Construction of Hurwitz Spaces and Application to the Regular Inverse Problem

0.4 MB·English

by  Kenji Sakugawa

Construction of Hurwitz Spaces and Applications
2 to the Regular Inverse Galois Problem
1
0
2 Kenji Sakugawa

n
a January 11, 2012
J

0
1
1 Introduction

]
T
A famous open problem in Number theory called Inverse Galois Problem is as
N
follows.
.
h
t Question 1.1. (IGP) Does every finite group occurs as the Galois group of a
a
finite Galois extension over Q?
m
[ The same statement as IGP is always expected to be true when we replace

Q by any algebraicnumber field which is a Hilbertian field( see [Se, Def. 3.1.3.]
2
for the definition Hilbertian fields).
v
1 We do not recall the definition of Hilbertian fields, but we recall that any
9 finitenumberfieldF isHilbertian. TherearesomealgebraicnumberfieldsF of
3 infinite degreeoverQ whichare Hilbertian (e.g. the maximalabelian extension
1 Qab of Q is a Hilbertian fields. We will denote by IGP the analogue of the
. F
1 inverse Galois problem for a Hilbertian field F.
0
We consider the following variant of the problem IGP.
2
1 Question 1.2. (RIGP ) Let F be any field. Does every finite group G occur
: F
v as geometrically connected G-covering of P1?
F
i
X
ForafinitegroupG,wecallafiniteflatmorphismofschemesf :X →W the
r G-coveringifGactsX overW andiff inducestheisomorphismX/G−∼→W. We
a
saythatGisregularoverF ifthereexistsageometricallyconnectedG-covering
over P1. The above question is called the regular inverse Galois problem over
F
F ( RIGP ).
F
The following proposition is an immediate consequence of Hilbert’s irre-
ducibility theorem.
Proposition 1.3. Let F be a Hilbertian field. Then, IGP holds for F if RIGP
holds for F.
The advantage of considering RIGP instead of IGP is that RIGP admits a
purelygrouptheoreticapproach. Inotherwords,wegetapurelygrouptheoretic
1
sufficient condition of regularity of G. In [Th], Thompson proved the following
theorem.
Let C =(C1,...,Cr) be an r-tuple of conjugacy classes of G. Put
Ein(C):={(g1,...,gr)∈Gr | gi ∈Ci, g1...gr =1, hg1,...,gri=G}/Inn(G)
and define an abelian extension Q of Q by
C
Gal(Q/QC):=StabGQ({C1,...,Cr})
(see Definition 2.9 for the action of GQ on {{C1n,...,Crn | n∈ Z>0 , (n,|G|)=
1}).
Proposition 1.4. [Th]( Rigidity method. ) If there exists a natural number
r and an r-tuple of conjugacy class C of G such that |Ein(C)| = 1, then G is
regular over Qab. Moreover G/Z(G) is regular over Q .
C
We will give the proof of the above proposition in 2.1 in a slightly different
way from that of [Th]. The proof of this proposition will be related to an
essential step of the proof of our main results explained below.
In the paper [Fr-V¨o], Fried and Vo¨lklein considered the set of equivalence
classes
Hin(G)(C):={G-coveringsover P1 ramified at exactly r-points onP1}/∼.
r C C
Here,wecallcoveringsf :X →P1 andg :Y →P1 areequivalentifthereexists
C C
an isomorphism for X to Y over P1.
C
Fried and Vo¨lklein obtain the following theorem.
Proposition 1.5. [Fr-Vo¨]Let G,r be as above .
(1) There exists an algebraic variety Hin(G) over Qwhose C-valued points are
r
canonically identified with Hin(G)(C).
r
(2) Let L be an extension of Q. Suppose that the center of G is trivial. Then,
thereexistsageometricallyconnectedGcoveringoverP1 ifandonlyifHin(G)(L)
L r
is not empty.
(3) Let L be an extension of Q whose absolute Galois group G has cohomlogi-
L
cal dimension≤1. Then, the same equivalence as that of (2) holds without any
assumption of G.
We call Hin(G) the Hurwitz space.
r
It isnotdifficult to seethatthere exists analgebraicvarietyHrin(G)C whose
C-valued points are identified with Hin(G)(C).
r
FriedandVo¨lkleintriedtoprovethatHrin(G)CisthemodulispaceofGcoverings
overaprojectivelineramifyingatexactlyr-points. Thisimpliesthatthefieldof
definition of Hrin(G)C is equal to Q. Actually, they proved the representability
ofacertainauxiliarymoduliproblemHab(G,U)inthespecialcase(see[Fr-V¨o,
Section 1.2.] ), but to solve RIGP for G, it is enough to prove that Hrin(G)C is
defined over Q.
In this paper,we give asimpler constructionofthe Hurwitz spaces. Thanks
to our new construction, we generalize Thompson’s rigidity method as follows:
2
Main Theorem A . (Theorem 3.2, Theorem 3.3, Theorem 3.4.)Let G be a
finite group and r be a positive integer.
(1) Assume that there exists an r-tuple of conjugacy classes C of G and a sub-
group H ⊂Aut (G) which satisfy the following conditions:
C
(a) The subgroup H ⊂Aut (G) acts on Ein(C) transitively .
C
(b) The subgroup H is an abelian group or a dihedral group D .
n
Then, G is regular over Qab.
(2) Assume the conditions (a) and (b). Let m be the number defined by
the exponent of H if H is abelian.
m:=
(n if H =Dn.
Then G/Z(G) is regular over Q (µ ).
C m
(3) Assume the conditions (a) and (b). If the inclusion from H to Aut (G) can
C
be extended to a group homomorphism from (Z/mZ)×⋉H into Aut (G) and
C
if the action of (Z/mZ)× on Ein(C) is trivial, then G/Z(G) is regular over Q .
C
(4)Assumetheconditions(a)and(b). IfH ⊂D4 andtheactionofH onEin(C)
factors through H ⊂D4 ։(Z/2Z)2 and the action of H can be extended to an
action of D , then G/Z(G) is regular over Q .
4 C
Next, we obtain an application of our first main theorem as follows.
Corolally A . (Proposition 3.35.)Let p be an odd prime and n be an even
natural number. If p≡7( mod 12),4|n and n≥12, then PSO+(F ) is regular
n p
over Q.
To prove Corollary A, we use middle convolutionfunctors and scalar multi-
plications whicharedefined in the paper [D-R, Section3.2]. Letr be a positive
integerandK beafield. LetFr beafreegroupofrankr andRepK(Fr)be the
categoryoffinite dimensionallinear representationsofFr overK. We use mid-
dle convolution functors MC(r) for λ ∈ K× and scalar multiplication functors
λ
M(r) for Λ∈(K×)r which are defined in the paper [D-R]:
Λ
(r) (r)
MCλ ,MΛ :RepK(Fr)→RepK(Fr).
The functors MC(r),M(r) satisfy the following conditions:
λ Λ
(*) Let R be a full subcategory of RepK(Fr) which is defined in Lemma 3.23.
Then MC(r) and M(r) induce category equivalences R∼=R and quasi -inverses
λ Λ
(r) (r)
are MC and M
λ−1 Λ−1
WeprovealsotheregularityofPSO+(F )whennisevenandnotdivided4.
n p
Howeverin this case, we can not apply Main TheoremA. We use the theory of
the action of braid groupswhich are also consideredin the paper [D-R, Section
4.1]. The most important property of the action of braid groups is that they
commutewithMC(r) andM(r). Insection4,2,weobtainthefollowingtheorem.
λ Λ
Main Theorem B . (Theorem 4.11.)Let T = (T1,...,Tr) be an r-tuple of
elementsofGLn(Fq)asLemma4.5. LetT˜ beanr-tuplewhich arises aniterated
3
application of middle convolutions and scalar multiplications to T. Denote T˜ by
(T˜ ,...,T˜ ). Assume that the r-tuple conjugacy classes (C(T˜ ),...,C(T )) of
1 r 1 r
hT˜i is rational. Here C(Ti) is the conjugacy class of hT˜i such that C(Ti)
contains Ti. If there exists a subgroup H ⊂NGLm(Fq)(hT˜i) such that the image
of H in NGLm(Fq)(hT˜i)/hT˜i is equal to NGLm(Fq)(hT˜i)/hT˜i and satisfies one
of the following conditions:
(a) The group H is isomorphic to a product of several copies of Z/2Z.
(b) The group H is isomorphic to (Z/2Z)2⋊S2 =D4 and the action of H on
Ein(C) factors through the canonical projection H ։H/h(1,1)i∼=(Z/2Z)2.
Then hT˜i is regular over Q.
We prove the following corollary in Section 4.2, as a consequence of Main
Theorem B.
Corolally B . Let p be an odd prime and n be an even natural number.
(1) (Proposition 4.12.) If p ≡ 7( mod 12),4 ∤ n and n ≥ 12, then PSO+(F )
n p
is regular over Q.
(2) (Proposition 4.14.) Let q be a power of p and n be an even natural number.
If n > max{ϕ(q−1),7},n ≡ ϕ(q−1) +1 ( mod 2) and q ≡ 3( mod 4), then
2 2
PSO+(F ) is regular over Q.
n q
Remark 1.6. The regularities over Q of PSO+(F ) are known if q =p , n≡
n q
2 (mod 4) and p ≡ 7 (mod 12) and p ∤ n ([M-M, Theorem 10.3. (i)] ) or if
n>2max{ϕ(q−1),7} and q ≡3(mod 4)([D-R, Lemma 9.5.]).
The plan of the paper is as follows:
Plan. In Section 2, we construct Hin(G) as an etale sheaf on the configuration
r
space Ur of r-points over P1Q. We regard a finite group G as the constant sheaf
on Ur. Therefore we will generalize the Hurwitz spaces by replacing constant
sheaves with locally constant sheaves. We do not need the scheme Hab(G,U)
which was used in [Fr-V¨o]. The key of our construction is the use of the theory
of the etale fundamental group of schemes in the sense of SGA1. We construct
modulispacesofG-coveringsnotonlyoverP1 but alsooversomeothersmooth
Q
algebraic varieties ( for example, elliptic curves, see Remark 2.21). The author
believesthatthisgeneralizationisusefultostudyIGPviatheanalogousproblem
to RIGP over algebraic varieties.
In Section 3.1, we prove Main Theorem A. In Section 3.2, we prepare some
group theoretic lemmas to prove Corollary A and B and define middle convo-
lution functors and scalar multiplications in Section 3.3. Next, we recall the
notion of the linearly rigidity in 3.4. Then we proveCorollaryA in Section 3.5.
In Section 4, we prove Main Theorem B and Corollary B. The Main tool is
the theory of the action of the braid groups. In section 4.1, we prove Theorem
B(Theorem4.11)byusingsomeresultsonthe actionofbraidgroupswhichare
proved by Dettweiler and Reiter in the paper [D-R]. We prove Corollary B in
Section 4.2.
Acknowledgements TheauthorwouldliketothankProfessorTadashiOchiai
forreadingthispapercarefullyandvariablediscussions. Also,theauthorwould
like to Professor Tetsushi Ito for some useful suggestions.
4
2 Construction
In Section 2, we define Hurwitz spaces and generalize Proposition 1.5.
2.1 The fundamental groups of schemes and examples
In this section, we recall the etale fundamental group of schemes and give im-
portant examples.
Theorem-definition 2.1. [SGA1, Expos´e X] Let X be a connected locally
Noetherian scheme. Then there exists the unique pro-finite group πet(X,x¯) and
1
the equivalence of categories as follows:
(Finite etale coverings ofX)−∼→(Finite sets which has an action of πet(X,x¯)).
1
f :W →X 7−→ f−1(x¯) .
Here x¯ is a geometric point of X i.e. a morphism from a spectrum of an al-
gebraically closed field to X. If we take another geometric point y¯, then there
exists an isomorphism πet(X,x¯)∼=πet(X,y¯). This isomorphism is unique up to
1 1
inner automorphisms. We call πet(X,x¯) the etale fundamental group of X.
1
So we alwaysidentify a finite etale coveringof X with a finite set which has
an action of πet(X,x¯).
1
2.1.1 Example of punctured projective line over C
Let r be a positive integer and X = P1C\{x1,...,xr} , xi ∈ P1(C). By the
Riemann’sexistencetheorem,everycompactRiemannsurfaceisidentifiedwith
algebraic curves over C. In particular, every finite topological covering of X
is identified with an algebraic curve over C. Thus πet(X,x), x ∈ X(C) is
1
isomorphic to the pro-finite completion of the topological fundamental group
πtop(X(C),x)ofX(C). Thegroupπtop(X(C),x)isisomorphictothefreegroup
1 1
ofrankr−1. Thisgroupisgeneratedbythehomotopyclassesǫ ofalooparound
i
x for each i with the relation ǫ ǫ ...ǫ =1.
i 1 2 r
Let G be a finite group. By the definition of the etale fundamental group, we
identifyisomorphismclassesofetaleG-coveringsofX withG-orbitofsurjective
group homomorphisms πet(X,x)→G. This proves the following lemma.
1
Lemma 2.2. Let X be P1C\{x1,...,xr}. For a finite group G, the following
sets are identified:
(1) G-coverings of X modulo isomorphisms.
(2) Surj(πet(X,x),G)/InnG.
1
(3) Erin(G):={(g1,...,gr)∈Gr|g1...gr =1,hg1,...,gri=G}/Inn(G)
where Inn(G) acts on Gr diagonally.
Remark 2.3. In general, for any algebraic variety X over C, πet(X,x) is
1
canonically isomorphic to the pro-finite completion of the πtop(X(C),x) (see
1
[SGA1, Expos´e 10]).
5
Remark2.4. IfX is an algebraic variety over Q andfixan embedding Q֒→C.
Then the morphism X →X induces an isomorphism πet(X ,x)−∼→πet(X,x).
C 1 C 1
This is proved by the technique of specialization ( see [Se, Chapter 6] ).
2.1.2 Example of punctured projective line over Q
Next, we consider an arithmetic case. Let X = P1Q\{x1,...,xr} where xi ∈
P1(Q) and the set {xi}i is GQ stable. For a field k, we denote by Gk the
absolute Galois group Gal(k/k) throughout this paper.
Lemma 2.5. ([SGA1] The fundamental exact sequence.) Let X be a geometri-
cally connected algebraic variety over Q. Then, there exists an exact sequence:
1→πet(X ,x¯)→πet(X,x¯)→G →1 .
1 Q 1 Q
This exact sequence induces an outer action of G on π (X ,x¯) and an
Q 1 Q
action of G on Surj(πet(X,x¯),G)/InnG for a finite group G. This action is
Q 1
mysterious in general. However if X is isomorphic to P1 minus r-points, the
Q
action of G on πet(X )ab is described as follows.
Q 1 Q
Lemma 2.6. [M-M, Chapter 1,Theorem 2.6.] Let X =P1Q\{x1,...,xr} where
xi ∈ P1(Q) and the set {xi}i is GQ-stable. Let π : GQ → Sr be the group
homomorphism so that τ(x ) = x for all τ ∈ G . Then, τ(C(ǫ )) =
i π(τ)(i) Q i
C(ǫ )χ(τ),τ ∈G . Here C(ǫ ) is the conjugacy class of ǫ ∈πet(X ,x¯) and
π(τ)(i) Q i i 1 Q
χ:GQ →Zˆ× is the cyclotomic character.
Let G be a finite group. The outer action of GQ on π1et(P1Q\{x1,...,xr},x)
which is induced by the fundamental exact sequence induces the action of G
Q
on Ein(G) via the identification of Lemma 2.2. This GQ-action on Erin(G) can
lift to Er(G) := {(g1,...,gr) ∈ Gr|g1...gr = 1,hg1,...,gri = G} as follows.
First, we take Q-rational point y of P1Q\{x1,...,xr}. Since the formalism of
the fundamental group is covariant, a Q-rational point y induces a splitting of
the following exact sequence:
1→πet(X ,x¯)→πet(X,x¯)→G →1 .
1 Q 1 Q
Thus, the outer action of G action on the fundamental group πet(X ,x¯) lifts
Q 1 Q
tothe actiononπet(X ,x¯). ThisactionofG onπet(X ,x¯)induces the action
1 Q Q 1 Q
of GQ on the set Surj((π1et(P1Q\{x1,...,xr},x¯),G), and induces the action on
Er(G). Note that this action is independent on a choice of a rational point up
to inner automorphisms of G.
Letf :π1et(PQ1 \{x1,...,xr},x¯)→Gbeasurjectivegrouphomomorphismwhich
corresponds to [g =(g1,...,gr)]∈Erin(G).
If [g] is fixed by G the absolute Galois group of a number field F, then for
F
all β ∈ G there exists g ∈ G such that β(g ) = g g g−1 for all 1 ≤ i ≤ r.
F β i β i β
6
Here, we fix a lift of the action of GQ on Er(G) as above.
Putc(α,β):=g g g−1. Thisisa2-cocycleofG whosevaluesareinthecenter
α β αβ F
of G and class of c in H2(G ,Z(G)) is independent of a choice of a lift of the
F
action of GQ on Er(G) because this lift is unique up to inner automorphisms of
G.
Lemma2.7. LetGbeafinitegroupandf ∈Surj((π1et(P1Q\{x1,...,xr},x¯),G).
Let c be the 2-cocycle associated a group homomorphism f as explained above.
Then f can be extended to a group homomorphism πet(X ,x¯)≃πet(X ,x¯)⋊
1 F 1 F
G →G if and only if c is a coboundary. Here, the isomorphism πet(X ,x¯)≃
F 1 F
πet(X ,x¯)⋊G is defined by the fixed splitting of the fundamental exact se-
1 F F
quence.
Proof. Assumethatcisacoboundary,thatis,thereexistsacontinuousmapξ :
G → Z(G) such that c(β,γ) = dξ(β,γ) = ξ(β)ξ(γ)ξ(βγ)−1. Put f˜((τ,β)) :=
F
f(τ)g ξ(β)−1,thenf˜isagrouphomomorphism. Infact,foranyelementτ,µ∈
β
πet(X ,x) ,β,γ ∈G , f˜is checked to be multiplicative as follows:
1 F F
f˜((τ,β)(µ,γ)) = f˜((τµβ,βγ))=f(τµβ)g ξ(βγ)−1
βγ
= f(τ)f(µβ)g ξ(βγ)−1 =f(τ)g f(µ)g−1g ξ(βγ)−1
βγ β β βγ
= f(τ)g f(µ)g ξ(β)−1ξ(γ)−1 =f˜((τ,β))f˜((µ,γ)).
β γ
Conversely,iff canbeextendedtoagrouphomomorphismf˜,cisacoboundary
because of the fact that f˜is multiplicative.
Lemma 2.8. Notation and assumptions are same as the above lemma. Let
pr:G։G/Z(G) be the canonical projection. Then pr◦f can be extended to a
group homomorphism πet(X ,x¯)≃πet(X ,x¯)⋊G →G/Z(G).
1 F 1 F F
Proof. Wewritethe imageofg ∈GinG/Z(G)byg¯. Since g¯ g¯ =g¯ ,themap
ǫ τ ǫτ
f˜((τ,β)) := f¯(τ)g¯ is a group homomorphism. Indeed, we have the following
β
equation:
f˜((τ,β)(µ,γ)) = f˜((τµβ,βγ))=f¯(τµβ)g¯
βγ
= f¯(τ)f¯(µβ)g¯ =f¯(τ)g¯ f¯(µ)g¯−1g¯
βγ β β βγ
= f¯(τ)g¯ f¯(µ)g¯ =f˜((τ,β))f˜((µ,γ)).
β γ
Definition 2.9. Let G be a finite group and C = (C1,...,Cr) be an r-tuple
of conjugacy classes of G. Put S = S(C) := {{Cn,...,Cn}|(n,|G|) = 1} and
1 r
7
define an action κ:G →Aut(S) of G on S as follows:
Q Q
κ(ǫ)({Cn,...,Cn}):={Cnχ(ǫ),...,Cnχ(ǫ)}.
1 r 1 r
where χ is the cyclotomic character. We define an abelian extension Q of Q
C
so that Stab ({C ,...,C })=Gal(Q/Q ).
GQ 1 r C
We say that C is rational if Q =Q.
C
Lemma2.10. ([Vo¨,Lemma3.16.]) LetGbeafinitegroupandC =(C1,...,Cr)
be an r-tuple of conjugacy classes of G. Then, there exists a G -stable set of
Q
points {xi}1≤i≤r ⊂P1(Q) such that
{C ,...,C }∼={x ,...,x }
1 r 1 r
as G -sets. Here the action of G on {C ,...,C } is defined by κ.
QC QC 1 r
Now, we can prove Thompson’s rigidity method .
Proof. (Proof of Proposition 1.4)
Let G be a finite groupand C =(C1,...,Cr) be anr-tuple of conjugacyclasses
of G. Put
Ein(C):={(g1,...,gr)∈Gr |gi ∈Ci}/InnG.
By Lemma 2.2, there exists an identification :
Ein(C)−∼→{f ∈Surj( π1(XQ,x),G) | f(ǫi)∈Ci }/ InnG .
Here we take {x } so that
i
{C ,...,C }∼={x ,...,x }
1 r 1 r
as GQ -sets by Lemma 2.10. By Lemma 2.6, GQ acts on Ein(C).
C C
Assumethatthereexistsanr-tupleofconjugacyclassesCsuchthat|Ein(C)|=
1 so [g]∈Ein(C) is fixed by GQ . Thus we have a 2-cocycle c:GQ →Z(G) as
C C
the above. Since the cohomologicaldimension of Qab is one, the restrictionof c
onG is a2-coboundary. Letf :πet(X ,x¯)→Gbe agrouphomomorphism
Qab 1 Q
andW →P1 beaG-coveringwhichcorrespondsto[g]. ByapplyingLemma2.7
Q
with F =Qab, f can be extended to a group homomorphism on πet(X ,x¯).
1 Qab
AccordingtoTheorem-Definition2.1,aG-coveringW →P1 isdefinedoverQab
Q
if and only if f can be extended to a group homomorphism πet(X ,x)→G.
1 Qab
Applying Lemma 2.8, the group homomorphism f can be extended to a
group homomorphism f˜ : πet(X ,x¯) → G. This completes the proof of
1 QC
Proposition 1.4.
8
2.1.3 Example of configuration spaces
Let Ur(C) := {{x1,...,xr}|xi ∈ P1(C),xi 6= xj if i 6= j}. The topological
fundamental group of Ur(C) is the Artin’s braid group Br.
B := Q ,1≤i≤r−1 Q satisfy thefollowing eqations (1),(2)
r i i
D (cid:12) E
(cid:12)
QiQi+1Qi =(cid:12)Qi+1QiQi+1,[Qi,Qj]=1|i−j|≥2, (1)
Q ···Q Q2 Q ···Q =1 (2)
1 r−2 r−1 r−2 1
We define the action of π1top(Ur(C),u) on Erin(G) by
Qi[g1,...,gr]:=[g1,...,gigi+1gi−1,gi,...,gr] , [g1,...,gr]∈Ein(G) .
This action is well-defined. By the definition of the fundamental group, there
exists the finite etale covering Hrin(G) → UrC whose fiber at u is identified
with Ein(G). This is the classical Hurwitz space which are the moduli space of
r
ramified G-coverings which are ramified at most r-points.
2.2 Hurwitz spaces
Inthissection,weconstructHurwitzspaces. LetF beasub-fieldofC. Consider
the following diagram of F-schemes:
j
U ❅❅❅❅❅❅❅❅❅❅❅f❅❅❅❅//(cid:8)P1F ×fF¯U(cid:9)oo ⑦⑦⑦⑦i⑦f⑦˜⑦⑦⑦⑦⑦⑦⑦⑦Y (∗)U,Y
❅ ⑦
❅ ⑦
❅ ⑦
❅ ⑦
❅ ⑦
(cid:15)(cid:15) ~~⑦
U
where Y is a relative normal crossing divisor of P1 × U over U and f˜is a
F F
finite etale morphism, U is the complement of Y.
LetGbeafinitegroup. ByTheorem-Definition2.1,agrouphomomorphism
φ : πet(U,u¯) → Aut(G) define a locally constant constructible sheaf G on U,
1 φ
where u¯ is a geometric point of U.
The following well-known proposition is the key of our construction.
Proposition2.11. [SGA1, Expos´e13 Theorem. 2.4] LetG bea locally constant
constructiblegroupsheaf on Uet. Then, R1f∗G is alocally constantconstructible
sheaf on U . Moreover R1f G is compatible witharbitrary base change by U′ →
et ∗
U.
According to the above proposition, R1f f∗G is a locally constant con-
∗ φ
structible sheaf whose fiber is isomorphic to
H1(P1 \{x ,...,x },(G ) )=Hom(πet(P1 \{x ,...,x }),G)/InnG.
F 1 r φ u¯ 1 F 1 r
9
Wedescribetheactionofπet(U,u¯)onthefiberofR1f f∗G atu¯.AssumethatU
1 ∗ φ
has an F-rationalpoint v, and we fix the isomorphismπet(U,u¯)∼=πet(U ,u¯)⋊
1 1 F
G which is induced by v →U.
F
The Action of G First, we determine the action of G on the fiber of
F F
R1f f∗G at u¯. According to Proposition 4.12, we may assume that U =
∗ φ
Spec(F). Bydefinition,thesheafR1f f∗G isthesheafificationofthepresheaf
∗ φ
:
L/F 7→H1(P \{x ,...,x },Gφ(GL)).
L 1 r
HereLisanF-algebraandGφ(GL) isthefixedpartofGbyφ(GL)the absolute
Galois group of L.
The transformation of an F-algebra homomorphism K → L by the above
presheaf
(K →L/F)7→H1(P1K\{x1,...xr},Gφ(GK))→H1(P1L\{x1,...,xr},Gφ(GL))
is decomposed as the following diagram:
H1(P1K \{x1,...,xr◗}◗,◗G◗◗φ◗(◗G◗K◗)◗)◗◗◗◗p◗◗◗//◗H◗◗1◗(◗P◗1L◗(cid:9)◗\◗{◗x1,.α..OO ,xr},Gφ(GL))
((
H1(P1L\{x1,...,xr},Gφ(GK))
where p is the pullback of torsors by
P1L\{x1,...,xr}−→P1K\{x1,...,xr}
and α is the map which is induced by a canonical morphism
Gφ(GK) →Gφ(GL) .
Hence the action of β ∈G is decomposed as follows:
F
H1(P1F \{x1,...,◆x◆r◆}◆◆,◆G◆)◆◆◆◆◆β◆∗◆◆// H◆◆1◆(◆P◆F1(cid:9)◆◆\◆φ{(xβ)1,OO...,xr},G)
&&
H1(P1F \{x1,...,xr},G)
where β∗ is the pullback of torsors by
β :P1 \{x ,...,x }−→P1 \{x ,...,x }
F 1 r F 1 r
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