Introduction to Galois Cohomology with Applications to Local Fields

A. Hayward

Spring 2001

Contents

0.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.1.1 Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . 5 0.1.2 Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.1.3 The Category of R-modules . . . . . . . . . . . . . . . . . . . 6 0.1.4 Abelianisation of Groups . . . . . . . . . . . . . . . . . . . . 8

1 Cohomology of Groups 9

1.1 Basic Definitions and Results . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 G-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 The Fixed-Point Functor . . . . . . . . . . . . . . . . . . . . 9 1.1.3 Some G-module Structures . . . . . . . . . . . . . . . . . . . 10 1.1.4 The Cohomology Groups of a G-module . . . . . . . . . . . . 11 1.1.5 Uniqueness of the Cohomology Groups . . . . . . . . . . . . . 12 1.1.6 Cohomology of Co-induced Modules . . . . . . . . . . . . . . 14

1.2 The Standard Resolution . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 The Inhomogeneous Cochain Complex . . . . . . . . . . . . . 16 1.2.2 Explicit Descriptions of the Cochain Complex . . . . . . . . . 17

1.3 Homology of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.1 The Standard Resolution . . . . . . . . . . . . . . . . . . . . 21 1.3.2 The Group H_{1}(G, ℤ) . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Change of Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4.1 The Restriction-Inflation Sequence . . . . . . . . . . . . . . . 24 1.5 Cohomology of Finite Groups . . . . . . . . . . . . . . . . . . . . . . 24

1.5.1 The Tate Groups . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5.2 Dimension Shifting of Tate Cohomology Groups . . . . . . . 28 1.5.3 Complete Resolutions . . . . . . . . . . . . . . . . . . . . . . 30 1.5.4 The Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5.5 Restriction and Corestriction . . . . . . . . . . . . . . . . . . 33

1.6 Tate’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Galois Cohomology 38 2.1 Finite Galois Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Infinite Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2.1 The Krull Topology . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.2 Open and Closed Subgroups . . . . . . . . . . . . . . . . . . . 40 2.2.3 Projective Limits . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.4 The Galois Group as a Projective Limit . . . . . . . . . . . . 42 2.2.5 The Fundamental Theorem . . . . . . . . . . . . . . . . . . . 43

2.3 Infinite Galois Cohomology . . . . . . . . . . . . . . . . . . . . . . . 44 2.3.1 Direct Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3.2 Properties of Direct Limits . . . . . . . . . . . . . . . . . . . 45 2.3.3 Cohomology of Profinite Groups . . . . . . . . . . . . . . . . 45 2.3.4 Continuous Cochains . . . . . . . . . . . . . . . . . . . . . . . 46 2.3.5 The Restriction-Inflation Sequence . . . . . . . . . . . . . . . 49

2.4 Infinite Galois Cohomology . . . . . . . . . . . . . . . . . . . . . . . 50 2.5 Functoriality of Galois Cohomology Groups . . . . . . . . . . . . . . 51

3 The Brauer Group 52 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 Characterization of Central Simple Algebras . . . . . . . . . . . . . . 52 3.3 Structure of the Classical Brauer Group . . . . . . . . . . . . . . . . 53 3.4 Relating the Brauer Group to a One-Dimensional Cohomology Set . 54 3.5 Brauer Groups as Two-Dimensional Cohomology Groups . . . . . . . 57

4 Local Class Field Theory 60 4.1 The Restriction-Inflation Sequence . . . . . . . . . . . . . . . . . . . 60 4.2 Local Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3 Extensions of Local Fields . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4 Unramified Extensions and Brauer Groups . . . . . . . . . . . . . . . 63 4.5 Quasi-Algebraically Closed Fields . . . . . . . . . . . . . . . . . . . . 65 4.6 Finite Fields are Quasi-Algebraically Closed . . . . . . . . . . . . . . 66 4.7 Unit Group of an Unramified Extension . . . . . . . . . . . . . . . . 67 4.8 Witt’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.9 The Group ℤ̂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.10 The Invariant Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.11 Properties of the Invariant Map . . . . . . . . . . . . . . . . . . . . . 73 4.12 The Reciprocity Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A Homology of Complexes 1 A.1 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A.2 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A.3 Morphisms of Complexes . . . . . . . . . . . . . . . . . . . . . . . . 2 A.4 The Connecting Homomorphism . . . . . . . . . . . . . . . . . . . . 3 A.5 The Exact Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 B Non-Abelian Cohomology of Groups 7

B.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 B.2 The Zero-Dimensional Connecting Morphism . . . . . . . . . . . . . 8 B.3 The One-Dimensional Connecting Morphism . . . . . . . . . . . . . 9 B.4 The Exact Cohomology Sequence . . . . . . . . . . . . . . . . . . . . 11

C Galois Theory 14 C.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 C.2 The Normal Basis Theorem . . . . . . . . . . . . . . . . . . . . . . . 16

D Topological Groups 19

E Central Simple Algebras 21 E.1 Simple Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 E.2 Structure of Simple Algebras . . . . . . . . . . . . . . . . . . . . . . 24 E.3 Matrix Rings and Tensor Products . . . . . . . . . . . . . . . . . . . 27 E.4 Extension of Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 E.5 Simple Algebras and Opposite Rings . . . . . . . . . . . . . . . . . . 30 E.6 Finite Dimensional Neutralising Fields . . . . . . . . . . . . . . . . . 32 E.7 Norms on Finite Dimensional Algebras . . . . . . . . . . . . . . . . . 33

E.7.1 The Absolute Norm . . . . . . . . . . . . . . . . . . . . . . . 33 E.7.2 The Reduced Norm . . . . . . . . . . . . . . . . . . . . . . . 34

Introduction

We consider the problem of investigating the Galois group of a field extension. In general, this is a very difficult problem. There are, however, certain fields, related very closely to the fields we really want to study (number fields and function fields), for which there is a theory which helps us in this task. This “Local Class Field Theory” is the subject of this report and tells us very explicitly about the abelianisation (maximal abelian quotient) of the Galois group. More precisely, our aim is to establish a homomorphism, called the Artin reciprocity map,

ψ : K^{∗} ⟶ Gal_{K}^{ab}.

Here K^{∗} is the multiplicative group of a local field and Gal_{K}^{ab} the maximal abelian quotient of the absolute Galois group of K. This map ψ is “almost” an isomorphism, in that it is bijective on finite quotients of the groups.

The definition of this function is far from trivial. We need a very powerful tool to compare these apparently dissimilar objects. That tool will be the cohomology of groups, and most of the text will be taken up with explaining it. Group cohomology plays an increasingly important role in Number Theory, although a number of Number Theory texts try to exclude it entirely because its exposition is necessarily a lengthy diversion. This approach is less than satisfactory, because it is not so clear and more technical. I hope that the present text gives a reasonable presentation of this deserving theory.

The structure of the presentation roughly follows that of the second half of the book [1] by Serre, which is the main reference for much of the material here. The presentation of cohomology theory comes from the expositions by Serre in [1] and Atiyah and Wall [3]. Other references may be found in the text and the bibliography.

This report contains nothing original, but I hope that I have managed in parts to give a clearer explanation than can be found in the standard literature.

My supervisor for this project was Dr. Mohamed Saidi, to whom I extend my thanks.

0.1 Preliminaries

0.1.1 Rings and Modules

I assume the reader is familiar with rings and fields. If G is a group, the integral group ring ℤ[G] is the set of formal finite ℤ-linear combinations of elements of G, ∑_{g∈G} n_{g}g. Multiplication is defined linearly by the multiplication in G. For ∑_{g∈G} n_{g}g ∈ ℤ[G], ∑_{g∈G} n_{g} ∈ ℤ is called its augmentation. Augmentation is a homomorphism ℤ[G] ⟶ ℤ.

If R is a ring (with a multiplicative identity 1 ∈ R), a (left-) R-module M is an abelian group (usually written additively) and a map

\begin{matrix}R × M & ⟶ & M \\ (r, m) & ⟼ & rm \end{matrix}

called scalar multiplication, satisfying

1. (rs)m = r(sm)

2. r(m + n) = rm + rn

3. 1m = m for all r, s ∈ R, m, n ∈ M.

A right R-module is defined similarly except we require (rs)m = s(rm) instead of the first property above. Then we usually write (r, m) ⟼ mr. If R is a field, a left R-module is called an R-vector space. If R = ℤ, a left R-module is an abelian group.

If M, N are R-modules, an R-homomorphism θ : M ⟶ N is a group homomorphism such that θ(rm) = rθ(m) for all r ∈ R, m ∈ M. We write Hom_{R}(M, N) for the additive group of R-homomorphisms M ⟶ N.

We define submodules and quotient modules in the obvious way. We have that ker θ is a submodule of M and im θ is a submodule of N. As for abelian groups, im θ ≅ M/ ker θ as R-modules. We define coker θ = N/im θ, the cokernel.

As usual, a bijective R-homomorphism M ⟶ M is called an R-automorphism of M. The group of all such maps under composition is written Aut_{R}(M).

An R-module M is called free if there is a subset B ⊆ M, called a basis, such that for any R-module N and any function f : B ⟶ N there is an R-homomorphism from M to N which restricts to f.

0.1.2 Category Theory

Category theory provides a useful language for discussing cohomology. A category C consists of a class of “objects” Obj C and, for each pair A, B ∈ Obj C, a set Mor(A, B) called the morphisms A ⟶ B.

In addition, for ϕ ∈ Mor(A, B), θ ∈ Mor(B, C) there is defined a composite θ ∘ ϕ ∈ Mor(A, C). This law must satisfy 1. For each A, Mor(A, A) ∋ id_{A} which is an element such that for all ϕ ∈

Mor(A, B), ψ ∈ Mor(B, A), we have ϕ ∘ id_{A} = ϕ and id_{A} ∘ ψ = ψ.

2. ∘ must be associative.

If in addition to this each Mor(A, B) is an additive group, and composition distributes over addition, then C is called a pre-additive category.

If ϕ ∈ Mor(A, B) is such that there exists ψ ∈ Mor(B, A) with ϕ ∘ ψ = id_{B} and ψ ∘ ϕ = id_{A} then ϕ is called an isomorphism. An isomorphism in Mor(A, A) is called an automorphism of A.

If C, D are categories, then a (covariant) functor F : C ⟶ D is a function F : Obj C ⟶ Obj D and, for each pair A, B ∈ Obj C, a function F : Mor(A, B) ⟶ Mor(F A, F B), such that F (ϕ ∘ ψ) = F ϕ ∘ F ψ. A contravariant functor is defined similarly except we have F : Mor(A, B) ⟶ Mor(F B, F A) such that F (ϕ ∘ ψ) = F ψ ∘ F ϕ.

If F, G : C ⟶ D are functors then they are called naturally equivalent if in each Mor(F A, GA) in D there is an isomorphism ι_{A} such that for any ϕ : A ⟶ B in C, ι_{B} ∘ F ϕ = Gϕ ∘ ι_{A}.

0.1.3 The Category of R-modules

If R is a ring, the class of all R-modules and the R-homomorphisms between them form a pre-additive category.

If f : M ⟶ N, g : N ⟶ P are R-homomorphisms between R-modules, we call the diagram

M ⟶^{f} N ⟶^{g} P  (1)

a sequence of R-modules. Longer “sequences” may be made up from parts like this. When ker g = im f, (1) is called an exact sequence. A longer sequence is called exact if each subsequence (1) is exact. A short exact sequence is a sequence of the form

0 ⟶ A ⟶^{i} B ⟶^{p} C ⟶ 0  (2)

where 0 denotes the trivial R-module. Then A is (isomorphic to) a submodule of B and C ≅ B/A.

(2) is called split if there is a morphism j : C ⟶ B such that p ∘ j = id_{C}. This is the case if and only if there is a morphism q : B ⟶ A such that q ∘ i = id_{A}. Equivalently, B ≅ A × C as R-modules, and then i, j are the inclusions of A and C into B, q, p the projections on the components. Note that for any subring S of R, A, B, C are S-modules, and (2) may be S-split but not R-split.

If F : C ⟶ D is a covariant functor, F is called left-exact if exactness of 0 → A → B → C implies exactness of 0 → F A → F B → F C. It is called rightexact if exactness of A → B → C → 0 implies exactness of F A → F B → F C → 0. If F is both left- and right-exact it is called exact. Then F takes any exact sequence to a corresponding exact sequence.

If F : C ⟶ D is contravariant, F is called left-exact if exactness of A → B → C → 0 implies exactness of 0 → F C → F B → F A. Given this, the definition of right-exact and exact contravariant functors is obvious.

Note that any functor preserves exactness of split short exact sequences.

Theorem 0.1.1 For any R-module M, Hom_{R}(?, M) is a contravariant left-exact functor^{1} and Hom_{R}(M,?) is a covariant left-exact functor.

For the proof, see [7].

If for an R-module M, Hom_{R}(?, M) is exact, M is called (R-)injective. If Hom_{R}(M,?) is exact, M is called (R-)projective. Free modules are both projective and injective.

If A is a right R-module and B is a left R-module, we define the tensor product over R of A and B, written A ⊗_{R} B, to be the quotient of the group of formal finite ℤ-linear combinations of elements of A × B by the subgroup generated by all elements of the form

(a + a′, b) − (a, b) − (a′, b)
(a, b + b′) − (a, b) − (a, b′)
(ar, b) − (a, rb)  for all a, a′ ∈ A, b, b′ ∈ B, r ∈ R

The class of (a, b) is written a ⊗ b and these generate the abelian group A ⊗_{R} B.

Theorem 0.1.2 (Universal Mapping Property of Tensor Products) If X is an abelian group and f : A × B ⟶ X any group homomorphism such that f(ar, b) = f(a, rb) for all a ∈ A, b ∈ B, r ∈ R, then there is a unique map f̄: A ⊗_{R} B ⟶ X such that f̄(a ⊗ b) = f(a, b) for all (a, b) ∈ A × B.

If R is commutative, A⊗_{R} B is an R-module. When R is a field, A⊗_{R} B is an Rvector space, and if U_{A} is a basis of A, U_{B} a basis of B, then {u⊗v : u ∈ U_{A}, v ∈ U_{b}} is a basis of A ⊗_{R} B.

For any left R-module M, ? ⊗_{R} M is a covariant right-exact functor. For any right R-module M, M⊗_{R}? is again a covariant right-exact functor.

For more details on tensor products, see Bourbaki’s Algèbre or [7].

A diagram is a system of objects of a category placed in Euclidean space and morphisms between them. A diagram is said to commute if any chain of morphisms from one point to another composes to give the same morphism. For example the diagram

\begin{matrix}A & ⟶^{ϕ} & B \\ ↓ψ & & ↓τ \\ C & ⟶^{θ} & D \end{matrix}

commutes if and only if θ ∘ ψ = τ ∘ ϕ. Then we may draw a □ in the middle.

1from the category of R-modules to the category of abelian groups

Proposition 0.1.3 (Snake Lemma) A commutative diagram with exact rows

A ⟶ B ⟶^{f} C ⟶ 0
↓α □ ↓β □ ↓γ
0 ⟶ D ⟶^{g} E ⟶ F

defines an exact sequence

ker α ⟶ ker β ⟶ ker γ ⟶^{δ} coker α ⟶ coker β ⟶ coker γ,

where δ := g^{−1} ∘ β ∘ f^{−1}.

The proof is basic diagram-chasing.

0.1.4 Abelianisation of Groups

If G is a group, its derived subgroup G′ is the subgroup generated by the commutators [g, h] for g, h ∈ G. By definition, [g, h] = ghg^{−1}h^{−1}.

G′ is a normal subgroup of G. G/G′ is the maximal abelian quotient of G. It is also called the abelianisation G^{ab} of G.

If H is any subgroup of G such that G/H is abelian, then G′ ⊆ H, for

gg′H = g′gH ⇒ g^{−1}g′^{−1}gg′ ∈ H′.

One final definition used in the text: almost all means all but a finite number.

Chapter 1

Cohomology of Groups

[The reader who is not familiar with homological algebra is referred to Appendix A for the “general nonsense” underlying this material.]

1.1 Basic Definitions and Results

1.1.1 G-modules

Let G be a group, either finite or infinite (although we will only use the finite case here). Let A be an abelian group, and suppose we have a homomorphism

G ⟶ Aut(A).

Then we say that A is an abelian (left-)G-module. What this means is that for each g ∈ G there is an automorphism a ⟼ ga in such a way that (gh)a = g(ha) for all g, h ∈ G, a ∈ A and g(a + a′) = ga + ga′ for all g ∈ G, a, a′ ∈ A. In this section we will write simple “G-module”.

Consider the integral group ring ℤ[G]. If A is a G-module, putting

\begin{matrix}ℤ[G] × A & ⟶ & A \\ (∑_{g∈G} n_{g}g, a) & ⟼ & ∑_{g∈G} n_{g}ga \end{matrix}

defines a ℤ[G]-module structure on A, extending the G-module structure linearly. On the other hand, a ℤ[G]-module structure on A is necessarily a G-module structure on A, the action of G on A being given by g : a ⟼ ga. Thus an abelian G-module is the same as a ℤ[G]-module. ℤ[G]-homomorphisms will also be called G-homomorphisms. If the homomorphism G ⟶ Aut(A) is trivial, G is said to act trivially on A.

1.1.2 The Fixed-Point Functor

Write A^{G} for the set of elements of A fixed by every g ∈ G; it is a ℤ[G]-submodule of A. A ⟼ A^{G} is a covariant functor on the category of G-modules and Ghomomorphisms.

Proposition 1.1.1 A ⟼ A^{G} is naturally equivalent to the functor A ⟼ Hom_{G}(ℤ, A) where G acts trivially on ℤ.

PROOF - An element a ∈ A^{G} can be made to correspond to a map

\begin{aligned}ϕ_{a} : & ℤ & ⟶ & A \\ & 1 & ⟼ & a \end{aligned}

because for such a map ga = gϕ_{a}(1) = ϕ_{a}(g1) = ϕ_{a}(1) = a. Then a homomorphism A^{G} ⟶ B^{G} induces a homomorphism Hom_{G}(ℤ, A) ⟶ Hom_{G}(ℤ, B). Natural equivalence is easy.

□

We know Hom_{G}(ℤ,?) is left-exact, so ^{G} is left-exact. That is, given an exact sequence, say

0 ⟶ A ⟶ B ⟶ C ⟶ 0

of G-modules, the sequence

0 ⟶ A^{G} ⟶ B^{G} ⟶ C^{G}  (1.1)

is exact. We are going to use the cohomology theory developed in Appendix A to continue this sequence to the right.

1.1.3 Some G-module Structures

In this section A, B will denote G-modules (for the same G, of course). X will denote an additive abelian group. X can be given the structure of a G-module by making G act trivially on X (ie. gx = x for all g ∈ G, x ∈ X).

Consider first the group Hom_{ℤ}(A, B) of group homomorphisms A ⟶ B.

Definition 1.1.2 The standard G-module structure on Hom_{ℤ}(A, B) is given by

(gϕ)(a) = g(ϕ(g^{−1}a)) for all g ∈ G, ϕ ∈ Hom_{ℤ}(A, B), a ∈ A

This can be checked to define a G-module structure. Notice that with this structure, Hom_{ℤ}(A, B)^{G} = Hom_{G}(A, B).

We must always be careful about the G-module structure of the arguments A, B. For example, when we talk about co-induced modules, we show how to embed any G-module A in Hom_{ℤ}(ℤ[G], A_{+}). Here A_{+} denotes A considered as a trivial G-module (abelian group).

Now consider the tensor product A ⊗_{ℤ} B.

Definition 1.1.3 The standard G-module structure on A ⊗_{ℤ} B is given by

g(a ⊗ b) = ga ⊗ gb for all g ∈ G, a ∈ A, b ∈ b

Again we must be careful when factors are to be considered with trivial G-action. If A is a left G-module then A is a right ℤ[G]-module under the action as := s^{−1}a

for each a ∈ A, s ∈ G. This allows us to define for each pair of G-modules A, B the tensor product over G

A ⊗_{G} B := A ⊗_{ℤ[G]} B.

1.1.4 The Cohomology Groups of a G-module

The first step is to choose a G-free resolution of ℤ. By this we mean an acyclic complex

P : … ⟶^{d_{2}} P_{2} ⟶^{d_{1}} P_{1} ⟶^{d_{0}} P_{0} ⟶^{ϵ} ℤ → 0.

So the d_{i} are G-homomorphisms and the sequence is exact. We will exhibit two such resolutions in the next section, so the reader may trust that they exist.

Now let A be a G-module we want to study, and apply the contravariant functor Hom_{G}(?, A) to the “deleted complex” P_{ℤ}

… ⟶^{d_{2}} P_{2} ⟶^{d_{1}} P_{1} ⟶^{d_{0}} P_{0} → 0

to obtain a cochain complex Hom_{G}(P_{ℤ}, A) =: K(A) of abelian groups

0 → Hom_{G}(P_{0}, A) →^{d_{0}} Hom_{G}(P_{1}, A) →^{d_{1}} Hom_{G}(P_{2}, A) →^{d_{2}} …

so K^{q}(A) = Hom_{G}(P_{q}, A) for q ≥ 0, K^{q}(A) = 0 for q < 0, and d_{q} acts on a cochain f ∈ Hom_{G}(P_{q}, A) by (d_{q}f)(x) := f(d_{q}(x)).

The cohomology groups of G with coefficients in A are now defined as the cohomology groups of this complex

H^{q}(G, A) := H^{q}(K(A))

Thus an element of the q-th cohomology group H^{q}(G, A) is an equivalence class of G-homomorphisms P_{q} ⟶ A. When we want to show a property of one of these elements, we will usually work with a representative G-homomorphism, and then show that our conclusion does not depend on which representative we choose. The groups H^{q}(G, A) for q > 0 are called the higher cohomology groups of G with coefficients in A.

The lowest cohomology group H^{0}(G, A) has a special meaning: it is (naturally equivalent to) A^{G}. This is because the sequence

P_{1} ⟶^{d_{0}} P_{0} ⟶^{ϵ} ℤ ⟶ 0

is exact, so applying the contravariant left-exact functor Hom_{G}(?, A) gives an exact sequence of abelian groups

0 ⟶ Hom_{G}(ℤ, A) ⟶ Hom_{G}(P_{0}, A) ⟶^{d_{0}} Hom_{G}(P_{1}, A)

so that ker d_{0} is naturally equivalent to Hom_{G}(ℤ, A), which in turn is naturally equivalent to A^{G} (by Proposition 1.1.1). So H^{0}(K(A)) = ker d_{0}/im d_{−1} = ker d_{0}/0 = A^{G}.

Suppose now we are given a short exact sequence of G-modules

0 ⟶ A ⟶^{i} B ⟶^{p} C ⟶ 0.  (1.2)

We apply for each q ∈ ℤ the covariant functor Hom_{G}(P_{q},?), which is exact because P_{q} is G-free and hence G-projective. We obtain the commutative diagram

0  0  0
↓  ↓  ↓
0 ⟶ Hom_{G}(P_{0}, A) ⟶^{i^{∗}} Hom_{G}(P_{0}, B) ⟶^{p^{∗}} Hom_{G}(P_{0}, C) ⟶ 0
d_{0}↓  □  d_{0}↓  □  d_{0}↓
0 ⟶ Hom_{G}(P_{1}, A) ⟶^{i^{∗}} Hom_{G}(P_{1}, B) ⟶^{p^{∗}} Hom_{G}(P_{1}, C) ⟶ 0
d_{1}↓  □  d_{1}↓  □  d_{1}↓
⋮  ⋮  ⋮

with exact rows and columns, in other words an exact sequence of complexes

0 ⟶ K(A) ⟶ K(B) ⟶ K(C) ⟶ 0.

We know from Appendix A that this gives rise to the long exact cohomology sequence, which in our new notation is

0 ⟶ H^{0}(G, A) ⟶^{i^{∗}} H^{0}(G, B) ⟶^{p^{∗}} H^{0}(G, C)
⟶^{δ} H^{1}(G, A) ⟶^{i^{∗}} H^{1}(G, B) ⟶^{p^{∗}} …  (1.3)

We have obtained the promised continuation of (1.1). This continuation is in a sense unique, as will soon be shown.

1.1.5 Uniqueness of the Cohomology Groups

Proposition 1.1.4 If A is G-injective then H^{q}(G, A) = 0 for q ≥ 1.

PROOF - recall that the condition means that Hom_{G}(?, A) is an exact functor. This means that it takes the exact sequence

… ⟶^{d_{1}} P_{1} ⟶^{d_{0}} P_{0} ⟶^{ϵ} ℤ ⟶ 0

to an exact sequence

0 ⟶ Hom_{G}(ℤ, A) ⟶^{ϵ} Hom_{G}(P_{0}, A) ⟶^{d_{0}} Hom_{G}(P_{1}, A) ⟶^{d_{1}} …

so that ker d_{q} = im d_{q−1} for q ≥ 1.

□

We need:

Theorem 1.1.5 For any ring R, every R-module is a submodule of an injective R-module.

For the proof, see [7], §3, theorem 3.26.

We can now establish

Proposition 1.1.6 The cohomological functor {H, δ} is characterised by the properties

• H^{0}(G, A) = A^{G}

• H^{q}(G,?) is a covariant functor for each q

• δ gives a long exact sequence for each short exact sequence 0 ⟶ A ⟶ B ⟶ C ⟶ 0

• If A is an injective G-module, H^{q}(G, A) = 0 for q > 0.

PROOF - We have seen that the functor {H, δ} defined as above has these properties. On the other hand, suppose {H, δ} is a functor with these properties, then by the theorem above there is, for a given A, an exact sequence

0 ⟶ A ⟶^{i} A^{∗} ⟶^{p} A^{∗}/A ⟶ 0

with A^{∗} injective; the long exact cohomology sequence then gives an exact sequence

0 → H^{0}(G, A) →^{i^{∗}} H^{0}(G, A^{∗}) →^{p^{∗}} H^{0}(G, A^{∗}/A) →^{δ} H^{1}(G, A) →^{i^{∗}} H^{1}(G, A^{∗})

But H^{1}(G, A^{∗}) = 0 so δ induces an isomorphism

\frac{H^{0}(G, A^{∗}/A)}{im p^{∗}} ⟶ H^{1}(G, A)

Which defines H^{1}(G, A) in terms of H^{0}. Similarly for q > 1, the long exact cohomology sequence gives an exact sequence

0 ⟶ H^{q−1}(G, A^{∗}/A) ⟶^{δ} H^{q}(G, A) ⟶ 0

so that H^{q−1}(G, A^{∗}/A) ≅ H^{q}(G, A). In this way H^{q} is always determined by H^{q−1} up to natural equivalence, and as we know H^{0}, the functor {H, δ} must be naturally equivalent to the one defined above.

□

Thus we see in particular that the cohomology groups and connecting homomorphisms as defined above do not depend on which free G-resolution of ℤ is chosen.

The trick used in the proof is called “dimension-shifting”: we use a short exact sequence with a module in the middle which has zero cohomology groups to relate a q-dimensional cohomology group to a (q − 1)-dimensional one.

1.1.6 Cohomology of Co-induced Modules

The condition of injectivity is excessively strong. We will show that a weaker condition suffices for the higher cohomology groups to be zero.

Definition 1.1.7 A G-module is co-induced if it is isomorphic to a module Hom_{ℤ}(ℤ[G], X), where X is an abelian group (see Definition 1.1.2). A G-module is called relatively injective if it is a direct factor of a co-induced module.

Proposition 1.1.8 A G-module A embeds in the co-induced G-module Hom_{ℤ}(ℤ, A_{+}), where A_{+} indicates the additive group of A. (cf. Theorem 1.1.5)

PROOF - The required monomorphism is

A ⟶ Hom_{ℤ}(ℤ[G], A_{+})
a ⟼  ϕ_{a} : ℤ[G] ⟶ A_{+}
g ⟼ g^{−1}a

the inverse ϕ ⟼ ϕ(1_{G}) showing injectivity.

□

It can be shown that injective modules are relatively injective.

Lemma 1.1.9 (Additivity of Cohomology Functors) Let A ⊕ B denote the direct sum of two G-modules A and B. Then H^{q}(G, A⊕B) = H^{q}(G, A)⊕H^{q}(G, B).

PROOF - Take cohomology of the split short exact sequence

0 ⟶ A ⟵^{⟶} A ⊕ B ⟵^{⟶} B ⟶ 0

noting that then the connecting homomorphisms are all the zero map.

□

We can now show the weaker condition, with the help of the following lemma.

Lemma 1.1.10 A free G-resolution of ℤ admits ℤ-homomorphisms

\begin{matrix}h_{0} : ℤ ⟶ P_{0} \\ h_{1} : P_{0} ⟶ P_{1} & etc. \end{matrix}

so that h_{n} ∘ d_{n−1} + d_{n} ∘ h_{n+1} is the identity map.

\begin{aligned}… & ⟶^{d_{2}} & P_{2} & ⟶^{d_{1}} & P_{1} & ⟶^{d_{0}} & P_{0} & ⟶^{ϵ} & ℤ & ⟶ & 0 \\ & ↖⧸_{h_{3}} & & ↖⧸_{h_{2}} & & ↖⧸_{h_{1}} & & ↖⧸_{h_{0}} \end{aligned}

This is proved by defining the maps inductively, using the fact that each P_{q} is a free ℤ-module.

The maps h_{n} form a “contracting homotopy” of ℤ-homomorphisms on the resolution.

Proposition 1.1.11 If A is relatively injective, H^{q}(G, A) = 0 for q > 0.

PROOF - By the lemma on additivity, if we show that co-induced modules have zero cohomology, their direct factors must also. So let A = Hom_{ℤ}(ℤ[G], X).

Then the functors on the category of G-modules, Hom_{G}(?, A) and Hom_{ℤ}(?, X) are naturally equivalent, the isomorphism for any G-module M being

\begin{matrix}Hom_{G}(M, A) & ⟶ & Hom_{ℤ}(M, X) \\ ϕ & ⟼ & (m ↦ ϕ(m)(1_{G})) \end{matrix}

Applying Hom_{G}(?, A) to the free resolution of ℤ now yields a sequence

0 ⟶ Hom_{ℤ}(ℤ, X) ⟶^{ϵ} Hom_{ℤ}(P_{0}, X) ⟶^{d_{0}} Hom_{ℤ}(P_{1}, X) ⟶^{d_{1}} …

We use the last lemma to show that this sequence is exact, and so the cohomology of A is zero. We have d_{q+1} ∘d_{q}, so im d_{q} ⊆ ker d_{q+1}. If ϕ : P_{q+1} ⟶ X is in ker d_{q+1}, so that ϕ ∘ d_{q+1} = 0, put ψ = ϕ ∘ h_{q+1} : P_{q} ⟶ X. Then

ψ(d_{q}(y)) = ϕ(h_{q+1} ∘ d_{q}(y))
= ϕ((h_{q+1} ∘ d_{q} + d_{q+1} ∘ h_{q+2})(y))
= ϕ(y)

so that ϕ ∈ im d_{q}.

□

All this works because ℤ is a free ℤ-module - note that it is not a free ℤ[G]- module, for nontrivial G.

1.2 The Standard Resolution

Recall (§1.1, Proposition 1.1.6) that any free G-resolution of ℤ will do to obtain the cohomology groups of G with coefficients in A. In practice, there is one which is particularly useful.

Define Q_{q} for q ≥ 0 to be the free ℤ-module on G^{q+1}, made into a G-module via the diagonal action

s(g_{0}, …, g_{q}) := (sg_{0}, …, sg_{q})

this is a free G-module with basis eg. {(1, g_{1}, …, g_{q}) : g_{j} ∈ G}. The boundary formula is given, for each ℤ-generator of Q_{q}, by

d_{q−1}(g_{0}, …, g_{q}) = ∑_{i=0}^{q}(−1)^{i}(g_{0}, …, g_{i−1}, g_{i+1}, …, g_{q})

which defines a G-homomorphism Q_{q} ⟶ Q_{q−1}. Note that d_{−q} : Q_{0} ⟶ ℤ is just the “augmentation map”

\begin{matrix}ℤ[G] & ⟶ & ℤ \\ ∑_{g∈G} n_{g}g & ⟼ & ∑_{g∈G} n_{g} \end{matrix}

Proposition 1.2.1 The sequence of G-modules

… ⟶^{d_{2}} Q_{2} ⟶^{d_{1}} Q_{1} ⟶^{d_{0}} Q_{0} ⟶^{d_{−1}} ℤ ⟶ 0

is exact.

PROOF - It is easy to check that d_{q−1} ∘ d_{q} = 0, and so im d_{q} ⊆ ker d_{q−1}. To show the reverse inclusion, we construct a contracting homotopy on the complex. Define

\begin{aligned}h_{i} : & P_{i−1} & ⟶ & P_{i} \\ & (g_{0}, …, g_{i−1}) & ⟼ & (1, g_{0}, …, g_{i−1}) \end{aligned}

Then it is easy to check that h_{q} ∘ d_{q−1} + d_{q} ∘ h_{q+1} = id_{Q_{q}}, so if x ∈ ker d_{q−1} then

x = (h_{q} ∘ d_{q−1} + d_{q} ∘ h_{q+1})(x) = d_{q}(h_{q+1}(x)) ∈ im d_{q}
□

Thus this resolution of ℤ gives the cochain complex

0 ⟶ Hom_{G}(Q_{0}, A) ⟶^{d_{0}} Hom_{G}(Q_{1}, A) ⟶^{d_{1}} Hom_{G}(Q_{2}, A) ⟶^{d_{2}} …

called the homogeneous cochain complex. A q-dimensional homogeneous cochain is then a G-homomorphism

ℤ[G^{q+1}] ⟶ A

As such a function is clearly defined by its values on the ℤ-basis G^{q+1}, we may say that a q-dimensional homogeneous cochain is a function

f : G^{q+1} ⟶ A

such that s(f(g_{0}, …, g_{q})) = f(sg_{0}, …, sg_{q})

1.2.1 The Inhomogeneous Cochain Complex

This is not a particularly efficient definition because one of the g_{i} is superfluous, by the G-homomorphism definition; for example, f(g_{0}, …, g_{q}) = g_{0}f(1, g_{0}^{−1}g_{1}, …, g_{0}^{−1}g_{q}) This leads us to rewrite the definition as follows. Let P_{q} be the free ℤ[G]-module on the basis G^{q}, so that P_{q} is the set of formal sums of products of an element s of G and a q-tuple^{1} [g_{1}, …, g_{q}] ∈ G^{q}.

Define G-homomorphisms

\begin{matrix}Q_{q} & ⟶ & P_{q} \\ (g_{0}, …, g_{q}) & ⟼ & g_{0}[g_{0}^{−1}g_{1}, g_{1}^{−1}g_{2}, …, g_{q−1}^{−1}g_{q}] \\ P_{q} & ⟶ & Q_{q} \\ [g_{1}, …, g_{q}] & ⟼ & (1, g_{1}, g_{1}g_{2}, …, g_{1}g_{2} … g_{q}) \end{matrix}

1We will always write square brackets when dealing with the P_{q}.

It is easy to check that these are mutually inverse.

These G-isomorphisms, along with the boundary formula for homogeneous cochains, yield a boundary homomorphism

P_{q+1} ⟶ Q_{q+1} ⟶^{d_{q}} Q_{q} ⟶ P_{q}

given by the formula

d_{q}[g_{1}, …, g_{q+1}] = g_{1}[g_{2}, …, g_{q+1}]

+ ∑_{j=1}^{q}(−1)^{j}[g_{1}, …, g_{j}g_{j+1}, …, g_{q+1}]

+(−1)^{q+1}[g_{1}, …, g_{q}]  (1.4)

So now we have a free G-resolution of ℤ

… ⟶^{d_{2}} P_{2} ⟶^{d_{1}} P_{1} ⟶^{d_{0}} P_{0} ⟶ ℤ

and the cochain complex obtained is

0 ⟶ Hom_{G}(P_{0}, A) ⟶^{d_{0}} Hom_{G}(P_{1}, A) ⟶^{d_{1}} Hom_{G}(P_{2}, A) ⟶^{d_{2}} …

called the inhomogeneous cochain complex. A q-dimensional inhomogeneous cochain is then a G-homomorphism P_{q} ⟶ A or, what amounts to the same, any function

f : G^{q} ⟶ A

The coboundary of f is df : G^{q+1} ⟶ A given by

df[g_{1}, …, g_{q+1}] = g_{1}f[g_{2}, …, g_{q+1}]

+ ∑_{j=1}^{q}(−1)^{j}f[g_{1}, …, g_{j}g_{j+1}, …, g_{q+1}]

+(−1)^{q+1}f[g_{1}, …, g_{q}]  (1.5)

This is the “bare hands” definition of cohomology which is most useful in practice. Although we always know the cohomology of a G-module exists and is unique, actually calculating the groups is in general very difficult. That is why it is useful to have this more concrete representation.

1.2.2 Explicit Descriptions of the Cochain Complex

Let’s examine more closely the inhomogeneous cochain complex. A q-cocycle is a function f : G^{q} ⟶ A satisfying df = 0, ie.

g_{1}f[g_{2}, …, g_{q+1}] + ∑_{j=1}^{q}(−1)^{j}f[g_{1}, …, g_{j}g_{j+1}, …, g_{q+1}]

+(−1)^{q+1}f[g_{1}, …, g_{q}] = 0

for all [g_{1}, …, g_{q+1}] ∈ G^{q+1}. A q-coboundary is dh : G^{q} ⟶ A for some (q − 1)- cochain h. If

0 ⟶ A ⟶^{i} B ⟶^{p} C ⟶ 0

is a short exact sequence of G-modules, then the connecting homomorphism may be evaluated as follows. Given a q-cocycle f in C representing a cohomology class, there is a q-cochain h : G^{q} ⟶ B with p ∘ h = f, because p is surjective.

Hom_{G}(P_{q}, B) ⟶^{p^{∗}} Hom_{G}(P_{q}, C) ⟶ 0
↓ d
0 ⟶ Hom_{G}(P_{q+1}, A) ⟶^{i^{∗}} Hom_{G}(P_{q+1}, B)

Applying d gives dh, a (q + 1)-coboundary in B and hence a (q + 1)-cocycle in B. But p^{∗} ∘ d = d ∘ p^{∗}, since p^{∗} is a morphism of complexes, so p ∘ dh = 0 and so dh takes values in ker p = im i = i(A). Thus i^{−1} ∘ dh is a (q + 1)-cocycle in A, whose cohomology class is by definition δ_{q}f.

Let us now look at some low-dimensional examples.

q = 0: We know H^{0}(G, A) ≅ A^{G}. Let us see how this fits with the inhomogeneous cochains. A 0-cocycle is a function

f : G^{0} = {1} ⟶ A

ie. an element of A, with df = 0. Since

df[g_{1}] = g_{1}f − f

This condition says that f ∈ A satisfies f = g_{1}f for all g_{1} ∈ G, so that f is an element of A^{G}.

q = 1: Now the coboundary formula takes a 1-cochain f : G ⟶ A to

\begin{aligned}df : & G^{2} & ⟶ & A \\ & [g_{1}, g_{2}] & ⟼ & g_{1}f[g_{2}] − f[g_{1}g_{2}] + f[g_{1}] \end{aligned}

so that a 1-cocycle is a function f : G ⟶ A with

f[g_{1}g_{2}] = f[g_{1}] + g_{1}f[g_{2}].

Such a function is called a crossed homomorphism. f is a 1-coboundary if there is a ∈ A such that

f[g_{1}] = g_{1}a − a.

From this we see that when G acts trivially on A, H^{1}(G, A) = Hom_{ℤ}(G, A). q = 2: This time a 2-cocycle is a function f : G^{2} ⟶ A satisfying

g_{1}f[g_{2}, g_{3}] − f[g_{1}g_{2}, g_{3}] + f[g_{1}, g_{2}g_{3}] − f[g_{1}, g_{2}] = 0

Such a function is called a factor set. Factor sets arise in the problem of group extensions (see [7] §5). f is a 2-coboundary if there is a function h : G ⟶ A with

f[g_{1}, g_{2}] = g_{1}h[g_{2}] − h[g_{1}g_{2}] + h[g_{1}]

1.3 Homology of Groups

There is a dual theory to the cohomology theory introduced above. Starting with a free G-resolution of ℤ,

… ⟶^{d_{2}} P_{2} ⟶^{d_{1}} P_{1} ⟶^{d_{0}} P_{0} ⟶^{ϵ} ℤ → 0

remove ℤ and apply the covariant functor ?⊗_{ℤ[G]}A, which is, in a sense, the “dual” to Hom_{G}(?, A). We obtain a chain complex

… ⟶^{d_{2}⊗1_{A}} P_{2} ⊗_{ℤ[G]} A ⟶^{d_{1}⊗1_{A}} P_{1} ⊗_{ℤ[G]} A ⟶^{d_{0}⊗1_{A}} P_{0} ⊗_{ℤ[G]} A ⟶ 0

where 1_{A} = id_{A}, the identity function on A. The q-th homology group of G with coefficients in A is now

H_{q}(G, A) := \frac{ker(d_{q−1} ⊗ 1)}{im (d_{q} ⊗ 1)}

a quotient of a subgroup of the additive group P_{q} ⊗_{ℤ[G]} A. Similarly to before, H_{0} has a special meaning. Applying the right-exact functor ? ⊗_{ℤ[G]} A to the exact sequence

P_{1} ⟶^{d_{0}} P_{0} ⟶ ℤ ⟶ 0

gives an exact sequence

P_{1} ⊗_{ℤ[G]} A ⟶^{d_{0}⊗1_{A}} P_{0} ⊗_{ℤ[G]} A ⟶ ℤ ⊗_{ℤ[G]} A ⟶ 0

so that

\frac{P_{0} ⊗_{ℤ[G]} A}{im (d_{0} ⊗ 1)} ≅ ℤ ⊗_{ℤ[G]} A

and we have only to identify ℤ⊗_{ℤ[G]}A. Let I_{G} be the “augmentation ideal” of ℤ[G], that is, the additive subgroup generated by all ∑_{g∈G} n_{g}g in ℤ[G] with augmentation ∑_{g∈G} n_{g} = 0. If we project an element of I_{G}A onto ℤ ⊗_{ℤ[G]} A then it maps to an element of ℤI_{G} ⊗_{ℤ[G]} A. But the action of G on ℤ is trivial, so ℤI_{G} = 0. Thus I_{G} projects to zero. In fact:

Proposition 1.3.1 We have the isomorphism of abelian groups

ℤ ⊗_{ℤ[G]} A ≅ \frac{A}{I_{G}A}

PROOF - The correspondence

ℤ ⊗_{ℤ[G]} A ⟷ \frac{A}{I_{G}A}
n ⊗ a ⟷ na + I_{G}A

gives well-defined inverse homomorphisms.

□

A/I_{G}A is written A_{G}, and is the largest quotient group of A on which G acts trivially^{2}. H_{q}(G, A) is the unique homological extension of the functor A ⟼ A_{G}, and is characterised by

• H_{0}(G, A) = A_{G}

• H_{q}(G,?) is a covariant functor for each q

• δ (see Appendix A) gives a long exact sequence

… ⟶ H_{1}(G, C) ⟶^{δ} H_{0}(G, A) ⟶ H_{0}(G, B)
⟶ H_{0}(G, C) ⟶ 0  (1.6)

for each short exact sequence 0 ⟶ A ⟶ B ⟶ C ⟶ 0

• If A is a projective G-module, H_{q}(G, A) = 0 for q > 0.

The proof is similar to that of the corresponding result in cohomology. Again, the last condition can be weakened.

A G-module A is called induced if it has the form ℤ[G] ⊗_{ℤ} X, where X can be any additive group, the G-action being as defined in §1.1.3. A is called relatively projective if it is a direct factor of such a module. Relatively projective modules also have all their higher homology groups zero. The proofs are similar to those for the corresponding results in cohomology.

We know that every module is a quotient of a free, and hence projective, module. Moreover, the surjective G-homomorphism

\begin{matrix}ℤ[G] ⊗_{ℤ} A_{+} & ⟶ & A \\ s ⊗ a & ⟼ & sa \end{matrix}

shows that every module is canonically a quotient of an induced module. Thus for any A there is always an exact sequence

0 ⟶ A′ ⟶ A_{∗} ⟶ A ⟶ 0

with A_{∗} induced, which allows dimension-shifting.

2Compare A^{G}, which was the largest subgroup on which G acts trivially.

1.3.1 The Standard Resolution

We can use our standard resolution to give a description of these groups as groups of homology classes of inhomogeneous chains. Recall that P_{q} is the free ℤ[G]-module on G^{q}. The properties of tensor products show that any (q + 1)-dimensional chain can be uniquely written

x = ∑[g_{1}, …, g_{q+1}] ⊗ a_{[g_{1},...,g_{q+1}]}

where the summation is over all (q + 1)-tuples [g_{1}, …, g_{q+1}] ∈ G^{q+1}, and only finitely many of the a are nonzero. So x is effectively a function

\begin{aligned}f : & G^{q+1} & ⟶ & A \\ & [g_{1}, …, g_{q+1}] & ⟼ & a_{[g_{1},...,g_{q+1}]} \end{aligned}

with almost all its values zero (meaning all but a finite number).

We can use the boundary formula d : P_{q+1} ⟶ P_{q} (1.4) to give a formula for the boundary

\begin{aligned}d : & P_{q+1} ⊗_{ℤ[G]} A & ⟶ & P_{q} ⊗_{ℤ[G]} A \\ & [g_{1}, …, g_{q+1}] ⊗ a_{[g_{1},...,g_{q+1}]} & ⟼ & (d[g_{1}, …, g_{q+1}]) ⊗ a_{[g_{1},...,g_{q+1}]} \end{aligned}

and the result is

df[g_{1}, …, g_{q}] = ∑_{g∈G} g^{−1}f[g, g_{1}, …, g_{q}]

+ ∑_{j=1}^{q}(−1)^{j} ∑_{g∈G} f[g_{1}, …, g_{j−1}, g_{j}g, g^{−1}, g_{j+1}, … g_{q}]

+(−1)^{q+1} ∑_{g∈G} f[g_{1}, …, g_{q}, g]

Note that a G-homomorphism ϕ : A ⟶ B induces ϕ : H_{q}(G, A) ⟶ H_{q}(G, B) by ϕ(f) = ϕ ∘ f.

1.3.2 The Group H_{1}(G, ℤ)

Our final task in this section is to identify the group H_{1}(G, ℤ), where G acts trivially on ℤ. This group will be of central importance later.

Consider the exact sequence

0 ⟶ I_{G} ⟶^{i} ℤ[G] ⟶^{ϵ} ℤ ⟶ 0

where

\begin{aligned}ϵ : & ℤ[G] & ⟶ & ℤ \\ & ∑_{g∈G} n_{g}g & ⟼ & ∑_{g∈g} n_{g} \end{aligned}

is the augmentation map and I_{G} its kernel. The exact homology sequence gives an exact sequence

H_{1}(G, ℤ[G]) ⟶^{ϵ_{∗}} H_{1}(G, ℤ) ⟶^{δ} H_{0}(G, I_{G}) ⟶^{i_{∗}} H_{0}(G, ℤ[G])

As ℤ[G] ≅ ℤ[G] ⊗_{ℤ} ℤ is induced, H_{1}(G, ℤ[G]) is zero. On the other hand,

\frac{I_{G}}{I_{G}I_{G}} ⟶^{i_{∗}} \frac{ℤ[G]}{I_{G}ℤ[G]}

is the zero map, so ker i_{∗} = im δ = H_{0}(G, I_{G}) = I_{G}/I_{G}^{2}. Thus δ is an isomorphism

H_{1}(G, ℤ) ⟶ I_{G}/I_{G}^{2}

We can further identify I_{G}/I_{G}^{2}. It is not hard to see that I_{G} is an abelian group generated by the set {g − 1 : g ∈ G}. Define a function ϕ from the multiplicative group G to the additive group I_{G}/I_{G}^{2} by ϕ(g) = (g − 1) + I_{G}^{2}. Then because

gh − 1 ≡ (g − 1)(h − 1) + (g − 1) + (h − 1)
≡ (g − 1) + (h − 1)  (mod I_{G}^{2})

ϕ is a homomorphism. ϕ is clearly surjective. If we can identify ker ϕ, we have found I_{G}/I_{G}^{2} as a quotient of G.

The key here is to note that G/ ker ϕ is abelian. Hence G′ ⊆ ker ϕ, where G′ is the derived subgroup of G. In fact, G′ = ker ϕ; to see the reverse inclusion, define

\begin{aligned}ψ : & I_{G}/I_{G}^{2} & ⟶ & G/G′ \\ & ∑ n_{g}(g − 1) + I_{G}^{2} & ⟼ & (∏ g^{n_{g}})G′ \end{aligned}

Let’s check ψ is well-defined: suppose x, x′ ∈ I_{G} represent the same class in I_{G}/I_{G}^{2}. Then

x′ = x + ∑_{i}(h_{i} − 1)(l_{i} − 1)  (for some h_{i}, l_{i} ∈ G)
= x + ∑_{i} [(h_{i}l_{i} − 1) − (h_{i} − 1) − (l_{i} − 1)]

and so

ψ(x′ + I_{G}^{2}) = ( ∏_{i} h_{i}l_{i}h_{i}^{−1}l_{i}^{−1})ψ(x + I_{G}^{2}) = ψ(x + I_{G}^{2})

since G/G′ is abelian. Clearly ψ is surjective.

Since G′ ⊆ ker ϕ, ϕ defines a homomorphism

ϕ̄: G/G′ ⟶ I_{G}/I_{G}^{2}

and clearly ϕ̄ ∘ ψ = id_{I_{G}/I_{G}^{2}}. The surjectivity of ψ then shows ϕ̄ is injective, so ker ϕ ⊆ G′.

So we obtain that H_{1}(G, ℤ) ≅ G/G′, so that we have a homology group equal to the abelianization of G.

1.4 Change of Group

Let G′ and G be groups, and

ϕ : G′ ⟶ G

a homomorphism. Denote the standard resolutions of ℤ by free G′- and G-modules (see §1.2) by

P(G′) and P(G)

respectively. Then ϕ induces a morphism of complexes

… → P_{2}(G′) → P_{1}(G′) → P_{0}(G′) → ℤ → 0
↓ϕ_{2} □ ↓ϕ_{1} □ ↓ϕ_{0} □ ∥
… → P_{2}(G) → P_{1}(G) → P_{0}(G) → ℤ → 0

by taking a typical ℤ-generator g′_{0}[g′_{1}, …, g′_{q}] of P_{q}(G′) to ϕ(g′_{0})[ϕ(g′_{1}), …, ϕ(g′_{q})].

Now suppose A is a G-module. Then A is also a G′ module via g′a := ϕ(g′)a for g′ ∈ G′, a ∈ A. Thus we can apply the contravariant functor Hom_{G}(?, A) to the induced maps ϕ_{q}:

Hom_{G}(P_{q}(G), A) ⟶ Hom_{G}(P_{q}(G′), A) ⊆ Hom_{G′}(P_{q}(G′), A)

where the last inclusion is a consequence of im ϕ ⊆ G. These together define a morphism of complexes^{3}

Hom_{G}(P_{ℤ}(G), A) ⟶ Hom_{G′}(P_{ℤ}(G′), A)

Hence it gives a morphism of cohomological functors

H^{q}(G, A) ⟶ H^{q}(G′, A)

A similar process can be done for homology, using the covariant functor ⊗_{ℤ[G]}A in place of Hom_{G}(?, A), giving morphisms H_{q}(G′, A) ⟶ H_{q}(G, A).

For the moment, we wish to define two morphisms:

Restriction: If H is a subgroup of G, and ϕ : H ↪ G is inclusion, then ϕ induces

Res_{G/H} : H^{q}(G, A) ⟶ H^{q}(H, A)

For a cochain f : G^{q} ⟶ A, Res f is simply the restriction

f|_{H^{q}} ⟶ A

In dimension 0, Res is the inclusion A^{G} ↪ A^{H}.

Inflation: If H is a normal subgroup of G, then A^{H} has the structure of a G/H module. Let ϕ : G ⟶ G/H be the quotient map, then ϕ induces a morphism

H^{q}(G/H, A^{H}) ⟶ H^{q}(G, A^{H})

3P_{ℤ} denoting the corresponding “deleted complex”, ie. with ℤ removed.

We know that the inclusion A^{H} ↪ A induces a morphism

H^{q}(G, A^{H}) ⟶ H^{q}(G, A)

So compose these to get

Inf : H^{q}(G/H, A^{H}) ⟶ H^{q}(G, A)

For a q-cocycle f : (G/H)^{q} ⟶ A^{H}, Inf f is the map G^{q} ⟶ A which takes [g_{1}, …, g_{q}] to f[g_{1}H, …, g_{q}H].

1.4.1 The Restriction-Inflation Sequence

Proposition 1.4.1 Let H be a normal subgroup of G, A a G-module. Then the sequence

0 ⟶ H^{1}(G/H, A^{H}) ⟶^{Inf} H^{1}(G, A) ⟶^{Res} H^{1}(H, A)

is exact.

Proposition 1.4.2 If H^{i}(H, A) = 0 for 1 ≤ i ≤ q − 1 then the sequence

0 ⟶ H^{q}(G/H, A^{H}) ⟶^{Inf} H^{q}(G, A) ⟶^{Res} H^{q}(H, A)

is exact

The first proposition is proved by direct verification on the inhomogeneous cochains. The second proposition follows from the first by dimension-shifting. For the proofs, see [1], Chapter 7, §6.

1.5 Cohomology of Finite Groups

1.5.1 The Tate Groups

If G is a group, then for an exact sequence of G-modules 0 ⟶ A ⟶ B ⟶ C ⟶ 0, we have defined the long exact homology and cohomology sequences:

\begin{aligned}… → & H_{1}(G, C) & →^{δ} & H_{0}(G, A) & → & H_{0}(G, B) & → H_{0}(G, C) & → & 0 \\ & 0 & → & H^{0}(G, A) & → & H^{0}(G, B) & → H^{0}(G, C) & →^{δ} & H^{1}(G, A) & → … \end{aligned}

When G is finite, these two sequences can be spliced together to form a doublyinfinite exact cohomology sequence. The tool for doing this is the element

N = ∑_{g∈G} g ∈ ℤ[G]

called the norm. Multiplication by N defines a G-homomorphism N : A ⟶ A^{G} because

Ng = gN = N for all g ∈ G

If I_{G} is the augmentation ideal of ℤ[G] (see §1.3), then NI_{G} = 0 because N(g−1) = N − N = 0 for each generator g − 1 of I_{G}. Hence NI_{G}A = 0, and N defines a G-homomorphism

N^{∗} : A/I_{G}A ⟶ A^{G}

ie. N^{∗} : H_{0}(G, A) ⟶ H^{0}(G, A).

Definition 1.5.1 Define^{4} Ĥ^{−1}(G, A) = ker N^{∗} and Ĥ^{0}(G, A) = cokerN^{∗}, in other words

\begin{aligned}Ĥ^{−1}(G, A) & = & ker N/I_{G}A & ⊆ & H_{0}(G, A) \\ Ĥ^{0}(G, A) & = & A^{G}/NA & = & H^{0}(G, A)/NA \end{aligned}

The next definition brings in the rest of the homology and cohomology groups:

Ĥ^{−q}(G, A) := H_{q−1}(G, A)  for q > 2
Ĥ^{q}(G, A) := H^{q}(G, A)  for q > 1

These groups Ĥ^{q}(G, A), q ∈ ℤ are called the Tate cohomology groups of G with coefficients in A.

Suppose now that

0 ⟶ A ⟶ B ⟶ C ⟶ 0

is an exact sequence of G-modules. Then the diagram

H_{0}(G, A) ⟶ H_{0}(G, B) ⟶ H_{0}(G, C) ⟶ 0
↓N_{A}^{∗} □ ↓N_{B}^{∗} □ ↓N_{C}^{∗}
0 ⟶ H^{0}(G, A) ⟶ H^{0}(G, B) ⟶ H^{0}(G, C)

is commutative and has exact rows. Thus the snake lemma (0.1.3) shows that it defines a homomorphism

δ : ker N_{C}^{∗} ⟶ coker N_{A}^{∗}

such that the sequence

ker N_{A}^{∗} ⟶ ker N_{B}^{∗} ⟶ ker N_{C}^{∗} ⟶^{δ} coker N_{A}^{∗}

⟶ coker N_{B}^{∗} ⟶ coker N_{C}^{∗}  (1.7)

is exact, ie.

\begin{matrix}Ĥ^{−1}(G, A) ⟶ Ĥ^{−1}(G, B) ⟶ Ĥ^{−1}(G, C) ⟶^{δ} Ĥ^{0}(G, A) \\ ⟶ Ĥ^{0}(G, B) ⟶ Ĥ^{0}(G, C) \end{matrix}

is exact.

4Ĥ^{−1} is also written Ĥ_{0}, and ker N is also written _{N}A

Theorem 1.5.2 Let G be a finite group and

0 ⟶ A ⟶ B ⟶ C ⟶ 0

an exact sequence of G-modules. Then the definitions of δ : Ĥ^{q−1}(G, C) ⟶ Ĥ^{q}(G, A) from the exact cohomology and homology sequences, and from (1.7) above, give a doubly-infinite exact sequence

… ⟶ Ĥ^{q−1}(G, C) ⟶^{δ} Ĥ^{q}(G, A) ⟶ Ĥ^{q}(G, B) ⟶ Ĥ^{q}(G, C) …

PROOF - In view of the exactness of the homology and cohomology sequences (1.6 and 1.3) we need only check that they fit together properly in the middle.

1. We need that the connecting homomorphism δ : H_{1}(G, C) ⟶ A/I_{G}A (in the notation of §1.3) is in fact a map into ker N_{A}/I_{G}A, ie. N_{A} ∘ δ = 0.

We represent an element of H_{1}(G, C) as a one-dimensional chain f : G ⟶ C (see §1.3). Calculate δf by lifting f to h : G ⟶ B and applying the boundary formula:

dh = ∑_{g∈G} g^{−1}f[g] − ∑_{g∈G} f[g] = ∑_{g∈G}(g^{−1} − 1)f[g]

Applying N ∈ ℤ[G] to this gives ∑_{g∈G}(N − N)f[g] = 0 so δf ∈ ker N_{A}^{∗}. This was what we needed to show, and now the exactness of

H_{1}(G, B) ⟶ H_{1}(G, C) ⟶^{δ} H_{0}(G, A) ⟶ H_{0}(G, B)

implies exactness of

Ĥ^{−2}(G, B) ⟶ Ĥ^{−2}(G, C) ⟶^{δ} Ĥ^{−1}(G, A) ⟶ Ĥ^{−1}(G, B)

2. We need exactness of

Ĥ^{0}(G, B) ⟶ Ĥ^{0}(G, C) ⟶^{δ} Ĥ^{1}(G, A) ⟶ Ĥ^{1}(G, B)

which follows as above if we can show that the connecting homomorphism in

H^{0}(G, B) ⟶ H^{0}(G, C) ⟶^{δ} H^{1}(G, A) ⟶ H^{1}(G, B)

is constant on the cosets of im N_{C}^{∗} in H^{0}(G, C), so that we may pass to the quotient. If c ∈ H^{0}(G, C) = C^{G} is in im N_{C}^{∗}, lift c to Nb ∈ B^{G}, then dc[g] = gNh − Nh = Nh − Nh = 0, which is what we needed.

□

Proposition 1.5.3 If G is finite, Hom_{ℤ}(ℤ[G], X) ≅ ℤ[G] ⊗_{ℤ} X for any abelian group X. Hence a G-module A is co-induced if and only if it is induced.

PROOF - The map

\begin{matrix}Hom_{ℤ}(ℤ[G], X) & ⟶ & ℤ[G] ⊗_{ℤ} X \\ f & ⟼ & ∑_{g} g ⊗ f(g^{−1}) \end{matrix}

sends sf, s ∈ G to

∑_{g∈G} g ⊗ f(s^{−1}g^{−1}) = ∑_{g′∈G} sg′ ⊗ f(g′^{−1}) = s( ∑_{g′∈G} g′ ⊗ f(g′^{−1}))

and so is a G-homomorphism. The inverse map takes a general element ∑_{g} g ⊗ x_{g} to the map ℤ[G] ⟶ X, g ⟼ x_{g^{−1}}.

□

Corollary 1.5.4 A is relatively injective if and only if it is relatively projective.

Remark 1.5.5 Note that if A is G-induced, then A is H-induced for any subgroup H of G. This is easy to see by considering ℤ[G] ⊗_{ℤ} X as a direct sum

⨁_{g∈G} g · X = ⨁_{h∈H} h · Y

where Y = ⨁_{t∈T} t · X for T a system of representatives of the cosets tH in G.

Definition 1.5.6 If G is a finite group, a G-module A is called cohomologically trivial if Ĥ^{q}(H, A) = 0 for every q ∈ ℤ and every subgroup H of G.

Proposition 1.5.7 If A is an induced G-module then A is cohomologically trivial.

PROOF - By Remark 1.5.5, it is sufficient to check that H^{q}(G, A) = 0 for all q ∈ ℤ. We know (§1.3) that Ĥ^{−2}(G, A), Ĥ^{−3}(G, A), … are all zero. By Proposition 1.5.3, A is co-induced, so §1.1 shows Ĥ^{1}(G, A), Ĥ^{2}(G, A), … are all zero. So we just have to show Ĥ^{0}(G, A) and Ĥ^{−1}(G, A) are zero.

Now A = ℤ[G] ⊗_{ℤ}X for some abelian group X, so every element can be written uniquely in the form ∑_{g∈G} g ⊗ x_{g} for some x_{g} ∈ X. Ĥ^{−1}(G, A) = ker N/I_{G}A so suppose a = ∑ g ⊗ x_{g} ∈ ker N. So

0 = ∑_{s∈G} s ∑_{g∈G} g ⊗ x_{g} = ∑_{g∈G} ∑_{s∈G} sg ⊗ x_{g}
= ∑_{g∈G} ∑_{s∈G} g ⊗ x_{s^{−1}g} = ∑_{g∈G} g ⊗ (∑_{t∈G} x_{t})

Hence ∑_{t∈G} x_{t} = 0 and so

a = ∑_{g∈G}(g − 1) ⊗ x_{g} ∈ I_{G}A

as required.

For the next group, Ĥ^{0}(G, A) = A^{G}/NA, take a general element a = ∑ g⊗x_{g} ∈ A^{G}. Then sa = ∑ sg ⊗ x_{g} = ∑ g ⊗ x_{g} for all s ∈ G so that x_{1} = x_{s} for all s ∈ G and consequently a ∈ NA.

□

1.5.2 Dimension Shifting of Tate Cohomology Groups

Let A^{∗} = Hom_{ℤ}(ℤ[G], A) be the canonical co-induced module associated to A (see Proposition 1.1.8), and A_{∗} = ℤ[G] ⊗_{ℤ} A_{+} the canonical induced module associated to A (see §1.3). If G is finite, A^{∗} = A_{∗} by Proposition 1.5.3. The dimension-shifting sequences

0 ⟶ A ⟶ A^{∗} ⟶ A′ ⟶ 0
0 ⟶ A′ ⟶ A_{∗} ⟶ A ⟶ 0

express A as a submodule or as a quotient module of a cohomologically trivial G-module, and so allow dimension-shifting up and down.

Let us now look at some results which show how dimension-shifting is a “nice” process, by which we mean that the forming of the two dimension-shift sequences above constitute functors from the category of G-modules to the category of short exact sequences of G-modules. These four results are needed for the proof of Tate’s Theorem (1.6.4).

Lemma 1.5.8 Let f : B ⟶ C be a G-homomorphism. Then there exists a commutative diagram

0 ⟶ B′ ⟶ B_{∗} ⟶ B ⟶ 0
↓f′ □ ↓f′′ □ ↓f
0 ⟶ C′ ⟶ C_{∗} ⟶ C ⟶ 0

PROOF - Recall B_{∗} ⟶ B is given by g ⊗ b ⟼ gb. Define f′′(g ⊗ b) = g ⊗ f(b). It is easy to check that f′′ is a G-homomorphism, the second square is commutative, and it follows from this that the restriction of f′′ to B′ ⊆ B_{∗} is a function into C′.

□

Lemma 1.5.9 Let ϕ : A ⊗_{ℤ} B ⟶ C be a G-homomorphism. Then there exists a commutative diagram with exact rows

0 ⟶ A ⊗_{ℤ} B′ ⟶ A ⊗_{ℤ} B_{∗} ⟶ A ⊗_{ℤ} B ⟶ 0
↓ϕ′  □  ↓ϕ′′  □  ↓ϕ
0 ⟶  C′  ⟶  C_{∗}  ⟶  C  ⟶ 0

PROOF - The top row is exact because

0 ⟶ B′ ⟶ B_{∗} ⟶ B ⟶ 0

is a ℤ-split exact sequence, as is shown by the ℤ-homomorphism B ⟶ B_{∗}, b ⟼ 1 ⊗ b.

Define

\begin{aligned}ϕ′′ : & A ⊗_{ℤ} B_{∗} & ⟶ & C_{∗} \\ & a ⊗ g ⊗ b & ⟼ & g ⊗ ϕ(g^{−1}a ⊗ b). \end{aligned}

Then ϕ′′(s(a ⊗ g ⊗ b)) = ϕ′′(sa ⊗ sg ⊗ b) = sg ⊗ ϕ(g^{−1}ss^{−1}a ⊗ b) = sϕ′′(a ⊗ g ⊗ b), so ϕ′′ is a G-homomorphism. Commutativity of the second square can easily be checked, and the rest follows.

□

Similar results apply for dimension-shifting in the other direction, when B is expressed as a submodule of a co-induced module.

Lemma 1.5.10 Let f : B ⟶ C be a G-homomorphism. Then there exists a commutative diagram

0 ⟶ B ⟶ B^{∗} ⟶ B′ ⟶ 0
↓f □ ↓f′′ □ ↓f′
0 ⟶ C ⟶ C^{∗} ⟶ C′ ⟶ 0

PROOF - If β ∈ B^{∗}, let

\begin{aligned}f′′(β) : & ℤ[G] & ⟶ & C \\ & g & ⟼ & f(β(g)) \end{aligned}

This is a G-homomorphism and the first square commutes. General properties of diagrams then show that f′′ defines f′ making the second square commute, since the composition

\begin{matrix}B^{∗} \\ ↓f′′ \\ C^{∗} & ⟶ & C′ \end{matrix}

carries the submodule B to zero.

□

Lemma 1.5.11 Let ϕ : A ⊗_{ℤ} B ⟶ C be a G-homomorphism. Then there exists a commutative diagram with exact rows

0 ⟶ A ⊗_{ℤ} B ⟶ A ⊗_{ℤ} B^{∗} ⟶ A ⊗_{ℤ} B′ ⟶ 0
↓ϕ  □  ↓ϕ′′  □  ↓ϕ′
0 ⟶  C  ⟶  C^{∗}  ⟶  C′  ⟶ 0

PROOF - The top row is exact because

0 ⟶ B ⟶ B^{∗} ⟶ B′ ⟶ 0

is a ℤ-split exact sequence, as is shown by the ℤ-homomorphism B^{∗} ⟶ B, β ⟼ β(1).

Define for β ∈ B^{∗}

\begin{aligned}ϕ′′(a ⊗ β) : & ℤ[G] & ⟶ & C \\ & g & ⟼ & ϕ(g^{−1}a ⊗ β(g)). \end{aligned}

This map carries s(a ⊗ β) = sa ⊗ (g ↦ β(s^{−1}g)) to

(g ↦ ϕ(g^{−1}sa ⊗ β(s^{−1}g))) = (g ↦ ϕ((s^{−1}g)^{−1}a ⊗ β(s^{−1}g)))

which is sϕ′′(a ⊗ β). So ϕ′′ is a G-homomorphism. It is easy to check the commutativity of the first square, and then the properties of ϕ′ follow.

□

1.5.3 Complete Resolutions

In §1.5.1 we defined the doubly-infinite sequence of Tate cohomology groups, by piecing together the cohomology and homology groups. There is another way to define this sequence, which is more elegant but less explicit. We start with a resolution of the G-module ℤ (with trivial G-action) by finitely-generated free Gmodules:

… ⟶ P_{1} ⟶ P_{0} ⟶^{ϵ} ℤ ⟶ 0  (1.8)

and apply the contravariant functor Hom_{ℤ}(?, ℤ) to obtain the dual sequence

0 ⟶ ℤ ⟶^{ϵ^{∗}} P_{0}^{∗} ⟶ P_{1}^{∗} ⟶ …

so P_{i}^{∗} = Hom_{ℤ}(P_{i}, ℤ), the sequence being exact because the P_{i} are ℤ-free (see Lemma 1.1.10 and the proof of the result which follows it).

Putting these together gives an exact sequence

… ⟶ P_{1} ⟶ P_{0} ⟶^{ϵ^{∗}∘ϵ} P_{0}^{∗} ⟶ P_{1}^{∗} ⟶ …

and we apply the functor Hom_{G}(?, A) to obtain the complex

K : … ⟶ Hom_{G}(P_{1}^{∗}, A) ⟶ Hom_{G}(P_{0}^{∗}, A) ⟶ Hom_{G}(P_{0}, A) ⟶ …

The Tate groups are now the cohomology groups of this complex, namely Ĥ^{q}(G, A) = H^{q}(K). For more details and proof, see [3].

1.5.4 The Cup Product

Theorem 1.5.12 Let G be a finite group. Then there exists a unique family of homomorphisms

Ĥ^{p}(G, A) ⊗_{ℤ} Ĥ^{q}(G, B) ⟶ Ĥ^{p+q}(G, A ⊗_{ℤ} B)

(denoted a ⊗ b ⟼ a · b) defined for all integers p, q and all G-modules A, B, such that

1. The homomorphisms are functorial in A and B (considering each side as a covariant bifunctor in (A,B)). Namely, if f : A ⟶ A′ and g : B ⟶ B′ are G-homomorphisms, the diagram

Ĥ^{p}(G, A) ⊗_{ℤ} Ĥ^{q}(G, B) ⟶ Ĥ^{p+q}(G, A ⊗_{ℤ} B)
↓f^{∗}⊗g^{∗}  ↓(f⊗g)^{∗}
Ĥ^{p}(G, A′) ⊗_{ℤ} Ĥ^{q}(G, B′) ⟶ Ĥ^{p+q}(G, A′ ⊗_{ℤ} B′)

is commutative.

2. For p = q = 0 they are induced by the inclusion

A^{G} ⊗_{ℤ} B^{G} ↪ (A ⊗_{ℤ} B)^{G}

3. If 0 → A → A′′ → A′ → 0 is an exact sequence of G-modules, and if 0 → A ⊗_{ℤ} B → A′′ ⊗_{ℤ} B → A′ ⊗_{ℤ} B → 0 is exact, then for a′ ∈ Ĥ^{p}(G, A′) and b ∈ Ĥ^{q}(G, B) we have

(δa′) · b = δ(a′ · b) (in Ĥ^{p+q+1}(G, A ⊗_{ℤ} B))

4. If 0 → B → B′′ → B′ → 0 is an exact sequence of G-modules, and if 0 → A ⊗_{ℤ} B → A ⊗_{ℤ} B′′ → A ⊗_{ℤ} B′ → 0 is exact, then for a ∈ Ĥ^{p}(G, A) and b′ ∈ Ĥ^{q}(G, B) we have

a · (δb′) = (−1)^{p}δ(a · b′) (in Ĥ^{p+q+1}(G, A ⊗_{ℤ} B))

The proof of this can be found in [3], p106. I will sketch it here.

Let P_{q} be a complete resolution of ℤ as described in the last section. We need to map a pair of cochains f ∈ Hom_{G}(P_{p}, A) and g ∈ Hom_{G}(P_{q}, B) to a cochain f · g ∈ Hom_{G}(P_{p+q}, A ⊗_{ℤ} B). The tensor product

\begin{aligned}f ⊗ g : & P_{p} ⊗_{ℤ} P_{q} & ⟶ & A ⊗_{ℤ} B \\ & a ⊗ b & ⟼ & f(a) ⊗ g(b) \end{aligned}

is the starting point, and we consider composing it with a G-homomorphism

ϕ_{p,q} : P_{p+q} ⟶ P_{p} ⊗_{ℤ} P_{q}

to obtain a G-homomorphism from P_{p+q} to A ⊗_{ℤ} B. Suppose we can find such a ϕ with the following two properties:

• a ϕ_{p,q} ∘ d = (d ⊗ 1) ∘ ϕ_{p+1,q} + (−1)^{p}(1 ⊗ d) ∘ ϕ_{p,q+1}, where d is the coboundary map in the resolution;

• b (ϵ ⊗ ϵ) ∘ ϕ_{0,0} = ϵ, where ϵ is as in (1.8).

Then we find that the cochain f · g has all the properties required for the cup product. Property (a) shows that a pair of cocycles maps to a cocycle, whose cohomology class depends only on their ones. Thus we do have a homomorphism Ĥ^{p}(G, A) ⊗_{ℤ} Ĥ(G, B) ⟶ Ĥ^{p+q}(G, A ⊗_{ℤ} B). That this satisfies the first condition of the theorem can be seen by looking at how maps are induced between the Ĥ^{q}.

Let us demonstrate the second requirement. Recall that the exact sequence

P_{1} ⟶^{d_{0}} P_{0} ⟶^{ϵ} ℤ ⟶ 0

gives an exact sequence

0 ⟶ Hom_{G}(ℤ, A) ⟶^{ϵ} Hom_{G}(P_{0}, A) ⟶^{d_{0}} Hom_{G}(P_{1}, A),

and similarly for B and A ⊗_{ℤ} B. So if α ∈ Hom_{G}(P_{0}, A) is a cocycle, then α ∈ im ϵ so that α = a ∘ ϵ for some a : ℤ ⟶ A. Similarly if β ∈ Hom_{G}(P_{0}, B) is a cocycle, β = b ∘ ϵ for b : ℤ ⟶ B. Thus

α · β = (a ∘ ϵ ⊗ b ∘ ϵ) ∘ ϕ_{0,0}
= (a ⊗ b) ∘ (ϵ ⊗ ϵ) ∘ ϕ_{0,0}
= (a ⊗ b) ∘ ϵ = α ⊗ β : P_{0} ⟶ A ⊗_{ℤ} B,

which shows that · is induced by the inclusion (A^{G} ⊗_{ℤ} B^{G}) ↪ (A ⊗_{ℤ} B)^{G}, remembering that a ∈ A^{G} is identified with a G-homomorphism ℤ ⟶ A, 1 ⟼ a.

Properties (3) and (4) are checked by verifying that the various stages in the evaluation of the connecting homomorphism in question are represented by the appropriate cup-products of cochains of A and B, using property (b).

Now we define ϕ, after Atiyah and Wall in [3]. We use the complete resolution by homogeneous cochains (see §1.2). So

Q_{q} = ℤ[G^{q+1}] for q ≥ 0
Q_{−q} = Q_{q−1}^{∗} = Hom_{ℤ}(ℤ[G^{q}], ℤ) for q ≥ 1

and a basis for Q_{−q} is given by homomorphisms of the form (g_{1}^{∗}, …, g_{q}^{∗}), which means the map which takes (g_{1}, …, g_{q}) ∈ G^{q} to 1 and all other basis elements to zero. The G-action is then s(g_{1}^{∗}, …, g_{q}^{∗}) = ((sg_{1})^{∗}, …, (sg_{q})^{∗}), and the coboundary

\begin{aligned}d : & Q_{−q} & ⟶ & Q_{−q−1} \\ & (g_{1}^{∗}, …, g_{q}^{∗}) & ⟼ & ∑_{s∈G} ∑_{i=0}^{q}(−1)^{i}(g_{1}^{∗}, …, g_{i}^{∗}, s^{∗}, g_{i+1}^{∗}, … g_{q}^{∗}) \end{aligned}

for q ≥ 1, and

\begin{aligned}d : & Q_{0} & ⟶ & Q_{−1} \\ & (g_{0}) & ⟼ & ∑_{s∈G}(s^{∗}). \end{aligned}

Then ϕ_{p,q} : Q_{p+q} ⟶ Q_{p} ⊗_{ℤ} Q_{q} is defined by

• If p ≥ 0 and q ≥ 0,

ϕ_{p,q}(g_{0}, …, g_{p+q}) = (g_{0}, …, g_{p}) ⊗ (g_{p}, …, g_{p+q})

• If p ≥ 1 and q ≥ 1,

ϕ_{−p,−q}(g_{1}^{∗}, …, g_{p+q}^{∗}) = (g_{1}^{∗}, …, g_{p}^{∗}) ⊗ (g_{p+1}^{∗}, …, g_{p+q}^{∗})

• If p ≥ 0 and q ≥ 1,

ϕ_{p,−p−q}(g_{1}^{∗}, …, g_{q}^{∗}) = ∑(g_{1}, s_{1}, …, s_{p}) ⊗ (s_{p}^{∗}, …, s_{1}^{∗}, g_{1}^{∗}, …, g_{q}^{∗})
ϕ_{−p−q,p}(g_{1}^{∗}, …, g_{q}^{∗}) = ∑(g_{1}^{∗}, …, g_{q}^{∗}, s_{1}^{∗}, …, s_{p}^{∗}) ⊗ (s_{p}, …, s_{1}, g_{q})
ϕ_{p+q,−q}(g_{0}, …, g_{p}) = ∑(g_{0}, …, g_{p}, s_{1}, …, s_{q}) ⊗ (s_{q}^{∗}, …, s_{1}^{∗})
ϕ_{−q,p+q}(g_{0}, …, g_{p}) = ∑(s_{1}^{∗}, …, s_{q}^{∗}) ⊗ (s_{q}, …, s_{1}, g_{0}, …, g_{p})

In the sums on the right-hand side, the s_{i} run independently through G.

1.5.5 Restriction and Corestriction

Recall from §1.4 that the restriction homomorphism was defined for the cohomology groups

Res : H^{q}(G, A) ⟶ H^{q}(H, A)

for any subgroup H of G. The dimension-shifting of §1.5.2 shows it can be extended to all the Tate groups Ĥ^{q}. Res is the unique morphism of Tate cohomology functors which is induced in dimension 0 by the inclusion A^{G} ⊆ A^{H}.

The corestriction is a morphism in the other direction

Cor : Ĥ^{q}(H, A) ⟶ Ĥ^{q}(G, A)

induced in dimension zero by the norm map

\begin{aligned}N_{G/H} : & A^{H} & ⟶ & A^{G} \\ & a & ⟼ & ∑_{i} s_{i}a \end{aligned}

where G/H = {s_{1}H, …, s_{(G:H)}H}. This extends by dimension shifting to all the Tate groups.

Proposition 1.5.13 Cor ∘ Res is multiplication by (G : H).

PROOF - This is clear in dimension zero. So if f denotes multiplication by (G : H), then Cor∘Res−f is a morphism of cohomological functors which is zero in dimension zero for all G-modules. By dimension-shifting, it is zero in all dimensions.

□

Corollary 1.5.14 If |G| = n then nĤ^{q}(G, A) = 0 for all G-modules A and all q ∈ ℤ.

PROOF - Set H = {1}, then Ĥ^{q}(G, A) = 0 because H-module means the same as ℤ-module, and so all H-modules are induced, and so cohomologically trivial. Now Res is the zero map, and so Cor∘Res is the zero map, and is equal to multiplication by n.

□

Corollary 1.5.15 If A is a finitely-generated G-module then the Ĥ^{q}(G, A) are finite groups.

PROOF - The definition in terms of the standard resolution shows they are finitelygenerated groups. By the previous corollary, the generators have order at most n, so that the groups are actually finite.

□

1.6 Tate’s Theorem

Tate’s Theorem is the central result of finite group cohomology which we want to apply to Number Theory. To prove this theorem, a lengthy and technical diversion into the cohomology of p-groups is necessary, and was carried out by Tate and Nakayama. Good accounts of this can be found in [1], Chapter 9, and [3]. Space considerations prevent us from presenting the discussion here, so we state only the conclusion.

By Sylow’s Theorem ([1], Chapter 9), a group of order ∏ p^{m_{p}} contains subgroups of order p^{m_{p}} for each p, called Sylow p-subgroups of G. For a given p, any two such subgroups are conjugate in G. We choose a system of Sylow subgroups {G_{p} : p prime} in G.

Theorem 1.6.1 Let A be a G-module, where G is a finite group. Suppose that for each p there is q_{p} ∈ ℤ such that

Ĥ^{q_{p}}(G_{p}, A) = Ĥ^{q_{p}+1}(G_{p}, A) = 0.

Then A is cohomologically trivial.

We use this result to prove an isomorphism criterion:

Theorem 1.6.2 Let G be a finite group, B and C two G-modules, and f : B ⟶ C a G-homomorphism. For each prime p, suppose there exists an integer n_{p} such that

f_{n}^{∗} : Ĥ^{n}(G_{p}, B) ⟶ Ĥ^{n}(G_{p}, C)

is surjective for n = n_{p}, bijective for n = n_{p} + 1 and injective for n = n_{p} + 2. Then for any subgroup H of G and any integer n,

f_{n}^{∗} : Ĥ^{n}(H, B) ⟶ Ĥ^{n}(H, C)

is an isomorphism.

PROOF - Let B^{∗} = Hom_{ℤ}(ℤ[G], B_{+}) and i : B ⟶ B^{∗} the injection of Proposition 1.1.8. Then f × i : B ⟶ C ⊕ B^{∗} is injective, giving an exact sequence

0 ⟶ B ⟶ C ⊕ B^{∗} ⟶ D ⟶ 0.

B^{∗} is cohomologically trivial so the cohomology of C ⊕ B^{∗} is the same as that of C (see Lemma 1.1.9).

Then one may check that the exact cohomology sequence for G_{p} shows Ĥ^{n_{p}}(G_{p}, D) = Ĥ^{n_{p}+1}(G_{p}, D) = 0. Hence by the previous theorem, D is a cohomologically trivial G-module, so the exact cohomology sequence for H < G gives the result.

□

Tate’s Theorem will follow from the next general result.

Theorem 1.6.3 Let A,B and C be three G-modules and

ϕ : A ⊗_{ℤ} B ⟶ C

a G-homomorphism. Fix an integer q and an element a ∈ Ĥ^{q}(G, A). Write · for the cup-product (see §1.5.4).

Suppose that for all p there is an integer n_{p} such that the function

\begin{matrix}Ĥ^{n}(G_{p}, B) & ⟶ & Ĥ^{n+q}(G_{p}, C) \\ x & ⟼ & ϕ^{∗}(Res_{G/G_{p}}(a) · x) \end{matrix}

is surjective for n = n_{p}, bijective for n = n_{p} + 1 and injective for n = n_{p} + 2. Then for all subgroups H of G and all n ∈ ℤ,

\begin{matrix}Ĥ^{n}(H, B) & ⟶ & Ĥ^{n+q}(H, C) \\ x & ⟼ & ϕ^{∗}(Res_{G/H}(a) · x) \end{matrix}

is an isomorphism.

PROOF - As the first step, we deal with the case q = 0. We choose α ∈ A^{G} representing a (so α also represents Res_{G/H}(a) for all subgroups H of G). Define

\begin{aligned}f_{ϕ} : & B & ⟶ & C \\ & b & ⟼ & ϕ(α ⊗ b). \end{aligned}

f_{ϕ} is a G-homomorphism, since α is G-invariant.

We claim that for all b ∈ Ĥ^{n}(H, B), and any ϕ,

ϕ^{∗}(Res_{G/H}(a) · b) = f_{ϕ}^{∗}(b).

The claim is clearly true for n = 0. It is proved for the other n by dimensionshifting. For example, suppose it is true for n − 1, for all modules and ϕ’s. From Lemma 1.5.10 we obtain a commutative diagram

\begin{matrix}Ĥ^{n−1}(H, B′) & \underrightarrow{⟶_{∼}^{δ}} & Ĥ^{n}(H, B) \\ ↓f′^{∗} & □ & ↓f_{ϕ}^{∗} \\ Ĥ^{n−1}(H, C′) & \underrightarrow{⟶_{∼}^{δ}} & Ĥ^{n−1}(H, C) \end{matrix}

and from Lemma 1.5.11 we obtain a map ϕ′ : A ⊗_{ℤ} B′ ⟶ C′. It is easy to check from the definitions that f′ = f_{ϕ′}. So the inductive hypothesis shows

ϕ′^{∗}(Res_{G/H}(a) · b) = f′^{∗}(b).

Then for each b′ in B′,

f^{∗}(δb′) = δ(f′(b′))
= δϕ′^{∗}(Res_{G/H}(a) · b′)
= ϕ^{∗}(δ(Res_{G/H}(a) · b′))  by Lemma 1.5.11
= ϕ^{∗}(Res_{G/H}(a) · δb′).

Since every b ∈ B is of the form δb′, this proves the claim.

One may carry out the dimension-shifting from n + 1 to n in an entirely similar way, using Lemmas 1.5.8 and 1.5.9.

Having proved the claim, we need only to apply the previous theorem to obtain the stated result for q = 0.

Step 2: We vary q. Suppose the statement is true for q + 1. Let

0 ⟶ A′ ⟶ A_{∗} ⟶ A ⟶ 0

be an exact sequence with A_{∗} = ℤ[G] ⊗_{ℤ} A_{+}. Then Ĥ^{q}(G, A) ⟶_{∼}^{δ} Ĥ^{q+1}(G, A′) is an isomorphism. Put a′ = δa, and note that Res_{G/H}(a′) = δRes_{G/H}(a) since Res is a morphism of cohomological functors.

From Lemma 1.5.9 we obtain a map

ϕ′^{∗} : Ĥ^{n+q+1}(G_{p}, A′ ⊗_{ℤ} B) ⟶ Ĥ^{n+q+1}(G_{p}, C′).

We have the commutative diagram

\begin{matrix}Ĥ^{n}(G_{p}, B) & = & Ĥ^{n}(G_{p}, B) \\ ↓⊗Res(a) & □ & ↓⊗Res(a′) \\ Ĥ^{n}(G_{p}, B) ⊗_{ℤ} Ĥ^{q}(G_{p}, A) & ⟶^{1⊗δ} & Ĥ^{n}(G_{p}, B) ⊗_{ℤ} Ĥ^{q+1}(G_{p}, A′) \\ ↓· & □ & ↓· \\ Ĥ^{n+q}(G_{p}, A ⊗_{ℤ} B) & ⟶^{δ} & Ĥ^{n+q+1}(G_{p}, A′ ⊗_{ℤ} B) \\ ↓ϕ^{∗} & □ & ↓ϕ′^{∗} \\ Ĥ^{n+q}(G, C) & ⟶^{δ} & Ĥ^{n+q+1}(G, C) \end{matrix}

In this diagram, the first square commutes because Res is a morphism of cohomological functors, the second by the properties of the cup product, and the third by Lemma 1.5.9.

This diagram shows that when the map Ĥ^{n}(G_{p}, B) ⟶ Ĥ^{n+q+1}(G, C′) given by the second column is surjective or injective, then so is the corresponding map given by the first column. This allows for the induction in the downwards direction. Lemma 1.5.11 works just the same to show induction in the other direction.

□

We want the following special case of this.

Theorem 1.6.4 (Tate) Let G be a finite group, A a G-module, and a an element of H^{2}(G, A). For each prime p let G_{p} be a Sylow subgroup of G, and assume

1. H^{1}(G_{p}, A) = 0,

2. H^{2}(G_{p}, A) is generated by Res_{G/G_{p}}(a) and has order equal to that of G_{p}.

Then for all subgroups H of G and all n ∈ ℤ, the map^{5}

\begin{matrix}Ĥ^{n}(H, ℤ) & ⟶ & Ĥ^{n+2}(H, A) \\ x & ⟼ & x · Res_{G/H}(a) \end{matrix}

is an isomorphism.

PROOF - Set B = ℤ, C = A, q = 2, n_{p} = −1 in Theorem 1.6.3. For n = −1, Ĥ^{−1}(G_{p}, ℤ) ⟶ Ĥ^{1}(G_{p}, A) is surjective by (1). For n = 0, we have a map defined on Ĥ^{0}(G_{p}, ℤ) = ℤ/|G_{p}|ℤ (see Definition 1.5.1). We claim that 1 · Res_{G/G_{p}}(a) = Res_{G/G_{p}}(a), so that the generator 1 maps to a generator of H^{2}(G_{p}, A), and then the equipotence condition shows the map is bijective. To see the claim, we use the cup-product formulae of §1.5.4. In the notation of that section, we have the G-homomorphism

\begin{aligned}ϕ_{0,2} : & Q_{2} & ⟶ & Q_{0} ⊗_{ℤ} Q_{2} \\ & (g_{0}, g_{1}, g_{2}) & ⟼ & (g_{0}) ⊗ (g_{0}, g_{1}, g_{2}) \end{aligned}

Represent x ∈ Ĥ^{0}(G_{p}, ℤ) by a homogeneous cochain x ∈ Hom_{G}(Q_{0}, ℤ) and Res_{G/G_{p}})(a) by α ∈ Hom_{G}(Q_{2}, A). Then the cup-product is represented by the cochain

(x · α)(g_{0}, g_{1}, g_{2}) = x(g_{0}) ⊗ α(g_{0}, g_{1}, g_{2})
= x(1) ⊗ α(g_{0}, g_{1}, g_{2}) ∈ Hom_{G}(Q_{2}, ℤ ⊗_{ℤ} A).

So if x = 1, x · α = α, as required.

Finally, consider n = 1. We have Ĥ^{1}(G_{p}, ℤ) = Hom_{ℤ}(G_{p}, ℤ) = 0 and so the map is injective for n = 1.

Hence we have satisfied all the conditions of Theorem 1.6.3 and so Tate’s Theorem is proved.

□

5identifying A with A ⊗_{ℤ} ℤ

Galois Cohomology

2.1 Finite Galois Cohomology

[We refer to Appendix C for basic ideas and results of Galois Theory.]

Now we apply the cohomology of groups to studying fields. If L/K is a finite Galois extension, then the Galois group G := Gal(L/K) makes the additive and multiplicative groups of L into abelian G-modules. Thus we have two cohomology sequences, Ĥ^{q}(G, L) and Ĥ^{q}(G, L^{∗}).

Theorem 2.1.1 For every q ∈ ℤ, Ĥ^{q}(G, L) = 0.

PROOF - By the normal basis theorem (C.2.6)

L = ⨁_{s∈G} s · K = ℤ[G] ⊗_{ℤ} K

is an induced G-module, hence cohomologically trivial (noting G is finite, so induced is the same as co-induced by Proposition 1.5.3).

□

Next we investigate the multiplicative group L^{∗}. Its cohomology groups are not all trivial. However, we have the very important result:

Theorem 2.1.2 H^{1}(G, L^{∗}) = 0, for any finite group G of automorphisms of L.

PROOF - Let f : G ⟶ L^{∗} be a 1-cocycle. For c ∈ L, form the “Poincaré series”

b = ∑_{s∈G} f[s] · sc

By Dedekind’s lemma (C.1.9), the map ∑_{s∈G} f[s] · s is not identically zero, so c can be chosen so that b ≠ 0. But sb = ∑_{t∈G} sf[t] · stc, which using the 1-cocycle condition f[st] = f[s] · sf[t] is the same as

∑_{t∈G} f[s]^{−1} · f[st] · stc = f[s]^{−1} · b

since st runs through G as t does. Thus f[s] = (sb)^{−1} · b is a coboundary.

□

Corollary 2.1.3 (Hilbert’s Theorem 90) If G = Gal(L/K) =< s > is finite cyclic, and x ∈ L^{∗} has N_{L/K}x = 1, then x = y/sy for some y ∈ L^{∗}.

PROOF - Define f : G ⟶ L^{∗} by f[1] = 1, f[s] = x, …, f[s^{n}] = x·sx·s^{2}x·. . .·s^{n−1}x. Then f[s^{n}] depends only on s^{n} and not n because the norm of x is 1. Furthermore, f[s^{n}s^{m}] = f[s^{n}] · s^{n}f[s^{m}], so that f is a 1-cocycle. Hence f is a 1-coboundary, so x = f[s] = y/sy.

□

Note that Theorem 2.1.2 is also sometimes called Hilbert’s Theorem 90. It can be shown by extending this argument that:

Proposition 2.1.4 H^{1}(G, GL(n, L)) = 0

See [1], Chapter 10, §1, Proposition 3 for more details. This result will be used when we study the Brauer group later on.

The remainder of this chapter will be spent showing how to extend the ideas of Galois correspondence and cohomology to infinite extensions.

2.2 Infinite Galois Theory

2.2.1 The Krull Topology

Let L/K be a Galois^{1} extension, and G = Gal(L/K) the group of K-automorphisms of L. If L/K is finite, the Fundamental Theorem of Galois Theory (C.1.8) says that the intermediate subfields are in bijective correspondence with the subgroups of G. In the more general case, this is not always true - there may be more subgroups than subfields.

The solution is to turn G into a topological group. The reader is referred to Appendix D for the facts about topological groups that we need.

For G = Gal(L/K) we define a canonical topology, called the Krull topology, as follows. The nonempty open sets are defined to be all unions of the cosets

σGal(L/E) : σ ∈ G, E/K is a finite Galois extension.

We claim that the subset

{Gal(L/E) : E/K is a finite Galois extension}

forms a fundamental system of neighbourhoods of the identity in the sense of Definition D.0.7. The required properties follow easily from finite Galois theory. For example:

1meaning normal and separable.

Proposition 2.2.1 If L/K is Galois, ⋂_{E} Gal(L/E) = {1}.

PROOF - Let σ ∈ Gal(L/E) − {1}. We want to show there is a finite Galois extension E′ of K contained in L such that σ ∉ Gal(L/E′), ie. σ does not fix E′ pointwise. But σ ≠ 1 so there is y ∈ L with σy ≠ y. By finite Galois theory, there is a finite Galois extension E′ of K contained in L with y ∈ E′.

□

Definition 2.2.2 A topological group which is compact, and has a fundamental system of open neighbourhoods of the identity consisting of normal subgroups is called a profinite group.

We will see later that a Galois group with the Krull topology is compact, so that Galois groups are profinite.

2.2.2 Open and Closed Subgroups

Proposition 2.2.3 For any topological group G, every open subgroup is also closed.

PROOF - Let H be an open subgroup. Then the cosets sH, s ∈ G partition G, and are open. Then

H = G − ⋃_{s∉H} sH

is the complement of an open set and so is closed.

□

Hence the open subgroups are special cases of the closed subgroups. Not all closed subgroups are necessarily open. If H is a closed subgroup of G with finite index in G, ie.

(G : H) = |G/H| < ∞

then H must be open in G. This is because, if {e, g_{2}, …, g_{n}} is a finite system of representatives of the set G/H of left-cosets, then

H = G − (g_{2}H ∪ … ∪ g_{n}H) = (G − g_{2}H) ∩ … ∩ (G − g_{n}H)

is a finite intersection of open sets and so is open. When G is compact this condition is also necessary:

Proposition 2.2.4 Let G be a compact topological group and H a closed subgroup of G. Then H is open if and only if it has finite index in G.

PROOF - One implication has been shown. So suppose H is open, then decompose G as

G = ⋃_{s∈G} sH

This is an open covering of G, so there is a finite set s_{1}, …, s_{n} ∈ G such that G = s_{1}H ∪ … ∪ s_{n}H. This shows that (G : H) is finite.

□

2.2.3 Projective Limits

To characterise the profinite groups differently, we make use of the concept of projective limit.

Definition 2.2.5 1. A directed set I is a set with a relation ≤ which is reflexive^{2} and transitive^{3} and has the property that, given x, y ∈ I, we can always find z ∈ I with x ≤ z and y ≤ z.

2. An inverse system of topological groups over a directed set I consists of an indexed collection of topological groups G^{i}, i ∈ I and continuous homomorphisms π_{i}^{j} : G^{j} ⟶ G^{i} for each pair i ≤ j. We require that π_{i}^{i} = id_{G^{i}} and, if i ≤ j ≤ k, π_{i}^{j} ∘ π_{j}^{k} = π_{i}^{k}.

To an inverse system {G^{i}, π_{i}^{j}} is associated a topological group called the projective limit (or sometimes the inverse limit) of the G^{i}, written \underleftarrow{lim}_{i} G^{i}.

To define it, start with the product group ∏_{i∈I} G^{i}. This is a topological group as described in Appendix D. We define Λ = \underleftarrow{lim}_{i} G^{i} to be the subset of ∏ G^{i} consisting

of those (x_{i}) such that π_{i}^{j}(x_{j}) = x_{i} whenever i ≤ j. Think of the π as projections and an element of the projective limit as a system of group elements compatible with these projections. Λ is a subgroup of ∏ G^{i} and inherits the induced topology from this group.

We assume from now on that the G^{i} are finite groups, since this will be our only application.

Proposition 2.2.6 Λ is closed in ∏ G^{i}.

PROOF - Suppose x = (x_{i}) ∉ Λ, then there are i ≤ j such that π_{i}^{j}x_{j} ≠ x_{i}. The set of all elements in ∏ G^{i} with i-th term x_{i} and j-th term x_{j} is open in ∏ G^{i} (see Appendix D) and disjoint from Λ. Λ is the complement of the union of all such open sets.

□

Now each of the groups G^{i} is compact, being finite. Tychonoff’s Theorem (D.0.9) shows that ∏ G^{i} is compact. We know that a closed subset of a compact space is always itself a compact space. Hence we see:

Proposition 2.2.7 \underleftarrow{lim} G^{i} is a compact group.

2This means x ≤ x ∀x ∈ I.

3This means that whenever x ≤ y and y ≤ z then x ≤ z.

2.2.4 The Galois Group as a Projective Limit

We now apply this construction to the Galois case. Let L/K be any Galois extension. Our directed set I will be the set of all finite Galois extensions E of K contained in L. The groups in the inverse system will be the Galois groups Gal(E/K) of these extensions, considered as topological groups with the discrete topology.

In the directed set, we say E′ ≤ E when E′ ⊆ E. When this is the case, a homomorphism

Gal(E/K) ⟶ Gal(E′/K)

is given by restricting^{4} an automorphism to the subset E′. This satisfies all the conditions of an inverse system. Hence we obtain the inverse limit Λ. We now identify this with the Galois group. In fact there is the general result

Proposition 2.2.8 A topological group G is profinite if and only if it is isomorphic to a projective limit of finite groups. These groups are then the finite quotients of G.^{5}

The proof can be found in [4]. I will demonstrate it here for the special case of a Galois group.

Proposition 2.2.9 If we ignore the topology, there is a group isomorphism Gal(L/K) ≅ Λ.

PROOF - As above, restriction gives a homomorphism Gal(L/K) ⟶ Gal(E/K) for each E ∈ I. These combine to give a homomorphism

θ : Gal(L/K) ⟶ ∏_{E∈I} Gal(E/K).

It is clear that im θ ⊆ Λ (θ and the π_{E′}^{E} are defined by restricting field automorphisms). So we need to show that θ is bijective.

If s ∈ G − {1} then there is y ∈ L with sy ≠ y. Then there exists E containing y. So s|_{E} ∈ Gal(E/K) is not the identity. Hence θ(y) ≠ 1, so θ is injective.

Now let (s_{E}) ∈ Λ. Define

s : L ⟶ L

by s(x) := s_{E}(x) for any E such that x ∈ E. The conditions on Λ mean that s is well-defined. Using the restrictions s_{E}, it is easy to show that s is a Kautomorphism of L. Hence s ∈ Gal(L/K) and (s_{E}) = θ(s). So θ is surjective.

□ Proposition 2.2.10 The group isomorphism defined above is a homeomorphism

4because E′/K is normal, we can be sure that the images of these restricted automorphisms will be contained in E′

5Yet another characterisation of a profinite group is that it is compact and totally disconnected.

PROOF - We have to show that θ and θ^{−1} are continuous.

First θ^{−1}. Let H be an open subset of Gal(L/K); by the existence of the fundamental system of neighbourhoods of the identity, it is enough to assume that H is an open, normal subgroup of G. We have to show that the preimage of H under θ^{−1}, ie. θ(H), is open in Λ.

We know that H = Gal(L/E) for some finite Galois extension E/K. So if σ ∈ Gal(L/E), then σ : L ⟶ L is a field automorphism with σ|_{E} = id_{E}. This shows that the image of σ under θ is an element (σ_{E′})_{E′∈I} with

σ|_{E∩E′} = id_{E∩E′}

for each E′ ∈ I. On the other hand, the proof of the previous proposition shows that every such element comes from an element of H. So θ(H) is equal to this subgroup. But this subgroup is precisely ker π_{E} ∩ Λ, where

π_{E} : ∏_{E′∈I} Gal(E′/K) ⟶ Gal(E/K)

is projection on the E-th component. So θ(H) is open. We have thus established that θ^{−1} is continuous.

To see θ is continuous, let X be a closed subset of Λ. We have to show the pre-image θ^{−1}(X) is closed. All that is needed here is elementary topology. X is a closed subset of the compact space Λ, so is a compact space. θ^{−1}(X) is the continuous image of a compact space so is also compact. Then θ^{−1}(X) is a compact subspace of a Hausdorff space and so is closed.

□

Corollary 2.2.11 If L/K is a Galois extension, then Gal(L/K) (with the Krull topology) is a compact group

PROOF - Gal(L/K) is homeomorphic to \underleftarrow{lim} Gal(E/K) as E runs through all finite Galois subextensions of L/K. By the comments at the end of §2.2.3, this group is compact.

□

2.2.5 The Fundamental Theorem

We can use the topology on the Galois group to extend Galois theory to infinite extensions.

Theorem 2.2.12 (Fundamental Theorem of Galois Theory) Let L/K be a Galois extension with group G, C the set of all closed subgroups of G and F the set of all fields between K and L. Then

E ⟼ Gal(L/E)

is a bijection F ⟶ C. Then inverse is H ⟼ E^{H}, the fixed-field function.

For the proof, see [4]

2.3 Infinite Galois Cohomology

2.3.1 Direct Limits

In order to proceed with the definition of infinite-dimensional Galois cohomology, we need the notion of direct limit. This is a dual to the definition of projective limit given in §2.2.3. The reader is asked to recall from that section the notion of a directed set. Note that in contrast to that section, we are not going to be interested in the topology on a direct limit, as all the ones we deal with will be given the discrete topology - so this has been omitted from the definition.

Definition 2.3.1 A direct system of groups over a directed set I is an indexed set of groups (G^{i})_{i∈I} and homomorphisms ι_{i}^{j} : G^{i} ⟶ G^{j} for i ≤ j. We require that ι_{i}^{i} = id_{G^{i}} and, if i ≤ j ≤ k, ι_{j}^{k} ∘ ι_{i}^{j} = ι_{i}^{k}.

To define the direct limit^{6} \underrightarrow{lim}_{i} G^{i} of a direct system, we begin with the disjoint

union of the G^{i}. This is the set-theoretic union of the G^{i} with no x_{i} ∈ G^{i} identified with any x_{j} ∈ G^{j} for i ≠ j. We define an equivalence relation on the disjoint union by saying that

• x_{i} ∈ G^{i} and x_{j} ∈ G^{j} are equivalent when i ≤ j and ι_{i}^{j}(x_{i}) = x_{j}

Let + denote the group law in each G^{i}. The set of equivalence classes is made into a group by defining the composition of the class of x_{i} ∈ G^{i} and x_{j} ∈ G^{j} to be the class of ι_{i}^{k}(x_{i}) + ι_{j}^{k}(x_{j}) for any k ≥ i, j. This is well-defined, and the resulting group is by definition the direct limit \underrightarrow{lim}_{i} G^{i}.

Remark 2.3.2 1. There are obvious injections ι_{i} : G^{i} ⟶ \underrightarrow{lim} G^{i} which are compatible with the ι_{i}^{j}.

2. When the G^{i} are all subgroups of some group G, with G^{i} ⊆ G^{j} ⟺ i ≤ j, and ι_{i}^{j} : G^{i} ⟶ G^{j} defined by inclusion, then the direct limit \underrightarrow{lim} G^{i} is the same as the union ⋃ G^{i}.

6Also called the injective limit

3. We can just as easily define the direct limit of a direct system of Rmodules, by replacing “group” by “R-module” and “homomorphism” by “Rhomomorphism” throughout.

2.3.2 Properties of Direct Limits

Let (G^{i}, ι) be a direct system.

Proposition 2.3.3 (Universal Mapping Property of Direct Limits) Given a system of homomorphisms f_{i} : G^{i} ⟶ X for some group X such that f_{i} = f_{j} ∘ ι_{i}^{j} : G^{i} ⟶ G^{j} ⟶ X for each i ≤ j, there exists a unique homomorphism f : \underrightarrow{lim} G^{i} ⟶ X such that f ∘ ι_{j} = f_{j} for all j. The picture is:

[figure: X; \underrightarrow{lim} G^{i}; f_{i}; f_{j}; ι_{i}; ι_{j}; G^{i}; G^{j}; ∃!f; ι_{i}^{j}]

The proof is a simple matter of applying the definition.

From this property we deduce the functoriality of direct limits, as follows. We make all direct systems of groups over a given directed set into a category by defining morphisms between them in the obvious way (collections of homomorphisms that commute appropriately with the ι_{i}^{j}). The proposition then shows that a morphism (G^{i}) ⟶ (H^{i}) induces a morphism of the direct limits \underrightarrow{lim} G^{i} ⟶ \underrightarrow{lim} G^{j} in an obvious way. Hence \underrightarrow{lim} is a functor. In fact, it is much more:

Proposition 2.3.4 \underrightarrow{lim} is an exact functor. That is, if

(G^{i}) ⟶ (H^{i}) ⟶ (M^{i})

is an exact sequence of direct systems over I (in an obvious sense) then the sequence

\underrightarrow{lim} G^{i} ⟶ \underrightarrow{lim} H^{i} ⟶ \underrightarrow{lim} M^{i}

is exact.

For the proof, see [7].

2.3.3 Cohomology of Profinite Groups

Let A be a G-module where G is a profinite group. Suppose A = ⋃_{H} A^{H} where H runs through all the open normal subgroups of G. The H form a directed set with respect to the ordering H_{1} ≤ H_{2} ⟺ H_{2} is a subgroup of H_{1}. As a first observation, this shows that A = \underrightarrow{lim}_{H} A^{H}.

Note next that if H_{1} ≤ H_{2} then H_{1}/H_{2} must be a normal subgroup of G/H_{2}. So the quotient map

G/H_{2} ⟶ G/H_{1}( ≅ (G/H_{2})/(H_{1}/H_{2}))

gives, by the results of §1.4, an inflation homomorphism

Inf_{q} : H^{q}(G/H_{1}, A^{H_{1}}) ⟶ H^{q}(G/H_{2}, A^{H_{2}})

for each q. The cohomology groups and inflation homomorphisms now form a direct system of groups, in the sense of §2.3.1. Hence we may define

H^{q}(G, A) := \underrightarrow{lim}_{H} H^{q}(G/H, A^{H})

which is a direct limit of finite groups.

We can use this definition to deduce certain properties of H^{q}(G, A) from the finite case. For example, if all the H^{q}(G/H, A^{H}) are trivial then the definition shows H^{q}(G, A) must also be.

Furthermore Proposition 2.3.4 shows that the same exact cohomology sequences we have for finite groups hold also for this new definition for infinite groups.

Note that we could simply apply our original definitions of cohomology from §1.1 to the G-module A. The groups H^{q}(G, A) thus obtained are not the same as the ones we have defined here. The reason is that it is important to consider the topology on the group, which was not done in that definition. However, our new definition exactly comes from considering the topology on the group, as will be demonstrated soon.

2.3.4 Continuous Cochains

Definition 2.3.5 Let G be a profinite group and A a G-module, given the discrete topology. A will be called a discrete G-module if the action

\begin{matrix}G × A & ⟶ & A \\ (g, a) & ⟼ & ga \end{matrix}

is continuous.

If A is a discrete G-module, a q-dimensional continuous cochain is a continuous map

G^{q} ⟶ A

Proposition 2.3.6 If G is a profinite group and A a discrete G-module, and f : G ⟶ A is a function, then the following are equivalent.

1. f is continuous; 2. For each g ∈ G, there is a neighbourhood of g on which f is constant (“f is

locally constant”);

3. f can be filtered through a finite quotient of G, ie. there exists an open normal subgroup H of G such that f is equal to the composition

G ⟶^{proj.} G/H ⟶^{f̄} A

PROOF - (1) ⇒ (2): Let g ∈ G, then f^{−1}({f(g)}) is open in G (since A has the discrete topology) and so there is a neighbourhood of g on which f takes only the value f(g).

(2) ⇒ (3): Take for each g an open neighbourhood gM_{g} of g on which f is constant. These form an open cover of the compact group G and so there is a finite subcover {gM_{g} : g ∈ S}, where S ⊆ G is a finite subset. The intersection ⋂_{g∈S} M_{g} is an open subgroup M of G such that g is constant on gM for each g ∈ G. M contains an open normal subgroup H of G by the definition of profinite group. Then f is constant on each gH, ie. the diagram

G ⟶^{f} A
↘  ↗f̄
G/H

commutes.

(3) ⇒ (1): If a ∈ A and f(g) = a then f^{−1}({a}) contains gH where H is an open normal subgroup of G, and so is open in G. Then for any open set U ∈ A (ie. any subset) we have f^{−1}(U) = ⋃_{u∈U} f^{−1}({u}) is open, and so f is continuous.

□

Corollary 2.3.7 A function G^{q} ⟶ A is continuous if and only if it is locally constant.

PROOF - The above discussion extends easily to a finite product of copies of G.

□

Note that a continuous map on G has finite image.

Let C^{n}(G, A) denote the group of continuous cochains G^{n} ⟶ A. Define a coboundary map

d : C^{n}(G, A) ⟶ C^{n+1}(G, A)

by the usual formula for inhomogeneous cochains (1.5):

df[g_{1}, …, g_{q+1}] = g_{1}f[g_{2}, …, g_{q+1}]

+ ∑_{j=1}^{q}(−1)^{j}f[g_{1}, …, g_{j}g_{j+1}, …, g_{q+1}]

+(−1)^{q+1}f[g_{1}, …, g_{q}].

It is clear that if f is continuous then so is df, using the previous proposition. We check as in §1.2.1 that

C(G, A) : 0 ⟶ C^{0}(G, A) ⟶^{d_{0}} C^{1}(G, A) ⟶^{d_{1}} C^{2}(G, A) ⟶ …

is a cochain complex. Hence we may define the cohomology groups

H^{q}(C(G, A)) := ker d_{q}/im d_{q−1}

Proposition 2.3.8 H^{q}(C(G, A)) is naturally equivalent to the H^{q}(G, A) defined in §2.3.3.

SKETCH PROOF - First we want an isomorphism of groups

θ : C^{q}(G, A) ⟶ \underrightarrow{lim}_{H} C^{q}(G/H, A^{H})

Where H runs through the open normal subgroups of G. We note that, using the usual definition of inhomogeneous cochains in terms of the standard free G/H resolution of ℤ, C^{q}(G/H, A^{H}) = Hom_{G}(P_{q}(G/H), A^{H}) because G/H is finite (and so continuity of maps on G/H is automatic). Given a map f ∈ C^{q}(G, A), we know f can be filtered through G^{q}/H′^{q} for some open normal subgroup H′ of G. Furthermore im f is finite, so there is H′′ with im f ⊆ A^{H′′} by Definition 2.3.5. Set H = H′ ∩ H′′, and map f to the class of f̄: G^{q}/H^{q} ⟶ A^{H} in \underrightarrow{lim} C^{q}(G/H, A^{H}). This gives a well-defined homomorphism θ.

On the other hand we use Proposition 2.3.3 to define an inverse. For h ∈ C^{q}(G/H_{1}, A^{H_{1}}) map h by inflation to a map in C^{q}(G, A). This commutes with the inflations C^{q}(G/H_{1}, A^{H_{1}}) ⟶ C^{q}(G/H_{2}, A^{H_{2}}) for H_{2} ⊆ H_{1} and so defines a map

\underrightarrow{lim} C^{q}(G/H, A^{H}) ⟶ C^{q}(G, A)

which, checking on the C^{q}(G/H, A^{H}), is inverse to θ. Hence we have the isomorphism.

Now the functoriality of direct limits (§2.3.2) defines another coboundary map d′, so that we have a diagram

C^{q}(G, A)  ⟶^{d}  C^{q+1}(G, A)
↓≀  ↓≀
\underrightarrow{lim} C^{q}(G/H, A^{H}) ⟶^{d′} \underrightarrow{lim} C^{q+1}(G/H, A^{H})

which is easily seen to be commutative since continuous maps can be filtered through finite quotients.

Finally we have to show that taking cohomology commutes with taking direct limits. For each q ≥ 0, the exact sequences

0 ⟶ ker f_{q} ⟶ C^{q}(G/H, A^{H}) ⟶^{d_{q}} im d_{q} ⟶ 0

make up an exact sequence of direct systems, and so

0 ⟶ \underrightarrow{lim}_{H} ker d_{q} ⟶ \underrightarrow{lim}_{H} C^{q}(G/H, A^{H}) ⟶^{d′_{q}} \underrightarrow{lim}_{H} im d_{q} ⟶ 0

is exact, giving natural equivalence

\frac{ker d′_{q}}{im d′_{q−1}} = \frac{\underrightarrow{lim}_{H} ker d_{q}}{\underrightarrow{lim}_{H} im d_{q−1}} = \underrightarrow{lim}_{H} \frac{ker d_{q}}{im d_{q−1}} .

2.3.5 The Restriction-Inflation Sequence

Recall the Restriction-Inflation sequence of §1.4.1. We show an analogous result for profinite groups. First we need some technical results about profinite groups. The quotient of a profinite group by a closed subgroup is again a profinite group, under the identification topology.

Let G be a profinite group, G = \underleftarrow{lim}_{U} G/U, where U runs through the open

normal subgroups of G.

Proposition 2.3.9 1. If H is a closed subgroup of G, H ≅ \underleftarrow{lim}_{U} H/H ∩ U.

2. If H is a closed, normal subgroup of G, G/H ≅ \underleftarrow{lim}_{U} G/UH.

For the proof, see [4].

Proposition 2.3.10 Suppose H^{n}(H/H ∩ U, A^{U}) = 0 for all U and all n with 1 ≤ n < q. Then the sequence

0 ⟶ H^{q}(G/H, A^{H}) ⟶^{Inf} H^{q}(G, A) ⟶^{Res} H^{q}(H, A)

is exact.

PROOF - In the finite restriction-inflation sequence (Proposition 1.4.2), take G/U as the main group “G” and HU/U ≅ H/H ∩ U as the subgroup “H”. The isomorphism is by the second isomorphism theorem. The quotient of these is G/HU by the third isomorphism theorem. Take A^{U} as the module “A”. We obtain an exact sequence

0 ⟶ H^{q}(G/HU, A^{HU}) ⟶^{Inf} H^{q}(G/U, A^{U}) ⟶^{Res} H^{q}(H/H ∩ U, A^{U})

This forms an exact sequence of direct systems, and we pass to the direct limit over U to obtain exact

0 ⟶ H^{q}(G/H, A^{H}) ⟶^{Inf} H^{q}(G, A) ⟶^{Res} \underrightarrow{lim}_{U} H^{q}(H/H ∩ U, A^{U}).

Let us see that

\underrightarrow{lim}_{U} H^{q}(H/H ∩ U, A^{U}) = H^{2}(H, A) = \underrightarrow{lim}_{U} H^{q}(H/H ∩ U, A^{H∩U}).

The exact sequence of H/H ∩ U-modules

0 ⟶ A^{U} ⟶ A^{H∩U} ⟶ A^{H∩U}/A^{U} ⟶ 0

gives exact

H^{q−1}(\frac{H}{H ∩ U} , \frac{A^{H∩U}}{A^{U}} ) ⟶ H^{q}(\frac{H}{H ∩ U} , A^{U}) ⟶ H^{q}(\frac{H}{H ∩ U} , A^{H∩U}). (2.1)

This is another exact sequence of direct systems. So we may pass to the direct limit over U to obtain an exact sequence of groups.

Let us show that \underrightarrow{lim}_{U} H^{n}(H/H ∩ U, A^{H∩U}/A^{U}) = 0. Let f be a cocycle. Choose

U′ so that A^{U′} ⊇ im f (a finite set because the domain of f is finite). Then

Inff : (H/H ∩ U′)^{n} ⟶ A^{H∩U′}/A^{U′}

is the zero map. So the class of f is zero. This shows the vanishing of the direct limits and then (2.1) gives us the identification we wanted.

□

2.4 Infinite Galois Cohomology

We apply this theory to the cohomology of infinite Galois extensions. Recall that the Galois group Gal(L/K) of a Galois field extension L/K acts on the multiplicative group L^{∗} and the additive group L, making them abelian G-modules. This action is continuous if L, L^{∗} are given the discrete topology, and G has the canonical (Krull) topology (see §2.2.1). To see this, suppose a ∈ L, then a is contained in a finite Galois extension M ⊆ L of K. Let (g, b) be an element of the preimage of a under the map

G × L ⟶ L

so that gb = a. Then b ∈ M because M is a normal extension. Hence the open set gGal(L/M) × {b} is also contained in the preimage of a, and so this preimage is open. Hence the action is continuous.

Therefore L and L^{∗} are discrete G-modules in the sense of Definition 2.3.5. We see

H^{q}(Gal(L/K), L^{∗}) = \underrightarrow{lim}_{H}(Gal(L/K)/H, L^{∗H}) = \underrightarrow{lim}_{E}(Gal(E/K), E^{∗})

where H runs through all open normal subgroups, or, equivalently^{7}, E runs through all finite Galois subextensions of L/K, ordered by E ≤ E′ ⟺ E ⊆ E′. Exactly the same result holds for the additive group.

7By the Fundamental Theorem of Galois Theory, 2.2.12!

Theorem 2.4.1 H^{q}(Gal(L/K), L) = 0 for all q ≥ 1.

PROOF - Pass to the limit on L in Theorem 2.1.1.

□

We write again H^{q}(L/K) for H^{q}(Gal(L/K), L^{∗}).

Theorem 2.4.2 H^{1}(L/K) = 0.

PROOF - Pass to the limit on L in Theorem 2.1.2.

□

2.5 Functoriality of Galois Cohomology Groups

We conclude this section by looking at the object H^{q}(L/K) more closely. It depends functorially on the pair (L, K) =: L/K. Take the class of Galois field extensions L/K and make it into a category C by defining morphisms

L/K ⟶ L′/K′

whenever there is a field homomorphism ι : L ↪ L′ with ι(K) ⊇ K′. If G := Gal(L/K), G′ := Gal(L′/K′) then there is a corresponding morphism G′ ⟶ G, where σ ∈ Gal(L′/K′) acts on L/K by

σ(λ) = ι^{−1} ∘ σ ∘ ι(λ),

(note σ(ι(λ)) ∈ ι(L) by normality). The restriction morphism then gives

Res : H^{q}(G, L^{∗}) ⟶ H^{q}(G′, ι(L^{∗})) ⟶ H^{q}(G′, L′^{∗})

ie. Res : H^{q}(L/K) ⟶ H^{q}(L′/K′). We see that H^{q}(L/K) is a covariant functor in the argument L/K from C to the category of abelian groups.

The Brauer Group

3.1 Definitions

If K is a field, let K_{s} denote the separable closure of K, which is the union of all finite Galois extensions of K. The usual Zorn’s lemma arguments shows that this field is well-defined and unique up to isomorphism. In the last chapter we saw how to define H^{2}(K_{s}/K) as a direct limit of H^{2}(L/K) over all finite Galois extensions L/K. This group^{1} H^{2}(K_{s}/K) is called the Brauer Group of K, and will also be written Br(K). It occupies a central place in Local Class Field Theory by dint of Tate’s Theorem (1.6.4). It also has a very interesting arithmetic interpretation, and the purpose of this chapter is to explore this.

It turns out that each element of Br(K) corresponds to a class of central simple K-algebras. A central simple K-algebra is a ring A with centre K and which is finite-dimensional over K, and furthermore is a simple ring. We refer to Appendix E for basic facts about central simple algebras.

3.2 Characterization of Central Simple Algebras

Proposition 3.2.1 Let A be a finite-dimensional algebra over a field K. Then the following properties are equivalent:

(a) A is central simple over K;

(b) A⊗_{K} K̄ ≅ M_{n}(K̄) as K̄-algebras, for some n, where K̄ is an algebraic closure of K;

(b’) There exists a finite Galois extension L of K such that A ⊗_{K} L ≅ M_{n}(L) as L-algebras, for some n;

(c) A ≅ M_{n}(D) as K-algebras, for some n, where D is a finite-dimensional central division algebra over K.

1This group is well-defined, ie. independent of our choice of separable closure. See [4].

PROOF - “(a) ⇒ (c)” is Wedderburn’s Theorem, proved in §E.2.

“(b’) ⇒ (b)”: Given (b’),

A ⊗_{K} K̄ ≅ A ⊗_{K} (L ⊗_{L} K̄) ≅ (A ⊗_{K} L) ⊗_{L} K̄ ≅ M_{n}(L) ⊗_{L} K̄ ≅ M_{n}(K̄)

“(b) ⟺ (a)” is shown in §E.5.

Then “(c) ⇒ (b’)” follows from Proposition E.6.1 and the results of §E.3.

□

3.3 Structure of the Classical Brauer Group

Consider the class of central simple of algebras over K, and the equivalence relation on this class given by isomorphism of K-algebras. Any such algebra is isomorphic as a K-algebra to the K-vector space K^{n} supplied with a ring multiplication, which is a function μ : K^{n} × K^{n} ⟶ K^{n}. This shows that the isomorphism relation has a set of representatives, consisting of the set^{2} of such pairs (K^{n}, μ).

Two central simple K-algebras will be called similar or equivalent if the division algebras associated to them by Proposition 3.2.1 are isomorphic (as K-algebras).

By the uniqueness up to isomorphism of the D (§E.2) this is an equivalence relation on the set of isomorphism classes of central simple K-algebras.

Write [A] for the class of A, and Br(K) for the set of equivalence classes. Br(K) is given the structure of an abelian group, and so becomes the classical Brauer Group of K, as follows. Define the product of [A] and [B] by

[A] · [B] := [A ⊗_{K} B]

We have to check this is well-defined. Well,

M_{n}(D) ⊗_{K} M_{n′}(D′) ≅ (M_{n}(K) ⊗_{K} M_{n′}(K)) ⊗_{K} (D ⊗_{K} D′)
≅ M_{nn′}(K) ⊗_{K} M_{m}(D′′)
≅ M_{nn′m}(D′′)

and then it follows from Proposition E.2.3 of §E.2 that D′′ is determined up to isomorphism by the isomorphism classes of D and D′, so that the class [A ⊗_{K} B] depends only on the similarity classes of A and B, and multiplication is well-defined.

The commutativity and associativity of ⊗_{K} give the same properties for the group law.

Since

[K] · [B] = [K ⊗_{K} B] = [B] = [B ⊗_{K} K] = [B] · [K]

[K] is an identity element for the group.

Now §E.5 shows A ⊗_{K} A^{∘} ≅ End_{K}(A) ≅ M_{n}(K), so that [A] · [A^{∘}] = [K], ie. [A^{∘}] is an inverse for [A]. Thus Br(K) is an abelian group.

2Recall that a class is a set if it has a cardinal number, that is, it is equipotent with a set which can be built up from the axioms of set theory, like this {(K^{n}, μ)}

3.4 Relating the Brauer Group to a One-Dimensional Cohomology Set

Let L ⊇ K be a field extension, and write A_{n}(L) for the set of elements [A] of the Brauer group such that A ⊗_{K} L ≅ M_{n}(L).^{3}

The strategy is to relate A_{n}(L) to the cohomology set H^{1} of a particular nonabelian G-module^{4}. Let A be a central simple K-algebra with A ⊗_{K} L ≅ M_{n}(L); let F : M_{n}(L) ⟶ A ⊗_{K} L be the isomorphism. For s ∈ Gal(L/K) =: G, let sF be defined by

\begin{aligned}sF : & M_{n}(L) & ⟶ & A ⊗_{K} L \\ & (λ_{ij})_{i,j} & ⟼ & s(F ((s^{−1}λ_{ij})_{i,j})) \end{aligned}

ie. sF = s ∘ F ∘ s^{−1}. Put θ(A)[s] := F^{−1} ∘ sF ∈ Aut(M_{n}(L)), ie. θ(A)[s] = F^{−1}∘s∘F ∘s^{−1}. Here Aut(M_{n}(L)) =: C is the group of L-linear ring automorphisms of M_{n}(L) under composition, made into a non-abelian G-module by sc := s∘c∘s^{−1} for s ∈ G, c ∈ C.

Proposition 3.4.1 θ(A) is a 1-cocycle of G with values in C, which depends only on the similarity class [A] of A. Choosing a different F changes θ(A) to a cohomologous cocycle. Thus θ determines a function A_{n}(K) ⟶ H^{1}(G, C).

PROOF - Recall from Appendix B that H^{1}(G, C) consists of 1-cocycles f : G ⟶ C, ie. f[st] = f[s] ∘ sf[t], and two such cocycles f, f′are cohomologous if there is c ∈ C such that f[s] = c^{−1} ∘ f′[s] ∘ sc for all s ∈ G. Let us check the 1-cocycle condition on θ(A):

θ(A)[st] = F^{−1} ∘ st ∘ F ∘ t^{−1}s^{−1}
= F^{−1} ∘ (s ∘ F ∘ s^{−1} ∘ s ∘ F^{−1} ∘ s^{−1}) ∘ st ∘ F ∘ t^{−1}s^{−1}
= θ(A)[s] ∘ s ∘ F^{−1} ∘ t ∘ F ∘ t^{−1} ∘ s^{−1}
= θ(A)[s] ∘ sθ(A)[t]

Next we check the independence of F. If F, F′ : M_{n}(L) ⟶ A ⊗_{K} L are isomorphisms defining θ(A) and θ′(A) then

θ(A)[s] = F^{−1} ∘ s ∘ F ∘ s^{−1}
= (F^{−1} ∘ F′) ∘ F′^{−1} ∘ s ∘ F′ ∘ s^{−1} ∘ (s ∘ F′^{−1} ∘ F ∘ s^{−1})
= c^{−1} ∘ θ′(A)[s] ∘ sc

3We might choose L = K̄ to be an algebraic closure of K, then Br(K) = ∪_{n=1}^{∞}A_{n}(L), since L neutralises all central simple K-algebras. Then we would relate it to the group H^{2}(K_{s}/K) by using the definition of this group in terms of continuous cochains. This is the approach taken by Weil in Basic Number Theory. Alternatively, we can take L/K to be a finite extension and pass to the limit later. The second approach is shown here.

4This is a special case of the general method of “Galois descent”; A_{n} is a set of K-vector spaces with a particular type of additional structure, which all become isomorphic to each other when the functor ⊗_{K}L is applied (isomorphic in the sense that a morphism must also preserve the additional structure). Such systems can be systematically related to cohomology sets H^{1}. See [1], Chapter 10 for more details.

where c = F^{−1} ∘ F′, so that θ(A) is cohomologous to θ′(A).

Now suppose A′ is a similar algebra to A, then A ≅ M_{m}(D) and A′ ≅ M_{m′}(D) for some m, m′. But comparing K-dimensions (since both A, A are in A_{n}(L)) shows that m = m′, so that in fact A ≅ A′. This induces an isomorphism μ : A ⊗_{K} L ⟶ A′ ⊗_{K} L which commutes with the action of Gal(L/K), s(a ⊗ λ) = a ⊗ sλ. Thus if F′ : M_{n}(L) ⟶ A′ ⊗_{K} L is an isomorphism then F′ = μ ∘ F for some isomorphism F : M_{n}(L) ⟶ A ⊗_{K} L, and so

θ(A′) = F^{−1} ∘ μ^{−1} ∘ s ∘ μ ∘ F ∘ s^{−1}
= F^{−1} ∘ (μ^{−1} ∘ μ) ∘ s ∘ F ∘ s^{−1}
= θ(A)

Also, θ(M_{n}(K)) = 1 because G acts trivially on K, which verifies that θ is a morphism of pointed sets.

□

Proposition 3.4.2 θ is injective.

PROOF - Suppose θ(A) = θ(B), ie. we have

\begin{matrix}F : & M_{n}(L) & ⟶ & A ⊗_{K} L \\ F′ : & M_{n}(L) & ⟶ & B ⊗_{K} L \end{matrix}

with F^{−1} ∘ s ∘ F ∘ s^{−1} = F′^{−1} ∘ s ∘ F′ ∘ s^{−1} for all s ∈ G. We want to show that A ≅ B. We have

F′ ∘ F^{−1} ∘ s = s ∘ F′ ∘ F^{−1} for all s ∈ G

so that f := F′ ∘ F^{−1} : A ⊗_{K} L ⟶ B ⊗_{K} L is an isomorphism that commutes with the action of the Galois group.

Take a⊗κ ∈ A⊗_{K} K; we show that f(a⊗κ) ∈ B⊗_{K} K. Any element x ∈ B⊗_{K} L can be uniquely written as:

x = ∑_{i=1}^{dim_{K} B} b_{i} ⊗ λ_{i}

where {b_{i}} is a K-basis for B, and the λ_{i} ∈ L. Recall s ∈ G takes x to sx = ∑ b_{i} ⊗ sλ_{i}. If G acts trivially x, then it follows that sλ_{i} = λ_{i} for each λ_{i}, which shows that λ_{i} ∈ K and so x ∈ B ⊗_{K} K. Applying this to the G-invariant element f(a ⊗ κ) shows that f restricts to an injective morphism of K-algebras

A ⊗_{K} K ⟶ B ⊗_{K} K

Comparing K-dimension shows this is an isomorphism, and then identifying A ⊗_{K} K ≅ A shows A ≅ B, so [A] = [B].

Proposition 3.4.3 θ is surjective

PROOF - Let f : G ⟶ C be a 1-cocycle. We want to show that there is a central simple K-algebra A such that A ⊗_{K} L ≅ M_{n}(L) and there is such an isomorphism F : M_{n}(L) ⟶ A ⊗_{K} L with f[s] = F^{−1} ∘ s ∘ F ∘ s^{−1}.

We build the required K-algebra from M_{n}(L). Let F be an L-automorphism of M_{n}(L), then choosing a basis of M_{n}(L) identifies F with a matrix X = (x_{ij})_{i,j} in GL(n^{2}, L). Recall sF is the L-automorphism of M_{n}(L) defined by s ∘ F ∘ s^{−1}. This L-automorphism has matrix sX = (sx_{ij})_{i,j}, for we can choose the basis (v_{i})_{i=1}^{n^{2}} of M_{n}(L) to be G-invariant, for example, the standard basis. Then

(sF)v_{i} = s(F v_{i}) = s(∑_{j} x_{ji}v_{j}) = ∑_{j}(sx_{ji})v_{j}

Applying this to the L-automorphisms f[st], f[s] and sf[t] shows that, if X[st] etc. are the associated matrices,

f[st] = f[s] · sf[t] ⇒ X[st] = X[s] · sX[t]

So this X : G ⟶ GL_{(}n^{2}, L) is a 1-cocycle. Now we can apply the result about the Galois cohomology of such a group (Proposition 2.1.4):

H^{1}(G, GL(n^{2}, L)) = 0

so that X is in fact a 1-coboundary, ie. there is Y ∈ GL(n^{2}, L) with

X[s] = Y^{−1} · sY for all s ∈ G.

Y determines an L-linear map

F : M_{n}(L) ⟶ M_{n}(L)

such that f[s] = F^{−1} · (sF) = F^{−1} · s · F · s^{−1}.

We can use this to define a composition law on M_{n}(L)

(A, B) ⟼ A ⊙ B := F (F^{−1}(A) · F^{−1}(B))

where · is usual matrix multiplication.

⊙ is associative because · is associative, and distributive because F is L-linear. F (1) is the identity for multiplication. So we have defined a new ring structure on M_{n}(L). In fact, this ring structure restricts to the subspace M_{n}(K). For:

s(A ⊙ B) = s(F (F^{−1}(A) · F^{−1}(B))
= (sF)(s(F^{−1}(A)) · s(F^{−1}(B)))
= (sF)((sF^{−1})(sA) · (sF^{−1})(sB))
= (F ∘ f[s])((f[s]^{−1} ∘ F^{−1})(sA) · (f[s]^{−1} ∘ F^{−1})(sB))
= (F ∘ f[s])f[s]^{−1}(F^{−1}(sA) · F^{−1}(sB))

(since f[s] is a ring homomorphism)

= (sA) ⊙ (sB)

So if A, B ∈ M_{n}(K) and so are fixed by G, then A ⊙ B is fixed by G and so is in M_{n}(K).

By looking at the standard basis under the identification M_{n}(K)⊗_{K} L ≅ M_{n}(L), we see that our ring multiplication ⊙ in M_{n}(L) is the same as the ring multiplication in M_{n}(K) ⊗_{K} L defined by

(A ⊗ λ, B ⊗ λ′) ⟼ (A ⊙ B) ⊗ λλ′

so that, if we put A for M_{n}(K) with multiplication defined by ⊙,

F : M_{n}(L) ⟶ A ⊗_{K} L

gives f[s] = F^{−1} ∘ sF

□

3.5 Brauer Groups as Two-Dimensional Cohomology Groups

So far we have an isomorphism of pointed sets

θ : A_{n}(L) ⟶ H^{1}(G, C)

where C is the G-module of L-linear ring automorphisms of M_{n}(L). The next task is to identify C and fit it in a short exact sequence, so we can use the cohomology sequence of Appendix B. We use the following result, a special case of the Theorem of Skolem-Noether:

Proposition 3.5.1 Let B be a central-simple L-algebra. Then every L-linear ring automorphism of B is of the form x ⟼ b^{−1}xb for some invertible b ∈ B, ie. is an inner automorphism.

For the proof, see [10], Chapter 9, §1, Prop. 4.

Applying this to M_{n}(L) shows that an element of C is an inner automorphism

X ⟼ Q^{−1}XQ

for some Q ∈ GL(n, L). Multiplying Q by an element of L^{∗} does not change the automorphism; on the other hand, if

Q^{−1}XQ = Q′^{−1}XQ′

for some invertible Q and Q′ in GL(n, L) and all X ∈ M_{n}(L) then Q′Q^{−1} is in the centre L of GL(n, L). Hence the exact sequence of non-abelian G-modules

1 ⟶ L^{∗} ⟶ GL(n, L) ⟶ C ⟶ 1

which shows that C is isomorphic as a G-module and L-algebra to PGL(n, L) = GL(n, L)/L^{∗}, a projective general linear group of L.

We now apply again the Proposition 2.1.4 that H^{1}(G, GL(n, L)) is trivial. The exact cohomology sequence of Appendix B gives an exact sequence

1 ⟶ H^{1}(G, C) ⟶^{δ} H^{2}(G, L^{∗})

That is, a morphism of pointed sets δ : H^{1}(G, C) ⟶ H^{1}(G, L^{∗}) with trivial “kernel” (preimage of the identity element). Thus we have a morphism of pointed sets

α_{n} = δ ∘ θ : A_{n}(L) ⟶ H^{2}(G, L^{∗})

which we now study.

Proposition 3.5.2 α_{nm}([A][B]) = α_{n}([A]) · α_{m}([B]), where · denotes composition in the abelian group H^{2}(G, L^{∗}) and A and B are central-simple K-algebras such that A ⊗_{K} L ≅ M_{n}(L) and B ⊗_{K} L ≅ M_{m}(L).

PROOF - Isomorphisms

\begin{matrix}M_{n}(L) & ⟶^{F} & A ⊗_{K} L \\ M_{m}(L) & ⟶^{F′} & B ⊗_{K} L \end{matrix}

induce an isomorphism

M_{nm}(L) = M_{n}(L) ⊗_{L} M_{m}(L) ⟶^{F ⊗F′} (A ⊗_{K} L) ⊗_{L} (B ⊗_{K} L) = (A ⊗_{K} B) ⊗_{K} L

Then θ([A]) is defined by θ([A])[s] = F^{−1} ∘ s ∘ F ∘ s^{−1}, θ([B]) similarly in terms of F′, and θ([A][B]) similarly in terms of F ⊗ F′. Lift these automorphisms of matrix algebras to matrices giving them by conjugation, so

θ([A])[s] ⇝ M[s]  such that θ([A])[s](X) = M[s]XM[s]^{−1}
θ([B])[s] ⇝ M′[s]
θ([A][B])[s] ⇝ M[s] ⊗ M′[s]

Applying d gives d(M ⊗ M′)[s, t] = (dM)[s, t] ⊗ (dM′)[s, t], and we know these matrices are now of the form λ1_{M} for some λ ∈ L^{∗} and for 1_{M} ∈ M_{nm}(L). So comparing the top-left entry (under the correspondences defined in §E.3) shows that

α_{mn}([A ⊗_{K} B])[s, t]1_{M} = d(M ⊗ M′)[s, t]
= dM[s, t] ⊗ dM′[s, t]
= α_{n}([A])[s, t]1_{M} ⊗ α_{m}([B])[s, t]1_{M}

so α_{mn}([A ⊗_{K} B]) = α_{n}([A])α_{m}([B]) in H^{2}(G, L^{∗}).

□

Now the group A(L/K) := ⋃_{n=1}^{∞} A_{n}(L) is a subgroup of the Brauer group Br(K). All the α_{n} together define a homomorphism

α : A(L/K) ⟶ H^{2}(G, L^{∗})

Proposition 3.5.3 α is injective

PROOF - If α([A]) = α([B]) then α_{n}([A][B]^{−1}) = 1 for some n. So [A][B]^{−1} ∈ θ^{−1}(δ^{−1}({1})) = {1}, ie. [A] = [B], so α is injective.

□

Lemma 3.5.4 (See [1], Chapter 10, §5) if n = [L : K], then

δ_{n} : H^{1}(G, PGL(n, L)) ⟶ H^{2}(G, L^{∗})

is surjective.

PROOF - We need to show that given a 2-cocycle

h : G^{2} ⟶ L^{∗}

there is a 1-cochain f : G ⟶ GL(n, L) such that h[s, t] = f[s] · sf[t] · f[st]^{−1}. Let V be the L-vector space with basis (e_{s})_{s∈G}, so dim_{L} V = n. Let f[s] ∈ Hom_{K}(V, V) be given by

\begin{aligned}f[s] : & V & ⟶ & V \\ & e_{t} & ⟼ & h[s, t]e_{st} \end{aligned}

Then

f[s] ∘ sf[t](e_{u}) = h[s, tu] · sh[t, u]e_{stu}
h[s, t]f[st](e_{u}) = h[s, t] · h[st, u]e_{stu}

and because h is a 2-cocycle, this gives h[s, t] = f[s] · sf[t] · f[st]^{−1}, as required.

□

Thus we have a group isomorphism

α : A(L/K) ⟶ H^{2}(L/K)

Now

Br(K) = ∪A(L/K) = \underrightarrow{lim} A(L/K)

as L runs through all finite Galois extensions of K, and similarly

H^{2}(K_{s}/K) = \underrightarrow{lim} H^{2}(L/K)

One checks that the maps in the direct systems commute with the α, so that the α define a group isomorphism

Br(K) ≅ H^{2}(K_{s}/K)

Local Class Field Theory

4.1 The Restriction-Inflation Sequence

Proposition 4.1.1 Let L ⊇ M ⊇ K be a tower of fields with L/M and M/K Galois extensions. Then there is an exact sequence

0 ⟶ H^{2}(M/K) ⟶ H^{2}(L/K) ⟶ H^{2}(L/M)

PROOF - Let G = Gal(L/K) and H = Gal(L/M) the subgroup corresponding to M. Note that Gal(M/K) = G/H by the Fundamental Theorem of Galois Theory. The result then follows from Theorem 2.4.2 and Proposition 2.3.10.

□

Corollary 4.1.2 With the hypotheses of the proposition, there is an exact sequence

0 ⟶ H^{2}(M/K) ⟶ Br(K) ⟶ Br(M)

PROOF - Set L = K_{s}.

4.2 Local Fields

There is a very special class of fields for which we can say enough about the Brauer group to use it to obtain the reciprocity isomorphism, our ultimate aim. In this section we describe these “local fields”. They are systematically studied in the first part of [1], and are also described in Anne Esslemont’s part of this project. We recall the basic definitions here.

Definition 4.2.1 Let K be a field and v : K ⟶ ℝ ∪ {+∞} be a map satisfying

1. v|_{K^{∗}} : K^{∗} ⟶ ℝ is a nonzero homomorphism

2. v(0) = +∞ 3. v(x + y) ≥ min{v(x), v(y)} for all x, y ∈ K

4. The image v(K^{∗}) is a discrete subgroup of ℝ.

Then v is called a discrete valuation on K.

v(K^{∗}) is an infinite cyclic group; if v(K^{∗}) = ℤ then v is called normalised. The set {x ∈ K : v(x) ≥ 0} is a subring of K called the valuation ring of v

and written O_{K}. The subset 𝔪_{K} := {x ∈ K : v(x) > 0} is the unique prime ideal of O_{K}. It is a principal ideal generated by any element π with minimal positive valuation (v(π) = 1 for a normalised valuation). The π are called uniformisers of K. All ideals of O_{K} are of the form π^{n}O_{K} = 𝔪_{K}^{n} for some n ≥ 0.

The set O_{K} − 𝔪_{K} comprises the elements of valuation zero; these are all the units, ie. the invertible elements of O_{K}. We write U_{K} = O_{K} − 𝔪_{K}.

Any principal ideal domain with a unique prime ideal is called a discrete valuation ring, and its field of fractions then has a discrete valuation defined on it by a uniformiser. Namely, every element is of the form π^{n}u for some unit u and integer n, which we take to be the valuation of the element.

Finally, we define the residue field of K to be

k := O_{K}/𝔪_{K},

which is a field because 𝔪_{K} is a maximal ideal^{1}.

The discrete valuation v defines an absolute value of K as follows. Pick any real number a with 0 < a < 1. Define for x ∈ K,

∥x∥_{v} := a^{v(x)}.

Then ∥ ∥_{v} satisfies

1. ∥xy∥_{v} = ∥x∥_{v}∥y∥_{v},

2. ∥0∥_{v} = 0,

3. ∥x + y∥_{v} ≤ max{∥x∥_{v}, ∥y∥_{v}}.

The third property is called the non-Archimedean triangle inequality; an absolute value which has it is called ultrametric or non-Archimedean. ∥ ∥_{v} defines a metric on K in the usual way,

d(x, y) = ∥x − y∥_{v},

and hence a topology on K , called the topology defined by v. This topology does not depend on the choice of the number a.

Definition 4.2.2 If K is a field and v a discrete valuation of K such that K is complete with respect to the topology defined by v, and the residue field k is finite^{2}, then K will be called a local field.

1for there certainly exists a maximal ideal of O_{K}, which must be prime and so is 𝔪_{K}

2More generally, for the purposes of local class field theory, it is sufficient to take k to be quasi-finite. For a treatment of the quasi-finite residue field case, see [1], chapter 8, §2.

4.3 Extensions of Local Fields

If L/K is a finite extension and K is complete under a discrete valuation v, then v extends uniquely to L. That is to say, there is a unique discrete valuation w of L with w|_{K} = v. L is again complete with respect to the topology defined by w. This is shown in [1] to follow from the structure of L as a topological vector space over K, using results about such spaces in [9].

It is clear that a finite extension of a local field is again a local field.

We classify these extensions according to their “ramification properties”. If π_{L}, π_{K} are uniformisers of L and K respectively then the ratio of valuations

\frac{w(π_{K})}{w(π_{L})} =: e_{L/K}

is called the ramification index of L/K. Note that if w′ is a normalised multiple of w then e_{L/K} = w(π_{K}). If e = 1 then π_{K} is also a uniformiser of L.

Note that 𝔪_{K} = 𝔪_{L} ∩ O_{K} - just look at the valuations under w. Hence there is a natural injective homomorphism

O_{K}/𝔪_{K} ⟶ O_{L}/𝔪_{L}

on the residue fields; ie. l = O_{L}/𝔪_{L} is an extension of k. It can be shown that O_{L} is always a finitely-generated module over O_{K} ([1], Chapter 2, §2, Proposition 3), hence l/k is a finite extension. Define

f_{L/K} := [l : k];

it is called the residue degree of L/K. e_{L/K} and f_{L/K} are connected by the fundamental identity

[L : K] = e_{L/K}f_{L/K}.

See [1], Chapter 1, §5, Proposition 10.

Definition 4.3.1 Let L/K be a finite extension of a field K which is complete with respect to a discrete valuation. If e_{L/K} = 1 and the residue extension l/k is separable^{3} then we say L/K is an unramified extension.

This is the simplest case, and it turns out to be the most important when discussing the Brauer groups. It is particularly easy to work with because any uniformiser π_{L} of L is actually an element of K, and so fixed by K-linear maps such as those in Gal(L/K). We will make extensive use of the following fundamental results about unramified extensions, for which we refer to [1], Chapter 3, §6, Theorem 3.

3If K is a local field then this is always the case, since then k is a finite field and so is perfect, namely won’t admit inseparable extensions.

Theorem 4.3.2 Let L/K be a finite unramified extension, where K is a field complete with respect to a discrete valuation. Let l/k be the corresponding residue extension. Let L′/K be any finite extension, with residue extension l′/k. Then there is an isomorphism from the group of K-linear ring homomorphisms L ⟶ L′ to the group of k-linear ring homomorphisms l ⟶ l′. The isomorphism is defined by restricting a map L ⟶ L′ to a map O_{L} ⟶ O_{L′} and then reducing modulo the respective maximal ideals.

Corollary 4.3.3 With the hypotheses of the theorem, Gal(L/K) is canonically isomorphic to Gal(l/k).

Corollary 4.3.4 If M/K is a finite extension where K is as in the theorem, and m/k the corresponding residue extension, then the subextensions L/K of M/K which are unramified are in 1-1 correspondence with the separable subextensions l/k of m/k.

The union of all the unramified extensions of K within an algebraic closure of K is called the maximal unramified extension of K and written K_{nr}. The unique extension of discrete valuations on the finite level shows that K_{nr} has a discrete valuation, and is unramified in the sense that it has π_{K} as a uniformiser. K_{nr} is also a Galois extension. Its residue field is the separable closure k_{s} of k (see §3.1). Since this is infinite, we note that K_{nr} is not a local field in our sense.

4.4 Unramified Extensions and Brauer Groups

We return to the question of determining the Brauer Group of a local field K. In Corollary 4.1.2, set M = K_{nr}. We get an exact sequence

0 ⟶ H^{2}(K_{nr}/K) ⟶ Br(K) ⟶ Br(K_{nr})

We will prove soon:

Theorem 4.4.1 Br(K_{nr}) = 0.

It follows then that

Theorem 4.4.2 Br(K) = H^{2}(K_{nr}/K).

Hence the unramified extensions of K are sufficient to determine its Brauer group H^{2}(K_{s}/K).

There are a number of possible approaches to proving this theorem. In [1], Chapter 12, §1, Theorem 1, Serre uses results about the norm map in finite, “totally ramified” extensions to show that only unramified extensions are needed to determine Br(K). Alternatively, in [5], §§1.4, 1.5 he takes a different approach by looking at subgroups of the unit group in a finite extension L/K, and applying a technical result.

The third method is to use the interpretation of the Brauer group in terms of central simple algebras. This is the method we use here.

Let K be a field complete under a (normalised) discrete valuation v and D a central division algebra of rank n^{2} over K. We wish to extend the discrete valuation on K to a “discrete valuation” on D.

For this we use the absolute norm map N_{D/K} : D ⟶ K (see Appendix E, §E.7). One can also use the reduced norm Nrd. For x ∈ D^{∗}, set

w(x) = \frac{1}{n^{2}} v(N_{D/K}(x))

and let w(0) = +∞. Then w : D^{∗} ⟶ \frac{1}{n^{2}} ℤ is a homomorphism and w|_{K} = v since

N_{D/K}(y) = y^{n^{2}} for y ∈ K.

Now consider the restriction of w to a commutative subfield L of D containing K. We know (Proposition E.7.2) that N_{D/K}(λ) = N_{L/K}(λ)^{[D:L]} for all λ ∈ L, so that w(λ) = \frac{[D:L]}{n^{2}}v(N_{L/K}(λ)).

Now it follows from unique extension of the valuation of K to a valuation of L (see §4.3 and [1], Chapter 2, §2, Proposition 3) that v ∘ N_{L/K} is a multiple of the valuation of L; in particular, w restricts to a discrete valuation of L. Apply this to the field K(x^{−1}y) for x, y ∈ D^{∗}; it shows that

w(1 + x^{−1}y) ≥ min{w(1), w(x^{−1}y)}.

Add w(x) to each side to see that w(x + y) ≥ min{w(x), w(y)}.

Hence w satisfies all the axioms (Definition 4.2.1) of a usual discrete valuation on a commutative field.

In particular, the ring of integers of a commutative subfield L of D containing K is {x ∈ L : w(x) ≥ 0} and the maximal ideal is {x ∈ L : w(x) > 0}. L is complete in the topology defined by w, since w extends v.

Theorem 4.4.3 Let K be a field complete under a discrete valuation and having perfect residue field. Then every nontrivial finite dimensional central division algebra D over K contains a proper unramified extension of K.

PROOF - Let D have rank n^{2} over K. Suppose n^{2} > 1 but K does not have a proper unramified extension in D. Then for each field L with D ⊇ L ⊇ K, the residue field l = k; otherwise by Corollary 4.3.4 there is an unramified extension of K in L, not equal to K.

Let b ∈ D have w(b) ≥ 0. Then b reduces modulo the maximal ideal of O_{K(b)} to an element ā of k, and lifting to a ∈ O_{K} shows that w(b − a) > 0. Choose π ∈ D with minimal positive valuation (a uniformiser of D), then b = a + πb_{1} with w(b_{1}) ≥ 0. Repeating this procedure for b_{1} gives a series for b:

b = a + πa_{1} + π^{2}a_{2} + … a, a_{i} ∈ O_{K}.

K(π) is complete with respect to the topology defined by w. Hence b ∈ K(π). Now any x ∈ D has w(π^{ν}x) ≥ 0 for some positive integer ν and so is equal to π^{−ν}b for some such b. Hence D ⊆ K(π) and so is commutative - contradicting the assumption that D has centre K and n^{2} > 1.

□

Corollary 4.4.4 When K is as in the proposition, Br(K_{nr}) = 0

PROOF - If D is a finite dimensional central division algebra over K_{nr} of rank > 1, then D contains a proper finite unramified extension L of K_{nr}. Hence by Corollary 4.3.4 the residue extension l/k_{s} (where k_{s} is the separable closure of k - see §4.3) is nontrivial and separable, which is impossible by the maximality of k_{s}.

□

This proves Theorem 4.4.1.

Soon we will calculate H^{2}(K_{nr}/K) = Br(K) for a local field K. This turns out to be related to the Brauer group of the residue field k, by Witt’s theorem (4.8.2). We therefore turn now to determining the Brauer group of a finite field. We show a general criterion for vanishing of the Brauer group, and show that finite fields have this property.

4.5 Quasi-Algebraically Closed Fields

Definition 4.5.1 A field K is called quasi-algebraically closed, or C_{1}, or a Chevalley field, if it has the following property.

• Every nonzero homogeneous polynomial f ∈ K[X_{1}, …, X_{n}] of degree < n has a nontrivial zero 0 ≠ x ∈ K^{n}.

Proposition 4.5.2 if K is C_{1} then Br(K) = 0.

PROOF - Let D be a finite dimensional central division algebra over K of rank n^{2}. Fix a K-basis (u_{1}, …, u_{n^{2}}) of D; then the reduced norm (Appendix E §E.7)

Nrd(x_{1}u_{1} + … + x_{n^{2}}u_{n^{2}}) = f(x_{1}, …, x_{n^{2}}) x_{i} ∈ K

for f homogeneous of degree n in K[X_{1}, …, X_{n^{2}}], by Proposition E.7.4. The same proposition shows that the only zero of Nrd in D is 0. Hence the only zero of f is K^{n^{2}} is 0. So by the C_{1} property, deg f = n ≥ n^{2}. Thus n = 1 and D = K.

So all central division algebras over K are trivial and Br(K) = 0.

□

Remark 4.5.3 In fact the fields K_{nr} are C_{1}. This was proved by S. Lang in “On Quasi-Algebraic Closure”, Annals of Mathematics 55 (1952), pp. 373-390.

4.6 Finite Fields are Quasi-Algebraically Closed

This proof may be found in [12], Chapters 2 and 3. Let F = 𝔽_{q}, the finite field of q elements.

Proposition 4.6.1 Define the d-th power sum for a finite field F by

S_{d} = ∑_{x∈F} x^{d} d = 1, 2, …

Then S_{d} = −1 if q − 1|d and 0 otherwise.

PROOF - Since x^{q} = x for all x ∈ F, S_{d} depends only on the congruence class of d mod. q − 1.

S_{q−1} = ∑_{x∈F −{0}} 1 = q − 1 = −1.

So assume 1 ≤ d ≤ q − 1, and choose b ∈ F^{∗} with b^{d} ≠ 1, ie. not one of the d roots of X^{d} − 1 = 0. Then since multiplication by b permutes F, we have S_{d} = b^{d}S_{d}, so that S_{d} must be zero.

□

Proposition 4.6.2 Let f ∈ F [X_{1}, …, X_{n}] have degree d < n(q − 1). Then

∑_{x∈F^{n}} f(x) = 0

PROOF - It suffices to prove this for each monomial g(X_{1}, …, X_{n}) = X_{1}^{a_{1}}X_{2}^{a_{2}} … X_{n}^{a_{n}} with ∑ a_{i} = d < n(q − 1). If, say, a_{1} = 0 then each value g(∗, x_{2}, …, x_{n}) appears q times in the sum and so the sum is zero. So assume no a_{i} = 0. Then

∑_{x∈F^{n}} g(x) = ∑_{(x_{1},...,x_{n})} x_{1}^{a_{1}}x_{2}^{a_{2}} … x_{n}^{a_{n}}
= ( ∑_{x∈F} x^{a_{1}})( ∑_{x∈F} x^{a_{2}}) … ( ∑_{x∈F} x^{a_{n}})
= S_{a_{1}}S_{a_{2}} … S_{a_{n}}

But a_{1} + … + a_{n} < n(q − 1) so 0 < a_{i} < q − 1 for at least one index i. Then the factor S_{a_{i}} = 0.

□

Theorem 4.6.3 Let f ∈ F [X_{1}, …, X_{n}] have degree d < n. Then the number of zeros of f in F^{n} is divisible by p = char F.

PROOF - Let V = f^{−1}({0}) ⊆ F^{n}. Put f̄ = 1 − f^{q−1}, then f̄ is the characteristic function^{4} of V, and

∑_{x∈F^{n}} f̄(x) = Card(V) · 1_{F}

But deg f̄ = d(q − 1) < n(q − 1) and so by Proposition 4.6.2 this sum is 0 ∈ F. Hence p|Card(V).

□

Corollary 4.6.4 F is C_{1}.

Remark 4.6.5 The fact that Br(𝔽_{q}) = 0 is a famous result called Wedderburn’s Theorem (but not to be confused with the other result (Theorem E.2.2) also called Wedderburn’s Theorem). It states that any finite division ring is commutative. We might compare it with the result that any finite integral domain is a field. The proof of that extends easily to the non-commutative case, and together these results show that any finite ring either has zero-divisors or is a field.

There are many proofs of Wedderburn’s theorem; an elementary one can be found in [10], Chapter 1, Theorem 1.

4.7 Unit Group of an Unramified Extension

Let K be a field complete under a discrete valuation and having perfect^{5} residue field. We continue in the calculation of Br(K) = H^{2}(K_{nr}/K). To this end, let L be a finite unramified extension of K. Let G = Gal(L/K), then also G = Gal(l/k), the Galois group of the residue extension, by Theorem 4.3.2. Consider the exact sequence

0 ⟶ U_{L} ⟶ L^{∗} ⟶^{v} ℤ ⟶ 0.  (4.1)

We claim this is an exact sequence of G-modules, where G acts trivially on ℤ. This is because any element of L^{∗} is of the form π_{K}^{n}u, where π_{K} is a uniformiser of K (on which G acts trivially) and u ∈ U_{L}. Moreover, this sequence splits: the choice of π_{K} establishes a G-isomorphism L^{∗} ≅ ℤ × U_{L} by π_{K}^{n}u ⟷ (n, u).

Applying the covariant functor H^{q}(G,?) for q ≥ 0 gives a split exact sequence

0 ⟶ H^{q}(G, U_{L}) ⟶^{⟵} H^{q}(G, L^{∗}) ⟶^{⟵} H^{q}(G, ℤ) ⟶ 0.

We want to show that H^{q}(G, U_{L}) = H^{q}(G, l^{∗}) = H^{q}(l/k), then this sequence will relate the cohomology of L to that of its residue field.

To do this, we define a filtration of the unit group. For integer n ≥ 0, we define

U_{L}^{(n)} := {x ∈ U_{L} : v(1 − x) ≥ n}

4Meaning it takes the value 1 ∈ F on V and 0 elsewhere. This follows from x^{q−1} = 1 for x ≠ 0.

5Note that we are most interested in the case where the residue field is finite, and then it is automatically perfect

Since

v((1 − x)(1 − y)) = v((1 − x) + (1 − y) − (1 − xy))
≥ min{v(1 − x), v(1 − y), v(1 − xy)},

each U_{L}^{(n)} is a subgroup of U_{L} = U_{L}^{(0)}. Let us look at the quotients of successive groups.

Proposition 4.7.1 If n ≥ 1, U_{L}^{(n)}/U_{L}^{(n+1)} ≅ l, the additive group of the residue field. If L/K is a finite unramified extension, this is as Gal(L/K)-modules.

PROOF - A general element of U_{L}^{(n)} is 1 + π_{L}^{n}y for y ∈ O_{L}; map this to ȳ, the class of y in l. Then (1 + π_{L}^{n}y)(1 + π_{L}^{n}y′) maps to ȳ + ȳ′ so this is a homomorphism. It is clearly onto, and its kernel is U_{L}^{(n+1)}. If L/K is unramified, it is a Gal(L/K)- homomorphism because for each s in the Galois group, s(1 + π_{L}^{n}y) = 1 + π_{L}^{n}s(y) maps to s(ȳ).

□

Corollary 4.7.2 If L/K is a finite unramified Galois extension with Galois group G, H^{q}(G, U_{L}^{(n)}/U_{L}^{(n+1)}) = 0 for each n ≥ 1, q ≥ 1.

PROOF - This follows from the above isomorphism and Theorem 2.4.1.

□

Proposition 4.7.3 H^{q}(G, U_{L}^{(1)}) = 0, for each q ≥ 1.

PROOF - This follows from a general result on the cohomology of filtered modules - see [1], Chapter 12, §3, Lemma 3. I will demonstrate the method. Let f : G^{q} ⟶ U_{L}^{(1)} be a q-cocycle. Since H^{q}(G, U_{L}^{(n)}/U_{L}^{(n+1)}) = 0, there exists a (q − 1)-cocycle h_{1} : G^{q−1} ⟶ U_{L}^{(1)} such that

f = δh_{1} + f_{1},

where f_{1} : G^{q} ⟶ U_{L}^{(2)} is a q-cochain with values in U_{L}^{(2)}. Applying this procedure to f_{1} and iterating gives a sequence

f = δh_{1} + f_{1}
f = δh_{1} + δh_{2} + f_{2}
⋮
f = δh_{1} + δh_{2} + … + δh_{n} + f_{n}.

The valuation topology on L shows that the sequence h_{1}, h_{2}, … converges to a (q − 1)-cochain h : G^{q−1} ⟶ U_{L}^{(1)}. Then the previous equations show that f = δh, so that H^{q}(G, U_{L}^{(1)}) = 0

□

It remains to calculate H^{q}(G, U_{L}). Let L/K be a finite unramified Galois extension with group G, and consider the exact sequence of G-modules^{6}

0 ⟶ U_{L}^{(1)} ⟶ U_{L} ⟶^{p} l^{∗} ⟶ 0

where p is reduction modulo the prime ideal of O_{L}. Then the exact cohomology sequence and Proposition 4.7.3 give an isomorphism

H^{q}(G, U_{L}) ≅ H^{q}(G, l^{∗}) = H^{q}(l/k).  (4.2)

4.8 Witt’s Theorem

The split short exact sequence 4.1 becomes, in view of 4.2,

0 ⟶ H^{q}(G, l^{∗}) ⟶ H^{q}(G, L^{∗}) ⟶ H^{q}(G, ℤ) ⟶ 0.

Take the direct limit over the finite unramified Galois extensions L to obtain a split exact sequence

0 ⟶ H^{q}(k_{s}/k) ⟶ H^{q}(K_{nr}/K) ⟶ H^{q}(G, ℤ) ⟶ 0,

with G = Gal(K_{nr}/K). So for q = 2, using Theorem 4.4.2, we have a split exact sequence

0 ⟶ Br(k) ⟶ Br(K) ⟶ H^{2}(G, ℤ) ⟶ 0.

Of course, we now wish to identify H^{2}(G, ℤ). For this, we use the exact sequence of G-modules (with trivial G-action):

0 ⟶ ℤ ⟶ ℚ ⟶ ℚ/ℤ ⟶ 0  (4.3)

Proposition 4.8.1 For finite G, the G-module ℚ (with trivial G-action) is relatively injective (see Definition 1.1.7).

PROOF - Consider the G-module Hom_{ℤ}(ℤ[G], ℚ) (see §1.1.3). Let Q be the subset of maps which are constant on elements of G. It is an additive subgroup and a G-submodule, on which G acts trivially. Let Q′ be the G-submodule of maps f such that ∑_{g∈G} f(g) = 0. Any element ϕ can be written as

ϕ(s) = \frac{1}{|G|} ∑_{g∈G} ϕ(g) + (ϕ(s) − \frac{1}{|G|} ∑_{g∈G} ϕ(g)) ∈ Q + Q′,

and since Q ∩ Q′ = 0, have Hom_{ℤ}(ℤ[G], ℚ) = Q ⊕ Q′ as G-modules. Since ℚ is clearly G-isomorphic to Q, we conclude that ℚ is relatively injective.

6Again, this is G-exact because G fixes the uniformisers of L

□

Hence H^{q}(G, ℚ) = 0 for all q ≥ 1 and finite G. Indeed if G is profinite (see Chapter 2), then passing to the direct limit of these cohomology groups shows again that^{7} H^{q}(G, ℚ) = 0.

Thus the connecting homomorphism of the long exact cohomology sequence gives an isomorphism

H^{1}(G, ℚ/ℤ) ⟶_{∼}^{δ} H^{2}(G, ℤ),

and we recall (§1.2.2) that the trivial G-action means that H^{1}(G, ℚ/ℤ) = Hom_{ℤ}(G, ℚ/ℤ), the group of continuous group homomorphisms. Hence we have an isomorphism

H^{2}(G, ℤ) ⟶_{∼}^{δ^{−1}} Hom_{Z}(G, ℚ/ℤ).

Hom_{Z}(G, ℚ/ℤ) is called the (1-dimensional) character group of G.^{8} We write it as X(G).

To summarise, we have proved the following theorem.

Theorem 4.8.2 (Witt) (reference: E. Witt, “Schiefkörper über diskret bewerteten Körpern”, J. reine ang. Math. 176, 1936, pp. 153-156)

Let K be a field complete under a discrete valuation and having perfect residue field k. Let G = Gal(k_{s}/k) = Gal(K_{nr}/K) and let X(G) be its character group. Then there is a split exact sequence

0 ⟶ Br(k) ⟶ Br(K) ⟶ X(G) ⟶ 0.

By the results of §4.6, we know that if k is finite, Br(k) = 0 and so then Br(K) = X(G).

4.9 The Group ℤ̂

We will need to use a “completion” of ℤ. Recall the definition of an inverse system (Definition 2.2.5). Consider the projective system of finite cyclic groups

ℤ/nℤ,

over the directed set of positive integers ordered by division, where the morphism ℤ/nℤ ⟶ ℤ/mℤ for m|n is given by the natural projection. We define

ℤ̂ := \underleftarrow{lim}_{n} ℤ/nℤ.

ℤ̂ is thus a profinite group, called the profinite completion of ℤ.

ℤ is included in ℤ̂ in an obvious way, so an integer M maps to (M mod n)_{n} ∈ ℤ̂. In fact:

7We have to be careful about G here. For example, recall (§1.2.2) that H^{1}(G, ℚ) = Hom_{ℤ}(G, ℚ)

because of the trivial G-action. If, say, G = ℤ, this group is not zero.

8It is the Pontrjagin Dual of G/G′, ie. Hom_{Z}(G/G′, ℚ/ℤ)

Proposition 4.9.1 ℤ̂ contains ℤ as a dense^{9} subset.

PROOF - Take an open set containing an element (m_{n})_{n} ∈ ℤ̂. So (§2.2.3) this is a finite intersection of sets of the form

S_{j} = ((m_{n})_{n} + 1 × ∏_{k≠j} ℤ/kℤ) ∩ ℤ̂.

Take J to be a common multiple of all the j, and consider the integer M = m_{J}. We see that under the projection ℤ/Jℤ ⟶ ℤ/jℤ, M maps to m_{j}, for each j. Hence the inclusion of M into ℤ̂ is in ⋂_{j} S_{j}.

Thus every nonempty open set of ℤ̂ contains an integer, as required.

□

The group ℤ̂ is important to us because it arises as the absolute Galois group of a finite field. We recall from Appendix C the Galois Theory of finite fields. Let k be a finite field, then for each n ≥ 1 there is a unique extension l of k of degree n. The extension is Galois, and Gal(l/k) is isomorphic to ℤ/nℤ, where the Frobenius automorphism corresponds to the generator 1. We note that if m/k is another extension containing l, we have

[m : k] = [m : l][l : k]

and the Frobenius of l/k is the restriction of the Frobenius of m/k. Hence the natural projection Gal(m/k) ⟶ Gal(l/k) corresponds under the isomorphism to the natural projection

ℤ/[m : k]ℤ ⟶ ℤ/[l : k]ℤ.

So we have an isomorphism of inverse systems^{10}.

The union of all the l is the separable closure of k. The results of Chapter 2 §2.2.4 give us

Gal(k_{s}/k) = \underleftarrow{lim}_{l} Gal(l/k) = \underleftarrow{lim}_{n} ℤ/nℤ = ℤ̂.

Now let K be a local field and K_{nr} its maximal unramified extension (see §§4.2, 4.3). It follows from Corollary 4.3.3 that also Gal(K_{nr}/K) = ℤ̂.

We are now ready to identify the character group X(G) in Theorem 4.8.2, where G = Gal(k_{s}/k) = ℤ̂. Recall this is the group of continuous homomorphisms from ℤ̂ into ℚ/ℤ. But because of Proposition 4.9.1, we see that a continuous homomorphism

ϕ : ℤ̂ ⟶ ℚ/ℤ

is defined completely by the value ϕ(1); ie.

9Meaning that the closure of ℤ in ℤ̂ is all of ℤ̂; every element of ℤ̂ can be “approximated arbitrarily closely” by an integer

10This means a system of isomorphisms between each element of the inverse systems which is compatible with the projections in each system.

Proposition 4.9.2 The map

\begin{aligned}γ : & Hom_{ℤ}^{cont}(ℤ̂, ℚ/ℤ) & ⟶ & ℚ/ℤ \\ & ϕ & ⟼ & ϕ(1) \end{aligned}

is an isomorphism.

4.10 The Invariant Map

We have proved:

Theorem 4.10.1 Let K be a local field. Then

Br(K) ≅ ℚ/ℤ.

PROOF - This follows from

• Theorem 4.4.2;

• Witt’s Theorem (4.8.2);

• Wedderburn’s Theorem (§4.6) applied to the residue field k;

• Proposition 4.9.2.

□

We write

inv_{K} : Br(K) ⟶^{∼} ℚ/ℤ.

Let us retrace our steps to see how inv_{K} is calculated. For each finite unramified Galois extension L/K, the sequence (4.1) and Witt’s Theorem (4.8.2) show that the valuation induces an isomorphism

H^{2}(L/K) ⟶ H^{2}(Gal(L/K), ℤ).

If M ⊇ L is another unramified extension of K, then the valuation of M restricts to the valuation of L, by the unramified condition. Hence we passed to the direct limit over L to obtain an isomorphism

H^{2}(K_{nr}/K) ⟶ H^{2}(ℤ̂, ℤ).

Now the connecting homomorphism which comes from the exact sequence

0 ⟶ ℤ ⟶ ℚ ⟶ ℚ/ℤ ⟶ 0

gives

H^{2}(ℤ̂, ℤ) ⟶_{∼}^{δ^{−1}} H^{1}(ℤ̂, ℚ/ℤ) = Hom_{ℤ}(ℤ̂, ℚ/ℤ).

Finally, we let γ represent the evaluation map at 1. We may describe inv_{K} by inv_{K} = γ ∘ δ^{−1} ∘ v:

Br(K) = H^{2}(K_{nr}/K) ⟶^{v} H^{2}(ℤ̂, ℤ) ⟶^{δ^{−1}} Hom_{ℤ}(ℤ̂, ℚ/ℤ) ⟶^{γ} ℚ/ℤ.

4.11 Properties of the Invariant Map

Proposition 4.11.1 Let K be a local field, L/K a finite extension of degree n. Then K_{nr} ⊆ L_{nr}. The diagram

H^{2}(K_{nr}/K) ⟶^{Res} H^{2}(L_{nr}/L)
↓inv_{K}  □  ↓inv_{L}
ℚ/ℤ  ⟶^{n}  ℚ/ℤ

commutes.

PROOF - Let G_{K} = Gal(K_{nr}/K), F_{K} ∈ G_{K} the Frobenius automorphism^{11}, G_{L} and F_{L} similarly for L. If f := [l : k] denotes the residue degree, then F_{L} = (F_{K})^{f}. Let e be the ramification index of L/K. Consider the diagram

H^{2}(G_{K}, K_{nr}^{∗}) ⟶^{v_{K}} H^{2}(G_{K}, ℤ) ⟶^{δ^{−1}} Hom_{ℤ}^{cont}(G_{K}, ℚ/ℤ) ⟶^{γ_{K}} ℚ/ℤ
↓Res  □  ↓eRes  □  ↓eRes  □ ↓n
H^{2}(G_{L}, L_{nr}^{∗}) ⟶^{v_{L}} H^{2}(G_{L}, ℤ) ⟶^{δ^{−1}} Hom_{ℤ}^{cont}(G_{L}, ℚ/ℤ) ⟶^{γ_{L}} ℚ/ℤ.

In this diagram γ_{K,L} are evaluation at the Frobenius. The diagram commutes by the basic results about ramification in §4.2 and the fundamental identity ef = n.

Because inv = γ ∘ δ^{−1} ∘ v (§4.10), we have the desired result.

□

Corollary 4.11.2 For any Galois extension L/K of degree n, H^{2}(L/K) is cyclic of degree n, and generated by the unique element u_{L/K} ∈ Br(K) such that inv_{K}(u_{L/K}) = 1/n.

Note that we are not assuming the extension to be unramified any more. PROOF - Recall that the inclusion H^{2}(L/K) ↪ Br(K) is given by inflation. Then the restriction-inflation sequence (Corollary 4.1.2)

0 ⟶ H^{2}(L/K) ⟶^{Inf} Br(K) ⟶^{Res} Br(L)

shows H^{2}(L/K) is the kernel of Res. By inv_{L} ∘ Res = ninv_{K}, and injectivity of inv, we have that inv_{K}(ker Res) consists of all those a ∈ ℚ/ℤ such that na = 0; ie. it is the subgroup \frac{1}{n}ℤ/ℤ. Hence ker Res is also cyclic of order n, generated by inv_{K}^{−1}(1/n).

□

Corollary 4.11.3 1. Res : H^{2}(K_{nr}/K) ⟶ H^{2}(L_{nr}/L) is surjective.

11This is the element of the Galois group which corresponds to the q-th power map in the residue extension k_{s}/k, where Card(k) = q. The correspondence is via Theorem 4.3.2. It is identified with the topological generator 1 ∈ ℤ̂.

2. Cor : H^{2}(L_{nr/L}) ⟶ H^{2}(K_{nr}/K) is injective.

3. inv_{K} ∘ Cor = inv_{L}.

PROOF - (1) follows because Res = inv_{L}^{−1} ∘ n ∘ inv_{K}, a composition of surjective maps. Then (3) is a consequence of Proposition 1.5.13, because

inv_{L} ∘ Res = ninv_{K} = inv_{K} ∘ Cor ∘ Res,

and Res is surjective. Finally, (3) implies (2) by the injectivity of inv_{L}.

□

We summarise the position of Inf relative to inv in the following proposition.

Proposition 4.11.4 Let L ⊇ M ⊇ K be a tower of finite extensions of local fields with L/K and M/K Galois. Let a = [L : M], b = [M : K]. Then the following diagram commutes.

0 ⟶ H^{2}(M/K) ⟶^{Inf} H^{2}(L/K) ⟶^{Res} H^{2}(L/M)
↓inv_{K}  □  ↓inv_{K}  □  ↓inv_{M}
0 ⟶  \frac{1}{b}ℤ/ℤ  ↪ \frac{1}{ab}ℤ/ℤ ⟶^{b}  \frac{1}{a}ℤ/ℤ.

PROOF - The first square commutes because inflations are the homomorphisms in the direct limit, and so we have the commutative diagram:

[figure: Inf; Inf; H^{2}(L/K); H^{2}(M/K); \underrightarrow{lim}_{E} H^{2}(E/K); ℚ/ℤ; Inf; inv_{K}]

The images of H^{2}(M/K), H^{2}(L/K) are \frac{1}{b}ℤ/ℤ, \frac{1}{ab}ℤ/ℤ by Corollary 4.11.2. The second square commutes by Proposition 4.11.1.

□

We shift our attention now to the canonical generators of the cohomology groups. If L/K is a finite extension of local fields of degree n, we write u_{L/K} := inv_{K}^{−1}(1/n) ∈ H^{2}(L/K).

Proposition 4.11.5 Let K be a local field and L ⊇ M ⊇ K be a tower of finite extensions with L/K Galois.

1. If M/K is also Galois then Inf(u_{M/K}) = [L : M]u_{L/K}.

2. Res(u_{L/K}) = u_{L/M}.

3. Cor(u_{L/M}) = [M : K]u_{L/K}.

All this follows easily from the results of this section.

4.12 The Reciprocity Map

Recall Tate’s Theorem (1.6.4). We have now established enough facts about Br(K), when K is a local field, to apply this theorem. If L/K is a finite Galois extension, we know that for each Galois subextension M, H^{2}(Gal(L/M), L^{∗}) is cyclic of order [L : M] = |Gal(L/M)| and generated by u_{L/M}. Furthermore, u_{L/M} = Res(u_{L/K}). We conclude by Tate’s Theorem:

Theorem 4.12.1 Let L/K be a finite Galois extension of local fields with group G. Then for all n ∈ ℤ, the map

\begin{aligned}θ_{n} : & Ĥ^{n}(G, ℤ) & ⟶ & Ĥ^{n+2}(G, L^{∗}) \\ & x & ⟼ & x · u_{L/K}, \end{aligned}

where · denotes cup product, is an isomorphism.

We are particularly interested in the case n = −2. Then

θ_{−2} : G/G′ ⟶ K^{∗}/NL^{∗}

is an isomorphism relating the abelianisation of the Galois group to the multiplicative group of the field. The inverse of this isomorphism will be called the norm residue symbol and denoted

\begin{matrix}K^{∗} & ⟶ & K^{∗}/NL^{∗} & ⟶ & G/G′ \\ a & ⟼ & & & (a, L/K) \end{matrix}

It is so called because (a, L/K) = 1 if and only if a is the norm of an element of L.

The next proposition lists some of the functorial properties of the norm residue symbol, which is denoted in the proposition by the vertical arrows.

Proposition 4.12.2 Let K be a local field, L ⊇ M ⊇ K a tower of finite extensions with L/K Galois. Then:

1. The morphism Cor gives a commutative diagram

M^{∗}  ⟶^{norm}  K^{∗}
↓  □  ↓
Gal(L/M)^{ab} ⟶^{can.} Gal(L/K)^{ab}.

Here “can.” denotes the canonical map induced by inclusion of Galois groups.

2. The morphism Res^{12} gives a commutative diagram

K^{∗}  ↪  M^{∗}
↓  □  ↓
Gal(L/K)^{ab} ⟶^{Res} Gal(L/M)^{ab}.

12The map G/G′ ⟶ H/H′, where H is a subgroup of G, defined by Res : Ĥ^{−2}(G, ℤ) ⟶ Ĥ^{−2}(H, ℤ), is classically called the transfer, written “Ver”.

3. If M/K is Galois, we have a commutative diagram

K^{∗}  =  K^{∗}
↓  □  ↓
Gal(L/K)^{ab} ⟶^{proj.} Gal(M/K)^{ab}.

The proof of this can be found in [1], Chapter 11. For (1) and (2) it follows from Proposition 4.11.5 and properties of Cor and Res relative to the cup product. For (3) it requires some more detailed cup-product calculations.

The property (3) shows that the norm residue symbols define together a homomorphism

ψ : K^{∗} ⟶ \underleftarrow{lim}_{E} Gal(E/K)^{ab},

where E runs through all finite Galois extensions, since the norm residue symbols form a consistent system with respect to the projections in the projective limit.

It can be shown that this projective limit is equal to the abelianisation of the absolute Galois group of K, Gal(K_{s}/K)^{ab}. This is the Galois group of the maximal abelian extension of K.

The map ψ is called the Artin Reciprocity Map. It is almost an isomorphism, in the sense that for any finite Galois extension L of K, we have the diagram

\begin{matrix}K^{∗} & ⟶^{ψ} & Gal_{K}^{ab} \\ ↓ & □ & ↓ \\ K^{∗}/NL^{∗} & ⟶_{∼}^{(?,L/K)} & Gal(L/K)^{ab}, \end{matrix}

and the bottom row is an isomorphism.

This shows that Gal_{K}^{ab} is isomorphic to \underleftarrow{lim}_{E} K^{∗}/NE^{∗}, the completion of K^{∗} with

respect to the norm topology. We see that for a local field, the Galois group of its maximal abelian extension is precisely related to its multiplicative structure.

It will not be possible to study the norm topology here. The main result is:

Theorem 4.12.3 (The Existence Theorem) Let H be a subgroup of K^{∗}. Then H = NE^{∗} for some finite Galois extension E/K if and only if H is closed and of finite index in K^{∗}.

E is called a class field for H. For more details, see [1], Chapters 11 and 14.

Conclusion

The general group cohomology theory of Chapter 1 allows us to build a bridge between field arithmetic and the Galois group, via the appearance as Tate cohomology groups of

Ĥ^{0}(Gal(L/K), L^{∗}) = K^{∗}/NL^{∗}
Ĥ^{−2}(G, ℤ) = G/G′.

We saw in Tate’s Theorem (1.6.4) that to do this we need certain properties of groups H^{2}(L/K). In Chapter 3 we investigated the arithmetical significance of these groups. Then in Chapter 4 we studied local fields and showed that they give a context in which Tate’s Theorem applies. As a result, the Artin reciprocity law could be established.

This is only the starting point of class field theory. I will discuss some of the possibilities for application of the theory, and for further investigation of problems suggested by this text.

The Brauer groups and central-simple algebras of Chapter 3 arise in problems of number-theory and algebraic geometry, and investigation of their properties is a subject of research. They are relevant to function fields of algebraic curves.

On the theme of investigating Galois groups, we note that we have only looked at the abelianised group. This is equivalent to the determination of all the onedimensional representations of G, ie. all homomorphisms

G ⟶ GL(n, ℂ),

where n = 1. For larger n, this is the problem of higher-dimensional Galois representation and is also the subject of research.

Artin’s reciprocity map itself is worthy of much more attention than it was possible to give it here. The definition is not very helpful when it comes to actually trying to evaluate this function, and the difficulty of doing this turns out to depend greatly on the properties of a given extension of local fields L/K. The discussion is phrased in terms of “local symbols”, which prove relatively easy to calculate in the case of a “tamely ramified” extension and much more complicated otherwise.

We find that Artin’s map is helpful in studying the ramification groups of an extension of local fields (see [1], Chapter 4, and Anne Esslemont’s part of this project). Under the reciprocity map, these correspond precisely to the filtration of the unit group in §4.7.

Local class field theory as introduced here is taken as the basis for global class field theory. Consider for example a Galois number field K (ie. K/ℚ is a finite Galois extension). All the absolute values on K are either Archimedean, and so related to the usual norms on ℝ and ℂ, or come from discrete valuations (a theorem of Ostrowski). The completions of K for each of these valuations may be studied via local class-field theory, with applications to studying the Galois group and investigating factorisation in the field.

Appendix A

Homology of Complexes

A.1 Complexes

Definition A.1.1 Let R be a ring. A complex A of R-modules is a doubly-infinite sequence of modules and homomorphisms

… ⟶^{d_{3}} A_{2} ⟶^{d_{2}} A_{1} ⟶^{d_{1}} A_{0} ⟶^{d_{0}} A_{−1} ⟶^{d_{−1}} …  (A.1)

such that d_{q} ∘ d_{q+1} = 0 for each q ∈ ℤ

For each q ∈ ℤ, im d_{q+1} is a submodule of ker d_{q}, because d_{q} ∘ d_{q+1} = 0. We define the q-th homology group of A to be the quotient:

H_{q}(A) := \frac{ker d_{q}}{im d_{q+1}}

Remember that if im d_{q+1} = ker d_{q} then (A.1) is an exact sequence, in which case it will also be called an acyclic complex, and we see that it has zero homology. So in a sense, the homology groups measure the “inexactness” of the sequence (A.1). Note that for each q ∈ ℤ, H_{q}(A) is a quotient group of a subgroup of A_{q}. We will be particularly concerned with the case R = ℤ, so that “R-module” means abelian group.

There is now a large amount of arcane jargon to introduce, which comes from the origination of homology in the study of topological surfaces.

Definition A.1.2 1. A complex of ℤ-modules (A.1) will also be called a chain complex. An element of A_{q} is a q-dimensional chain or q-chain.

2. The q-dimensional chains in ker d_{q} are called q-dimensional cycles or q-cycles. Those q-chains which are in im d_{q+1} are called q-boundaries; they are necessarily q-chains because d_{q} ∘ d_{q+1} = 0. d_{q} is called the q-dimensional boundary operator.

3. Two q-cycles are called homologous if their difference is a q-boundary, ie. they give the same coset in H_{q}(A). An element of H_{q}(A) is called a homology class. A q-cycle is homologous to zero if it is homologous to the zero chain, ie. it is a q-boundary.

A.2 Cohomology

Now let us write the sequence (A.1) differently, putting A^{q} := A_{−q} and d_{−q} instead of d_{q}:

… ⟶^{d_{−3}} A^{−2} ⟶^{d_{−2}} A^{−1} ⟶^{d_{−1}} A^{0} ⟶^{d_{0}} A^{1} ⟶^{d_{1}} …  (A.2)

so d_{q} ∘ d_{q−1} = 0. The (−q)-th homology group of this complex will now be written

H^{q}(A) = \frac{ker d_{q}}{im d_{q−1}}

and called the q-th cohomology group of A. A q-dimensional cochain will be a (−q)-dimensional chain; (A.2) will now be called a cochain complex. similarly we define “cocycles”, “coboundaries”, “cohomologous cocycles”, etc.

This distinction at this stage seems artificial and confusing, since a cochain complex is just a chain complex differently written down. However, we will see later that the different numberings tend to arise in different ways, so that the distinction is useful in practice.

From now on we discuss only “cohomology”, since this is the notation that will be used the most later on. There are obvious corresponding results for “homology”, by renumbering.

A.3 Morphisms of Complexes

Definition A.3.1 A morphism of complexes A ⟶ B where

A = … ⟶^{d_{−1}} A^{0} ⟶^{d_{0}} A^{1} ⟶^{d_{1}} …
B = … ⟶^{d_{−1}} B^{0} ⟶^{d_{0}} B^{1} ⟶^{d_{1}} …

is a sequence of homomorphisms

ϕ_{q} : A^{q} ⟶ B^{q} q ∈ ℤ

such that for each q ∈ ℤ, the diagram

A^{q} ⟶^{d_{q}} A^{q+1}
↓ϕ_{q}  ↓ϕ_{q+1}
B^{q} ⟶^{d_{q}} B^{q+1}

commutes (ie. ϕ commutes with the coboundary operators).

Morphisms of complexes are also called chain maps.

This definition of morphism makes the complexes into a pre-additive category. For any q ∈ ℤ, the procedure A ⟼ H^{q}(A) is then a covariant functor from the category of complexes to the category of abelian groups. For if f : A ⟶ B is a morphism of complexes then for each q ∈ ℤ the homomorphism

f_{q} : A^{q} ⟶ B^{q}

carries ker d_{q} ⊆ A^{q} into ker d_{q} ⊆ B^{q} because d_{q} ∘ f_{q} = f_{q+1} ∘ d_{q}, and carries im d_{q−1} ⊆ A^{q} into im d_{q−1} ⊆ B^{q} because d_{q−1} ∘ f_{q−1} = f_{q} ∘ d_{q−1}.

Thus f_{q} defines a homomorphism

f_{q}^{∗} : ker d_{q}/im d_{q−1} ⟶ ker d_{q}/im d_{q−1}
ie. f_{q}^{∗} :  H^{q}(A)  ⟶  H^{q}(B),

called the homomorphism induced by f in dimension q. Since

(f_{q} ∘ g_{q})^{∗} = f_{q}^{∗} ∘ g_{q}^{∗}  and  (id_{A^{q}})^{∗} = id_{H^{q}(A)}

H^{q} is a covariant functor. Since (f_{q} + g_{q})^{∗} = f_{q}^{∗} + g_{q}^{∗} it is pre-additive. H^{q} is called the q-dimensional cohomology functor.

Definition A.3.2 Let f : A ⟶ B be a morphism of complexes. Then ker f and im f are the subcomplexes

… ⟶^{d_{−2}} ker f_{−1} ⟶^{d_{−1}} ker f_{0} ⟶^{d_{0}} ker f_{1} ⟶^{d_{1}} …

… ⟶^{d_{−2}} im f_{−1} ⟶^{d_{−1}} im f_{0} ⟶^{d_{0}} im f_{1} ⟶^{d_{1}} …

Note that the relation f_{q+1} ∘ d_{q} = d_{q} ∘ f_{q} shows that the coboundary maps restrict as required.

Thus a sequence of complexes

… A ⟶^{α} B ⟶^{β} C ⟶ …

is exact at B if im α = ker β, ie. im α_{q} = ker β_{q} for all q ∈ ℤ, and an exact sequence if it is exact at every place.

A.4 The Connecting Homomorphism

Consider now the “short exact sequence of complexes”

0 ⟶ A ⟶^{i} B ⟶^{p} C ⟶ 0  (A.3)

where 0 denotes the zero complex, so that i is injective, p surjective and ker p = im i. In this situation there is a “canonical homomorphism”

δ_{q} : H^{q}(C) ⟶ H^{q+1}(A)

which will now be defined.

What we want to do is get from the top-right to the bottom-left in the following diagram

\begin{matrix}A^{q} & ⟶^{i_{q}} & B^{q} & ⟶^{p_{q}} & C^{q} \\ ↓d_{q} & □ & ↓d_{q} & □ & ↓d_{q} \\ A^{q+1} & ⟶^{i_{q+1}} & B^{q+1} & ⟶^{p_{q+1}} & C^{q+1} \end{matrix}

Since p ∘ i = 0, we will not have much luck trying to go too far horizontally. So omit some of the diagram to get

B^{q}  ⟶^{p_{q}} C^{q}
↓d_{q}
A^{q+1} ⟶^{i_{q+1}} B^{q+1}

The necessary path is now clear. So starting with a cocycle c ∈ C^{q}, representing a cohomology class in H^{q}(C), choose b ∈ B^{q} with p_{q}(b) = c. This is possible because p_{q} is surjective. Now b is a cochain but not necessarily a cocycle. Applying now d_{q} to b gives a (q + 1)-coboundary d_{q}b in B^{q+1}; in particular, d_{q}b is a (q + 1)-cocycle. Now if we were to apply p_{q+1} to d_{q}b:

\begin{matrix}B^{q} & ⟶^{p_{q}} & C^{q} \\ ↓d_{q} & □ & ↓d_{q} \\ B^{q+1} & ⟶^{p_{q+1}} & C^{q+1} \end{matrix}

We would get (p_{q+1} ∘ d_{q})b = (d_{q} ∘ p_{q})b = d_{q}c = 0 because c is a cocycle. So in fact d_{q}b ∈ ker p_{q+1} = im i_{q+1}.

Now i_{q+1} is injective so there is a unique (q + 1)-cochain a = i_{q+1}^{−1}(d_{q}b) in A^{q+1}. a is a cocycle because (i_{q+2} ∘ d_{q+1})a = (d_{q+1} ∘ i_{q+1})a = (d_{q+1} ∘ d_{q})b = 0, and then the injectivity of i_{q+2} shows that d_{q+1}a = 0. Thus a defines a cohomology class in H^{q+1}(A), which we propose to write δ_{q}c.

What do we need to check to be sure that δ_{q} is a well-defined function H^{q}(C) ⟶ H^{q+1}(A)? We have to show that δ_{q}c does not depend on the arbitrary choices we made in defining it. These were two: we chose the representative c ∈ C^{q} of the cohomology class in H^{1}(C), and we chose b ∈ p_{q}^{−1}({c}).

Independence of b: If b, b′ ∈ p_{q}^{−1}({c}) then p_{q}(b − b′) = c − c = 0 so b − b′ ∈ ker p_{q} = im i_{q}, so that b − b′ = i_{q}(a) for some a ∈ A^{q}. Then

d_{q}b = d_{q}(b′ + i(a))
= d_{q}b′ + (d_{q} ∘ i_{q})a
= d_{q}b′ + (i_{q+1} ∘ d_{q})a

so that (i_{q+1}^{−1} ∘ d_{q})b = (i_{q+1}^{−1} ∘ d_{q})b′ + d_{q}a is cohomologous to (i_{q+1}^{−1} ∘ d_{q})b′.

Independence of c: Let c′ be a q-cocycle cohomologous to c, so c′ = c + d_{q−1}c′′ for some (q − 1)-cochain c′′. Let b ∈ B^{q} be such that p_{q}b = c and b′′ ∈ B^{q−1} be such that p_{q−1}b′′ = c′′, then p_{q}(b + d_{q−1}b′′) = c + d_{q−1}c′′ = c′ so that we choose (as we now know we may) b′ = b + d_{q−1}b′′ ∈ p_{q}^{−1}({c′}). Then, going to the next step in the evaluation of δ_{q}c, d_{q}b′ = d_{q}b + (d_{q} ∘ d_{q−1})b′′ = d_{q}b so that we lose any dependence on c′′, as required.

This shows that δ_{q} is a well-defined function. Lifting a sum of cocycles c+c′ ∈ C^{q} to b + b′ ∈ B^{q} shows δ_{q}(c + c′) = i_{q+1}^{−1}(d_{q}b) + i_{q+1}^{−1}(d_{q}b) = δ_{q}c + δ_{q}c′ so that δ_{q} is a group homomorphism. We write δ = i^{−1} ∘ d ∘ p^{−1} to mean “δ_{q} associates to the cohomology class of a cocycle c ∈ C^{q} the cohomology class of i_{q+1}^{−1}(d_{q}b) where b ∈ p_{q}^{−1}({c}) may be chosen arbitrarily”. δ_{q} is called the connecting homomorphism in dimension q.

A.5 The Exact Triangle

We can now show the central result of cohomology:

Theorem A.5.1 (Exact Triangle) Let

0 ⟶ A ⟶^{i} B ⟶^{p} C ⟶ 0  (A.4)

be an exact sequence of complexes. Define for each q ∈ ℤ the connecting homomorphism

δ_{q} : H^{q}(C) ⟶ H^{q+1}(A)

as above. Then the doubly infinite sequence of abelian groups

… →^{p_{q−1}^{∗}} H^{q−1}(C) →^{δ_{q−1}} H^{q}(A) →^{i_{q}^{∗}} H^{q}(B) →^{p_{q}^{∗}} H^{q}(C) →^{δ_{q}} H^{q+1}(A) →^{i_{q+1}^{∗}} … (A.5)

is exact.

PROOF - For q ∈ ℤ there are three places of exactness to check. We work with cocycle representatives of the cohomology classes for brevity.

1. H^{q}(A): im δ_{q−1} = ker i_{q}^{∗}: a ∈ A^{q−1} represents a cohomology class in ker i_{q}^{∗} ⟺ i_{q}a is cohomologous to zero in B^{q−1} ⟺ i_{q}a = d_{q−1}b for some b ∈ B^{q−1} ⟺ a = (i_{q}^{−1} ∘ d_{q−1})b = δ_{q−1}(p_{q−1}b) ⟺ a represents a cohomology class in im δ_{q−1}.

2. H^{q}(B): im i_{q}^{∗} ⊆ ker p_{q}^{∗}: p_{q} ∘ i_{q} = 0 by exactness of (A.4). im i_{q}^{∗} ⊇ ker p_{q}^{∗}: Suppose a q-cocycle b has p_{q}b cohomologous to zero in C^{q}, ie. p_{q}b = d_{q−1}c. Let b′ ∈ p_{q−1}^{−1}({c}) so that p_{q}(b − d_{q−1}b′) = 0, so b − d_{q−1}b′ ∈ ker p_{q} = im i_{q}, ie. b − d_{q−1}b′ = i_{q}a for some a ∈ A^{q}. Now

(i_{q+1} ∘ d_{q})a = (d_{q} ∘ i_{q})a = d_{q}b − (d_{q} ∘ d_{q−1}b′ = d_{q}b = 0

This shows d_{q}a = 0 because i_{q+1} is injective. Thus b is cohomologous to i_{q}(a).

3. H^{q}(C): im p_{q}^{∗} ⊆ ker δ_{q}: If a q-cocycle c = p_{q}b for some q-cocycle b then δ_{q}c is represented by (i_{q+1}^{−1} ∘ d_{q})b = i^{−1}(0) = 0. im p_{q}^{∗} ⊇ ker δ_{q}: if a q-cocycle c can be lifted to b ∈ p_{q}^{−1}({c}) such that (i_{q+1} ∘ d_{q})b is cohomologous to zero in A^{q+1} then (i_{q+1} ∘ d_{q})b = d_{q}a for some a ∈ A^{q}. So

d_{q}b = (i_{q+1} ∘ d_{q})a = (d_{q} ∘ i_{q})a

Thus c = p_{q}(b − i_{q}a) and d_{q}(b − i_{q}a) = 0 so that b − i_{q}a is a q-cocycle.

□

This result is called the exact triangle because of the diagram

H(A) ⟶^{i^{∗}} H(B)
↖^{δ} ▽ ↙_{p^{∗}}
H(C)

The pair {H^{q}, δ} is said to form a cohomological functor. By this we mean that whenever we have two short exact sequences of complexes 0 → A′ → A → A′′ → 0 and 0 → B′ → B → B′′ → 0 and a morphism between them, ie. a commutative diagram

0 ⟶ A′ ⟶ A ⟶ A′′ ⟶ 0
↓ f′ □ ↓ f □ ↓ f′′
0 ⟶ B′ ⟶ B ⟶ B′′ ⟶ 0

then the morphisms f′, f, f′′ induce a commutative diagram

… → H^{q−1}(A′′) →^{δ} H^{q}(A′) → H^{q}(A) → H^{q}(A′′) →^{δ} …
↓ f′′_{q−1}^{∗} □ ↓ f′_{q}^{∗} □ ↓ f_{q}^{∗} □ ↓ f′′_{q}^{∗}
… → H^{q−1}(B′′) →^{δ} H^{q}(B′) → H^{q}(B) → H^{q}(B′′) →^{δ} …

called a morphism of cohomology functors. This property is called “naturality of δ”; its verification is routine.

Appendix B

Non-Abelian Cohomology of Groups

[This appendix follows [1], Chapter 7, appendix]

B.1 Definitions

We consider the case of a non-abelian G-module A. This means that A is a group and we have a homomorphism

G ⟶ Aut(A)

so that G acts on A. It turns out that it is still reasonable to define objects H^{0}(G, A) and H^{1}(G, A) and obtain an “exact sequence”, shorter than the one we had before. The definition is a “hands-dirty” job, using the inhomogeneous cochains directly.

Throughout this section, we will denote composition in A by (a, b) ↦ a · b and the G-action by (g, a) ↦ ga.

Everything that we will say also makes sense in the context of the continuous cochains of §2.3.4.

Definition B.1.1 Let A be a non-abelian G-module.

1. H^{0}(G, A) := A^{G}

2. A 1-cochain of G with values in A is a map

f : G ⟶ A

3. A 1-cocycle of G with values in A is a 1-cochain with f[st] = f[s] · sf[t].

4. Two 1-cochains f, g are cohomologous if there is a ∈ A such that

g[s] = a^{−1} · f[s] · sa

for all s ∈ G. f is called cohomologous to zero if f[s] = a^{−1} · sa, ie. f is cohomologous to the trivial cochain s ↦ 1.

Proposition B.1.2 (3) defines an equivalence relation on the set of 1-cocycles.

PROOF - Since s1 = s(1 · 1) = s1 · s1, s1 = 1, and so f[s] = 1^{−1} · f[s] · s1 for all s ∈ G, ie. f is cohomologous to itself. If g[s] = a^{−1} · f[s] · sa then f[s] = (a^{−1})^{−1} · g[s] · s(a^{−1}) so the relation is commutative. If g[s] = a^{−1} · f[s] · sa and h[s] = a′^{−1} · g[s] · sa′ then h[s] = (a · a′)^{−1} · f[s] · s(a · a′) showing transitivity.

□

Definition B.1.3 The 1-dimensional cohomology set of G with values in A is the set of equivalence classes of 1-cocycles, and is written H^{1}(G, A).

H^{1}(G, A) has the structure of a pointed set – it is not necessarily a group, but a set with one element singled out, called the unit element. In this case the unit is the class of the trivial cocycle s ↦ 1.

Note that the definitions coincide with the earlier ones if A is abelian, except then H^{1}(G, A) also has a group law.

H^{0}(G, A) and H^{1}(G, A) are functors in A: If f : A ⟶ B is a G-homomorphism, ie. f(sa) = sf(a)∀s ∈ G, then f induces morphisms

f^{∗0} = f|_{A^{G}} : H^{0}(G, A) ⟶ H^{0}(G, B)
f^{∗1} : H^{1}(G, A) ⟶ H^{1}(G, B)
g  ⟼ f ∘ g : G ⟶ B
s ⟼ f(g[s])

satisfying (f ∘ f′)^{∗} = f^{∗} ∘ f′^{∗} and id_{A}^{∗} = id_{H(G,A)}. The map f^{∗0} is a group homomorphism while f^{∗1} is a morphism of pointed sets, which means a function which maps the unit element to the unit element. By the kernel of a morphism of pointed sets is meant the preimage of the unit element. Thus we can talk about an exact sequence of pointed sets.

B.2 The Zero-Dimensional Connecting Morphism

Let

1 ⟶ A ⟶^{i} B ⟶^{p} C ⟶ 1  (B.1)

be a short exact sequence of non-abelian G-modules. Then im i = ker p is a normal subgroup of B. Recall the definition of the connecting homomorphism δ_{q} : H^{q}(G, C) ⟶ H^{q+1}(G, A) in the abelian case; if K^{q}(A) denotes the cochain complex underlying the cohomology groups, we have the situation

K^{q}(B) ⟶^{p} K^{q}(C)

↓ d

K^{q+1}(A) ⟶^{i} K^{q+1}(B)

and δ_{q}(f) := i^{−1} ∘ d ∘ p^{−1}(f); that is, given a q-dimensional inhomogeneous cocycle f : G^{q} ⟶ C, choose a cochain g : G^{q} ⟶ B such that p(g[s_{1}, … s_{q}]) = f[s_{1}, … s_{q}]. Then apply the coboundary map d to g to obtain dg : G^{q+1} ⟶ B We show that dg is a (q + 1)-cocycle taking values in i(A), and that its cohomology class depends only on the class of f in H^{q}(C).

This construction is adapted to the non-abelian case as follows. In (B.1) assume for simplicity that A is a subgroup of B, so that i is inclusion of sets.

Let c ∈ H^{0}(G, C) = C^{G} and choose b ∈ B so that p(b) = c. Now the coboundary db is defined as

\begin{aligned}db : & G & ⟶ & B \\ & s & ⟼ & b^{−1} · sb \end{aligned}

Note that

p(b^{−1} · sb) = p(b^{−1}) · p(b) = c^{−1} · sc = c^{−1} · c = 1

so that b^{−1} · sb ∈ ker p = im i = A for each s ∈ G. So put

\begin{aligned}δ_{0}c : & G & ⟶ & A \\ & s & ⟼ & b^{−1} · sb \end{aligned}

We have to show:

• δ_{0}c is a 1-cocycle:

δ_{0}c[st] = b^{−1} · st(b) = b^{−1} · sb · sb^{−1} · st(b) = δ_{0}c[s] · s(δ_{0}c[t])

• The cohomology class of δ_{0}c depends only on the cohomology class of c, not on the choice of b ∈ p^{−1}({c}): Choose b′ ∈ p^{−1}({c}), then b′^{−1} · b ∈ ker p = im i = A so b = b′a for some a ∈ A. Then b^{−1} · sb = a^{−1} · b′^{−1} · sb′ · sa which is cohomologous to b′^{−1} · sb′.

So δ_{0} gives a well-defined function H^{0}(G, C) ⟶ H^{1}(G, A). Since δ_{0}1 = 1, δ_{0} is a morphism of pointed sets. δ_{0} is called the connecting morphism in dimension 0.

B.3 The One-Dimensional Connecting Morphism

Now suppose A is in the centre Z(B) of B. Then A is particular is an abelian G-module and so we have already defined the 2-dimensional homology group of G with values in A, H^{2}(G, A).

Remembering the long exact cohomology sequence, we would like a morphism

δ_{1} : H^{1}(G, C) ⟶ H^{2}(G, A)

We can define this similarly to δ_{0}, as follows.

Let f : G ⟶ C be a 1-dimensional cocycle with values in C and choose a 1-cochain g : G ⟶ B such that p(g[s]) = f[s] for each s ∈ G. Now remembering^{1} the 1-dimensional coboundary formula

d[s, t] = [s] + s[t] − [st]

define

\begin{aligned}dg : & G^{2} & ⟶ & B \\ & [s, t] & ⟼ & g[s] · sg[t] · g[st]^{−1} \end{aligned}

Then p(dg[s, t]) = f[s] · sf[t] · f[st]^{−1} = 1 because f is a 1-cocycle. Thus for each [s, t], dg[s, t] ∈ ker p = im i = A. So dg gives a function G^{2} → A. There are three checks to be done to complete the definition of δ_{1}:

Proposition B.3.1 dg is a 2-cocycle, ie. a factor set. That is,

sdg[t, u] · dg[s, tu] = dg[st, u] · dg[s, t]

for all s, t, u ∈ G

Note that dg measures how far the lifting g differs from a 1-cocycle, in that

dg[s, t] · g[st] = g[s] · sg[t]

The associative law in A shows

g[s] · (sg[t] · stg[u]) = (g[s] · sg[t]) · stg[u]

ie.

g[s] · (sdg[t, u] · sg[tu]) = (dg[s, t] · g[st]) · stg[u]

so, remembering im dg ⊆ Z(B),

sdg[t, u] · (g[s] · sg[tu]) = dg[s, t] · (g[st] · stg[u])
sdg[t, u] · dg[s, tu] · g[stu] = dg[s, t] · dg[st, u] · g[stu]

and cancelling g[stu] gives

sdg[t, u] · dg[s, tu] = dg[st, u] · dg[s, t]

as required.

Proposition B.3.2 The cohomology class of dg does not depend of the choice of g given f.

1remembering it and choosing a certain order in which to write its terms down. This order is chosen because it makes the rest of the construction work; not all orderings do.

Choose a cochain g′ : G ⟶ B with p ∘ g′ = f. Then p(g′[s]g[s]^{−1}) = 1 so g′[s]g[s]^{−1} ∈ ker p = im i = A for all s ∈ G and g′[s] = g[s] · a[s] for some cochain a : G ⟶ A.

Thus

dg′[s, t] = g′[s] · sg′[t] · g′[st]^{−1}
= g[s] · a[s] · sg[t] · sa[t] · a[st]^{−1} · g[st]^{−1}
= a[s] · sa[t] · a[st]^{−1} · dg[s, t]

since A is in the centre of B, which shows that dg′ is cohomologous to dg in the abelian group H^{2}(G, A).

Proposition B.3.3 dg depends only on the cohomology class of f.

Let f′ be cohomologous to f, so f′[s] = c^{−1}·f[s]·sc for some c ∈ C. Let g : G ⟶ B be a lifting of f, then choosing b ∈ p^{−1}({c}) gives a lifting g′[s] = b^{−1} · g[s] · sb of f′.

Then

dg′[s, t] = g′[s] · sg′[t] · g′[st]^{−1}
= b^{−1} · g[s] · (sb · sb^{−1}) · sg[t] · (stb · stb^{−1}) · g[st]^{−1} · b
= b^{−1} · dg[s, t] · b
= dg[s, t] · b^{−1} · b  since im dg ⊆ A ⊆ Z(B)
= dg[s, t]
□

Thus the function which assigns to the cohomology class of f in H^{1}(G, C) the cohomology class of dg in H^{2}(G, A) is well-defined, and will be written δ_{1} and called the 1-dimensional connecting morphism. It is easy to check that it is a morphism of pointed sets.

B.4 The Exact Cohomology Sequence

It is now routine to check the exactness of the following “cohomology sequences”.

Proposition B.4.1 Let

1 ⟶ A ⟶^{i} B ⟶^{p} C ⟶ 1

be an exact sequence of non-abelian G-modules. Then the sequence of pointed sets

1 → H^{0}(G, A) →^{i^{∗}} H^{0}(G, B) →^{p^{∗}} H^{0}(G, C)

→^{δ_{0}} H^{1}(G, A) →^{i^{∗}} H^{1}(G, B) →^{p^{∗}} H^{1}(G, C)

is exact. If in addition i(A) is in the centre of B, the sequence of pointed sets

1 → H^{0}(G, A) →^{i^{∗}} H^{0}(G, B) →^{p^{∗}} H^{0}(G, C)

→^{δ_{0}} H^{1}(G, A) →^{i^{∗}} H^{1}(G, B) →^{p^{∗}} H^{1}(G, C) →^{δ_{1}} H^{2}(G, A)

is exact.

PROOF - There are six places of exactness to check. Again we assume A is a subgroup of B, and i is inclusion, for simplicity of writing.

1. H^{0}(G, A): A^{G} ↪^{i^{∗}} B^{G}.

2. H^{0}(G, B): im i^{∗} ⊆ ker p^{∗} because p ∘ i = 0 so p^{∗} ∘ i^{∗} = 0. ker p^{∗} ⊆ im i^{∗}: if b ∈ ker p^{∗} ⊆ B^{G} then b ∈ ker p = im i = A and sb = b so b ∈ B^{G} ∩ A = A^{G} = im i^{∗}.

3. H^{0}(G, C): im p^{∗} ⊆ ker δ_{0}: If c = p(b) then δ_{0}c = db which is cohomologous to zero. ker δ_{0} ⊆ im p^{∗}: if c ∈ H^{0}(G, C) = C^{G} is in ker δ_{0} then there is a lifting b ∈ B of c such that db is cohomologous to zero in H^{1}(G, A), ie. there is a ∈ A with b^{−1} · sb = a^{−1} · sa f or all s ∈ G. Then b · a^{−1} = s(b · a^{−1}) so b · a^{−1} ∈ A^{G} ⊆ B^{G}. But p(b · a^{−1}) = c · 1 = c so c ∈ im p^{∗} : B^{G} ⟶ C^{G}.

4. H^{1}(G, A): im δ_{0} ⊆ ker i^{∗}: Let b ∈ p^{−1}({c}), then

\begin{matrix}G & ⟶^{δ_{0}c=db} & A & ⟶^{i} & B \\ s & ⟼ & b^{−1} · sb & ⟼ & b^{−1} · sb \end{matrix}

is cohomologous to zero in H^{1}(G, B). ker i^{∗} ⊆ im δ_{0}: if f ∈ ker i^{∗} then there is b ∈ B such that f[s] = b^{−1} · sb for all s ∈ G. So f = db and so f ∈ im δ_{0}

5. H^{1}(G, B): im i^{∗} ⊆ ker p^{∗} because p ∘ i = 0 so p^{∗} ∘ i^{∗} = 0. ker p^{∗} ⊆ im i^{∗}: if f ∈ ker p^{∗} then p(f[s]) = c^{−1} · sc for some c ∈ C and all s ∈ G. Choose b ∈ p^{−1}({c}), then p(b · f[s] · sb^{−1}) = 1 for all s, so b · f[s] · sb^{−1} = a[s] for some cochain a : G ⟶ A.

a[st] = b · f[s] · (sb^{−1} · sb) · sf[t] · stb^{−1}
= a[s] · sa[t]

so a is a 1-cocycle and so f[s] = b^{−1} · a[s] · s[b] is cohomologous to a ∈ im i^{∗}.

6. H^{1}(G, C), assuming A ⊆ Z(B). im p^{∗} ⊆ ker δ_{1}: if f[s] = p(g[s]) for a 1cocycle g : G ⟶ B then δ_{1}f = dg which is cohomologous to zero in H^{2}(G, A). ker δ_{1} ⊆ im p^{∗}: Let f ∈ ker δ_{1}, f : G ⟶ C. So there is a lifting g : G ⟶ B such that dg[s, t] = a[s] · sa[t] · a[st]^{−1} for some cochain a : G ⟶ A, ie.

g[s] · sg[t] · g[st]^{−1} = a[s] · sa[t] · a[st]^{−1}

Which rearranges, because A is in the centre of B, to give

(a[s]^{−1} · g[s]) · s(a[t]^{−1} · g[t]) = a[st]^{−1} · g[st]

so that g′[s] := a[s]^{−1} · g[s] : G ⟶ B is a 1-cocycle with p^{∗}g′ = p^{∗}(a^{−1} · g) = p^{∗}g = f, and so f ∈ im p^{∗}.

□

Appendix C

Galois Theory

In this section I list the ideas and results of Galois Theory needed for the discussion of Galois Cohomology in Chapter 2. Most of the results are fundamental and are stated without proof; for more details, see any introductory text on Galois Theory, for example Stewart, [8]. Two particularly relevant results (the Normal Basis Theorem, linear independence of automorphisms) are proved here.

C.1 Basic Results

If L is a field and K a subfield of L, then L is called an extension of K. In this situation we write L/K to denote the pair (L, K). The group of K-automorphisms of L, ie. field automorphisms of L which leave every element of K fixed, under composition of functions, is called the Galois Group of L over K, written in this text as Gal(L/K). Elsewhere it is variously written as G(L/K), G_{L/K}, Γ(L/K), Γ_{L/K} or Aut_{K}(L).

Definition C.1.1 L/K is called a normal extension if every polynomial in K[X] with one root in L factorises into linear factors in L.

Proposition C.1.2 L/K is a normal extension if and only if it has the following property: every K-homomorphism from L into an extension M of L has its image contained in L.

Definition C.1.3 L/K is called a separable extension if, for every λ ∈ L, the polynomial of least degree in K[X] having λ as a root is not divisible by (X − λ)^{2} in K̄[X]. If L/K is not separable it is called inseparable.

Some fields do not have any inseparable extensions; such fields are called perfect.

Proposition C.1.4 A field K is perfect if and only if one of the following conditions holds.

1. K has characteristic zero 2. K has characteristic p > 0, and the field homomorphism

\begin{matrix}K & ⟶ & K \\ a & ⟼ & a^{p} \end{matrix}

is bijective; ie. every element has a unique p^{th} root.

If L/K is a field extension, then L is naturally a K-vector space. Write [L : K] := dim_{K} L. If [L : K] is finite, L/K is called a finite extension.

Definition C.1.5 L/K is called a Galois Extension if the fixed field of its Galois group (ie. {a ∈ L : σa = a ∀σ ∈ Gal(L/K)}) is equal to K.

Proposition C.1.6 A finite extension is a Galois extension if and only if it is normal and separable.

The next proposition makes no assumptions of finiteness. In the finite case it forms a component of the usual Fundamental Theorem of Galois Theory.

Proposition C.1.7 If L/K is any extension and L ⊇ E ⊆ K such that E/K is normal, then Gal(L/E) is a normal subgroup of Gal(L/K).

PROOF - If s ∈ Gal(L/K) then normality of E/K implies sE = E. Let t ∈ Gal(L/E) and x ∈ E. Then (s^{−1}ts)x = s^{−1}t(sx) = s^{−1}(sx) since sx ∈ E, and this is again x. So s^{−1}ts ∈ Gal(L/E), as required.

□

Theorem C.1.8 (Fundamental Theorem of Galois Theory) Let L/K be a finite Galois extension with group G = Gal(L/K). Then Card(G) = [L : K]. Let C be the set of subgroups of G and F the set of subextensions M (ie. L ⊇ M ⊇ K). The map

\begin{matrix}F & ⟶ & C \\ M & ⟼ & Gal(L/M) \end{matrix}

is bijective. Then inverse is given by taking a subgroup H of G to its fixed field inside L. M/K is a normal extension if and only if Gal(L/M) is a normal subgroup of G, in which case Gal(M/K) ≅ Gal(L/K)/Gal(L/M).

One stage in the proof of this is the following lemma of Dedekind.

Lemma C.1.9 (Linear Independence of Monomorphisms) Let L and M be fields. Then any set of distinct injective homomorphisms (“monomorphisms”) L ⟶ M is linearly independent over M.

PROOF - Let {ψ_{i}}_{i∈I} be a set of distinct monomorphisms and suppose

a_{1}ψ_{1} + … + a_{n}ψ_{n} = 0  a_{i} ∈ M  (C.1)

is an equation of linear dependence between some of the ψ_{i}, with all a_{i} ≠ 0 and n smallest possible. Then n > 1 because each ψ_{i} is a monomorphism.

Now let y ∈ L be such that ψ_{1}(y) ≠ ψ_{n}(y). Evaluate (C.1) at yx to get

a_{1}ψ_{1}(y)ψ_{1}(x) + … + a_{n}ψ_{n}(y)ψ_{n}(x) = 0

for all x ∈ L. Now multiply (C.1) by ψ_{n}(y) and evaluate at x:

a_{1}ψ_{n}(y)ψ_{1}(x) + … + a_{n}ψ_{n}(y)ψ_{n}(x) = 0

for all x ∈ L. Subtracting these equations gives a nonzero relation of linear dependence with less nonzero terms than (C.1), a contradiction.

□

The Galois theory of finite fields is particularly simple. If L/K is a finite extension of finite fields, let p be the characteristic of K (and hence also of L). Then elementary linear algebra shows that K has q = p^{m} elements and L has q^{n} elements, where m = [K : 𝔽_{p}] is the degree of K over the base field 𝔽_{p} = ℤ/p and n = [L : K]. L/K is always a Galois extension - K is perfect, so that L/K is separable, and one can show that L is a splitting field for the polynomial X^{q^{n}} − X over K, so that it is normal.

In fact, there is a unique field of cardinality p^{m} for each prime p and positive integer m.

The map σ : L ⟶ L which maps x ∈ L to x^{q} is a K-automorphism of L, called the Frobenius automorphism. In fact, any K-automorphism of L is a power of the Frobenius. Thus Gal(L/K) is cyclic of order n.

C.2 The Normal Basis Theorem

It is Theorem C.2.6. This proof is taken from David J. Winter, The Structure of Fields (Springer), page 51.

Lemma C.2.1 Let L/K be a cyclic extension. Then L has a K-basis of the form {σ(y) : σ ∈ G = Gal(L/K)} for some y ∈ L.

PROOF - Dedekind’s lemma shows that

e, τ, τ^{2}, …, τ^{n−1}

are L-linearly independent. Hence the minimum polynomial for τ ∈ End_{K}(L) is X^{n} − 1. If we make L into a K[X]-module via τ in the usual way (f(X)λ := f(τ)(λ)), then the structure theorem for modules over principle ideal domains shows that L decomposes as a direct sum of cyclic K[X]-modules, one of which is then K[X]/(X^{n} − 1). Counting K-dimension shows this is the only one, so that L is a cyclic K[G]-module, which is what we wanted to show.

□

Corollary C.2.2 if L/K is a finite extension of a finite field then L has a normal K-basis.

PROOF - in this case L/K is a Galois extension and Gal(L/K) is cyclic.

□

That disposes of the case L finite.

Theorem C.2.3 Let K be an infinite field, let L_{1} and L_{2} be two extensions of K and let σ_{1}, …, σ_{n} be distinct elements of Hom_{K}(L_{1}, L_{2}). Then there is a ∈ L_{1} such that σ_{1}(a), …, σ_{n}(a) are pairwise distinct.

PROOF - Choose some elements a_{0}, …, a_{m} ∈ L_{1} such that

(σ_{i}a_{0}, σ_{i}a_{1}, …, σ_{i}a_{m}) = (σ_{j}a_{0}, σ_{j}a_{1}, …, σ_{j}a_{m}) ⟺ i = j

Let h(X) = ∑_{r=0}^{m} a_{r}X^{r} and note that

σ_{i}(h(X)) = σ_{j}(h(X)) in L_{2}[X] ⟺ i = j

Thus

∏_{i≠j} ((σ_{i}h)(X) − (σ_{j}h)(X))

is a nonzero polynomial in L_{2}[X]. Since K is infinite, there is t ∈ L such that

∏_{i≠j} ((σ_{i}h)(t) − (σ_{j}h)(t)) ≠ 0

Put a = h(t), then σ_{1}(a), …, σ_{n}(a) are distinct.

□

Proposition C.2.4 Let L/K be a finite extension with Galois group G = {1 = σ_{1}, …, σ_{n}}. Assume K is infinite. Then there is y ∈ L such that det(σ_{i}^{−1}σ_{j}(y)) ≠ 0.

PROOF - By the theorem, we can choose z ∈ L such that σ_{1}(z), …, σ_{n}(z) are distinct. Define a polynomial f ∈ L[X] such that f(z) = 1, f(σ_{i}(z)) = 0 for i ≠ 1, by

f(X) = ∏_{i≠1} \frac{X − σ_{i}(z)}{z − σ_{i}(z)}

Then f(σ_{j}^{−1}σ_{i}(z)) = δ_{ij} (Kronecker delta). Apply σ_{i}^{−1}σ_{j} to this equation:

(σ_{j}^{−1}σ_{i}(f))(z) = δ_{ij}

where (σ_{j}^{−1}σ_{i}(f)) means the polynomial obtained from f by applying σ_{j}^{−1}σ_{i}(f) to each coefficient. Hence

det ((σ_{j}^{−1}σ_{i}(f))(z))_{i,j} = 1

so the polynomial in L[X]

det ((σ_{j}^{−1}σ_{i}(f))(X))_{i,j}

is not identically zero. Now because K is infinite, there is x ∈ K such that the value of this polynomial at x is not zero. But because x is fixed by G, this means that

det (σ_{j}^{−1}σ_{i}(f(x)))_{i,j} ≠ 0.

Setting y = f(x) gives the desired result.

□

Corollary C.2.5 Let L/K be as in the proposition, and assume furthermore that it is a Galois extension. Then there is y ∈ L such that σ_{1}(y), …, σ_{n}(y) are K-linearly independent (and hence a K-basis for L).

PROOF - Take y as given by the proposition. If a K-linear dependence of σ_{1}(y), …, σ_{n}(y) existed, it would give a K-linear dependence of the columns of the n × n matrix (σ_{i}^{−1}σ_{j}(y))_{i,j}.

□

Theorem C.2.6 (Normal Basis Theorem) Let L/K be a finite Galois extension with group G. Then there exists y ∈ L such that {σ(y) : σ ∈ G} is a K-basis for L, ie. L is a cyclic K[G]-module.

PROOF - This is corollaries C.2.5 and C.2.2

□

Appendix D

Topological Groups

The purpose of this appendix is to outline the basic ideas of topological groups needed in Chapter 2. The reader is assumed to have knowledge of the ideas of topologies, open and closed sets, compactness, induced topology on subspaces, the discrete topology and complete metric spaces. Our reference for these ideas is M. A. Armstrong, Basic Topology (Springer). For the properties of topological groups we refer to [6].

A topological group G is a group and a Hausdorff topological space, and these two structures must be compatible in that the multiplication and inversion maps

\begin{matrix}G × G & ⟶ & G & G & ⟶ & G \\ (x, y) & ⟼ & xy & x & ⟼ & x^{−1} \end{matrix}

are continuous.

The topology on a group is essentially the same at each point, because a set X ⊆ G is open if and only if its translation sX is open for each s ∈ G. Therefore we may restrict our attention to the neighbourhoods of the identity element e. We will study topological groups via the following definition:

Definition D.0.7 A set S of subsets of a group G is said to form a fundamental system of open neighbourhoods of the identity if the following conditions hold.

1. ⋂ S = {e};

2. S is closed under finite intersection, ie. if U, V ∈ S then U ∩ V ∈ S;

3. For U ∈ S, there is V ∈ S with V V^{−1} ⊆ U;

4. If U ∈ S and a ∈ U there is V ∈ S with V a ⊆ U;

5. If U ∈ S and a ∈ G then there is V ∈ S with aV a^{−1} ⊆ U.

For such an S we have the result Theorem D.0.8 The set S defines a topology on G consisting of unions of sets of the form {V a : V ∈ S, a ∈ G} making G into a topological group.

For the proof see [6], page 26.

We will also need to use the topology on a product of topological groups. Let {G^{i}}_{i∈I} be a possibly infinite collection of topological groups. Then the Cartesian product ∏ G^{i} is a group. The product topology on ∏ G^{i} is by definition the smallest^{1} topology on the set such that for each j ∈ I, the projection on the j-th component map

π_{j} : ∏ G^{i} ⟶ G^{j}

is continuous. The open sets are then of the form

{(x_{i}) : x^{j} ∈ U_{j} for j ∈ J, x_{i} ∈ G^{i} for i ∈ I − J}

for some finite subset J ⊆ I and open sets U_{j} ⊆ G^{j} chosen for each j ∈ J. Now in the case where we will use this, each G^{i} will be a finite group, with the discrete topology. In this case the open sets in the product can be built up in a very natural way from the sets

ker π_{j} = ∏_{i≠j} G^{i} × {e} = {(x_{i}) : x_{j} = e}

by taking a finite intersection of them, translating by an element of ∏ G^{i}, and then taking a union of such sets.

We will also need:

Theorem D.0.9 (Tychonoff’s Theorem) The product of a collection of compact topological spaces is compact.

The proof, which relies on the Axiom of Choice, can be found in [6]

1“smallest topology” with a certain property means that it is a subset of every other topology with this property.

Appendix E

Central Simple Algebras

Let K be a field. We discuss here K-algebras, simple rings and their structures.

Definition E.0.10 A K-algebra A is a ring which is also a K-vector space, and whose ring-multiplication is K-bilinear in the sense that

(κa) · b = a · (κb) = κ(a · b)

for all a, b ∈ A, κ ∈ K.

There is a natural inclusion

\begin{matrix}K & ↪ & A \\ κ & ↦ & κ1_{A} \end{matrix}

so that K may be regarded as a subfield of the ring A. Since (κ1) · a = κ(1 · a) = κ(a · 1) = a · (κ1), K is then in the centre Z(A) of A, the set of all the elements of A which commute with all the other elements of A. Conversely, a ring containing K in its centre is necessarily a K-algebra.

A morphism of K-algebras is simply a K-linear ring homomorphism.

The K-algebras that we can study with the Brauer group have to satisfy three conditions, which will now be described.

A K-algebra A is called finite-dimensional if it is finite-dimensional as a Kvector space. It is called central if its centre is K.

We will study K-algebras which satisfy these two conditions, and are also simple rings.

E.1 Simple Rings

Definition E.1.1 Let R be a ring.

1. A left-R-module is called simple if it has no R-submodules other than itself and 0.

2. A left-R-module is called semisimple if it decomposes as a direct sum of simple submodules.

3. R is called semisimple if it is semisimple as a left-R-module.

4. a left-ideal of R is called simple if it is simple as a left-R-module

5. R is called simple if it is semisimple and all its simple left-ideals are isomorphic as R-modules.

Proposition E.1.2 A simple ring has no proper nonzero double-sided ideals.

PROOF - Write R = ∑_{i∈I} 𝔞_{i} where 𝔞_{i} are simple left-R-modules, and let 𝔟 be a proper nonzero double-sided ideal of R.

So 𝔟 = 𝔟R (because 1 ∈ R) = ∑_{i∈I} 𝔟𝔞_{i}, so 𝔟𝔞_{i} ≠ 0 for some i ∈ I, say 𝔟𝔞_{1} ≠ 0. Thus 𝔟𝔞_{1} = 𝔞_{1}, being a nonzero submodule of 𝔞_{1}.

R being simple, there are R-module isomorphisms σ_{i} : 𝔞_{1} → 𝔞_{i}. So 𝔞_{i} = σ_{i}(𝔞_{1}) = σ_{i}(𝔟𝔞_{1}) = 𝔟σ_{i}(𝔞_{1}) = 𝔟𝔞_{i}. Thus

𝔟 = ∑_{i∈I} 𝔟𝔞_{i} = ∑_{i∈I} 𝔞_{i} = R

□

There is a converse of this result, using the following lemma:

Lemma E.1.3 Let R be a ring, 𝔞 a simple left-ideal of R and M a simple left-R-module. If 𝔞 ≇ M then 𝔞M = 0.

PROOF - Suppose 𝔞M ≠ 0, so there are a ∈ 𝔞, m ∈ M with am ≠ 0. Define

\begin{aligned}ϕ : & 𝔞 & ⟶ & M \\ & x & ⟼ & xm \end{aligned}

for this m. Then ϕ is an R-module homomorphism. im ϕ is an R-submodule of M and is not 0, so it is all of M. ker ϕ is an R-submodule of 𝔞 which is not all of 𝔞, so ker ϕ = 0. Hence ϕ is an isomorphism. So 𝔞M ≠ 0 ⇒ 𝔞 ≅ M, as required.

□

Proposition E.1.4 If R is a semisimple ring with no proper nonzero double-sided ideals then R is simple.

PROOF - Suppose R is not simple, then either R is not semisimple or R = ⨁_{i∈I} 𝔞_{i}, a direct sum of simple left-ideals. Let I′ := {i ∈ I : 𝔞_{i} ≅ 𝔞_{1}}, a proper subset of I, and 𝔟 := ∑_{i∈I′} 𝔞_{i}, which is a nonzero left-ideal of R because R is not simple. If we had 𝔟 = R then for any simple left ideal 𝔞 of R, 𝔞 ∩ 𝔞_{i} would be nonzero for some i ∈ I′, giving 𝔞 = 𝔞_{i} because 𝔞 ∩ 𝔞_{i} is a nonzero submodule of each, and contradicting I ≠ I′.

So 𝔟 is a proper nonzero left-ideal of R. Now

𝔟R = (∑_{i∈I′} 𝔞_{i})(∑_{j∈I} 𝔞_{j}) = ∑_{i∈I′} 𝔞_{i}(∑_{j∈I′} 𝔞_{j} + ∑_{j∈I−I′} 𝔞_{j}) = 𝔟^{2} + 0 = 𝔟^{2} ⊆ 𝔟

so 𝔟 is a proper nonzero double-sided ideal of R.

□

Proposition E.1.5 If a Noetherian ring R has a simple left-ideal, and R has no proper nonzero double-sided ideals, then R is simple.

PROOF - Consider the set of direct sums of simple left-ideals of R ordered by containment (𝔞 ≤ 𝔟 ⟺ 𝔞 ⊆ 𝔟). By the ascending chain condition, this set has a maximal element, ie. a direct sum 𝔟 = ⨁_{i∈I} 𝔞_{i} which is not contained properly in any other such direct sum. (a priori the set I may be empty, in which case 𝔟 = 0).

Now for any j ∈ I, 𝔞_{j}t is either simple or zero. This is because the map

\begin{matrix}𝔞_{j} & ⟶ & 𝔞_{j}t \\ m & ⟼ & mt \end{matrix}

is clearly a surjective R-module homomorphism, and its kernel, being an Rsubmodule of 𝔞_{j}, is either all of 𝔞_{j} or 0. Therefore it is either an isomorphism or the zero map, and so 𝔞_{j}t is either simple or zero.

If 𝔞_{j}t = 0 for given j and t, then 𝔞_{j}t ⊆ 𝔟. If on the other hand 𝔞_{j}t ≠ 0, the sum

∑_{i∈I} 𝔞_{i} + 𝔞_{j}t

is not direct, so 𝔞_{j}t ∩ 𝔞_{i} ≠ 0 for some i ∈ I. But then 𝔞_{j}t = 𝔞_{j}t ∩ 𝔞_{i} = 𝔞_{i}, the intersection being a nonzero submodule of both of the simple modules.

This shows 𝔟t ⊆ 𝔟 for all t ∈ R, so 𝔟 is a double-sided ideal of R. Since R has a simple left-ideal, 𝔟 ≠ 0, so 𝔟 = R and R = ⨁_{i∈I} 𝔞_{i} is semisimple.

Now the previous proposition shows R is simple.

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Proposition E.1.6 A finite-dimensional K-algebra A is a simple ring if and only if it has no proper nonzero double-sided ideals.

PROOF - “⇒” was shown above.

“⇐”: Because A is a finite-dimensional K-vector space, and a left-ideal of A a vector subspace, A must satisfy both the ascending and descending chain conditions on left-ideals, ie. A is both Noetherian and Artinian. This is because an ascending or descending sequence of ideals gives a monotonic and bounded sequence of integers under the map dim_{K}. Such a sequence must be constant and so linear algebra shows the original sequence of ideals is constant (ie. V ⊆ W and dim_{K} V = dim_{K} W ⇒ V = W).

By the descending chain condition it is possible to choose a nonzero left-ideal of A which is minimal in the set of all such, ie. which has no proper nonzero submodules, ie. is simple.

Thus A is a Noetherian ring with a simple left-ideal and no proper nonzero double-sided ideals, so by proposition E.1.5, A is simple.

□

It can be shown^{1} that if A is a simple algebra like this, so A = ⨁_{i=1}^{n} 𝔞, then all finite-dimensional left-A-modules are also A-module-isomorphic to a direct sum ⨁_{i=1}^{N} 𝔞; in particular, all simple left-A-modules are isomorphic.

Definition E.1.7 A finite-dimensional K-algebra is called central simple if it is simple and has centre K.

These algebras will be described by the elements of the Brauer Group. There is a general structure theorem for simple rings which will be necessary, and which is described next.

E.2 Structure of Simple Algebras

[This part follows Weil, ”Basic Number Theory”, Chapter IX, §1]

We will be interested in comparing our central simple algebras to matrix algebras.

Definition E.2.1 If R is a ring, write

\begin{aligned}μ : & R × R & ⟶ & R \\ & (x, y) & ⟼ & x · y \end{aligned}

for the multiplication table of R. Then the additive group (R, +) combined with the multiplication

\begin{aligned}μ^{∘} : & R × R & ⟶ & R \\ & (x, y) & ⟼ & μ(y, x) \end{aligned}

is a ring, called the opposite ring or inverse ring of R and written R^{∘}.

Theorem E.2.2 (Wedderburn’s Theorem) If R is a simple ring then R is isomorphic to a matrix ring M_{n}(D) where D is a division ring.

1Weil, Basic Number Theory, ch. 9, §1, Prop. 1

PROOF - Let 𝔞 be a simple left-ideal of R and put D := End_{R}(𝔞), the ring of R-endomorphisms of the R-module 𝔞 under pointwise addition, and composition of endomorphisms. If δ ∈ D then ker δ and im δ are R-submodules of 𝔞; if ker δ ≠ 0 then ker δ = 𝔞 so that δ = 0; otherwise im δ ≠ 0 so im δ = 𝔞, so that δ is an R-automorphism of 𝔞 and so is invertible in D.

Thus every nonzero element of D is invertible, so D is a division ring. Now consider End_{R}(R); if ψ ∈ End_{R}(R) then ψ(x) = xψ(1) ∀x ∈ R so that ψ

is a right-homothety. Define the homothety map

h : R ⟶ End_{R}(R)
x ⟼ h(x) : R ⟶ R
y ⟼ yx

h satisfies h(x + x′) = h(x) + h(x′) and h(xx′) = h(x′)h(x) so that h is a ring anti-homomorphism. h is surjective by the comment above, and, since h(x)(1) = x, injective. Thus h is a ring anti-isomorphism.

But End_{R}(R) = End_{R}(⨁_{i=1}^{n} 𝔞) ≅ M_{n}(End_{R}(𝔞)) as follows. If ψ ∈ End_{R}(⨁_{i=1}^{n} 𝔞) then, writing 𝔞_{i} for the i-th summand,

π_{i} ∘ ψ|_{𝔞_{j}} : 𝔞_{j} ⟶ ⨁_{i=1}^{n} 𝔞 ⟶^{π_{i}} 𝔞_{i}

is an R-endomorphism of 𝔞, ie. an element of D. Putting

\begin{aligned}μ : & End_{R}(R) & ⟶ & M_{n}(D) \\ & ψ & ⟼ & (π_{i} ∘ ψ|_{𝔞_{j}})_{i,j} \end{aligned}

we have

μ(ψ ∘ ϕ) = (π_{i} ∘ (ψ ∘ ϕ)|_{𝔞_{j}})_{i,j}
= (π_{i} ∘ ( ∑_{k} (ψ|_{𝔞_{k}} ∘ π_{k}) ∘ ϕ|_{𝔞_{j}})_{i,j}
= ( ∑_{k} (π_{i} ∘ ψ|_{𝔞_{k}}) ∘ (π_{k} ∘ ϕ|_{𝔞_{j}}))_{i,j}
= (π_{i} ∘ ψ|_{𝔞_{j}})_{i,j}(π_{i} ∘ ϕ|_{𝔞_{j}})_{i,j} = μ(ψ)μ(ϕ)

and it is easy to check that μ is additive and bijective, so that μ is an isomorphism of rings. Thus R is anti-isomorphic to M_{n}(D).

Define

\begin{aligned}τ : & M_{n}(D) & ⟶ & M_{n}(D^{∘}) \\ & (d_{ij})_{i,j} & ⟼ & (d_{ji})_{i,j} \end{aligned}

so that τ is given by transposition of matrices. τ is clearly an isomorphism of additive groups, and (writing all products in the ring D),

τ((d_{ij})_{i,j}(d′_{ij})_{i,j}) = τ((∑_{k} d_{ik}d′_{kj})_{i,j}) = (∑_{k} d_{jk}d′_{ki})_{i,j}

while

τ((d′_{ij})_{i,j})τ((d_{ij})_{i,j}) = (d′_{ji})_{i,j}(d_{ji})_{i,j} = (∑_{k} d_{jk}d′_{ki})_{i,j}

so that τ is an anti-isomorphism of rings. Then τ ∘ μ ∘ h : R ⟶ M_{n}(D^{∘}) is an isomorphism of rings. Since D^{∘} is a division ring, the theorem is proved.

□

Remark - one may avoid the use of anti-homomorphisms by considering End_{R}(R) as operating on the right, so that (x)ϕ ∘ ψ = ((x)ϕ)ψ is the product ϕ ∘ ψ ∈ End_{R}(R). The effect of this is to replace End_{R}(R) by its opposite ring, to which it is anti-isomorphic.

If A is a central simple K-algebra, then D = End_{A}(𝔞) is also a central-simple algebra over K, and the isomorphism given by Wedderburn’s Theorem is an isomorphism of K-algebras (it is easy to check that each of h, μ, τ is K-linear). Let us check the assertions about D:

• Finite-dimensional: End_{A}(𝔞) ⊆ End_{K}(𝔞) ≅ M_{m}(K).

• Central: since A ≅ M_{n}(D) as K-algebras, K ⊆ Z(M_{n}(D)) ≅ Z(A) = K which is a 1-dimensional K-vector space over K ⊆ A, so Z(D) = K.

• Simple: D is simple just like a field is simple - any nonzero ideal contains an invertible element, and so is all of D.

Proposition E.2.3 A determines n uniquely, and D up to isomorphism.

PROOF - It is sufficient to show that the K-algebra M_{n}(D) determines D up to isomorphism. For then if A ≅ M_{n}(D) ≅ M_{n′}(D′) we have D ≅ D′ and, considering the dimension of the free modules M_{n}(D) and M_{n′}(D′) over the principal ideal domain D, n^{2} = n′^{2} so n is also uniquely determined.

Let E_{kl} := (δ_{ik}δ_{jl})_{i,j} be the standard D-basis for M_{n}(D), where δ denotes the Kronecker δ-symbol. Then

M := M_{n}(D)E_{11}

is the left-M_{n}(D)-submodule of matrices with 0 outside the first column. If X = (x_{ij})_{i,j} ∈ M is nonzero then x_{k1} ≠ 0 for some k and x_{ij} = 0 for j > 1. So

x_{k1}^{−1}E_{mk}X = x_{k1}^{−1}(∑_{p} δ_{im}δpkx_{pj})_{i,j} = x_{k1}^{−1}(δ_{im}x_{kj})_{i,j} = E_{m1} ∈ M

So M_{n}(D)X = M, ie. M is a simple-left-M_{n}(D)-module.

Now consider End_{M_{n}(D)}(M). If f ∈ End_{M_{n}(D)}(M) then put

X = (x_{ij})_{i,j} := f(E_{11})

X has zeros outside the first column because X ∈ M; in fact M_{n}(D)-linearity implies that X also has zeros outside the first row, for

\begin{matrix}0 = f(0) = f(E_{1k}E_{11}) (fork ≠ 1) \\ = E_{1k}f(E_{11}) = (∑_{p} δ_{i1}δ_{jk}x_{pj})_{i,j} = (δ_{i1}x_{kj})_{i,j} \end{matrix}

Considering the (1, 1)-th element of this matrix gives x_{k1} = 0 for k > 1. So X = (δ_{1i}δ_{1j}d) for some d ∈ D, and f((y_{ij})_{i,j}) = (y_{ij}d)_{i,j}. f ⟼ d defines a K-linear anti-isomorphism End_{M_{n}(D)} ⟶ D, and so shows that M_{n}(D) determines D up to isomorphism.

□

Proposition E.2.4 If K is algebraically closed and A is a central simple K-algebra then

A ≅ M_{n}(K)

as K-algebras for some n.

PROOF - Wedderburn’s Theorem shows that A ≅ M_{n}(D) for a finitedimensional division algebra over K. But K admits no such algebras other than itself. For if D ≠ K is such an algebra, let d ∈ D − K, then K(d) is an algebraic extension of K, other than K, which is impossible. So in fact,

A ≅ M_{n}(K)

E.3 Matrix Rings and Tensor Products

It will be useful to have the following two observations on the interplay of the functor M_{n} and the bifunctor ⊗_{K}.

Proposition E.3.1 If B is a K-algebra then

M_{n}(B) ≅ M_{n}(K) ⊗_{K} B

as K-algebras.

PROOF - Define

\begin{aligned}ϕ : & M_{n}(K) × B & ⟶ & M_{n}(B) \\ & ((κ_{ij})_{i,j}, b) & ⟼ & (κ_{ij}b)_{i,j} \end{aligned}

ϕ is K-bilinear so defines

\begin{aligned}ϕ̃: & M_{n}(K) ⊗_{K} B & ⟶ & M_{n}(B) \\ & (κ_{ij})_{i,j} ⊗ b & ⟼ & (κ_{ij}b)_{i,j} \end{aligned}

a K-linear map.

ϕ̃(((κ_{ij})_{i,j} ⊗ b)((κ′_{ij})_{i,j} ⊗ b′)) = ϕ̃((∑_{p} κ_{ip}κ′_{pj})_{i,j} ⊗ bb′)

= (∑_{p} κ_{ip}κ′_{pj}bb′)_{i,j} = (κ_{ip}bκ′_{pj}b′)_{i,j}

= ϕ̃((κ_{ij})_{i,j} ⊗ b)ϕ̃((κ′_{ij})_{i,j} ⊗ b′)

so ϕ̃ is a ring homomorphism. ϕ̃ is surjective for, given (b_{ij})_{i,j} ∈ M_{n}(B), (b_{ij})_{i,j} = ϕ̃(∑_{i,j}[(δ_{ki}δ_{lj})_{k,l} ⊗ b_{ij}]). Now compare the K-dimensions:

dim_{K} M_{n}(B) = n^{2} dim_{K} B = dim_{K}(M_{n}(K) ⊗_{K} B)

Thus ϕ̃ is an isomorphism.

□

Proposition E.3.2 M_{n}(M_{n′}(K)) ≅ M_{nn′}(K) as K-algebras.

PROOF - Define

\begin{aligned}ϕ : & M_{n}(M_{n′}(K)) & ⟶ & M_{nn′}(K) \\ & ((κ_{i′j′}^{i j})_{i′,j′})_{i,j} & ⟼ & (κ_{i′j′}^{i j})_{s,t} \end{aligned}

where we write s, t ∈ {1, …, nn′} in the unique expression as s = n′i+i′, t = n′j+j′ to obtain i, j, i′, j′. ϕ is a K-linear map. Define typical matrices in M_{n}(M_{n′}(K)) by:

A := ((a_{i′j′}^{i j})_{i′,j′})_{i,j}, B := ((b_{i′j′}^{i j})_{i′,j′})_{i,j}

Then

AB = ( ∑_{p} (a_{i′j′}^{i p})_{i′,j′}(b_{i′j′}^{p j})_{i′,j′})_{i,j} = ( ∑_{p} ( ∑_{p′} a_{i′p′}^{i p}b_{p′j′}^{p j})_{i′,j′})_{i,j}

so

ϕ(AB) = ( ∑_{p,p′} a_{i′p′}^{i p}b_{p′j′}^{p j})_{s,t} = (a_{i′j′}^{i j})_{s,t}(b_{i′j′}^{i j})_{s,t} = ϕ(A)ϕ(B)

So ϕ is a ring homomorphism. Since ϕ is obviously bijective, the proposition is proved.

□

Corollary E.3.3 M_{n}(K) ⊗_{K} M_{n′}(K) ≅ M_{nn′}(K).

E.4 Extension of Scalars

We look at how, given a field extension L ⊇ K, we can obtain L-algebras from K-algebras.

If V is a K-vector space and L ⊇ K a field extension, then V ⊗_{K} L is made into an L-vector space by

λ(v ⊗ λ′) := v ⊗ λλ′ ∀λ, λ′ ∈ L, v ∈ V.

Proposition E.4.1 If V is finite-dimensional with K-basis (u_{i})_{i=1}^{n} then V ⊗_{K} L is a finite-dimensional L-vector space with L-basis (u_{i} ⊗ 1)_{i=1}^{n}.

PROOF - (u_{i} ⊗ 1)_{i=1}^{n} spans V ⊗_{K} L, for every ℤ-generator v ⊗ λ of V ⊗_{K} L may be written

v ⊗ λ = (∑_{i=1}^{n} κ_{i}u_{i}) ⊗ λ = ∑_{i=1}^{n} u_{i} ⊗ κ_{i}λ = ∑_{i=1}^{n} κ_{i}λ(u_{i} ⊗ 1)

for some κ_{i} ∈ K. To see L-linear independence, consider the function

\begin{aligned}ψ : & V × L & ⟶ & L^{n} \\ & (∑ κ_{i}v_{i}, λ) & ⟼ & (κ_{i}λ)_{i=1}^{n}. \end{aligned}

This is clearly K-bilinear so defines a function ψ̃ : V ⊗_{K} L → L^{n} with ψ̃(v⊗λ) = ψ(v, λ) by the universal property of tensor products. But then if ∑_{i=1}^{n} λ_{i}(u_{i}⊗1) = 0 for some (λ_{i})_{i=1}^{n} ∈ L^{n} then

0 = ψ̃(∑_{i=1}^{n} λ_{i}(u_{i} ⊗ 1)) = ψ̃(∑_{i=1}^{n} u_{i} ⊗ λ_{i}) = (λ_{i})_{i=1}^{n},

so that (λ_{i})_{i=1}^{n} = 0

□

If A is a K-algebra, then A ⊗_{K} L is an L-algebra, with the product of two ℤ-generators defined by

(a ⊗ λ)(a′ ⊗ λ′) := aa′ ⊗ λλ′

If A is finite-dimensional over K, the proposition above shows that A ⊗_{K} L is finite-dimensional over L. The next result shows that A ⊗_{K} L is central over L if and only if A is central over K.

Proposition E.4.2 Let Z(R) denote the centre of a ring R. Then

Z(A ⊗_{K} L) = Z(A) ⊗_{K} L

PROOF - “⊇”: Let z = ∑_{i=1}^{n} a_{i}λ_{i} ∈ Z(A)⊗_{K} L, a_{i} ∈ Z(A) be a typical element of the RHS. Then for any ℤ-generator a ⊗ λ of A ⊗_{K} L,

(a ⊗ λ)z = (a ⊗ λ)(∑_{i=1}^{n} a_{i} ⊗ λ_{i}) = ∑_{i=1}^{n} aa_{i} ⊗ λλ_{i}

= ∑_{i=1}^{n} a_{i}a ⊗ λ_{i}λ = (∑_{i=1}^{n} a_{i} ⊗ λ_{i})(a ⊗ λ) = z(a ⊗ λ)

so z ∈ Z(A ⊗_{K} L).

“⊆”: Let z = ∑_{i=1}^{n} a_{i} ⊗λ_{i} ∈ Z(A⊗_{K} L). Using the relations in tensor products, this may be written with the λ_{i} K-linearly independent.

Then for any a ∈ A, z(a ⊗ 1) = (a ⊗ 1)z, that is ∑_{i=1}^{n} a_{i}a ⊗ λ_{i} = ∑_{i=1}^{n} aa_{i} ⊗ λ_{i} or

∑_{i=1}^{n}(a_{i}a − aa_{i}) ⊗ λ_{i} = 0  (E.1)

Consider now the sub-K-vector space of A ⊗_{K} L with basis (u_{j} ⊗ λ_{i})_{i=1,j=1}^{n m}, where (u_{j})_{j=1}^{m} is a K-basis for A. The element (E.1) belongs to this subspace, which shows that a_{i}a − aa_{i} = 0 in A for all i and hence a_{i} ∈ Z(A) for all i. Thus z ∈ Z(A) ⊗_{K} L.

□

Corollary E.4.3 A ⊗_{K} L is central over L if and only if A is central over K.

PROOF - This follows from the fact that K ⊗_{K} L is equal to {1 ⊗ λ : λ ∈ L}, the inclusion of L into A ⊗_{K} L.

□

We let Gal(L/K) act on A ⊗_{K} L on the second factor.

E.5 Simple Algebras and Opposite Rings

Proposition E.5.1 Let A be a finite-dimensional central K-algebra, and C = A ⊗_{K} A^{∘}. Define a K-linear map

F : C ⟶ End_{K}(A)
a ⊗ b ⟼ F (a ⊗ b) : A ⟶ A
x ⟼ axb

by the universal property of tensor products.

Then A is simple if and only if F is surjective. When this is the case, F is a K-vector space isomorphism.

For the proof, see [10], Chapter 9.

Corollary E.5.2 If L ⊇ K is a field extension then A is simple if and only if A ⊗_{K} L is simple.

PROOF - The proposition says that A ⊗_{K} L is simple if and only if

F_{L} : (A ⊗_{K} L) ⊗_{L} (A^{∘} ⊗_{K} L) ⟶ End_{L}(A ⊗_{K} L)

is surjective. But the properties of tensor products give canonical isomorphisms

(A ⊗_{K} L) ⊗_{L} (A^{∘} ⊗_{K} L) ≅ A ⊗_{K} A^{∘} ⊗_{K} (L ⊗_{L} L) ≅ (A ⊗_{K} A^{∘}) ⊗_{K} L

so that the map becomes F_{L} : (A ⊗_{K} A^{∘}) ⊗_{K} L ⟶ End_{L}(A ⊗_{K} L) given by F_{L}(a ⊗ b ⊗ λ)(x) = axb ⊗ λ. But if n = dim_{K} A = dim_{L} A ⊗_{K} L then

End_{L}(A ⊗_{K} L) ≅ M_{n}(L) ≅ M_{n}(K) ⊗_{K} L ≅ End_{K}(A) ⊗_{K} L

where the correspondence is

\begin{matrix}End_{K}(A) ⊗_{K} L & ≅ & End_{L}(A ⊗_{K} L) \\ ϕ ⊗ λ & ↦ & ϕ ⊗ λ \end{matrix}

Thus F_{L} is surjective if and only if F is, ie. A is simple if and only if A ⊗_{K} L is.

□

Corollary E.5.3 If L = K̄ is an algebraic closure of K, then A is simple if and only if A ⊗_{K} L ≅ M_{n}(L) for some n.

PROOF - A is simple if and only if A ⊗_{K} L is simple, and this is the case if and only if A ⊗_{K} L ≅ M_{n}(L) (see §E.2).

□

Thus there is always a field which, when tensored with A, gives a matrix algebra. Such a field is called a neutralising field for A. We will see soon that there is always a neutralising field which is a finite extension of K. But first there is one more corollary to the proposition

Corollary E.5.4 If A and B are central simple K-algebras then so is A ⊗_{K} B.

PROOF - Let L = K̄ be an algebraic closure of K. Then A ⊗_{K} L is simple with centre L so A ⊗_{K} L ≅ M_{n}(L). Similarly B ⊗_{K} L ≅ M_{n′}(L). Using this and the properties of tensor products,

(A ⊗_{K} B) ⊗_{K} L ≅ A ⊗_{K} (L ⊗_{K} B)
≅ A ⊗_{K} ((L ⊗_{L} L) ⊗_{K} B)
≅ (A ⊗_{K} L) ⊗_{L} (B ⊗_{K} L)
≅ M_{n}(L) ⊗_{L} M_{n′}(L)
≅ M_{nn′}(L)

This shows that (A ⊗_{K} B) ⊗_{K} L is central over L, so by §E.4 A ⊗_{K} B is central over K, and then by the previous corollary A ⊗_{K} B is simple.

□

E.6 Finite Dimensional Neutralising Fields

Proposition E.6.1 Let D be a finite-dimensional central division algebra over a field K. Then there exists a finite Galois extension L of K such that D ⊗_{K} L ≅ M_{n}(L) as L-algebras, for some n.

PROOF - We know that D is central simple over K, so by §E.5 we have D ⊗_{K} K̄ ≅ M_{n}(K̄), so dim_{K} D = dimK̄ D ⊗_{K} K̄ = n^{2} is a square. The desired field L will be constructed from a subfield L′ of D, whose existence follows from the following result, proved in [11], §10, no.3.

• Any maximal commutative subring of D is a field L′ with dim_{K} L′ = n; L′ can be chosen so that L′/K is a separable extension.

We show that L′ is a neutralising field for D, ie. D ⊗_{K} L′ is isomorphic to a matrix algebra over L′. To see this, define in the usual way, using the universal property of tensor products,

ϕ : D ⊗_{K} L′ ⟶ End_{L′}(D)
d ⊗ λ ⟼ ϕ(d ⊗ λ) : D ⟶ D
x ⟼ λxd

ϕ is clearly L′-linear, given the L′-vector space structure on D ⊗_{K} L′. Now

ϕ((d ⊗ λ)(d′ ⊗ λ′))(x) = ϕ(dd′ ⊗ λλ′)(x)
= λλ′xdd′ = ϕ(d′ ⊗ λ′)(λxd) = ϕ(d′ ⊗ λ′) ∘ ϕ(d ⊗ λ)(x)

so ϕ is a ring anti-homomorphism. We know (§E.5) that D ⊗_{K} L′ is simple, and since ϕ is not the zero map and ker ϕ is a double-sided ideal of D ⊗_{K} L′, ker ϕ = 0, so ϕ is injective.

Now consider the L′-dimension of each side to show that ϕ is also surjective. For the left-hand side, dim_{L′}(D ⊗_{K} L′) = dim_{K} D = n^{2}. For the right-hand side,

dim_{K} D = (dim_{L′} D)(dim_{K} L′)  by considering bases, so
n^{2} = (dim_{L′} D)n

so dim_{L′} D = n and hence dim_{L′} End_{L′}(D) = n^{2}

Thus the L′-dimensions are equal, and linear algebra shows that the L′-linear map ϕ is injective if and only if it is surjective, hence it is surjective.

But End_{L′}(D) ≅ M_{n}(L′) is anti-isomorphic to itself under the transposition map, so D ⊗_{K} L′ ≅ M_{n}(L′) as L′-algebras.

Recall that it is possible to choose L′ so that L′/K is a separable extension. With this L′, its normal closure in a chosen K̄ is a finite Galois extension of K, and

D ⊗_{K} L ≅ (D ⊗_{K} L′) ⊗_{L′} L ≅ M_{n}(L′) ⊗_{L′} L ≅ M_{n}(L).
□

E.7 Norms on Finite Dimensional Algebras

These results are needed for our study of Brauer groups of local fields in Chapter 4.

E.7.1 The Absolute Norm

Definition E.7.1 Let K be a field, A a finite dimensional K-algebra. For a ∈ A, define θ_{a} to be the left-homothety by a in A, ie.

\begin{aligned}θ_{a} : & A & ⟶ & A \\ & x & ⟼ & ax, \end{aligned}

which is a K-linear map. Define the norm map of A/K by

N_{A/K}(a) := det θ_{a}.

The norm map has the following properties.

Proposition E.7.2 1. N_{A/K}(a) ∈ K;

2. N_{A/K}(ab) = N_{A/K}(a)N_{A/K}(b);

3. If a is invertible, N_{A/K}(a) ≠ 0;

4. If one fixes a K-basis of A, then N_{A/K}(a) is a homogeneous polynomial of degree [A : K] in the [A : K] coordinates of a, with coefficients in K; 5. If L is a commutative subfield of A containing K, and λ ∈ L, then

N_{A/K}(λ) = N_{L/K}(λ)^{[A:L]}.

PROOF -

1. Obvious.

2. θ_{ab} = θ_{a} ∘ θ_{b} so that det θ_{ab} = (det θ_{a})(det θ_{b}).

3. id_{A} = θ_{a^{−1}a} = θ_{a} ∘ θ_{a^{−1}} so 1 = (det θ_{a})(det θ_{a^{−1}}).

4. Obvious.

5. Write m = [A : L]. A is an L-vector space by scalar multiplication on the

left. Choose an L-basis u_{1}, …, u_{m} of A so

A = Lu_{1} ⊕ … ⊕ Lu_{m}.

Now λ ∈ L takes an element ∑ c_{i}u_{i} ∈ A to ∑(λc_{i})u_{i}, so if Θ_{λ} is the matrix for multiplication by λ in the K-vector space L, then the matrix for multiplication by λ in the K-vector space A is the block-diagonal matrix

( \begin{matrix}Θ_{λ} \\ & Θ_{λ} \\ & & . & . & . \\ & & & & & Θ_{λ} \end{matrix} )

with determinant (det Θ_{λ})^{m} = N_{L/K}(λ)^{[A:L]}.

□

E.7.2 The Reduced Norm

Let A be a central simple K-algebra of dimension n^{2}. Let L be a neutralising field for A (see §E.5) and h : A ⊗_{K} L ⟶ M_{n}(L) an isomorphism. Definition E.7.3 The reduced norm of A is

\begin{aligned}Nrd : & A & ⟶ & L \\ & x & ⟼ & det h(x ⊗ 1). \end{aligned}

Proposition E.7.4 1. Nrd does not depend on the choice of L or h. 2. Nrd(xy) = Nrd(x)Nrd(y) for all x, y ∈ A.

3. Nrd(x) ∈ K.

4. If x ∈ A is invertible, Nrd(x) ≠ 0.

5. If one fixes a K-basis of A then Nrd(x) is a homogeneous polynomial of degree n in the n^{2} coordinates of x, with coefficients in K.

6. N_{A/K}(x) = Nrd(x)^{n}.

PROOF -

1. Independence of h: Suppose h′ : A ⊗_{K} L ⟶ M_{n}(L) is another isomorphism. Then h′ ∘ h^{−1} : M_{n}(L) ⟶ M_{n}(L) is an L-algebra automorphism. By the Skolem-Noether Theorem (Proposition 3.5.1) this is an inner automorphism, ie. there is B ∈ M_{n}(L)^{∗} such that h′ ∘ h^{−1}(X) = B^{−1}XB for all X ∈ M_{n}(L). So if X = h(x ⊗ 1),

h′(x ⊗ 1) = B^{−1}h(x ⊗ 1)B,

and so det h′(x ⊗ 1) = det h(x ⊗ 1).

Independence of L: Take two neutralising fields L, L′ and embed them into a common field L′′. Let

\begin{matrix}h : & A ⊗_{K} L & ⟶ & M_{n}(L) \\ h′ : & A ⊗_{K} L′ & ⟶ & M_{n}(L′) \end{matrix}

be isomorphisms. Consider the diagram

A ⊗_{K} L  ⟶_{∼}^{h} M_{n}(L)  M_{n}(L′) ⟵^{h′}  A ⊗_{K} L′
↓id⊗1  □  ↓inc.  ↓inc.  □  ↓id⊗1
A ⊗_{K} L ⊗_{L} L′′  ⟶^{h⊗id} M_{n}(L′′) = M_{n}(L′′) ⟵^{h′⊗1} A ⊗_{K} L′ ⊗_{L′} L′′.

We have det h(x ⊗ 1) = det 1 ⊗ h(x ⊗ 1) and by independence of the isomorphism, this is det(h′(x ⊗ 1) ⊗ 1) = det h′(x ⊗ 1).

2. Obvious.

3. Recall (Proposition 3.2.1) there exists a Galois neutralising field for A. If

h : D ⊗_{K} L ⟶ M_{n}(L)

is an isomorphism, so is s ∘ h for any s ∈ Gal(L/K), and det(s ∘ h(x ⊗ 1)) = s det h(x ⊗ 1) because s is a ring homomorphism. Hence by (1) above, s det h(x ⊗ 1) = det h(x ⊗ 1) for all such s and so Nrd(x) ∈ K (see Definition C.1.5).

4. Follows from (2) as in Proposition E.7.2.

5. If (u_{1}, …, u_{n^{2}}) is a K-basis for A then a general element of A comes from an n^{2}-tuple of coordinates (x_{1}, …, x_{n^{2}}) ∈ K^{n^{2}}, and the Nrd map is then given by the homogeneous polynomial form

Nrd(x) = det(∑_{i} x_{i}h(u_{i} ⊗ 1)) ∈ L[x_{1}, …, x_{n^{2}}].

As in (3), take L to be a Galois neutralising field for A. We apply the Skolem-Noether Theorem (Proposition 3.5.1) again to see that for each s ∈ Gal(L/K) there is a matrix B ∈ M_{n}(L) such that

sh(x ⊗ 1) = B^{−1}h(x ⊗ 1)B for all x ∈ A.

Hence

sNrd(x) = det(∑(x_{i})sh(u_{i} ⊗ 1))
= det(∑(x_{i})B^{−1}h(u_{i} ⊗ 1)B)
= det(∑(x_{i})h(u_{i} ⊗ 1)) = Nrd(x)

as polynomials in L[x_{1}, …, x_{n^{2}}]. So each coefficient is Gal(L/K)-invariant, and so belongs to K.

6. Write B = h(x ⊗ 1), so Nrd(x) = det B and N_{A/K}(x) = det θ_{x} where θ_{x} is left-homothety by x in the K-vector space A. We note that in the extension of scalars (Chapter 3 §E.4) a K-basis of A becomes an L-basis of A ⊗_{K} L, so that the matrix of θ_{x} is also the matrix for left-homothety by x ⊗ 1 in the L-vector space A ⊗_{K} L.

Applying h to this is equivalent to a basis change, so N_{A/K}(x) = det θ_{B} where θ_{B} is left-homothety by B in the L-vector space M_{n}(L).

M_{n}(L) is the direct sum of n copies of the column space L^{n} as an M_{n}(L)- module. Left-multiplication by B induces an L-endomorphism of this column space with determinant det B. Hence

det θ_{B} = (det B)^{n}

ie. N_{A/K}(x) = (Nrd(x))^{n}.

□

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