Math. J. Okayama Univ. 52 (2010), 29–43

TRADING DEGREE FOR DIMENSION
IN THE SECTION CONJECTURE:
THE NON-ABELIAN SHAPIRO LEMMA
Jakob Stix
Abstract. This note aims at providing evidence for the section conjecture of anabelian geometry by establishing its behaviour under Weil
restriction of scalars. In particular, the étale fundamental group of theWeil restriction is determined by means of a Shapiro Lemma for nonabelian group cohomology.

Contents
1. Introduction and results
2. Non-abelian cohomology
3. Weil restriction of scalars
References

29
32
41
43

1. Introduction and results
Let K be a field with fixed separable closure K sep . The étale fundamentalgroup π1 (X) of a geometrically connected variety X/K forms naturally a
non-abelian extension of pro-finite groups
1 → π1 (X ⊗ K sep ) → π1 (X) → Gal(K sep /K) → 1,that we abbreviate by π1 (X/K).
1.1. The section conjecture. A K-rational point x ∈ X(K) yields byfunctoriality a section Gal(K sep /K) → π1 (X) of π1 (X/K), with image thedecomposition group of a point x̃ above x in the universal pro-étale coverof X. Having neglected base points and due to the choice of x̃, only theclass of a section up to conjugation by elements from π1 (X ⊗ K sep ) is welldefined. Let us denote by Sπ1 (X/K) the set of π1 (X ⊗K sep )-conjugacy classesof sections of π1 (X/K). The section conjecture of Grothendieck’s anabeliangeometry [Gr83] speculates the following.
Mathematics Subject Classification. 14H30 (Primary) ; 14G05, 20E22 (Secondary) .
Key words and phrases. Section Conjecture, Rational points, Anabelian Geometry,Non-abelian Shapiro Lemma.
The author acknowledges support provided by DFG grant STI576/1-(1+2).
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JAKOB STIX

Conjecture 1 (Grothendieck). The map X(K) → Sπ1 (X/K) which sends
a rational point to the section given by its conjugacy class of decomposition groups is bijective if K is a number field and X/K is a geometricallyconnected, smooth, projective curve of genus at least 2.
There is also a version of the section conjecture for affine curves. Hererational points at infinity will lead to an abundance of additional cuspidal
sections, see [EH08] and [Sx08b]. But apart from the obvious modification
forced upon us by cuspidal sections the conjecture remains the same. The
condition on the genus gets replaced by asking the Euler-characteristic to
be negative.
A birational version of the section conjecture over p-adic local fields was
successfully addressed by Koenigsmann in [Ko05], and later by Pop [Po07]
in a truncated version that exploits spectacularly modest pro-finite data torecover rational points.
Only recently evidence for the section conjecture could be found throughthe first examples of curves which satisfy the conjecture, see [Sx08a] and[HS08], though for the reason of having neither points nor sections. Anothersource of evidence has been the study of the cycle class of a section as
pioneered by [EW08].
The goal of the present paper is to provide evidence for the section conjecture from a different direction.
1.2. Results — trading degree for dimension. The evidence for thesection conjecture presented in this note consists in its compliance with
Weil restriction of scalars, see Section 3. Of course, in order for this to makesense, we widen the applicability of the conjecture beyond the case of curves.
Let L/K be a finite separable field extension within K sep , so that GalL =Gal(K sep /L) is a subgroup of GalK = Gal(K sep /K). Let X/L be a quasiprojective, geometrically connected variety and RL/K X its Weil restrictionof scalars as a geometrically connected variety over K. In Section 2 we will
construct an induction functor for extensions which turns out to describe thefundamental group of the Weil restriction in characteristic 0 as an extension
as follows.
Theorem 2 (Theorem 17 in Section 3). Let K be a field of characterisitc
0 or let X/L be projective. Then the fundamental group π1 (RL/K X/K)of the Weil restriction RL/K X of scalars is isomorphic to the non-abelianK
induction IndGal
GalL π1 (X/L).
Next, a non-abelian analogue of Shapiro’s Lemma yields a description of
the set of conjugacy classes of sections for an induction.

TRADING DEGREE FOR DIMENSION IN THE SECTION CONJECTURE

31

Theorem 3 (Corollary 15 in Section 3). Let E = [1 → N → E → H → 1]G
be an extension with induction IndG
H (E) = [1 → M → IndH (E) → G → 1]with respect to a subgroup H ⊆ G.
Then N -conjugacy classes of sections of E ։ H are naturally in bijection
with M -conjugacy classes of sections of IndG
H (E) ։ G.
Combining Theorem 2 and Theorem 3 above, we obtain our piece ofevidence for the section conjecture. We note in passing, that no special
assumption on the geometry of the quasi-projective variety X is used.
Theorem 4. Let L/K be a finite separable field extension and let X/L be aquasi-projective, geometrically connected variety. Let K have characteristic
0 or let X/L be projective.
(1) Let s ∈ Sπ1 (X/L) and t ∈ Sπ1 (RL/K X/K) be sections that correspond toeach other under the bijection of Theorem 3. Then s is the section associatedto a rational point x ∈ X(L) if and only if t is the section associated to arational point y ∈ RL/K X(K). In this case we may choose x and y so thatthey correspond to each other via the identification RL/K X(K) = X(L).
(2) Applying the functor π1 yields a bijective map X(L) → Sπ1 (X/L) if
and only if it yields a bijective map RL/K X(K) → Sπ1 (RL/K X/K) .
Proof: Theorem 2 and Theorem 3 show that there is a natural commutative diagram
RL/K X(K)Sπ1 (RL/K X/K)
whose horizontal maps are bijections.

∼
=

∼
=

/ X(L)
/ Sπ1 (X/L)

Corollary 5. The section conjecture holds for smooth projective curves overnumber fields if it holds for smooth, projective algebraic K(π, 1) spaces overQ (see [Sx02] Appendix A), which embed into their Albanese variety andhave non-vanishing Euler-Poincaré characteristic.
This corollary explains the title of the article. We have lowered the degreeof the number field in the section conjecture to 1 at the expense of working
with varieties of dimension exceeding 1. Of course, these higher dimensionalvarieties of interest are simply Q-forms of products of smooth, projective
hyperbolic curves and so the trade might be marginal. But on the one hand,
we are not required to limit the section conjecture to curves or products ofcurves, and secondly, the arithmetic of Q and so presumably also sections
over GalQ are arithmetically much simpler than for more general algebraic
number fields. For exampe, the recent modularity results were first provenfor representations of the full GalQ . So the base has become simpler.

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JAKOB STIX

Another argument in favour of the improvement of our situation is thefollowing weak analogue of [Sx08a] Theorem 17.
Proposition 6. Let X/Q be a smooth, projective and geometrically connected variety which is an algebraic K(π, 1) space such that π1 (X/Q) admits
a section. Then any Galois invariant line bundle L ∈ Pic0X (Q) has vanishing Brauer obstruction b(L) = 0 ∈ Br(Q), hence belongs to a genuine line
bundle on X.
Proof: The local components of b(L) are obtained by base change to Qp
or R which preserves the assumptions. The real section conjecture as in[Sx08a] Theorem 24 applies also for higher dimensional K(π, 1) spaces andyields the existence of a real point, and so b(L)R = 0.
Over a p-adic field the argument of [Sx08a] Proposition 12 and Corollary
14 still show that the order of b(L)Qp is a power of p.
The result now follows from the global reciprocity which states that the
local invariants of b(L) sum up to 0. A sum of summands of prime power
order where every prime occurs at most once can only vanish if all thesummands vanish. Hence b(L) vanishes by the local global principle for
Brauer groups of number fields.

2. Non-abelian cohomology
2.1. Twisted generalized wreath products. We recall some group theory in order to fix notations and to put it into a form useful for the sequel.
Although we will ultimately apply the results in the context of pro-finitegroups, we choose to neglect the topology in the presentation, because thisfrees us from adding an abundance of ”continuous” everywhere. However wenote, that in the pro-finite case the subgroups in question should be closed
subgroups.
2.1.1. Wreath products. The wreath product of two groups G and N along
a right G-set A is the semidirect productY
N ≀ G := (N) ⋊ Gα∈A

with respect to the action of G on
see [Hu67] §15.6.

Q

α∈A N given by g.(nα )α∈A = (nαg )α∈A ,2.1.2. Induction for groups acting on groups. Let G be a group. A G-groupis a group N together with an action ϑ : G → Aut(N ). For a subgroup H
in G, induction is a functor indG
H from H-groups to G-groups defined asfollows. As a groupindG
H (N ) = {f : G → N ; f (hg) = ϑ(h)(f (g)) all h ∈ H}

TRADING DEGREE FOR DIMENSION IN THE SECTION CONJECTURE

33

with pointwise multiplication. The G-action on f ∈ indG
H (N ) comes fromright translation of the argument, so (g.f )(α) = f (αg).
2.1.3. Twisted generalized wreath products. We recall that the twisted generalized wreath product of the group G with subgroup H relative to the
H-group N is the semidirect productN ≀H G := (N, ϑ) ≀H G := indG
H (N ) ⋊ Gwith respect to the natural G-action of the induction as a G-group, see[Ne63] §2, [Me95] I.9.3, [Hu67] §15.10 and [Ha99] §1.
2.1.4. Sections and H1 . Let N be a G-group. A 1-cocycle of G with values
in N is a map a : G → N such that for all s, t ∈ G we haveast = as (s.at ).
The first non-abelian cohomology H1 (G, N ) is the set of equivalence classesof 1-cocycles, where cocycles a and b are equivalent if there is c ∈ N with
as = cbs (s.c)−1 for all s ∈ G.
Two sections of N ⋊ G ։ G are equivalent if they differ by conjugation
with an element of N . The following lemma is well known and straightforward.
Lemma 7. The map which sends a 1-cocycle s 7→ as to the section s 7→ as ·s,
where we have identified G with the second factor in N ⋊ G establishes anatural bijection of H1 (G, N ) with the set of equivalence classes of sectionsof N ⋊ G ։ G.
2.1.5. The non-abelian Shapiro Lemma in degree 1. Let N be an H-groupfor a subgroup H ⊂ G. The restriction of the G-group indG
H (N ) to an
H-group admits an H-equivariant map
ev1 : indG
H (N )|H → N
by evaluating at 1. The composition of restriction and evaluation at 1 defines
the Shapiro map
1
sh1 : H1 (G, indG
H (N )) → H (H, N ).
1
Proposition 8. The Shapiro map sh1 : H1 (G, indG
H (N )) → H (H, N ) isbijective.

Proof: A 1-cocycle s 7→ bs for G with values in indG
H (N ) is given by
bs,t = bs (t) ∈ N for all s, t ∈ G such that (i) bs,ht = ϑ(h)(bs,t ) for all s, t ∈ G
and h ∈ H and (ii) bst,g = bs,g bt,gs for all s, t, g ∈ G. The map sh1 on thelevel of cocycles maps b to h 7→ bh,1 .

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JAKOB STIX

Surjectivity. We choose a set of representatives Y ⊂ G for H\G andobtain maps γ : G → H and y : G → Y such that g = γg yg for all g ∈ G.
Let a : H → N be a 1-cocycle, in particular a1 = 1. We set
−1bs,t := aγt ϑ(γt )(ayt )aγts ϑ(γts )(ayts )and a routine calculation shows that b is a 1-cocycle maping to a. The
cocycle condition (ii) is best checked by noting that our definition of b implies
bs,t = (bt,1 )−1 bts,1 .
Injectivity. Let b and b′ be cocycles with sh1 (b) ∼ sh1 (b′ ). The function
f (s) = (b′s,1 )−1 bs,1 for all s ∈ G is an element of indG
H (N ). It follows thatb′s,t = f (t)bs,t f (ts)−1 for all s, t ∈ G which translates into b ∼ b′ .
Remark 9. (1) An alternative proof is given in [Ho78] Theorem 4 using theinterpretation as conjugacy classes of complements in semidirect products.
(2) Proposition 8 speaks about conjugacy classes of sections of a twisted
wreath product N ≀H G = indG
H (N ) ⋊ G. The assertion appears in [PQ03]
Thm 2.6 in the case of wreath products, i.e., trivial action of H on N . The
introduction of [PQ03] contains the observation that this is a non-abelianversion of Shapiro’s Lemma but does not elaborate on this idea further.
2.2. Extensions after Eilenberg and MacLane. We recall the theoryof non-abelian extensions of Eilenberg and MacLane from [EMcL47].
2.2.1. Kernel. A kernel or more precisely a G-kernel is a group N together
with an exterior action ρ : G → Out(N ) by a group G, where Out(N ) =Aut(N )/ Inn(N ) is the group of exterior automorphisms of N . We denotethe set of all G-kernels on N by K(G, N ) = Hom(G, Out(N )).
2.2.2. Center. Let ρ : G → Out(N ) be a G-kernel. The center Z of thekernel is the center of N together with its inherited G-actionχ = χρ : G → Aut(Z).
For distinction purposes we may denote by Z(χ) the G-module Z withmodule structure given by χ : G → Aut(Z). The set of kernels ρ withcenter equal to χ is denoted by K(G, N )χ .
2.2.3. Extensions. An extension E of a group G by a group N is a short
exact sequence
1 → N → E → G → 1.
Isomorphisms of extensions of G by N respect both G and N identically.
We denote the set of isomorphism classes of extensions by Ext(G, N ). An

TRADING DEGREE FOR DIMENSION IN THE SECTION CONJECTURE

35

extension E leads to a kernel ρ via the restriction to N of the conjugationby preimages:EGe( )e−1 |N/ Aut(N )ρ/ Out(N )The set of isomorphy classes of extensions, whose kernel has center χ isdenoted by Ext(G, N )χ . The map Ext(G, N )χ → K(G, N )χ that assigns toeach extension its kernel is well defined.
2.2.4. Obstruction theory. A kernel which is the kernel of an extension iscalled extendible. By pullback of
1 → N/Z → Aut(N ) → Out(N ) → 1
a kernel ρ : G → Out(N ) determines an extension
1 → N/Z → Eρ → G → 1such that a presumptive extension E with kernel ρ by conjugation canonically sits in a diagram
1

/N

/E

/G

/1

1


/ N/Z


/ Eρ

/G

/1

Let χ be the center of ρ. If N were an abelian group with a G-submodule
Z, then the existence of E lifting Eρ were controlled by the coboundaryδ(Eρ ) under δ : H2 (G, N/Z) → H3 (G, Z(χ)). The set of such lifts would
receive a transitive action of the group H2 (G, Z(χ)) via the homomorphism
H2 (G, Z(χ)) → H2 (G, N ). In the non-abelian case discussed here (see[EMcL47] §7+8), as Z is central in N , the same formulas with inhomogeneous cocycles which prove the assertions in the abelian case succeed togive the following result, except for the 0 on both sides which follows from
[EMcL47] §9+11.
Proposition 10 (essentially [EMcL47]). The following is exact
δ0 → H2 (G, Z(χ)) → Ext(G, N )χ → K(G, N )χ −→ H3 (G, Z(χ)) → 0
in the sense that H2 (G, Z(χ)) acts freely on Ext(G, N )χ with quotient set
equal to the set of extendible kernels δ −1 (0) and δ is surjective.
The action of a ∈ H2 (G, Z(χ)) on Ext(G, N )χ can be constructed on
extensions as follows. Let 1 → Z → Za → G → 1 be an extension realizing
the cohomology class a ∈ H2 (G, Z(χ)), and let 1 → N → E → G → 1

36

JAKOB STIX

be an extension with center of its kernel equal to χ. Then a.E equals theisomorphism class of the extension
inclusion,0

1 → N −−−−−−→ (E ×G Za )/∆(Z) → G → 1,
where ∆ : Z → N × Z is the antidiagonal z 7→ (z, −z).
2.2.5. Categories of extensions. Let G be a group. The category Ext[G] has
as objects extensions of G with arbitrary kernel and morphisms are maps ofextensions up to composition by inner automorphisms from elements of thekernel.
Pushing an extension 1 → N → E → G → 1 by an automorphism of N
determines an action of Out(N ) on Ext(G, N ) such that the set of orbits
equals the set Ext[G, N ] of isomorphism classes in the category Ext[G] of
extensions of G by N . The map Ext(G, N ) → K(G, N ) becomes Out(N )equivariant when Out(N ) acts on K(G, N ) = Hom(G, Out(N )) by composition with inner automorphisms of Out(N ).
2.3. Wreath product type extensions. In this section we built on thework of Holt [Ho78].
2.3.1. Wreath kernels. Let N be a group and H ⊆ G a subgroup. On
Y
M = indG
N
H (N, 1) =α∈H\Gwe have an action of G byg.((nα )α∈H\G ) = (nαg )α∈H\GQ
and an outer action indG
H (Out(N ), 1) =α∈H\G Out(N ) → Out(M ) givenby(fα )α∈H\G (nα )α∈H\G = (fα (nα ))α∈H\G .
The two actions are compatible as follows. For g ∈ G andY(fα )α = (fα )α∈H\G ∈Out(N )
α∈H\G

we haveg · (fα )α · g −1 (nα )α∈H\G = fαg (nα ) α∈H\G = g.(fα )α (nα )α∈H\Gresulting in a homomorphism
R : (Out(N ), 1) ≀H G → Out(M ).
A wreath kernel of G on N is a kernel ρ : G → Out(M ) together with alift along R to a homomorphism ρ̃ : G → (Out(N ), 1) ≀H G which is a sectionof the projection to G. Such a lift is unique if it exists, as two lifts differ at

TRADING DEGREE FOR DIMENSION IN THE SECTION CONJECTURE

37

Q
most by elements in α∈H\G Out(N ) which injects into Out(M ). The setof wreath kernels is thus a subset Kwreath (G, H; N ) ⊆ K(G, M ) of the set of
all G-kernels on M .
2.3.2.
The Center of a wreath kernel. The center of M equals indG
H (Z, 1) =Q
α∈H\G Z where Z is the center of N . Thus the center of a wreath kernel
lifts to a homomorphism
G → (Aut(Z), 1) ≀H G,
which is a section of the projection to G. The construction in Section 2.1.5
performed on the level of cocycles, namely restriction to H and then evaluation at 1, yields the two surjective maps sh as in the following diagram.
(2.1)

Kwreath (G, H; N )centersh

{Sections of (Aut(Z), 1) ≀H G ։ G}

/ / K(H, N )sh center/ / Hom(H, Aut(Z))

The vertical maps associate to a kernel its center. In particular, the centerof M as a G-module under the center of the wreath kernel ρ is nothing but
GindG
H (χ) = indH (Z(χ)) where χ is the center of the H-kernel sh(ρ).
2.3.3. Wreath product type extensions. A wreath producttype extenQsion, see [Ho78] p.464, is an extension of G by M = α∈H\G N , the kernelof which is a wreath kernel as above. The set of isomorphism classes ofwreath product type extensions is denoted by Extwreath (G, H; N ). We denote by Extwreath (G, H; N )χ (resp. by Kwreath (G, H; N )χ ) the set of thosewreath type extensions whose kernel maps (resp. those wreath kernels whichmap) under sh to a kernel with center χ.
pr

2.3.4. The Shapiro map for extensions. Let E = [1 → M → E −→ G → 1]
be a wreath product type extension. The kernel of the map ev1 : M ։
N , which evaluates at 1 is a normal subgroup of E|H = pr−1 (H). We
may therefore push the restriction of E to H by the map ev1 to obtain an
extension of H by N , that will be denoted sh2 (E).
2.3.5. Non-abelian Shapiro Lemma in degree 2.
Proposition 11. Let H be a subgroup of G. Let N be a group with center
Z and H-action χ. We have a commutative ladder with exact columns in

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JAKOB STIX

the sense as in Proposition 10
0

0



2


/ H2 (H, Z(χ))

sh2/ / Ext(H, N )χsh
/ / K(H, N )χsh
H (G, indG
H (χ)) ∼=2



Ext(G, H; N )χ

Kwreath (G, H; N )χ

δ

H3 (G, indG
H (χ))

sh3
∼
=

δ



/ H3 (H, Z(χ))





0

0

where the horizontal maps are induced by the respective Shapiro map, are all
surjective and the two extremal ones are isomorphisms.
Proof: Exactness of the right column is Proposition 10. The commutativity of the diagram follows by tedious but elementary calculations oncochains. The abelian Shapiro Lemma shows that the two extremal horizontal maps are isomorphisms.
The surjectivity of sh was discussed in (2.1). Exactness of the left column
follows again from Proposition 10 besides the surjectivity of δ which follows
from a diagram chase. Now the surjectivity of the remaining horizontal map
follows again by diagram chase.
Q
2.3.6. Split extensions. The group α∈H\G Out(N ) acts naturally by pushing on isomorphism classes Extwreath (G, H; N ) of wreath product type extensions. Let Extwreath [G, H; N ] denote the set of orbits.
Theorem 12 ([Ho78] Thm 3). The map sh2 yields a bijection∼

Extwreath [G, H; N ] −→ Ext[H, N ].
This theorem by Holt has the following immediate corollary.
Corollary 13. Let E ∈ Extwreath (G, H; N )χ be aQwreath product type extension. Then E splits as an extension of G by α∈H\G N if and only if
sh2 (E) splits as an extension of H by N .

TRADING DEGREE FOR DIMENSION IN THE SECTION CONJECTURE

39

2.4. Non-abelian induction of extensions.
2.4.1. Adjoint to restriction. Let H be a subgroup of G. We are going toconstruct a right adjoint to the functor restriction from G to H
resG
H : Ext[G] → Ext[H],E 7→ resG
H = E|H

which by analogy is called induction from H to G and denoted by IndG
H . ByYoneda, all we have to achieve is a proof of the following theorem.
Theorem 14. Let F ∈ Ext[H] be an extensionof H. The functor whichG
sends T ∈ Ext[G] to HomExt[H] resH (T ), F is representable.
A representing object as in the theorem is denoted by IndG
H (F ). By
definition, adjointness


G
(2.2)
HomExt[H] resG
(T
),
F
=
Hom
T,
Ind
(F
)
Ext[G]
H
H
holds naturally, and F 7→ IndG
H (F ) is the sought for non-abelian induction
functor.
id
If we apply adjointness to the extension 1 → 1 → G −→ G → 1 we get thefollowing immediate corollary.
Corollary 15. Let H be a subgroup of G and let
E = [1 → N → E → H → 1]
be an extension in Ext[H] with induction
G
IndG
H (E) = [1 → M → IndH (E) → G → 1]

in Ext[G].
(1) The adjointness map (2.2) describes a bijection between N -conjugacy
classes of sections of E ։ H and M -conjugacy classes of sectionsof IndG
H (E) ։ G.
(2) In particular, E splits if and only if IndG
H (E) splits.
2.4.2. Proof of adjointness. Let F ∈ Ext[H] be an extension of H. By
Theorem 12 there exist an extension E ∈ Ext[G] which is a wreath producttype extension such that sh2 (E) ∼= F . We are done if we can find a bijection

 ∼
2
(T
),
sh
(E)
→ HomExt[H] resG
(2.3)
sh : HomExt[G] T, E −Hwhich is natural in T = [1 → S → T → G → 1].
Let ϕ : T → E be a morphism of extensions. We define the map sh(ϕ)
as the composition ev1 ◦ resG
H (ϕ) which is the vertical map in the following

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JAKOB STIX

diagram.
resG
H (T )=[1/S/ T |H

/H

/ 1]


/ E|H

/H

/ 1]


/ sh2 (E)

/H

/ 1]

resG
H (ϕ)



resG
H (E)
ev1

=

[1


/M

sh2 (E)

=

[1


/N



We denote the projection T → G by pr and its restriction to H by pr |H :T |H → H. A map ϕ : T → E is nothing but an M -conjugacy class ofsections ofpr∗1 E = [1 → M → E ×G T → T → 1],
which is a wreath product type extension for the subgroup T |H ⊆ T . On
the other hand, a map T |H → sh2 (E) is an N -conjugacy class of sections of(pr |H )∗ (sh2 (E)) = [1 → N → sh2 (E) ×H T |H → T |H → 1].
Because of the isomorphism sh2 (pr∗1 E) ∼= (pr |H )∗ (sh2 (E)), we get that infact Theorem 14 is equivalent to Corollary 15 with F = IndG
H (E) and E =sh2 (F ), which is what we are going to prove in the sequel. Part (2) has
already been shown in Corollary 13. We may therefore assume that thewreath product type extension F splits.
We fix a splitting σ, which allows to lift the kernel ρF : G → Out(M )to a homomorphism ϑF : G → Aut(M ). By means of ϑF the extension F
is isomorphic to the semidirect product M ⋊ϑF G. Because the following
diagram is a fibre product diagram
Aut(N ), 1 ≀H G/


Out(N ), 1 ≀H G
R

R̃





Aut(M )/ Out(M )where the map R̃ is constructed analogously to the map R, the wreath kernelρ̃, which lifts ρF , also lifts to a homomorphism ϑ̃ : G → Aut(N ), 1 ≀H G.
Restriction of the section σ to H and evaluating at 1 induce the corresponding section of sh2 (F ) and a true action ϑ : H → Aut(N ) by meansof which sh2 (F ) ∼= N ⋊ϑ H. But ϑ also equals the evaluation at 1 of therestriction to H of ϑ̃. Hence M as a G-group via ϑ̃ is nothing but indG
H (N )
for the H-group N via ϑ. The result now follows from Proposition 8.
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41

2.5. The anabelian case. Although still a mystery in general, it is widelybelieved that a group only qualifies to be anabelian when its center is trivial.
In this section we will work out under the assumption, that the center Z of
N is trivial, how the content of the Sections 2.1 - 2.4 specialises. Proposition
10 yields a bijection
Ext(G, N ) → K(G, N )with inverse assigning to a kernel ρ : G → Out(N ) the pullback under ρof the extension 1 → N → Aut(N ) → Out(N ) → 1. Hence all kernels areextendible and each extension is determined by its kernel up to isomorphism.
It follows that an N -conjugacy class of sections of an extension with kernelρ : G → Out(N ) is canonically the same as a lift of ρ to an actual actionρ̃ : G → Aut(N ) up toQ
uniform conjugation by elements of N .
With N also M = α∈H\G N has trivial center. The fact that sections
of Out(N ), 1 ≀H G ։ G and sections of Out(N ) × H ։ H correspond toeach other under sh1 up to conjugation by elements from M (resp. N ) thus
explains Theorem 12 in this case. The content of Corollary 15 follows by
applying the same argument to the semi-direct productsAut(N ), 1 ≀H G ։ G
and Aut(N ) × H ։ H.
3. Weil restriction of scalars
We content ourselves with a discussion of Weil restriction of scalars relative a finite separable field extension for quasi-projective varieties.
3.1. Properties of the Weil restriction. Let L/K be a finite separable
field extension. Weil restriction of scalars relative L/K is a functor RL/K
from quasi-projective varieties over L to quasi-projective varieties over K
that is right adjoint to scalar extension − ⊗K L and thus defined by an
identification

HomL Y ⊗K L, X = HomK Y, RL/K (X)which is natural for schemes X (resp. Y ) quasi-projective over L (resp.
K), see [BLR90] VII.6. If X has dimension d then RL/K (X) has dimensiond[L : K].
Q
Let K ′ /K be another field extension and L′ = L ⊗K K ′ = α L′α thedecomposition of the tensor product in separable K ′ extensions. Then thefollowing Mackey-formula holds
Y
RL/K (X) ⊗K K ′ =RL′α /K ′ X ⊗L L′α .
α

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JAKOB STIX

In particular, with an algebraic closure K alg of K we haveY
RL/K (X) ⊗K K alg =X ⊗L,σ K alg ,σwhere the product is over all K-embeddings σ : L → K alg . If K alg contains
L a priori, then we have a prefered σ1 = id which gives a projectionpr1 : RL/K (X) ⊗K K alg → X ⊗L K alg
onto the corresponding factor.
3.2. The fundamental group of the Weil restriction. From now on
we work with fields of characteristic 0 or ask the variety X/L to be projective. Under these conditions the fundamental group satisfies a Künneth
formula [SGA1] Exp X Cor 1.7, Exp XII Cor 5.2, i.e, the projections yield
an isomorphism
Y

 ∼ Y

(3.1) π1 RL/K (X)⊗K K alg = π1
X⊗L,σ K alg −→π1 X⊗L,σ K alg .
σσIn the extension π1 RL/K (X)/K , the outer action by conjugation withlifts of g ∈ GalK on (3.1) acts as π1 (1 ⊗ g −1 ) and therefore permutes thefactors by mapping σ ∈ HomK (L, K alg ) = GalK / GalL to g −1 σ. Reindexing
the product with α = σ −1 transforms this to the right translation action on
GalL \ GalK . Thus we have established the following proposition.
Proposition 16. The extension π1 RL/K (X)/K is a wreath product typeextension with respect to the subgroup GalL ⊆ GalK .
In the identification of the index set of the product with GalL \ GalK wehave chosen a distinguished embedding of L in K alg . Evaluation at 1 thenis nothing but the π1 (pr1 ) for
pr1 : RL/K (X) ⊗K L → X
which is the adjoint map for the pair of adjoint functors RL/K and − ⊗K L
and an L-form of the map pr1 from above. Consequently we find
sh2 (π1 RL/K (X)/K = π1 (X/L)
and the following structure theorem for the fundamental group of a Weil
restriction of scalars holds.
Theorem 17. The fundamental group π1 (RL/K X/K) is isomorphic to theK
non-abelian induction IndGal
GalL π1 (X/L).

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43

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Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288,
69120 Heidelberg
e-mail address: stix@mathi.uni-heidelberg.de
URL: http://www.mathi.uni-heidelberg.de/∼stix/(Received February 16, 2009 )(Revised July 12, 2009 )