
0, y = t(y)
=
(y,t(y),x,t(x))+(y,t(y),t(x),x), y (cid:44) t(y)
Furthermore,theresultingmapisindependentoftheparticularchoiceof x andy.
3. SomealgebrainRP(F)
In this section we study certain key elements and submodules of the refined pre-Bloch group
ofafield F.
3.1. Theelementsψ (x)andthemodulesK(i). Werecalltheelements
i F
(cid:104) (cid:105)
{x} := [x]+ x−1 ∈ P(F)
(for x ∈ F×). A straightforward calculation - see Suslin [11] - shows that these symbols allow
ustodefineagrouphomomorphism
F× → P(F), x (cid:55)→ {x}
8 KEVINHUTCHINSON
whosekernelcontains(F×)2;i.e. wehave
(cid:110) (cid:111)
x2 = 0and{xy} = {x}+{y} forall x,y.
Inparticular,theseelementssatisfy2{x} = 0forall x.
WenowconsidertwoliftingsoftheseelementsinRP(F): For x ∈ F× welet
(cid:104) (cid:105)
ψ (x) := [x]+(cid:104)−1(cid:105) x−1
1
and
(cid:40) (cid:16) (cid:104) (cid:105)(cid:17)
(cid:104)1− x(cid:105) (cid:104)x(cid:105)[x]+ x−1 , x (cid:44) 1
ψ (x) :=
2 0, x = 1
(If F = F ,weinterpretthisasψ (1) = 0fori = 1,2. For F = F ,wehaveψ (−1) = ψ (−1) =
2 i 3 1 2
[−1]+(cid:104)−1(cid:105)[−1]. )
Themaps F× → RP(F),x (cid:55)→ ψ (x)arenolongerhomomorphisms,butarederivations:
i
Lemma3.1. Let F beafield. Fori ∈ {1,2},themap
F× → RP(F),x (cid:55)→ ψ (x)
i
isa1-cocycle;i.e. wehave
ψ (xy) = (cid:104)x(cid:105)ψ (y)+ψ (x) forall x,y ∈ F×.
i i i
Proof. Thestatementistrivialfor F = F orF . Wecanthusassume F hasatleast4elements.
2 3
If x = 1 or y = 1, the required identities are clear. If x (cid:44) 1 and y (cid:44) x−1 the relation 0 =
S +(cid:104)−1(cid:105)S inRP(F)yieldstheidentity
x,xy x−1,x−1y−1
ψ (x)−ψ (y)+(cid:104)x(cid:105)ψ (xy) = 0.
1 1 1
(cid:16) (cid:17)
Thus we must also prove that (cid:104)x(cid:105)ψ x−1 + ψ (x) = 0 for all x (cid:44) 1. Fix x (cid:44) 1 and choose
1 1
y (cid:60) {1,x−1}(hereweusethat F hasatleast4elements). Then
(cid:16) (cid:17)
(cid:104)y(cid:105)ψ (x) = ψ (xy)−ψ (y) = −(cid:104)xy(cid:105)ψ x−1
1 1 1 1
andmultiplyingby(cid:104)y(cid:105)givestherequiredidentity.
Now,for x,y ∈ F×,let
(cid:34) (cid:35) (cid:20) (cid:21)
x y
Q(x,y) := (cid:104)x(cid:105) +(cid:104)y(cid:105) ∈ RP(F).
y x
Then
(cid:32)(cid:42) (cid:43)(cid:34) (cid:35) (cid:20) (cid:21)(cid:33) (cid:42) (cid:43) (cid:32) (cid:33) (cid:32) (cid:33)
x x y x x x
Q(x,y) = (cid:104)y(cid:105) + = (cid:104)y(cid:105) 1− ψ = (cid:104)y− x(cid:105)ψ .
2 2
y y x y y y
Fora,b (cid:44) 1,therelation0 = S +S inRP(F)givestheidentity
a,b a,b
Q(a−1 −1,b−1 −1) = Q(a−1,b−1)+Q(1−a,1−b).
Thus
(cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33)
(cid:68) (cid:69) a−1 −1 (cid:68) (cid:69) b 1−a
b−1 −a−1 ψ = b−1 −a−1 ψ +(cid:104)a−b(cid:105)ψ
2 b−1 −1 2 a 2 1−b
andhence
(cid:32) (cid:33) (cid:32) (cid:33) (cid:42) (cid:43) (cid:32) (cid:33)
b−1 −1 b b 1−a
ψ = ψ + ψ .
2 a−1 −1 2 a a 2 1−b
Nowifwefix x,y (cid:44) 1with xy (cid:44) 1,wecansolvetheequations
b 1−a
x = , y =
a 1−b
RefinedBlochgroupandH ofSL 9
3 2
foraandbandprovetherequiredidentityforψ (). (cid:3)
2
Corollary3.2. Fori ∈ {1,2}wehave:
(cid:16) (cid:17) (cid:16) (cid:17)
(1) ψ xy2 = ψ (x)+ψ y2 forall x,y
i i i
(cid:16) (cid:17)
(2) (cid:104)(cid:104)x(cid:105)(cid:105)ψ y2 = 0forall x,y
i
(3) 2·ψ (−1) = 0foralli
(cid:16) i(cid:17)
(4) ψ x2 = −(cid:104)(cid:104)x(cid:105)(cid:105)ψ (−1)forall x
i i
(cid:16) (cid:17) (cid:16) (cid:17)
(5) 2·ψ x2 = 0forall x andif−1isasquarein F thenψ x2 = 0forall x.
i i
(6) (cid:104)(cid:104)x(cid:105)(cid:105)(cid:104)(cid:104)y(cid:105)(cid:105)ψ (−1) = 0forall x,y
i
(7) (cid:104)−1(cid:105)(cid:104)(cid:104)x(cid:105)(cid:105)ψ (y) = (cid:104)(cid:104)x(cid:105)(cid:105)ψ (y)forall x,y
i i
(8) Let
(cid:40)
1, −1 ∈ (F×)2
(cid:15)(F) :=
2, −1 (cid:60) (F×)2
ThemapG → RP(F),(cid:104)x(cid:105) (cid:55)→ (cid:15)(F)ψ (x)isawell-defined1-cocycle.
F i
(cid:16) (cid:17)
Proof. The identities ψ (1) = 0 and ψ x−1 = (cid:104)−1(cid:105)ψ (x) follow from the definition of ψ ().
i i i i
Moregenerally,let M beanR -moduleandletψ : F× → M bea1-cocyclesatisfying
F
ψ(1) = 0andψ(x−1) = (cid:104)−1(cid:105)ψ(x)forall x ∈ F×.
(1) Forall x,ywehave
(cid:68) (cid:69)
ψ(xy2) = y2 ψ(x)+ψ(y2) = ψ(x)+ψ(y2)
(cid:68) (cid:69)
since y2 = 1inR .
F
(2) Thecocycleconditionimpliesthat(cid:104)(cid:104)x(cid:105)(cid:105)ψ(y) = (cid:104)(cid:104)y(cid:105)(cid:105)ψx forall x,y. Thus
(cid:68)(cid:68) (cid:69)(cid:69)
(cid:104)(cid:104)x(cid:105)(cid:105)ψ(y2) = y2 ψ(x) = 0.
(3) Wehave(cid:104)−1(cid:105)ψ(−1) = ψ(−1)andthus
0 = ψ(1) = ψ(−1·−1) = ψ(−1)+(cid:104)−1(cid:105)ψ(−1) = 2ψ(−1).
(4) Forall x wehave
(cid:32) (cid:33) (cid:32) (cid:33)
1 1
ψ(x) = ψ · x2 = ψ +ψ(x2) = (cid:104)−1(cid:105)ψ(x)+ψ(x2).
x x
Thus
ψ(x2) = −(cid:104)(cid:104)−1(cid:105)(cid:105)ψ(x) = −(cid:104)(cid:104)x(cid:105)(cid:105)ψ(−1).
(5) The first statement follows from (3) and (4). For the second, observe that for any x we
have
ψ(x2) = −(cid:104)(cid:104)−1(cid:105)(cid:105)ψ(x)
and(cid:104)(cid:104)−1(cid:105)(cid:105) = 0if−1isasquare.
(6) Thisstatementfollowsfrom(2)and(4).
(7) Thisisarestatementof(6);namely
(cid:104)(cid:104)−1(cid:105)(cid:105)(cid:104)(cid:104)x(cid:105)(cid:105)ψ(y) = (cid:104)(cid:104)x(cid:105)(cid:105)(cid:104)(cid:104)y(cid:105)(cid:105)ψ(−1) = 0.
(8) By (5), (cid:15)(F)ψ(x2) = 0 in M for all x and thus (cid:15)(F)ψ(xy2) = (cid:15)(F)ψ(x) for all x,y. Thus
theproposedmapiswell-defined(andisthusclearlya1-cocycle).
(cid:3)
NowletK(i) denotetheR -submoduleofRP(F)generatedbytheset{ψ (x) | x ∈ F×}.
F F i
10 KEVINHUTCHINSON
Lemma3.3. Let F beafield. Thenfori ∈ {1,2}
(cid:16) (cid:17)
λ K(i) = p+ (I ) ⊂ I2
1 F −1 F F
andKer(λ | )isannihilatedby4.
1 (i)
K
F
Proof. Weusetheidentities
(cid:68)(cid:68) (cid:69)(cid:69)
(cid:104)(cid:104)a(cid:105)(cid:105)(cid:104)(cid:104)b(cid:105)(cid:105) = (cid:104)(cid:104)ab(cid:105)(cid:105)−(cid:104)(cid:104)a(cid:105)(cid:105)−(cid:104)(cid:104)b(cid:105)(cid:105), (cid:104)−1(cid:105)(cid:104)(cid:104)a(cid:105)(cid:105) = (cid:104)(cid:104)−a(cid:105)(cid:105)−(cid:104)(cid:104)−1(cid:105)(cid:105), ab2 = (cid:104)(cid:104)a(cid:105)(cid:105)
inI .
F
Thus
(cid:104) (cid:105)
λ (ψ (x)) = λ ([x])+(cid:104)−1(cid:105)λ ( x−1 )
1 1 1 1
= (cid:104)(cid:104)x(cid:105)(cid:105)(cid:104)(cid:104)1− x(cid:105)(cid:105)+(cid:104)−1(cid:105)(cid:104)(cid:104)x(cid:105)(cid:105)(cid:104)(cid:104)x(x−1)(cid:105)(cid:105)
= (cid:104)(cid:104)x(1− x)(cid:105)(cid:105)−(cid:104)(cid:104)x(cid:105)(cid:105)−(cid:104)(cid:104)1− x(cid:105)(cid:105)+(cid:104)−1(cid:105)((cid:104)(cid:104)x−1(cid:105)(cid:105)−(cid:104)(cid:104)x(cid:105)(cid:105)−(cid:104)(cid:104)x(x−1)(cid:105)(cid:105))
= (cid:104)(cid:104)x(1− x)(cid:105)(cid:105)−(cid:104)(cid:104)x(cid:105)(cid:105)−(cid:104)(cid:104)1− x(cid:105)(cid:105)+(cid:104)(cid:104)1− x(cid:105)(cid:105)−(cid:104)(cid:104)−x(cid:105)(cid:105)−(cid:104)(cid:104)x(1− x)(cid:105)(cid:105)+(cid:104)(cid:104)−1(cid:105)(cid:105)
= (cid:104)(cid:104)−1(cid:105)(cid:105)−(cid:104)(cid:104)x(cid:105)(cid:105)−(cid:104)(cid:104)−x(cid:105)(cid:105) = (cid:104)(cid:104)−x(cid:105)(cid:105)·(cid:104)(cid:104)x(cid:105)(cid:105)
= −p+ ·(cid:104)(cid:104)x(cid:105)(cid:105)
−1
Thusλ (K(1)) = p+ (I ).
1 F −1 F
(cid:104) (cid:105)
For x (cid:44) 1wehaveψ (x) = (cid:104)x(1− x)(cid:105)[x]+(cid:104)1− x(cid:105) x−1 andthus
2
λ (ψ (x)) = (cid:104)x(1− x)(cid:105)(cid:104)(cid:104)x(cid:105)(cid:105)(cid:104)(cid:104)1− x(cid:105)(cid:105)+(cid:104)1− x(cid:105)(cid:104)(cid:104)x(cid:105)(cid:105)(cid:104)(cid:104)x(x−1)(cid:105)(cid:105)
1 2
= (cid:104)x(1− x)(cid:105)((cid:104)(cid:104)x(1− x)(cid:105)(cid:105)−(cid:104)(cid:104)x(cid:105)(cid:105)−(cid:104)(cid:104)1− x(cid:105)(cid:105))+(cid:104)1− x(cid:105)((cid:104)(cid:104)x−1(cid:105)(cid:105)−(cid:104)(cid:104)x(x−1)(cid:105)(cid:105)−(cid:104)(cid:104)x(cid:105)(cid:105))
= −(cid:104)(cid:104)1− x(cid:105)(cid:105)−(cid:104)(cid:104)x(cid:105)(cid:105)+(cid:104)(cid:104)x(1− x)(cid:105)(cid:105)+(cid:104)(cid:104)−1(cid:105)(cid:105)−(cid:104)(cid:104)−x(cid:105)(cid:105)−(cid:104)(cid:104)x(1− x)(cid:105)(cid:105)+(cid:104)(cid:104)1− x(cid:105)(cid:105)
= (cid:104)(cid:104)−1(cid:105)(cid:105)−(cid:104)(cid:104)x(cid:105)(cid:105)−(cid:104)(cid:104)−x(cid:105)(cid:105) = (cid:104)(cid:104)x(cid:105)(cid:105)·(cid:104)(cid:104)−x(cid:105)(cid:105) = −p+ ·(cid:104)(cid:104)x(cid:105)(cid:105).
−1
Thusλ (K(2)) = p+ (I )also.
1 F −1 F
For the second statement, recall that for any group G and any Z[G]-module M a 1-cocycle
ρ : G → M gives rise to Z[G]-homomorphism I → M, g−1 (cid:55)→ ρ(g). Thus, for i ∈ {1,2}, we
G
haveawell-definedR -homomorphism
F
I → RP(F), (cid:104)(cid:104)x(cid:105)(cid:105) (cid:55)→ 2ψ (x).
F i
Combining this with the inclusion p+ (I ) → I we obtain an R -module homomorphism
−1 F F F
µ : p+ (I ) → K(i) sendingp+ (cid:104)(cid:104)x(cid:105)(cid:105) = (cid:104)(cid:104)x(cid:105)(cid:105)+(cid:104)−1(cid:105)(cid:104)(cid:104)x(cid:105)(cid:105)to2ψ (x)+(cid:104)−1(cid:105)2ψ (x) = 4ψ (x).
−1 F F −1 i i i
(cid:18) (cid:19)
Itfollowsthatµ◦ λ | isjustmultiplicationby4,andtheresultisproved. (cid:3)
1 (i)
K
F
Remark 3.4. Since p+ I is a free abelian group, it follows that, as an abelian group, K(i)
−1 F (cid:16) (cid:17) F
decomposes as a direct sum A ⊕ K(i) where A is a free abelian group and 4 annihilates
(cid:16) (cid:17) F tors
K(i) = Ker(λ | ).
F tors 1K(i)
F (cid:16) (cid:17)
Furthermore, if (cid:104)(cid:104)−1(cid:105)(cid:105)ψ (x) = 0 for all x (for example, if −1 ∈ (F×)2) then ψ x2 = 0 for
i i
all x and the map G → RP(F),(cid:104)x(cid:105) (cid:55)→ ψ (x) is already a well-defined 1-cocycle. The above
F (cid:16) (cid:17) i
argumentsthenshowthat K(i) = Ker(λ | )isannihilatedby2.
F tors 1K(i)
F
Foranyfield F wewilllet
R(cid:103)P(F) := RP(F)/K(1).
F