- Home
- Documents
*Galois representations in fundamental groups and their Lie ...swc.math. the absolute Galois group GK*

If you can't read please download the document

View

0Download

0

Embed Size (px)

Galois representations in fundamental groups and

their Lie algebras

Makoto Matsumoto∗

March 9, 2005

Contents

1 Algebraic fundamental group 2 1.1 The classic-topological fundamental group . . . . . . . . . . . 2 1.2 Unramified covering . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Galois groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Galois category . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Finite etale coverings . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Comparison theorem . . . . . . . . . . . . . . . . . . . . . . . 11

2 Algebraic fundamental group as Galois groups 12 2.1 Rephrase by Galois theory . . . . . . . . . . . . . . . . . . . . 12 2.2 Analytic continuation . . . . . . . . . . . . . . . . . . . . . . 13

3 Galois representation on fundamental groups, as monodromy 14

4 Computation of Galois actions 16 4.1 Taking a section . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Case of P10,1,∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Lie algebraization 23 5.1 Group rings (discrete case) . . . . . . . . . . . . . . . . . . . 23 5.2 Group rings (pro-� case) . . . . . . . . . . . . . . . . . . . . . 24 5.3 Lie Algebraization of Galois representation . . . . . . . . . . 25

∗Dept. Math. Hiroshima Univ. m-mat@math.sci.hiroshima-u.ac.jp

1

6 Soulé’s cocycle 25

Abstract

Let X be a geometrically connected scheme over a field K. Then, the absolute Galois group GK of K acts on the algebraic fundamental group πalg1 (X ⊗ K, x).

This lecture explains the following:

1. This action is an analogy of “geometric monodromy of deforma- tion family” in the topology.

2. Lie algebraization of the fundamental group is effective to extract some information from this action.

1 Algebraic fundamental group

1.1 The classic-topological fundamental group

Let X be an arcwise connected topological space, and x be a point on X . Then, the (classical) fundamental group of X with base point x is defined by

π1(X , x) := {paths from x to x}/homotopy with x fixed. For the ordering in composing two paths, we define γ ◦ γ′ to be the path first going along γ′ and then γ. (Some papers adopt the converse ordering.)

It is well-known that the fundamental group of a (real two-dimensional) sphere or a sphere minus one point is trivial, and that of the sphere minus two points is isomorphic to the additive group Z. The fundamental group of a sphere minus three points is a free group of two generators, and this is a main subject of this lecture.

1.2 Unramified covering

The fundamental group is an important (homotopy) invariant of a topolog- ical space. The importance may be justified by the following theorem.

Theorem 1.1. Let X be an arcwise connected and locally simply connected topological scheme, and x be a point on it. Then, there is an equivalence of categories

Fx : {unramified coverings of X} → {π1(X , x)-sets},

given by taking the inverse image of x.

2

We shall give precise definitions of the above two categories.

Definition 1.2. A continuous map f : Y → X is an unramified covering, if for every x ∈ X there is an open neighborhood U ⊂ X of x such that every connected component of f−1(U) is isomorphic to U through f .

Morphisms between f : Y → X , f ′ : Y ′ → X are defined to be the continuous maps from h : Y → Y ′ satisfying f = f ′ ◦ h. (By abuse of language, we simply say that h is compatible with f .)

An important example of unramified covering of X is its universal cov- ering X̃ . This is constructed as follows: take a point x ∈ X . Then consider the set

X̃ := {γ : path on X starting from x}/homotopy where the homotopy is considered with the both ends of the path fixed.

3

Consider the morphism

p : X̃ → X , [γ] �→ the end point of γ.

Then, X̃ is equipped with a topology as follows. Take [γ] ∈ X̃ , and set z := p([γ]) to be the end point of γ. Take a simply connected open neighborhood U of z. For each point u of U , there is a unique path (zu) (modulo homotopy) connecting z to u in U . By mapping u �→ [(zu) ◦ γ], we have a mapping U → X . Let Ũ denote its image. We can check that by taking Ũ as an open neighborhood of [γ], X̃ is equipped with a topology such that p is an unramified covering. The fundamental group π1(X , x) acts on X̃ (over X ) from right, by

X̃ × π1(X , x) → X̃ , ([γ], [β]) �→ [γ ◦ β].

We define the right hand side of Theorem 1.1.

Definition 1.3. For a group G, a G-set means a set S with a left action of G is specified, i.e.,

ρ : G → Aut(S) is given. A morphism between two G-sets S1 → S2 is a mapping compatible with G-actions. (I.e. ρ1(g)(h(s)) = h(ρ2(g)(s)) holds.)

There is an obvious functor from the left category to the right one in Theorem 1.1:

Fx : (f : Y → X ) �→ f−1(x). The action of π1(X , x) on f−1(x) is given by the monodromy: for γ ∈ π1(X , x) and z ∈ f−1(x), there is a unique lift γ′ in Y of γ starting from z. Then γ(z) is defined as the endpoint of γ′. 1 Clearly this gives a left action of π1(X , x) on f−1(x). Thus, we defined

Fx : {unramified coverings of X} → {π1(X , x)-sets}, (f : Y → X ) �→ f−1(x).

1To be precise, we take open sets Uλ (λ ∈ Λ) of X whose union contains γ (here a path and its homotopy class are both denoted as γ), and each Uλ satisfies the property in Definition 1.2. By compactness, we may assume that U1, U2, . . . , Un cover γ and U1 ∩ γ, U2 ∩ γ, . . . , Un ∩ γ are paths composable in this order. Then, there is a unique lift of U1 ∩ γ in Y starting from z ∈ f−1(x) ⊂ Y. Then, there is a unique lift of U2 ∩ γ starting from the end point of the previous lift. Thus, there is a unique lift γ′ of γ in Y starting from z.

4

In the converse direction, we can construct an unramified covering of X from a π1(X , x)-set S as follows. For each orbit O ⊂ S of the action of π1(X , x), we take a point o ∈ O and consider YO := X̃/Go where Go is the stabilizer of o in π1(X , x). It follows from the construction that YO → X corresponds to the transitive π1(X , x)-set O. By taking direct sum over all orbits O, we have the desired covereing of X . From the above, it is seen that the subcategory of the connected covering corresponds to the subcategory of the transitive π1(X , x)-sets.

We may define the fundamental group without using paths, in the fol- lowing manner. Consider the functor

Fx : {unramified coverings of X} → {sets}, (πY : Y → X ) �→ π−1Y (x).

This is called a fiber functor. Any path γ ∈ π1(X,x) acts on Fx(Y) = π−1Y (x) by the monodromy. This action is compatible with any unramified maps Y → Y ′, i.e.,

Fx(Y) γ→ Fx(Y) ↓ ↓

Fx(X ) γ→ Fx(X ) commutes. This means that the action of γ is a natural (invertible) trans- formation from the functor Fx to itself.

This amounts to saying that we have a group homomorphism

π1(X,x) → Aut(Fx).

5

This is proved to be an isomorphism. It is injective since Fx(X̃ ) is one to one with π1(X,x). It is surjective since any element σ of Aut(Fx) is the image of the path ∈ π1(X,x) that lifts to a path x̃ → σ(x̃) in X̃ .

This is the way to define the algebraic fundamental group in SGA1[3]: we consider a suitable analogue to the unramified coverings and fiber func- tors. Then, the algebraic fundamental group is defined as the automorphism group of a fiber functor.

The suitable analogue to the unramified coverings in the algebraic situ- ation is étale.

1.3 Galois groups

Before stating the definition of étale morphisms, we treat another concrete example: the absolute Galois groups. Here, we assume the reader to have the basic knowledge on (finite) Galois theory of field extension.

Let K be a field. A good analogy of a finite connected unramified cov- ering is a finite field extension L of K. To be precise, we consider the following category (SpecK)et: its object is a ring R with injective homo- morphism K → R, where R is a finite direct product of finite separable extension field of K:

K → R = ∏

i

Li,

and morphisms are those ring homomorphisms R → R′ compatible with K → R.

To compare with the unramified coverings, it is better to reverse the direction of the morphism. So, to each ring (commutative with unit) R, we associate an object SpecR (the spectrum of R) and to each ring homomor- phism R → R′ we associate a morphism SpecR′ → SpecR. Now,

SpecR = Spec( ∏

i

Li) → SpecK

is the analogue of a finite unramifed covering. We denote by

(SpecK)et := category of SpecR as above,

where the morphisms are just ring homomorphisms R → R′ compatible with K → R, K → R′. Why one can say that this is a good analogue? Because the following category equivalence holds:

SpecKet ∼= {GK = π1(SpecK,x)-finite sets}, (1)

6

which we shall explain soon below (but π1(SpecK,x) in the next section, in Example 1.5).

Take an algebraically closed field Ω with injective homomorphism

K → Ω, in other words, x : SpecΩ → SpecK. This is called a geometric point of SpecK. Let Ksep ⊂ Ω be the separable closure of K in Ω. Then, SpecKsep → SpecK surves as the universal covering of SpecK, with a fiber x̃ : SpecΩ → SpecKsep above