A Bloch-Wigner complex for SL_2

0.32 MB·

by  Kevin Hutchinson

A BLOCH-WIGNER COMPLEX FOR SL
2
KEVINHUTCHINSON
Abstract. WeintroducearefinementoftheBloch-WignercomplexofafieldF.Itiscomplexof
modulesoverthemultiplicativegroupofthefield. InsteadofcomputingtheK (F)andKind(F)-
2 3
astheclassicalBloch-Wignercomplexdoes-itcalculatesthesecondandthirdintegralhomology
1 ofSL (F). Onpassingto F×-coinvariantswerecovertheclassicalBloch-Wignercomplex. We
1 2
includethecaseoffinitefieldsthroughoutthearticle.
0
2

ul Contents
J

1 1. Introduction 1

] 2. BlochGroupsandtheBloch-Wignermap 4
T
3. ThehomologyofSL (F ) 11
K 2 q
h. 4. ThethirdhomologyofSL2(F) 18
t 5. TherefinedBlochgroup,theclassicalBlochgroupandindecomposable K 29
a 3
m 6. ThemapH (G,Z) → RB(F)forsubgroupsG ofSL (F) 30
3 2
[
7. Blochgroupsoffinitefields 34

1
References 38
v
4
6
2
0 1. Introduction
.
7
0 What is now usually referred to the as the Bloch group of a field F arose first in the work of S.
1
Bloch as an explicitly-presented approximation to indecomposable of the field which could be
1
: used to define a regulator map based on the dilogarithm (see the notes [3]). When F = C (and,
v
Xi more generally, when F× = (F×)2) there is a natural identification K3ind(C) = H3(SL2(C),Z),
and this latter group is a natural target for invariants of hyperbolic 3-manifolds. It was because
r
a of this connection with hyperbolic geometry that Dupont and Sah ([6] and [22]) explored the
properties of the Bloch group. In particular, they wrote down a proof of the so-called Bloch-
Wigner Theorem ([6], Theorem 4.10): The pre-Bloch group of the field F is the group, P(F),
withgenerators[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
(These relations correspond to the 5-term functional equation satisfied by the classical dilog-
arithm. See Zagier [27] for a beautiful exposition of these and related matters.) We will let
S2(F×)denotetheantisymmetricproduct
Z
F× ⊗ F×
Z
.
< x⊗y+y⊗ x|x,y ∈ F× >
Date:July4,2011.
1991MathematicsSubjectClassification. 19G99,20G10.
Keywordsandphrases. K-theory,GroupHomology.
1
2 KEVINHUTCHINSON
Thenthereisawell-definedgrouphomomorphism
λ : P(F) → S2(F×), [x] (cid:55)→ (1− x)⊗ x
Z
andthetheoremofBlochandWignersaysthatthereisanexactsequence
0 (cid:47)(cid:47) µC (cid:47)(cid:47) H (SL (C),Z) (cid:47)(cid:47) P(C) λ (cid:47)(cid:47) S2(C) (cid:47)(cid:47) H (SL (C),Z) (cid:47)(cid:47) 0.
3 2 Z 2 2
The argument of Dupont and Sah works equally well for any algebraically closed field and
more generally for any quadratically closed field (i.e. satisfying F× = (F×)2). When the field
F is quadratically closed then the homology groups can be interpreted in terms of K-theory:
H (SL (F),Z) = Kind(F) and H (SL (F),Z) = K (F) = KM(F). Thus the homology groups of
3 2 3 2 2 2 2
theBloch-Wignercomplex
P(F) λ (cid:47)(cid:47) S2(F)
Z
are(essentially)theK-theorygroupsKind(F)andKM(F). ThegroupB(F) = Ker(λ)istheBloch
3 2
groupofthe F.
Suslin showed that, interpreted in this way, the Bloch-Wigner theorem extends to all (infinite)
fields. He proved (see [25], Theorem 5.2) that for any infinite field F there is a natural exact
sequence
0 (cid:47)(cid:47) TorZ(µ(cid:93),µ ) (cid:47)(cid:47) Kind(F) (cid:47)(cid:47) P(F) λ (cid:47)(cid:47) S2(F) (cid:47)(cid:47) KM(F) (cid:47)(cid:47) 0.
1 F F 3 Z 2
whereTorZ(µ(cid:93),µ )istheuniquenontrivialextensionofTorZ(µ ,µ )byZ/2ifthecharacteristic
1 F F 1 F F
of F isnot2,andTorZ(µ(cid:93),µ ) = TorZ(µ ,µ )incharacteristic2.
1 F F 1 F F
The purpose of the current article is to extend the original sequence of Bloch-Wigner-Dupont-
Sah in another direction: namely, to construct a complex, which coincides with the one above
when F is quadratically closed, but which calculates in the general case - instead of K-theory -
the homology groups H (SL (F),Z) and H (SL (F),Z). Our main goal is to understand better
3 2 2 2
thestructureoftheunstablehomologygroupH (SL (F),Z)anditsrelationto Kind(F).
3 2 3
To put this project in context, we recall some of what is known about the relationship between
thehomologygroupsandthe K-theorygroups. Ingeneral,thegroupextension
1 → SL (F) → GL (F) → F× → 1
n n
defines an action of F× on the homology groups H (SL (F),Z). Since the determinant of a
k n
scalar matrix is an n-th power, the subgroup (F×)n acts trivially. In the particular, the groups
H (SL (F),Z) are modules over the integral group ring R := Z[F×/(F×)2]. The natural map
k 2 F
H (SL (F),Z) → K (F) (via stabilization and an inverse Hurewicz map) is surjective and in-
2 2 2
ducesanisomorphismon F×-coinvariants
H (SL (F),Z) (cid:27) K (F).
2 2 F× 2
However, the action of F× on H (SL (F),Z) is in general nontrivial. The action of R factors
2 2 F
through the Grothendieck-Witt ring GW(F) of the field, and the kernel of the surjective map
H (SL (F),Z) → K (F) is isomorphic, as a GW(F)-module, to I(F)3, the third power of the
2 2 2
fundamental ideal I(F) of the augmented ring GW(F). (See Suslin [24], Appendix for the
details.) TobemoreexplicitH (SL (F),Z)canbeexpressedasafibreproduct
2 2
H (SL (F),Z) (cid:27) I(F)2 × KM(F).
2 2 I(F)2/I(F)3 2
H (SL (F),Z) is of interest in its own right to K-theorists and geometers because it coincides
2 2
with the second Milnor-Witt K-group, KMW(F), of the field F (see, for example, [18] or [19]).
2
Bloch-Wigner-Suslincomplex 3
More generally, the calculation of the groups H (SL (F),Z), which are at the boundary of the
n n
homologystabilityrange,involvestheMilnor-Witt K-groups KMW(F)([11]).
n
The group H (SL (F),Z) is of interest, among other reasons, because it is strictly below the
3 2
range of homology stability. However there is, for any field F, a natural homomorphism
H (SL (F),Z) → Kind(F)whichinducesasurjectivehomomorphism(see[10])
3 2 3
H (SL (F),Z) (cid:47)(cid:47) (cid:47)(cid:47) Kind(F) .
3 2 F× 3
Suslinhasaskedthequestionwhetherthisisanisomorphism,anditisknown(seeMirzaii[17])
thatthekernelconsistsof-atworst-2-primarytorsion.
In order to refine the Bloch-Wigner sequence to a sequence which captures the homology of
SL (F),itisnecessarytobuildintheR -modulestructuresateachstage. Thus,inthisarticlewe
2 F
introducefirsttherefinedpre-BlochgroupRP(F)ofafield F. ThisistheR -modulegenerated
F
bysymbols[x], x ∈ F× \{1}subjecttotherelations
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
where (cid:104)x(cid:105) denotes the class of x in F×/(F×)2. Similarly we introduce an R -module RS2(F×)
F Z
which has natural generators [x,y], x,y ∈ F×. The ‘refined Bloch-Wigner complex’ is then the
complexofR -modules
F
RP(F) Λ (cid:47)(cid:47) RS2(F) , [x] (cid:55)→ [1− x,x].
Z
Ontaking F×-coinvariantsthisreducestotheclassicalBloch-Wignercomplex. Ourmainresult
(Theorem4.3)isthatthereis,foranyfield F,anaturalcomplexofR -modules
F
0 (cid:47)(cid:47) TorZ(µ ,µ ) (cid:47)(cid:47) H (SL (F),Z) (cid:47)(cid:47) RP(F) Λ (cid:47)(cid:47) RS2(F×) (cid:47)(cid:47) H (SL (F),Z) (cid:47)(cid:47) 0
1 F F 3 2 Z 2 2
which is exact at every term except possibly at the term H (SL (F),Z), where the homology of
3 2
thecomplexisannihilatedby4.
The refined Bloch group of the field F is the R -module RB(F) := Ker(Λ). The main theorem
F
tells us that it is a good approximation to the H (SL (F),Z). In particular, we show that - up to
3 2
some2-primarytorsion-RB(F) (cid:27) B(F)and
F×
Ker(RB(F) → B(F)) (cid:27) Ker(H (SL (F),Z) → Kind(F)).
3 2 3
In a subsequent article we will develop further the algebraic properties of the refined Bloch
group(see[8],forexample). Inparticular,whenthefield F hasavaluationwithresiduefieldk,
thereareusefulspecializationhomomorphismsfromRB(F)toRP(k). Thus,wewillshowthat
if F isanondyadiclocalfieldwith(finite)residuefieldk thereisanaturalisomorphism
H (SL (F),Z[1/2]) (cid:27) (cid:16)Kind(F)⊕R(cid:91)B(k)(cid:17)⊗Z[1/2]
3 2 3
(cid:91)
whereRB(k)isacertainexplicitlycomputablequotientoftheBlochgroupofk.
Similarly,wewillshow(see[8]fordetails)thatforanyglobalfield F thekernel
Ker(H (SL (F),Z[1/2]) → Kind(F)⊗Z[1/2])
3 2 3
mapshomomorphicallyontotheinfinitedirectsum
(cid:16)⊕ R(cid:91)B(k )(cid:17)⊗Z[1/2],
v v
thesumbeingoverallfiniteplacesvof F.
Thusthe(refined)Blochgroupsoffinitefieldswillplayanimportantroleinfutureapplications.
Because of this, and unlike most of the references above, throughout the paper we include the
4 KEVINHUTCHINSON
case of finite fields. At times, they require separate treatment and methods. For this reason we
include a separate section - section 3- recalling the results we need on the homology of SL (F)
2
for finite fields F. In the last section of the paper we combine our main theorem with these
homology calculations to give a proof of Suslin’s theorem in the case of finite fields and to
makesomeusefulcalculationsinBlochgroupsoffinitefields.
Remark 1.1. Several authors (W. Neumann [20], W. Nahm, S. Goette and C. Zickert [7]) have
introducedandstudiedanextendedBlochgroup,whichisexactlyisomorphictothe Kind(F)-at
3
least for some fields F. This is a quite different object from the refined Bloch group introduced
here,whicheffectivelybearsthesamerelationshiptoH (SL (F),Z)astheclassicalBlochgroup
3 2
doesto Kind(F).
3
2. BlochGroupsandtheBloch-Wignermap
In this section, we review the definition of the classical Bloch group and pre-Bloch group of a
field,andwedefineourbasicobjectsofstudyinthisarticle,therefinedBlochgroupandrefined
pre-Blochgroup.
2.1. TheclassicalBlochgroup. Let F beafieldwithatleast4elementsandlet X denotethe
n
setofallorderedn-tuplesofdistinctpointsofP1(F). PGL (F),andhencealsoGL (F),actson
2 2
P1(F)viafractionallineartransformations. Thusthesegroupsacton X viaadiagonalaction.
n
NowletA(F)bethecokernelofthehomomorphismofGL (F)-modules
2
(cid:88)5
δ : ZX → ZX , (x ,...,x ) (cid:55)→ (−1)j+1(x ,...,xˆ,...,x ).
5 4 1 5 1 j 5
j=1
Thenthepre-Blochgroupof F isthegroup
P(F) := A(F) = Coker(δ¯ : (ZX ) → (ZX ) ).
GL2(F) 5 GL2(F) 4 GL2(F)
Now the orbits of GL (F) on X are classified by the cross-ratio: i.e., in general, (x ,...,x ) is
2 4 1 4
intheorbitof(0,∞,1,x)where x ∈ P1(F)\{∞,0,1} = F× \{1}isthecross-ratio
(x − x )(x − x )
4 1 3 2
(x − x )(x − x )
3 1 4 2
of x ,...,x .
1 4
Thus
(cid:77)
(ZX ) (cid:27) Z·(0,∞,1,x)
4 GL2(F)
x∈F×\{1}
and,similarly,
(cid:77)
(ZX ) (cid:27) Z·(0,∞,1,x,y).
5 GL2(F)
x,y∈F×\{1}
x(cid:44)y
andfor x (cid:44) yin F× \{1},δ¯(0,∞,1,x,y)
= (∞,1,x,y)−(0,1,x,y)+(0,∞,x,y)−(0,∞,1,y)+(0,∞,1,x)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
= 0,∞,1, 1−x − 0,∞,1, 1−x−1 + 0,∞,1, y −(0,∞,1,y)+(0,∞,1,x)
1−y 1−y−1 x
Thus, if we let [x] denote the class of the orbit of (0,∞,1,x) in P(F) then P(F) is the group
generatedbytheelements[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
Bloch-Wigner-Suslincomplex 5
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.
2.2. The refined pre-Bloch group. Let F be a field with at least 4 elements. The refined
pre-Blochgroupof F isthegroup
RP(F) := A(F) = Coker(δ¯ : (ZX ) → (ZX ) ).
SL2(F) 5 SL2(F) 4 SL2(F)
For a field F, we let G denote the multiplicative group, F×/(F×)2, of nonzero square classes
F
of the field. For x ∈ F×, we will let (cid:104)x(cid:105) ∈ G denote the corresponding square class. Let R
F F
denotetheintegralgroupringZ[G ]ofthegroupG . Wewillusethenotation(cid:104)(cid:104)x(cid:105)(cid:105)forthebasis
F F
elements,(cid:104)x(cid:105)−1,oftheaugmentationidealI ofR .
F F
Sinceforanyfield F wehaveashortexactsequenceofgroups
1 → PSL (F) → PGL (F) → G → 1
2 2 F
itfollowsthatif X isanyPGL (F)-set,thenSL (F)\X isaG -setand
2 2 F
(ZX) (cid:27) Z[SL (F)\X]
SL2(F) 2
isanR -module.
F
ThestabilizerinSL (F)of(0,∞)isthesubgroupT consistingofalldiagonalmatrices
2
(cid:34) (cid:35)
a 0
D(a) :=
0 a−1
For x ∈ P1(F), D(a)· x = a2x.
Given x (cid:44) yinP1(F),letT ∈ SL (F)bethematrix
x,y 2
 (cid:34) (cid:35)
T =  (cid:34) 1x1−1y−x−x−−yx(cid:35)y yx,=y (cid:44)∞∞
x,y  (cid:34) 001 −−11y (cid:35) x = ∞
Then T (x) = 0, T (y) = ∞, and, by the preceding remarks, if S ∈ SL (F) satisfies S(x) = 0
x,y x,y 2
andS(y) = ∞,thenS = D(a)·T forsomea ∈ F×. Inparticular,if A ∈ SL (F),itfollowsthat
x,y 2
T = D(a)·T ·A−1 forsomea = a(x,y,A) ∈ F×.
Ax,Ay x,y
For x,y,zdistinctpointsofP1(F),wedefine
 (z− x)(x−y)(z−y)−1, x,y,z (cid:44) ∞
φ(x,y,z) := T (z) =  (y−z)−1, x = ∞
x,y  zx−−xy,, zy==∞∞
6 KEVINHUTCHINSON
Thusφ(x,y,z) ∈ P1(F)\{∞,0} = F×,andφ(0,∞,z) = zforz ∈ F×. Furthermore,ifA ∈ SL (F),
2
then
φ(Ax,Ay,Az) = T (Az) = D(a)·T ·A−1(Az) = a2φ(x,y,z)forsomea ∈ F×.
Ax,Ay x,y
Now,forn ≥ 1,letY denotethesetoforderedn-tuplesofdistinctpointsofF×. Y isanF×-set
n n
viathediagonalaction.
Lemma2.1. Forn ≥ 3,themap
Φ : X → Y
n n n−2
(x ,x ,...,x ) (cid:55)→ (φ(x ,x ,x ),φ(x ,x ,x ),...,φ(x ,x ,x ))
1 2 n 1 2 3 1 2 4 1 2 n
inducesabijectionofG -sets
F
SL (F)\X ←→ (F×)2\Y .
2 n n−2
Proof. BytheremarksaboveΦ descendstoawell-definedmap
n
Φ¯ : SL (F)\X → (F×)2\Y . Furthermore,themap
n 2 n n−2
Ψ : Y → X
n n−2 n
(y ,...,y ) (cid:55)→ (0,∞,y ,...,y )
1 n−2 1 n−2
givesaset-theoreticsectionofΦ whichdescendstoaninverseofΦ¯ .
n n
Since,foranya ∈ F×,
(cid:40)
aφ(x ,x ,y), x (cid:44) ∞
φ(ax ,ax ,ay) = 1 2 1
1 2 a−1φ(x ,x ,y), x = ∞
1 2 1
italsofollowsthatΦ¯ isamapofG -sets. (cid:3)
n F
Corollary2.2. Forn ≥ 0,letZ denotethesetoforderedn-tuples,[z ,...,z ],ofdistinctpoints
n 1 n
of F× \{1}. Thenforalln ≥ 3thereisanisomorphismofR -modules
F
(ZX ) (cid:27) R [Z ].
n SL2(F) F n−3
Proof. ByLemma2.1wehaveR -isomorphisms
F
(ZX ) (cid:27) Z[SL (F)\X ] (cid:27) Z[(F×)2\Y ].
n SL2(F) 2 n n−2
Finally,wehaveanR -isomorphism
F
Z[(F×)2\Y ] (cid:27) R [Z ]
n−2 F n−3
viathemap
(cid:34) (cid:35)
y y
(y ,...,y ) (cid:55)→ (cid:104)y (cid:105) 2,..., n−2 .
1 n−2 1
y y
1 1
(cid:3)
ItfollowsthattheR -isomorphism(ZX ) (cid:27) R [Z ]isgivenby
F n SL2(F) F n−3
(cid:34) (cid:35)
φ(x ,x ,x ) φ(x ,x ,x )
(x ,...,x ) (cid:55)→ (cid:104)φ(x ,x ,x )(cid:105) 1 2 4 ,..., 1 2 n .
1 n 1 2 3
φ(x ,x ,x ) φ(x ,x ,x )
1 2 3 1 2 3
Inparticular,wehaveR -isomorphisms
F
(ZX ) (cid:27) R , (ZX ) (cid:27) R [F× \{1}], (ZX ) (cid:27) R [Z ]
3 SL2(F) F 4 SL2(F) F 5 SL2(F) F 2
NotethattakingG -coinvariantsofthetermsinCorollary2.2weobtain
F
Bloch-Wigner-Suslincomplex 7
Corollary2.3. Foralln ≥ 3thereisanisomorphismofgroups
(ZX ) (cid:27) Z[Z ].
n GL2(F) n−3
Inparticular,forn = 4,theisomorphism(ZX ) (cid:27) Z[F× \{1}]isgivenby
4 GL2(F)
(cid:34) (cid:35)
φ(x ,x ,x )
(x ,x ,x ,x ) (cid:55)→ 1 2 4
1 2 3 4
φ(x ,x ,x )
1 2 3
whichisjusttheclassicalcross-ratiomap.
Nowitfollowsfromthecalculationsabovethatthemap
R [Z ] (cid:27) (cid:47)(cid:47) (ZX ) δ¯ (cid:47)(cid:47) (ZX ) (cid:27) (cid:47)(cid:47) R [Z ]
F 2 5 SL2(F) 4 SL2(F) F 1
istheR -modulehomomorphism
F
[x,y] (cid:55)→ [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)
(sinceφ(∞,1,a) = (1−a)−1,φ(0,1,a) = (a−1 −1)−1 andφ(0,∞,a) = a.) Thuswehave:
Lemma2.4. Therefinedpre-BlochgroupRP(F)istheR -modulewithgenerators[x], x ∈ F×
F
subjecttotherelations[1] = 0and
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,bydefinition,wehaveP(F) = (RP(F)) = H (F×,RP(F)).
F× 0
2.3. ThemoduleRS2(F×)andtherefinedBlochgroupofafield.
Z
Lemma 2.5. Let G be an abelian group. There is a natural short exact sequence of Z[G]-
modules
0 → I3 → I2 → Sym2(G) → 0
G G Z
(whereG actstriviallyonthefourthterm).
Proof. In fact if R is any commutative ring and I an ideal in R, then there is a natural exact
sequence
Symn+1(I) η (cid:47)(cid:47) Symn(I) (cid:47)(cid:47) Symn (I/I2) (cid:47)(cid:47) 0
R R R/I
whereη(a ∗a ∗···∗a ) = a ·(a ∗···∗a ) = (a a )∗···∗a .
0 1 n 0 1 n 0 1 n
Intheparticularcasen = 2,R = Z[G], I = I thisgivesanexactsequence
G
Sym3 (I ) → Sym2 (I ) → Sym2(G) → 0
Z[G] G Z[G] G Z
sinceI /I2 (cid:27) G.
G G
NowthereisanaturalsurjectivehomomorphismofgradedZ[G]-algebras
Sym• (I ) → I•,
Z[G] G G
a ∗···∗a (cid:55)→ (cid:104)(cid:104)a (cid:105)(cid:105)···(cid:104)(cid:104)a (cid:105)(cid:105).
1 n 1 n
This is an isomorphism in dimension 2 (and,for trivial reasons, in dimensions 0 and 1). To see
this,applythefunctor−⊗ I totheshortexactsequence
Z[G] G
0 → I → Z[G] → Z → 0
G
toobtaintheexactsequence
0 → TorZ[G](Z,I ) → T2 (I ) → I2 → 0.
1 G Z[G] G G
8 KEVINHUTCHINSON
ButTorZ[G](Z,I ) = H (G,I ) (cid:27) H (G,Z) (cid:27) ∧2(G). Astraightforwardcalculationnowshows
1 G 1 G 2 Z
thatthemap
∧2(G) (cid:27) TorZ[G](Z,I ) → T2 (I )
Z 1 G Z[G] G
sendsg ∧g to(cid:104)(cid:104)g (cid:105)(cid:105)⊗(cid:104)(cid:104)g (cid:105)(cid:105)−(cid:104)(cid:104)g (cid:105)(cid:105)⊗(cid:104)(cid:104)g (cid:105)(cid:105).
1 2 1 2 2 1
Finally,weobservethattheimageofthemap
Sym3 (I ) → Sym2 (I ) (cid:27) I2
Z[G] G Z[G] G G
isclearlyI3.
G
(cid:3)
Remark2.6. ItisastraightforwardmattertoverifythatSym• (I )hasthefollowingpresen-
Z[G] G
tation as a graded ring: It is generated in degree 1 by the elements (cid:104)(cid:104)g(cid:105)(cid:105), g ∈ G, subject to the
relations
(N) (cid:104)(cid:104)1(cid:105)(cid:105) = 0
(R) (cid:104)(cid:104)g (cid:105)(cid:105)∗(cid:104)(cid:104)g g (cid:105)(cid:105)+(cid:104)(cid:104)g (cid:105)(cid:105)∗(cid:104)(cid:104)g (cid:105)(cid:105) = (cid:104)(cid:104)g g (cid:105)(cid:105)∗(cid:104)(cid:104)g (cid:105)(cid:105)+(cid:104)(cid:104)g (cid:105)(cid:105)∗(cid:104)(cid:104)g (cid:105)(cid:105)forallg ,g ,g ∈ G.
1 2 3 2 3 1 2 3 1 2 1 2 3
(S) (cid:104)(cid:104)g (cid:105)(cid:105)∗(cid:104)(cid:104)g (cid:105)(cid:105) = (cid:104)(cid:104)g (cid:105)(cid:105)∗(cid:104)(cid:104)g (cid:105)(cid:105)forallg ,g ∈ G
1 2 2 1 1 2
Forabeliangroupsthesurjectivehomomorphismofgradedringsα : Sym• (I ) → I• isnot
Z[G] G G
generallyinjectiveindimensionsgreaterthan2. However,thefollowingisknown:
If G is either torsion-free or cyclic then α is an isomorphism (see Bak and Tang [1]). On the
otherhand,ifG isanelementaryabelian2-group,thenthekernelofαistheidealgeneratedby
the degree 3 terms (cid:104)(cid:104)g (cid:105)(cid:105)∗(cid:104)(cid:104)g (cid:105)(cid:105)∗(cid:104)(cid:104)g g (cid:105)(cid:105) for g ,g ∈ G (see Bak and Vavilov [2]). It is easy to
1 2 1 2 1 2
seethattheselattertermsarenonzero(byconsideringtheirimageinSym3 (G),forexample).
Z/2
ApplyingLemma2.5tothecaseG = G gives:
F
Corollary2.7. Let F beafield. ThereisanaturalexactsequenceofR -modules
F
0 → I3 → I2 → Sym2 (G ) → 0.
F F F2 F
Ontheotherhand,clearlythereisalsoanaturalhomomorphismofadditivegroups
S2(F×) (cid:47)(cid:47) (cid:47)(cid:47) S2(F×)⊗Z/2 (cid:27) (cid:47)(cid:47) Sym2 (G ).
Z Z F F
2
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
Givena,b ∈ F×,welet[a,b]denotetheelement
[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
Lemma2.8. Let F beafield.
(1) I RS2(F×) (cid:27) I3
F Z F
(2) RS2(F×) (cid:27) S2(F×)
Z F× Z
(3) RS2(F×)isgeneratedasanR -modulebytheelements[a,b],a,b ∈ F×.
Z F
Proof. (1) Wehaveaninjectivehomomorphism
I3 → I2 ⊕S2(F×), x (cid:55)→ (x,0).
F F Z
But,since x mapsto0inSym2 (G ) = I2/I3,infactI3 ⊂ RS2(F×).
F2 F F F F Z
Bloch-Wigner-Suslincomplex 9
Ontheotherhand,ifa,b,c ∈ F×,then(cid:104)(cid:104)a(cid:105)(cid:105)[b,c] = ((cid:104)(cid:104)a(cid:105)(cid:105)(cid:104)(cid:104)b(cid:105)(cid:105)(cid:104)(cid:104)c(cid:105)(cid:105),0). So
I3 ⊂ I RS2(F×).
F F Z
Conversely, if (x,y) ∈ RS2(F×) and a ∈ F×, then (cid:104)(cid:104)a(cid:105)(cid:105)(x,y) = ((cid:104)(cid:104)a(cid:105)(cid:105)x,0) ∈ I3; i.e.
Z F
I RS2(F×) ⊂ I3.
F Z F
(2) Suppose that (x,y) lies in the kernel of the surjective R -homomorphism RS2(F×) →
F Z
S2(F×). Then y = 0 and thus x maps to 0 in Sym2 (G ). By Corollary 2.7, x ∈ I3 =
Z F2 F F
andhence(x,y) ∈ I RS2(F×).
F Z
ObservethatitfollowsthatthereisanaturalshortexactsequenceofR -modules
F
0 → I3 → RS2(F×) → S2(F×) → 0.
F Z Z
(3) Let K(F) be the R -submodule of RS2(F×) generated by the elements [a,b]. Since
F Z
(cid:104)(cid:104)a(cid:105)(cid:105)[b,c] = (cid:104)(cid:104)a(cid:105)(cid:105)(cid:104)(cid:104)b(cid:105)(cid:105)(cid:104)(cid:104)c(cid:105)(cid:105) ∈ I3 ⊂ RS2(F×), it follows that I3 ⊂ K(F). On the other
F Z F
hand the homomorphism RS2(F×) → S2(F×) maps K(F) onto S2(F×), since the latter
Z Z Z
isgeneratedbytheelementsa◦b. Thus K(F) = RS2(F×)asrequired.
Z
(cid:3)
ObservethattheR -modulestructureonRS2(F×)isgivenbytheformula
F Z
(cid:104)b(cid:105)[a,c] = [ab,c]−[b,c] = [a,bc]−[a,b].
WedefinetherefinedBloch-WignerhomomorphismΛtobetheR -modulehomomorphism
F
Λ : RP(F) → RS2(F), [x] (cid:55)→ [1− x,x].
Z
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
Itisatediouscalculationtoverifydirectlyλ isawell-definedhomomorphismofR -modules.
1 F
However,wewillseebelowthatλ arisesnaturallyasadifferentialinaspectralsequence.
1
RecallthatthehomologygroupsH (SL (F),Z)arenaturallyR -modulesforallk.
k 2 F
Theorem2.9. ForanyfieldF withatleast10elements,thereisanaturalsurjectiveR -module
F
homomorphism
RS2(F×) → H (SL (F),Z)
Z 2 2
inducinganisomorphismCoker(Λ) (cid:27) H (SL (F),Z).
2 2
Proof. Suppose first that F is finite. Then H (SL (F),Z) = 0. The statement of the theorem
2 2
amountstothesurjectivityofΛ.
Forafinitefield,since F× iscyclic,S2(F×) = Sym2 (G )andthusRS2(F×) = I2.
Z Z/2 F Z F
Recall that the Grothendieck-Witt ring of the field F is the ring GW(F) = R /J , where J
F F F
is the ideal generated by the elements (cid:104)(cid:104)1− x(cid:105)(cid:105)(cid:104)(cid:104)x(cid:105)(cid:105). Thus Coker(Λ) is I(F)2, where I(F) is the
fundamental ideal I /J in GW(F). However, it is well-known that I(F)2 = 0 for any finite
F F
field F (see,forexample,[16],sectionIII.5).
Thuswecansupposethat F isinfinite. Inthiscase,thesymplecticcaseofthetheoremofMat-
sumotoandMoore([13]),givesapresentationofthegroupH (SL (F),Z). Ithasthefollowing
2 2
form: Thegeneratorsaresymbols(cid:104)a ,a (cid:105),a ∈ F×,subjecttotherelations:
1 2 i
(i) (cid:104)a ,a (cid:105) = 0ifa = 1forsomei
1 2 (cid:68) i (cid:69)
(ii) (cid:104)a ,a (cid:105) = a−1,a
1 2 2 1
10 KEVINHUTCHINSON
(cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69)
(iii) a ,a a(cid:48) + a ,a(cid:48) = a a ,a(cid:48) +(cid:104)a ,α (cid:105)
1 2 2 2 2 1 2 2 1 2
(iv) (cid:104)a ,a (cid:105) = (cid:104)a ,−a a (cid:105)
1 2 1 1 2
(v) (cid:104)a ,a (cid:105) = (cid:104)a ,(1−a )a (cid:105)
1 2 1 1 2
Furthermore, Suslin has shown ([24], appendix) that for an infinite field F, there is an isomor-
phismofR -modules
F
H (SL (F),Z) (cid:27) I(F)2 × KM(F), (cid:104)a,b(cid:105) ↔ ((cid:104)(cid:104)a(cid:105)(cid:105)(cid:104)(cid:104)b(cid:105)(cid:105),{a,b}).
2 2 I(F)2/I(F)3 2
NowwehaveamapofdiagramsofR -modules
F
S2(F×) KM(F)
Z 2
(cid:47)(cid:47)
(cid:15)(cid:15) (cid:15)(cid:15)
I2 (cid:47)(cid:47) Sym2(G ) I(F)2 (cid:47)(cid:47) I(F)2/I(F)3
F Z F
whichinducesamapofpullbacks
RS2(F×) → I(F)2 × KM(F) (cid:27) H (SL (F),Z)
Z I(F)2/I(F)3 2 2 2
sendingtheelements[a,b]totheelements(cid:104)a,b(cid:105).
This map is surjective, since the elements (cid:104)a,b(cid:105) generate H (SL (F),Z), and the image of Λ is
2 2
containedinitskernelsince{1− x,x} = 0in KM(F)and(cid:104)(cid:104)1− x(cid:105)(cid:105)(cid:104)(cid:104)x(cid:105)(cid:105) = 0inI(F)2.
2
To complete the proof of the theorem we must show that there is an R -homomorphism
F
H (SL (F),Z) → Coker(Λ) sending (cid:104)a,b(cid:105) to [a,b] (mod Im(Λ)); i.e. we must show that the
2 2
elements[a,b] ∈ RS2(F×)satisfytheMatsumoto-MoorerelationsmodulotheimageofΛ.
Z
Now the elements [a,b] are easily seen to satisfy relations (i), (ii) and (iii). On the other hand,
since[a,1−a] ≡ 0 (mod Im(Λ))foralla ∈ F×,andsinceΛisanR -homomorphism,itfollows
F
that0 ≡ (cid:104)b(cid:105)[a,1−a] ≡ [a,(1−a)b]−[a,1−a] (mod Im(Λ))foralla,b ∈ F×.
(cid:104) (cid:105)
Now,foranya ∈ F×,Λ([a]+(cid:104)−1(cid:105) a−1 ) = [−a,a],since
(cid:104) (cid:105)
λ ([a]+(cid:104)−1(cid:105) a−1 ) = (cid:104)(cid:104)1−a(cid:105)(cid:105)(cid:104)(cid:104)a(cid:105)(cid:105)+(cid:104)−1(cid:105)(cid:104)(cid:104)a(a−1)(cid:105)(cid:105)(cid:104)(cid:104)a(cid:105)(cid:105)
1
= ((cid:104)(cid:104)(1−a)a(cid:105)(cid:105)−(cid:104)(cid:104)1−a(cid:105)(cid:105)−(cid:104)(cid:104)a(cid:105)(cid:105))+(cid:104)−1(cid:105)((cid:104)(cid:104)a−1(cid:105)(cid:105)−(cid:104)(cid:104)a(a−1)(cid:105)(cid:105)−(cid:104)(cid:104)a(cid:105)(cid:105))
= (cid:104)(cid:104)(1−a)a(cid:105)(cid:105)−(cid:104)(cid:104)1−a(cid:105)(cid:105)−(cid:104)(cid:104)a(cid:105)(cid:105)+(cid:104)(cid:104)1−a(cid:105)(cid:105)−(cid:104)(cid:104)a(1−a)(cid:105)(cid:105)−(cid:104)(cid:104)−a(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)a(cid:105)(cid:105)−(cid:104)(cid:104)−a(cid:105)(cid:105)
= (cid:104)(cid:104)a(cid:105)(cid:105)(cid:104)(cid:104)−a(cid:105)(cid:105)
and
(cid:104) (cid:105)
λ ([a]+(cid:104)−1(cid:105) a−1 ) = (1−a)◦a+(1−a−1)◦a−1
2
(cid:32) (cid:33)
1−a
= (1−a)◦a− ◦a = (−a)◦a.
−a
Thus [a,−a] ≡ 0 (mod Im(Λ)) for all a ∈ F× and hence 0 ≡ (cid:104)b(cid:105)[a,−a] ≡ [a,−ab] − [a,−a]
(mod Im(Λ)) for all a,b ∈ F×. Thus relations (iv) and (v) also hold in Coker(Λ), and the
theoremisproven. (cid:3)
Remark 2.10. The restriction to fields with at least 10 elements is to rule out the exceptional
casesofthefieldwith4andthefieldwith9elementsforwhichH (SL (F),Z) = Z/pisnonzero
2 2
(seeLemma3.14below)