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Galois theory

Contents

Idea

Classical Galois theory classifies field extensions. It is a special case of a classification of locally constant sheaves in a topos by permutations representations? of the fundamental groupoid/fundamental group.

Even more generally one can define a Galois group associated to a presentable symmetric monoidal stable (infinity,1)-category. There is an analogue of the Galois correspondence in this setting, see Mathew 14.

Classical Galois theory for fields

We discuss the classical/traditional case of Galois theory, which concerns the classification of field extensions. Below in Galois theory for schemes and then in Galois theory in a topos we discuss how this is a special case of a more general concept of Galois theory in a topos.

Galois theory of fields

Definition

We call a field extension KLK \subset L a Galois extension if KLK \subset L is algebraic and there exists a subgroup GAut(L)G \subset Aut(L) of the automorphism group such that KL GK \simeq L^G is the field of elements that are invariant under GG.

Definition

If KLK \subset L is a Galois extension, we define the Galois group to be

Gal(L/K):=Aut K(L). Gal(L/K) := Aut_K(L) \,.

This means that we have

KL Gal(L/K). K \simeq L^{Gal(L/K)} \,.
Definition

Let K¯\bar K be a fixed algebraic closure of KK. If FK[X]{0}F \subset K[X] - \{0\} is any collection of non-zero polynomials, the splitting field of FF over KK is the subfield of K¯\bar K generated by KK and the zeros of the polynomials in FF.

We call fK[X]{0}f \in K[X]- \{0\} separable if it has no multiple zero in K¯\bar K.

We call αK¯\alpha \in \bar K separable over KK if the irreducible polynomial f K αf^\alpha_K of α\alpha over KK is separable.

A subfield LKL \subset K is called separable over KK if each αL\alpha \in L is separable over KK.

Definition

We call LL normal over KK if for each αL\alpha \in L the polynomial f K αf^\alpha_K splits completely in linear factors in L[X]L[X].

Theorem

Let KK be a field and LL a subfield of K¯\bar K

KLK¯. K \subset L \subset \bar K \,.

Denote by II the set of subfields EE of LL for which EE is a finite Galois extension of KK. Then II, when partially ordered by inclusion is a directed poset.

The following assertions are equivalent:

  1. LL is a Galois extension of KK.

  2. LL is normal and separable over KK.

  3. There is a set FK[X]{0}F \subset K[X] - \{0\} of separable polynomials such that LL is the splitting field of FF over KK.

  4. EIEL\coprod_{E \in I} E \simeq L

If these conditions are satisfied, then there is a group isomorphism

Gal(L/K)lim EIGal(E/K), Gal(L/K) \simeq {\lim_{\leftarrow}}_{E \in I} Gal(E/K) \,,

where on the right we have the limit over the poset of subfield of the contravariant functor EGal(E/K)E \mapsto Gal(E/K).

Since each group Gal(E/K)Gal(E/K) is finite, the above isomorphism can be used to equip Gal(L/K)Gal(L/K) with a profinite topology (i.e. take the limit in the category of topological groups, where each Gal(E/K)Gal(E/K) has the discrete topology), making it into a profinite group. We henceforth consider Gal(L/K)Gal(L/K) as a profinite group in this way.

Theorem

(main theorem of classical Galois theory)

Let KLK \subset L be a Galois extension of fields with Galois group GG. Then the intermediate fields of KLK \subset L correspond bijectively to the closed subgroups of GG.

More precisely, the maps

{E|EisasubfieldofLcontainingK}ψϕ{H|HisaclosedsubgroupofG} \{E | E\;is\;a\;subfield\;of\;L\;containing\;K\} \stackrel{\overset{\phi}{\to}}{\underset{\psi}{\leftarrow}} \{H|H\;is\;a\;closed\;subgroup\;of\;G\}

defined by

ϕ(E)=Aut E(L) \phi(E) = Aut_E(L)

and

ψ(H)=L H \psi(H) = L^H

are bijective and inverse to each other. This correspondence reverses the inclusion relation: KK corresponds to GG and LL to {id L}\{id_L\}.

If EE corresponds to HH, then we have

  1. KEK \subset E is finite precisely if HH is open (in the profinite topology on GG)

    [E:K]index[G:K][E:K] \simeq index[G:K] if HH is open;

  2. ELE \subset L is Galois with Gal(L/E)HGal(L/E) \simeq H (as topological groups);

  3. for every σG\sigma \in G we have that σ[E]\sigma[E] corresponds to σHσ 1\sigma H \sigma^{-1};

  4. LEL \subset E is Galois precisely if HH is a normal subgroup of GG;

    Gal(E/K)G/HGal(E/K) \simeq G/H (as topological groups) if KEK \subset E is Galois.

This appears for instance as Lenstra, theorem 2.3.

This suggests that more fundamental than the subgroups of a Galois group GG are its quotients by subgroups, which can be identified with transitive GG-sets. This naturally raises the question of what corresponds to non-transitive GG-sets.

In terms of separable algebras

Free modules

Let AA be a commutative ring and NN a module over AA.

A collection of elements (w i) iI(w_i)_{i \in I} of NN is called a basis of NN (over AA) if for every xNx \in N there is a unique collection (a i) iI(a_i)_{i \in I} of elements of AA such that a i=0a_i = 0 for all but finitely many iIi \in I and x= iIa iw ix = \sum_{i \in I} a_i w_i.

If NN has a basis it is called free (over AA). If NN is free with basis a finite set of cardinality nn, then we say that NN is free with rank nn (over AA). In this case, NN is a finitely generated free module.

Let NN be a finitely generated free AA-module with basis w 1,w 2,,w nw_1, w_2, \cdots, w_n and let f:NNf\colon N \to N be AA-linear. Then

f(w i)= j=1 na ijw j(1in) f(w_i) = \sum_{j = 1}^n a_{i j} w_j \;\;\; (1 \leq i \leq n)

for certain a ijAa_{i j} \in A, and the trace Tr(f)Tr(f) of ff is defined by

Tr(f)= i=1 na ii. Tr(f) = \sum_{i = 1}^n a_{i i} \,.

This is an element of AA that only depends on ff, and not on the choice of basis. It is easily checked that the map Tr:Hom A(N,N)ATr : Hom_A(N,N) \to A is AA-linear.

Separable algebras

Let AA be a ring, BB an AA-algebra, and suppose that BB is free with finite rank nn as an AA-module. For every bBb \in B the map mult b:BBmult_b\colon B \to B defined by mult b:xbxmult_b\colon x \mapsto b x is AA-linear, and the trace Tr(b)Tr(b) or Tr B/A(b)Tr_{B/A}(b) is defined to be Tr(mult b)Tr(mult_b). The map Tr:BATr\colon B \to A is easily seen to be AA-linear and to satisfy Tr(a)=naTr(a) = n a for aAa \in A.

The AA-module Hom A(B,A)Hom_A(B,A) (underlying which is the hom-set in the category of modules) is clearly free over AA with the same rank as BB. Define the AA-linear map ϕ:BHom A(B,A)\phi\colon B \to Hom_A(B,A) by

ϕ(x):yTr(xy), \phi(x) : y \mapsto Tr(x y) \,,

for x,yBx, y \in B.

Definition

If for an AA-algebra BB the the morphism ϕ\phi is an isomorphism we call BB separable over AA, or a free separable AA-algebra if we wish to stress the condition that BB is finitely generated and free as an AA-module.

Separable closure

Recall the notion of separable elements

Definition

Let KK be a field and K¯\bar K an algebraic closure of KK. The separable closure K SK_S of KK is defined by

K S{xK¯|xisseparableoverK}. K_S \simeq \{x \in \bar K | x \; is \; separable \; over \; K\} \,.
Remark

We have that K SK_S is a subfield of K¯\bar K and that K SK¯K_S \simeq \bar K precisely if KK is a perfect field, in particular if the characteristic of KK is 0.

From xyz it follows that the inclusion KK SK \subset K_S is Galois.

Definition

The Galois group Gal(K S/K)Gal(K_S/K) is called the absolute Galois group of KK.

Galois theory for separable algebras

Theorem

Let KK be a field and π 1(SpecK)\pi_1(Spec K) its absolute Galois group. Then there is an equivalence of categories

SAlg K opπ 1(SpecK)Set. SAlg_K^{op} \simeq \pi_1(Spec K) Set \,.

Galois theory for schemes

The classical Galois theory for fields is a special case of a general geometric/topos theoretic statement about locally constant sheaves and the action of the fundamental group on their fibers.

Statement of the main theorem

Definition

A morphism f:YXf : Y \to X of schemes is a finite étale morphism if there exists a covering of XX by affine open subsets U i=SpecA iU_i = Spec A_i, such that

  • for each ii the open subschemes f 1(U i)f^{-1}(U_i) of YY is affine,

  • and equal to SpecB iSpec B_i, where B iB_i is a free separable A iA_i-algebra.

In this situation we also say that f:YXf : Y \to X is a finite étale covering of XX.

A morphism from a finite étale covering f:YXf : Y \to X to a finite étale covering g:ZXg : Z \to X is a morphism of schemes h:YZh : Y \to Z such that f=ghf = g \circ h.

This defines the category FEt XFEt_X of finite étale covers of XX.

Theorem

Let XX be a connected scheme. Then there exists a profinite group π 1(X)\pi_1(X) – the fundamental group of XX – uniquely determined up to isomorphism, such that the category of finite étale coverings FEexFEex is equivalent to the category Finπ 1(X)SetFin \pi_1(X) Set of finite permutation representations of π 1(X)\pi_1(X) (finite sets, with the discrete topology, on which π 1(X)\pi_1(X) acts continuously).

This appears for instance as Lenstra, main theorem 1.11. It is fully discussed in SGA1.

The profinite group, π 1(X)\pi_1(X), is often called the étale fundamental group of the connected scheme XX. In SGA1, Grothendieck also considers coverings with profinite fibres, and a profinitely enriched fundamental groupoid. In the above the actual group π 1(X)\pi_1(X) depends on the choice of a fibre functor given by a geometric point of XX. Different choices of fibre functor produce isomorphic groups. Taking two such fibre functors yields a π 1(X)\pi_1(X)-torsor for either version of π 1(X)\pi_1(X). This is important in attacks on Grothendieck's section conjecture.

Examples
  • The disjoint union of nn copies of XX corresponds, under this theorem, to a finite set of nn elements on which π 1(X)\pi_1(X) acts trivially.

  • The fact that for X=SpecX = Spec \mathbb{Z} there are no other finite étale coverings of XX is thus expressed by the group π 1(Spec)\pi_1(Spec \mathbb{Z}) being trivial .

  • The same is true for π 1(SpecK)\pi_1(Spec K), where KK is an algebraically closed field.

  • If KK is an arbitrary field, then π 1(SpecK)\pi_1(Spec K) is the absolute Galois group of KK; i.e. the Galois group of the separable closure K SK_S over KK. In this case theorem 4 is a reformulation of classical Galois theory.

  • In particular, if KK is a finite field, then π 1(SpecK)^\pi_1(Spec K) \simeq \hat \mathbb{Z}.

  • Let X=SpecAX = Spec A, where AA is the ring of integers in an algebraic number field KK. Let NN be the maximal algebraic extension of NN that is unramified at all non-zero prime ideals of AA. Then π 1(X)\pi_1(X) is the Galois group of NN over KK.

  • More generally, if aAa \in A, a0a \neq 0, then π 1(SpecA[1a])\pi_1(Spec A[\frac{1}{a}]) is the Galois group, over KK, of the maximal algebraic extension of KK that is unramified at all non-zero prime ideals of AA not dividing aa.

Reproducing classical Galois theory of field extensions

In this section we explain the connection between the main theorem of Galois theory for schemes, theorem 4, and classical Galois theory.

We denote by kk a field. It is our purpose to show that the opposite category of the category of free separable KK-algebras is equivalent to the category of finite π 1(X)\pi_1(X)-sets, for a certain profinite group π 1(X)\pi_1(X). This is a special case of the main theorem 4, with X=SpecK X = Spec K. In the general proof we shall use the contents of this section only for algebraically closed KK. In that case, which is much simpler, the group π 1(X)\pi_1(X) is trivial, so that the category of finite π 1(X)\pi_1(X)-sets is just the category of finite sets.

Theorem

(…)

Grothendieck’s Galois Theory

In SGA1, Grothendieck introduced an abstract formulation of the above theory in terms of Galois categories. A Galois category is a category, 𝒞\mathcal{C}, satisfying a small number of properties together with a fibre functor F:𝒞FinSetF: \mathcal{C}\to FinSet, preserving those properties. The theory is more fully described in the entry on Grothendieck's Galois theory.

Galois theory in topos theory

One notices that classical Galois theory has an equivalent reformulation in topos theory. That puts it into a wider general abstract context and leads to a topos-theoretic general Galois theory.

Reformulation of classical Galois theory

Observation

The étale morphisms f:YXf : Y \to X corresponds precisely to the locally constant sheaves on XX with respect to the etale topology, in that it is equivalently a morphism for which there is an etale cover {U iX}\{U_i \to X\} such that ff is a constant sheaf on each U iU_i.

For KK a field let Et(K)Et(K) be its small étale site. And

:=Sh(Et(K)) \mathcal{E} := Sh(Et(K))

the sheaf topos over it. This topos is a

Then Galois extensions of KK correspond precisely to the locally constant objects in \mathcal{E}. The full subcategory on them is the Galois topos Gal()Gal(\mathcal{E}) \hookrightarrow \mathcal{E}.

The Galois group is the fundamental group of the topos.

Topos-theoretic Galois theory

Accordingly in topos theory Galois theory is generally about the classification of locally constant sheaves. The Galois group corresponds to the fundamental group of the topos .

(…)

Higher topos theoretic Galois theory

In the context of higher topos theory, there are accordingly higher analogs of Galois theory.

For aspects see

‘La Longue Marche à travers la Théorie de Galois’

Between January and June 1981, Grothendieck wrote about 1600 manuscript pages of a work with the above title. The subject is the absolute Galois group, Gal(¯,)Gal(\overline{\mathbb{Q}},\mathbb{Q}) of the rational numbers and its geometric action on moduli spaces of Riemann surfaces. This will (one day be) discussed at Long March. Other entries that relate to this include anabelian geometry, children's drawings (in other words Dessins d’enfants, which is the study of graphs embedded on surfaces, their classification and the link between this and Riemann surfaces) and the Grothendieck-Teichmuller group.

The anabelian question is: how much information about the isomorphism class of an algebraic variety, XX is contained in the étale fundamental group of XX? Grothendieck calls varieties which are completely determined by their étale fundamental group, anabelian varieties. His anabelian dream was to classify the anabelian varieties in all dimensions over all fields. This can be seen to relate to questions of the étale homotopy types of varieties.

Tim: I have a feeling that this anabelian question should have a form that generalises to higher dimensions. (Not that I can shed much light on progress in dimension one.) Perhaps there is an anabelian version of the homotopy hypothesis or something of that nature.

(…)

References

Lecture notes on the Galois theory for schemes are in

  • Hendrik Lenstra, Galois theory for schemes , Mathematisch Instituut Universiteit van Amsterdam (1985) (pdf)

Some of the material above is taken from this.

A comprehensive textbook is

The locally simply connected case is discussed for instance in

  • Marco Robalo, Galois Theory towards Dessins d’Enfants (pdf)

Galois theory in a presentable symmetric monoidal stable (infinity,1)-category is developed in

See also

category: Galois theory

Revised on June 16, 2014 03:51:47 by Urs Schreiber (88.128.80.80)