QUOTIENTS OF ABSOLUTE GALOIS GROUPS WHICH
DETERMINE THE ENTIRE GALOIS COHOMOLOGY
SUNIL K. CHEBOLU, IDO EFRAT, AND J'AN MIN'A~C
Abstract.For prime power q = pd and a field F containing a root of unity *
*of
order q we show that the Galois cohomology ring H*(GF, Z=q) is determined*
* by a
quotient G[3]Fof the absolute Galois group GF related to its descending q*
*-central
sequence. Conversely, we show that G[3]Fis determined by the lower cohomo*
*logy
of GF. This is used to give new examples of pro-p groups which do not occ*
*ur as
absolute Galois groups of fields.
1.Introduction
A main open problem in modern Galois theory is the characterization of the pr*
*ofinite
groups which are realizable as absolute Galois groups of fields F . The torsion*
* in such
groups is described by the Artin-Schreier theory from the late 1920's, namely, *
*it consists
solely of involutions. More refined information on the structure of absolute Ga*
*lois groups
is given by Galois cohomology, systematically developed starting the 1950's by *
*Tate,
Serre, and others. Yet, explicit examples of torsion-free profinite groups whic*
*h are not
absolute Galois groups are rare. In 1970, Milnor [Mil70] introduced his K-ring *
*functor
KM*(F ), and pointed out close connections between this graded ring and the mod*
*-2
Galois cohomology of the field. This connection, in a more general form, became*
* known
as the Bloch-Kato conjecture: it says that for all r 0 and all m prime to cha*
*rF , there
is a canonical isomorphism KMr(F )=m ! Hr(GF , ~mr) ([GS06]; see notation below*
*).
The conjecture was proved for r = 2 by Merkurjev and Suslin [MS82], for r arbit*
*rary
and m = 2 by Voevodsky [Voe03a], and in general by Rost, Voevodsky, with a patc*
*h by
Weibel ([Voe03b], [Wei09], [Wei08], [HW09]).
In this paper we obtain new constrains on the group structure of absolute Gal*
*ois
groups of fields, using this isomorphism. We use these constrains to produce ne*
*w ex-
amples of torsion-free profinite groups which are not absolute Galois groups. W*
*e also
demonstrate that the maximal pro-p quotient of the absolute Galois group can be*
* char-
acterized in purely cohomological terms. The main object of our paper is a rema*
*rkable
small quotient of the absolute Galois group, which, because of the above isomor*
*phism,
already carries a substantial information about the arithmetic of F .
More specifically, fix a prime number p and a p-power q = pd, with d 1. All*
* fields
which appear in this paper will be tacitly assumed to contain a primitive qth r*
*oot of
____________
2000 Mathematics Subject Classification. Primary 12G05; Secondary 12F10, 12E*
*30.
Key words and phrases. absolute Galois group, Galois cohomology, descending *
*central sequence,
W-group.
J'an Min'a~c was supported in part by National Sciences and Engineering Coun*
*cil of Canada grant
R0370A01.
1
2 SUNIL K. CHEBOLU, IDO EFRAT, AND J'AN MIN'A~C
unity. Let F be such a field and let GF = Gal(Fsep=F ) be its absolute Galois g*
*roup,
where Fsepis the separable closure of F . Let H*(GF ) = H*(GF , Z=q) be the Ga*
*lois
cohomology ring with the trivial action of GF on Z=q. Our new constraints relat*
*e the
descending q-central sequence G(i)F, i = 1, 2, 3, . .,.of GF (see x4) with H*(G*
*F ). Setting
G[i]F= GF =G(i)F, we show that the quotient G[3]Fdetermines H*(GF ), and vice v*
*ersa.
Specifically, we prove:
Theorem A. The inflation map gives an isomorphism
H*(G[3]F)dec-~!H*(GF ),
where H*(G[3]F)decis the decomposable part of H*(G[3]F) (i.e., its subring gene*
*rated by
degree 1 elements).
We further have the following converse results.
Theorem B. G[3]Fis uniquely determined by Hr(GF ) for r = 1, 2, the cup product
[: H1(GF ) x H1(GF ) ! H2(GF ) and the Bockstein homomorphism fi :H1(GF ) !
H2(GF ) (see x2 for the definition of fi).
Theorem C. Let F1, F2 be fields and let ss :GF1 ! GF2 be a (continuous) homomor-
phism. The following conditions are equivalent:
(i)the induced map ss*: H*(GF2) ! H*(GF1) is an isomorphism;
(ii)the induced map ss[3]:G[3]F1! G[3]F2is an isomorphism.
Theorems A-C show that G[3]Fis a Galois-theoretic analog of the cohomology ri*
*ng
H*(GF ). Its structure is considerably simpler and more accessible than the ful*
*l absolute
Galois group GF (see e.g., [EM07]). Yet, as shown in our theorems, these small*
* and
accessible quotients encode and control the entire cohomology ring. Results si*
*milar
to Theorems A-C are valid in a relative pro-p setting, where one replaces GF by*
* its
maximal pro-p quotient GF (p) = Gal(F (p)=F ) (here F (p) is the compositum of *
*all
finite Galois extensions of F of p-power order; see Remark 8.2).
In the case q = 2 the group G[3]Fhas been extensively studied under the name *
*"W -
group", in particular in connection with quadratic forms ([Spi87], [MSp90], [MS*
*p96],
[AKM99], [MMS04]). In this special case, Theorem A was proved in [AKM99, Th.
3.14]. It was further shown that then G[3]Fhas great arithmetical significance:*
* it encodes
large parts of the arithmetical structure of F , such as its orderings, its Wit*
*t ring, and
certain non-trivial valuations. Theorem A explains this surprising phenomena, a*
*s these
arithmetical objects are known to be encoded in H*(GF ) (with the additional kn*
*owledge
of the Kummer element of -1).
First links between these quotients and the Bloch-Kato conjecture, and its sp*
*ecial
case the Merkurjev-Suslin theorem, were already noticed in [Spi87] and in Bogom*
*olov's
paper [Bog92]. The latter paper was the first in a remarkable line of works by *
*Bogomolov
and Tschinkel ([Bog92], [BT08], [BT09]), as well as by Pop (unpublished), focus*
*ing on
the closely related quotient GF =[GF , [GF , GF ]] (the analog of G[3]Ffor q = *
*0), where
F is a function field over an algebraically closed field. There the viewpoint *
*is that
of "birational anabelian geometry": namely, it is shown that for certain impor*
*tant
classes of such function fields, F itself is determined by this quotient. Our w*
*ork, on the
QUOTIENTS OF ABSOLUTE GALOIS GROUPS 3
other hand, is aimed at clarifying the structure of the smaller Galois group G[*
*3]Fand
its connections with the Galois cohomology and arithmetic of almost arbitrary f*
*ields,
focusing on the structure of absolute Galois groups.
Our approach is purely group-theoretic, and the main results above are in fac*
*t proved
for arbitrary profinite groups which satisfy certain conditions on their cohomo*
*logy (The-
orem 6.5, Proposition 7.3 and Theorem 6.3). A key point is a rather general gr*
*oup-
theoretic approach, partly inspired by [GM97], to the Milnor K-ring constructio*
*n by
means of quadratic hulls of graded algebras (x3). The Rost-Voevodsky theorem on*
* the
bijectivity of the Galois symbol shows that these cohomological conditions are *
*satisfied
by absolute Galois groups as above. Using this we deduce in x9 Theorems A-C in *
*their
field-theoretic version.
We thank L. Avramov, T. Chinburg, J.-L. Colliot-Th'el`ene, W.-D. Geyer, P. Go*
*erss,
M. Jarden, P. May, and T. Szamuely for their comments related to talks on this *
*work
given at the 2009 Field Arithmetic meeting in Oberwolfach, the University of Ch*
*icago,
Northwestern University, and the University of Nebraska, Lincoln.
2. Cohomological preliminaries
We work in the category of profinite groups. Thus subgroups are always tacit*
*ly
assumed to be closed and homomorphism are assumed to be continuous. For basic f*
*acts
on Galois cohomology we refer e.g., to [NSW08], [Ser02], or [Koc02].L We abbrev*
*iate
Hr(G) = Hr(G, Z=q) with the trivial G-action on Z=q. Let H*(G) = 1r=0Hr(G)
be the graded cohomology ring with the cup product. We write res, inf, and trg*
*for
the restriction, inflation, and transgression maps. Given a homomorphism ss :G1*
* ! G2
of profinite groups, we write ss*: H*(G2) ! H*(G1) and ss*r:Hr(G2) ! Hr(G1) for
the induced homomorphisms. The Bockstein homomorphism fi :H1(G) ! H2(G)
of G is the connecting homomorphism arising from the short exact sequence of tr*
*ivial
G-modules
0 ! Z=q ! Z=q2 ! Z=q ! 0.
When q = 2 one has fi(_) = _ [ _ [EM07, Lemma 2.4].
Given a normal subgroup N of G, there is a natural action of G on Hr(N). For *
*r = 1
it is given by ' 7! 'g, where 'g(n) = '(g-1ng) for ' 2 H1(N), g 2 G and n 2 N.
Let Hr(N)G be the group of all G-invariant elements of Hr(N). Recall that there*
* is a
5-term exact sequence
trgG=N 2 infG 2
0 ! H1(G=N) infG---!H1(G) resN---!H1(N)G ----! H (G=N) ---!H (G),
which is functorial in (G, N) in the natural sense.
3.Graded rings
L 1
Let R be a commutative ring and A = r=0Ar a graded associative R-algebra
with A0 = R. Assume that A is either commutative or graded-commutative (i.e.,
ab = (-1)rsba for a 2 Ar, b 2 As). For r 0 let Adec,rbe the R-submodule of Ar
generated by all productsLof r elements of A1 (by convention Adec,0= R). The gr*
*aded
R-subalgebra Adec= 1r=0Adec,ris the decomposable part of A. We say that Ar
(resp., A) is decomposable if Ar = Adec,r(resp., A = Adec).
4 SUNIL K. CHEBOLU, IDO EFRAT, AND J'AN MIN'A~C
Motivated by the Milnor K-theory of a field [Mil70], we define the quadratic *
*hull A^
of the algebra A as follows. For r 0 let Tr be the R-submodule of A1r genera*
*ted
by all tensors a1 . . .ar such that aiaj = 0 2 A2 for some distinct 1 i, j *
* r
(byLconvention, A10 = R, T0 = 0). We define A^to be the graded R-algebra A^=
1 r
r=0A1 =Tr with multiplicative structure induced by the tensor product. Becaus*
*e of
the commutativity/graded-commutativity, there is a canonical graded R-algebra e*
*pi-
morphism !A :A^! Adec, which is the identity map in degree 1. We call A quadrat*
*ic
if !A is an isomorphism. Note that
^A= (A^)dec= ([Adec).
These constructions are functorial in the sense that every graded R-algebra m*
*orphism
' = ('r)1r=0:A ! B induces in a natural way graded R-algebra morphisms
'dec= ('dec,r)1r=0:Adec! Bdec, '^= ('^r)1r=0:^A! ^B
with a commutative square
^'
(3.1) ^A_______//_^B
!A || !B||
fflffl|'defflffl|c
Adec_____//Bdec.
The proof of the next fact is straightforward.
Lemma 3.1. ^'is an isomorphism if and only if '1 is an isomorphism and 'dec,2is*
* a
monomorphism.
Remark 3.2. When G is a profinite group, R = Z=2, and A = H*(G, Z=2) the ring
^Acoincides with the ring Mil(G) introduced and studied in [GM97]. In the case *
*where
G = GF for a field F as before, this ring is naturally isomorphic to KM*(F )=2.*
* Thus in
this way one can construct KM*(F )=p for any p in a purely group-theoretic way.
4.The descending central sequence
Let G be a profinite group and let q be either a p-power or 0. The descending
q-central sequence of G is defined inductively by
G(1,q)= G, G(i+1,q)= (G(i,q))q[G(i,q), G], i = 1, 2, . ...
Thus G(i+1,q)is the closed subgroup of G topologically generated by all powers *
*hq and
all commutators [h, g] = h-1g-1hg, where h 2 G(i,q)and g 2 G. Note that G(i,q)*
*is
normal in G. For i 1 let G[i,q]= G=G(i,q). When q = 0 the sequence G(i,0)is c*
*alled
the descending central sequence of G. Usually q will be fixed, and we will abbr*
*eviate
G(i)= G(i,q), G[i]= G[i,q].
Any profinite homomorphism (resp., epimorphism) ss :G ! H restricts to a homo-
morphism (resp., an epimorphism) ss(i):G(i)! H(i). Hence ss induces a homomorph*
*ism
(resp., an epimorphism) ss[i]:G[i]! H[i].
Lemma 4.1. For i, j 1 one has canonical isomorphisms
(a)(G[j])(i)~=G(i)=G(max{i,j});
(b)(G[j])[i]~=G[min{i,j}].
QUOTIENTS OF ABSOLUTE GALOIS GROUPS 5
Proof.(a) Consider the natural epimorphism ss :G ! G[j]. Then ss(i):G(i)! (G[j*
*])(i)
is an epimorphism with kernel G(i)\ Ker(ss) = G(i)\ G(j)= G(max{i,j}).
(b) By (a), there is a canonical isomorphism
(G[j])[i]~=G[j]=(G[j])(i)~=(G=G(j))=(G(i)=G(max{i,j})) ~=G[min{i,j}].
Lemma 4.2. Let ss :G1 ! G2 be a homomorphism of profinite groups. If ss[j]is an
epimorphism (resp., isomorphism), then ss[i]is an epimorphism (resp., isomorphi*
*sm)
for all i j.
Proof.The assumption implies that (ss[j])[i]:(G[j]1)[i]! (G[j]2)[i]is an epimor*
*phism
(resp., isomorphism). Now use Lemma 4.1(b).
Lemma 4.3. Let i 1 and let N be a normal subgroup of G with N G(i)and
ss :G ! G=N the natural map. Then ss[i]is an isomorphism.
Proof.Apply Lemma 4.1(b) with respect to the composed epimorphism G ! G=N !
G[i]to see that ss[i]is injective.
Lemma 4.4. Let ss :G1 ! G2 be an epimorphism of profinite groups. Then Ker(ss[i*
*]) =
Ker(ss)G(i)1=G(i)1for all i 1.
Proof.The map G1 ! G[i]2= ss(G1)=ss(G1)(i)induced by ss has kernel Ker(ss)G(i)1,
whence the assertion.
We will also need the following result of Labute [Lab66, Prop. 1 and 2] (see *
*also
[NSW08, Prop. 3.9.13]).
Proposition 4.5. Let S beLa free pro-p group on generators oe1, . .,.oen. Consi*
*der the
Lie Zp-algebra gr(S) = 1i=1S(i,0)=S(i+1,0), with Lie brackets induced by the *
*commu-
tator map. Then gr(S) is a free Lie Zp-algebra on the images of oe1, . .,.oen i*
*n gr1(S).
InQparticular, S(2,0)=S(3,0)has a system of representatives consisting of all p*
*roducts
1 k 2, [BLMS07, Th. A.3] or [EM07, Prop. 12.3] imply that in thi*
*s case
as well, G2 is not a maximal pro-p Galois group.
Proposition 9.6. Let G be a pro-p group such that dimFpH1(G) < cd(G). When p = 2
assume also that G is torsion-free. Then G is not a maximal pro-p Galois group *
*of a
field as above.
Proof.Assume that p 6= 2. Let d = dimFpH1(G). Then also d = dimFpH1(G[3,p]).
Since the cup product is graded-commutative, Hd+1(G[3,p])dec= 0. On the other h*
*and,
Hd+1(G) 6= 0 [NSW08, Prop. 3.3.2]. Thus H*(G[3,p])dec6~=H*(G). By Theorem A and
Remark 8.2, G is not a maximal pro-p Galois group.
When p = 2 this was shown in [AKM99, Th. 3.21], using Kneser's theorem on the
u-invariant of quadratic forms, and a little later (independently) by R. Ware i*
*n a letter
to the third author.
Example 9.7. Let K, L be finitely generated pro-p groups with 1 n = cd(K) < 1,
cd(L) < 1, and Hn(K) finite. Let ss :L ! Symm , x 7! ssx, be a homomorphism such
QUOTIENTS OF ABSOLUTE GALOIS GROUPS 13
that ss(L) is a transitive subgroup of Sym m. Then L acts on Km from the left*
* by
x(y1, . .,.ym ) = (yssx(1), . .,.yssx(m)). Let G = Km oL. It is generated by th*
*e generators
of one copy of K and of L. Hence dimFpH1(G) = dimFpH1(K) + dimFpH1(L).
On the other hand, a routine inductive spectral sequence argument (see [NSW08,
Prop. 3.3.8]) shows that for every i 0 one has
(1)cd(Ki) = in;
(2)Hin(Ki) = Hn(K, H(i-1)n(Ki-1)), with the trivial K-action, is finite.
Moreover, cd(G) = cd(Km ) + cd(L). For m sufficiently large we get dimFpH1(G) <
mn + cd(L) = cd(G), so by Proposition 9.6, G is not a maximal pro-p Galois grou*
*p as
above. When K, L are torsion-free, so is G.
For instance, one can take K to be a free pro-p group 6= 1 on finitely many g*
*enerators,
and let L = Zp act on the direct product of ps copies of K via Zp ! Z=ps by cyc*
*licly
permuting the coordinates.
Remark 9.8. Absolute Galois groups which are solvable (with respect to closed s*
*ub-
groups) were analyzed in [Gey69], [Bec78], [W"ur85, Cor. 1], [Koe01]. In partic*
*ular one
can give examples of such solvable groups which are not absolute Galois groups *
*(compare
[Koe01, Example 4.9]). Our examples here are in general not solvable.
References
[AKM99]A. Adem, D. B. Karagueuzian, and J. Min'a~c, On the cohomology of Galois*
* groups determined
by Witt rings, Adv. Math. 148 (1999), 105-160.
[Bec74]E. Becker, Euklidische K"orper und euklidische H"ullen von K"orpern, J. *
*reine angew. Math.
268/269 (1974), 41-52.
[Bec78]E. Becker, Formal-reelle K"orper mit streng-aufl"osbarer absoluter Galoi*
*sgruppe, Math. Ann.
238 (1978), 203-206.
[BLMS07]D. J. Benson, N. Lemire, J. Min'a~c, and J. Swallow, Detecting pro-p-gr*
*oups that are not
absolute Galois groups, J. reine angew. Math. 613 (2007), 175-191.
[Bog91]F. A. Bogomolov, On two conjectures in birational algebraic geometry, Pr*
*oc. of Tokyo Satel-
lite conference ICM-90 Analytic and Algebraic Geometry, 1991, pp. 26-52.
[Bog92]F. A. Bogomolov, Abelian subgroups of Galois groups, Math. USSR, Izv. 38*
* (1992), 27-67
(English; Russian original).
[BT08] F. Bogomolov and Y. Tschinkel, Reconstruction of function fields, Geom. *
*Func. Anal. 18
(2008), 400-462.
[BT09] F. A. Bogomolov and Y. Tschinkel, Milnor K2 and field homomorphisms (200*
*9), preprint.
[Efr99]I. Efrat, Finitely generated pro-p absolute Galois groups over global fi*
*elds, J. Number Theory
77 (1999).
[EM07] I. Efrat and J. Min'a~c, On the descending central sequence of absolute *
*Galois groups (2007),
to appear.
[FJ05] M. D. Fried and M. Jarden, Field Arithmetic, 2nd ed., Springer-Verlag, B*
*erlin, 2005.
[GM97] W. Gao and J. Min'a~c, Milnor's conjecture and Galois theory I, Fields I*
*nstitute Communi-
cations 16 (1997), 95-110.
[Gey69]W.-D. Geyer, Unendlische Zahlk"orper, "uber denen jede Gleichung aufl"os*
*bar von beschr"ankter
Stufe ist, J. Number Theory 1 (1969), 346-374.
[GS06] P. Gille and T. Szamuely, Central simple algebras and Galois cohomology,*
* Cambridge Uni-
versity Press, Cambridge, 2006.
[Gil68]D. Gildenhuys, On pro-p groups with a single defining relator, Inv. math*
*. 5 (1968), 357-366.
[HW09] C. Haesemeyer and C. Weibel, Norm Varieties and the Chain Lemma (after M*
*arkus Rost),
Proc. Abel Symposium 4 (2009), to appear.
[Koc02]H. Koch, Galois theory of p-extensions, Springer-Verlag, Berlin, 2002.
14 SUNIL K. CHEBOLU, IDO EFRAT, AND J'AN MIN'A~C
[Koe01]J. Koenigsmann, Solvable absolute Galois groups are metabelian, Inv. mat*
*h. 144 (2001),
1-22.
[Lab66]J. P. Labute, Demu~skin groups of rank @0, Bull. Soc. Math. France 94 (1*
*966), 211-244.
[Lab67]J. P. Labute, Alg`ebres de Lie et pro-p-groupes d'efinis par une seule r*
*elation, Inv. math. 4
(1967), 142-158.
[MMS04]L. Mah'e, J. Min'a~c, and T. L. Smith, Additive structure of multiplicat*
*ive subgroups of fields
and Galois theory, Doc. Math. 9 (2004), 301-355.
[MS82] A. S. Merkurjev and A. A. Suslin, K-cohomology of Severi-Brauer varietie*
*s and the norm
residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011-104*
*6 (Russian);
English transl., Math. USSR Izv. 21 (1983), 307-340.
[Mil70]J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969*
*/1970), 318-344.
[MSp90]J. Min'a~c and M. Spira, Formally real fields, Pythagorean fields, C-fie*
*lds and W-groups,
Math. Z. 205 (1990), 519-530.
[MSp96]J. Min'a~c and M. Spira, Witt rings and Galois groups, Ann. of Math. (2)*
* 144 (1996), 35-60.
[NSW08]J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of Number Fields, S*
*econd edition,
Springer, Berlin, 2008.
[OVV07]D. Orlov, A. Vishik, and V. Voevodsky, An exact sequence for KM*=2 with *
*applications to
quadratic forms, Ann. Math. 165 (2007), 1-13.
[Rom86]N. S. Romanovskij, Generalized theorem on the freedom for pro-p-groups, *
*Sib. Mat. Zh. 156
(1986), 154-170.
[Ser65]J.-P. Serre, Sur la dimension cohomologique des groupes profinis, Topolo*
*gy 3 (1965), 413-
420.
[Ser02]J.-P. Serre, Galois cohomology, Springer, Berlin, 2002. Translated from *
*the 1964 French
Edition by Patrick Ion and revised by the author.
[Spi87]M. Spira, Witt rings and Galois groups, Ph.D. thesis, University of Cali*
*fornia, Berkeley,
1987.
[Voe03a]V. Voevodsky, Motivic cohomology with Z=2-coefficients, Publ. Math. Ins*
*t. Hautes 'Etudes
Sci. 98 (2003), 59-104.
[Voe03b]V. Voevodsky, On motivic cohomology with Z=l-coefficients (2003), avail*
*able at http://
www.math.uiuc.edu/K-theory/0639/. revised in 2009.
[Wei08]C. A. Weibel, The proof of the Bloch-Kato conjecture, ICTP Lecture Notes*
* series 23 (2008),
1-28.
[Wei09]C. A. Weibel, The norm residue isomorphism theorem (2009), available at*
* http://www.
math.uiuc.edu/K-theory/0934/. to appear in J. Topology.
[W"ur85]T. W"urfel, On a class of pro-p groups occurring in Galois theory, J. P*
*ure Appl. Algebra 36
(1985), 95-103.
Department of Mathematics, Illinois State University, Campus box 4520, Normal*
*, IL
61790, USA
E-mail address: schebol@ilstu.edu
Mathematics Department, Ben-Gurion University of the Negev, P.O. Box 653, Be'*
*er-Sheva
84105, Israel
E-mail address: efrat@math.bgu.ac.il
Mathematics Department, University of Western Ontario, London, Ontario, Canad*
*a N6A
5B7
E-mail address: minac@uwo.ca