Topological Methods in Galois Representation Theory

By Victor P. Snaith

An advanced monograph on Galois representation theory by one of the world's leading algebraists, this volume is directed at mathematics students who have completed a graduate course in introductory algebraic topology. Topics include abelian and nonabelian cohomology of groups, characteristic classes of forms and algebras, explicit Brauer induction theory, and much more. 1989 edition.

• Language: English
• Publisher: Dover Publications
• Release date: Dec 3, 2013
• ISBN: 9780486782270

Book preview

THEORY

Chapter One

Abelian Cohomology of Groups

It’s like a book, this bloomin’ world. Which you can read and care for just so long, But presently you feel that you will die Unless you get the page you’re readin’ done, An’ turn another—likely not so good; But what you’re after is to turn ’em all.

—RUDYARD KIPLING,

Sestina of the Tramp-Royal (1896)

In this chapter we first review the basic definitions of group cohomology with abelian coefficients, both continuous and discrete. Then we consider explicit formulae in low dimensions for applications such as products and the transfer (or corestriction) map. We introduce the usual basic concepts, for example, the long exact sequence and the homology/cohomology relationship. Our primary goal is to come away from this chapter with a few specific cohomology rings at our disposal—as well as transfer techniques such as the double coset formula. Transfer techniques are not only technically useful to us at this point; we will need to depend on them when, in Chapter 5, we encounter the double coset formula in stable homotopy theory.

After we have computed the cohomology rings of the cyclic groups, the dihedral group of order eight (with mod 2 coefficients) and (additively) the cohomology of the generalized quaternion groups, certain acts of faith will be required. These take the form of belief in the Stiefel-Whitney classes, Chern classes, and in the properties of spectral sequences. I have taken the view that faith knows no limits! Accordingly, with the briefest review of such things we are able to conclude this chapter with the computation of the mod 2 cohomology rings of dihedral and generalized quaternion groups and the integral cohomology ring of the dihedral group of order eight.

1. BASIC DEFINITIONS

(1.1) Definition

Let G be a group acting upon an abelian group, M. That is, we have a homomorphism

The associated (left) action

for g ∈ G, m ∈ M, then

The set of maps f: Gn = G × … × G → M is denoted by Cn(G; M) and is called the n-cochains on G with values in M. Define [Ser, p. 1–9]

by (g1, …, gn + 1 ∈ G),

One readily verifies that dd = 0 and the nth cohomology group (n ≥ 0) of G with coefficients in M, Hn(G; M), is defined by

where we set Cn(G, M) = 0 if n < 0.

(1.5) Remark

I have given the combinatorial definition first because it is the most appropriate one for generalization to continuous cohomology, to which we will turn our attention later in this chapter.

However,for many purposes the homological algebra definition of Hn(G; Mis very useful. We will recall that definition before proceeding further.

as follows. Set Bn G [G]-module on the set Gn. Hence, if g1, …, gn ∈ G, we may denote the n-tuple (g1, …, gn) ∈ Gn by [g1 | ··· | gn]. This is a free generator of Bn G [G], and an arbitrary element of Bn G is a sum of elements denoted X [g1 | ··· | gn], where X [G]. The generator of B0 G is simply written, [ ].

Define ε and di [G]-module homomorphisms given by

[G, is given by the integers with trivial action g · m = m (g ∈ G, m ).

If we define η, Si in

to be the (abelian group) homomorphisms given by

One easily checks that the following identities are true:

Without much difficulty, we obtain the following lemma:

(1.11) Lemma

  (i) (1.7) is an exact sequence of free [G]-modules. That is, ε is onto, ker ε = image (d0), and ker (dn) = image (dn + 1) for all n ≥ 0.

(ii)There is a natural isomorphism

given by

(iii) Furthermore

(1.12) Corollary

There is a natural isomorphism

(1.13)

[G]-module is called a projective [G]-module[G]-modules

in which each Pi(i to equal the cohomology of (1.14) with coefficients in M:

(1.14) is called a projective resolution [G. Of course, (1.7) is a particular example. However, given a partial commutative diagram in which the lower sequence is exact,

one can find fn+1:Pn+1 → Qn= fn dn. If we have constructed two sequences {fn} and {hn} to make (1.16) commute, we may inductively construct {σn: Pn → Qn+1} so that (σ–1 = 0)

Hence the chain maps {fn} and {hn} induce the same map from the cohomology of the upper sequence of (1.16) to that of the lower sequence. This map is denoted simply by f*, since it depends only on f, which are isomorphic by a canonical isomorphism, namely, 1*.

is a functor of M, and there is a natural isomorphism (n ≥ 0),

(1.18) CONTINUOUS COHOMOLOGY

Suppose that G is a topological group. The example that will occupy us is that of a profinite group. That is,

where {hα,β:Gα → Gβ} is an inverse system of homomorphisms of finite groups, Gα, as α If β ≥ α , there is one homomorphism hα,β, and if α ≤ β ≤ γ, then hα,γ = hβγ° hαβ. In these circumstances (such that hαβ(gα) = gβ. Each Gα has the discrete topology, ∏α Gα product topology, and (1.19) has the resulting subspace topology. It is a compact, totally disconnected group.

(1.20) Example

For 0 < n, set Gn /n set n less than m if m divides n, and set hn,m:Z/n /m equal to the canonical surjection, hn,m(1) = 1. The resulting group is the adic-integers

Similarly, the l-adic integers, for a prime l, are defined by

(1.21) Example

Let M be an abelian group, with the discrete topology, on which the topological group, G, acts continuously. This means that for m ∈ M

is open in G. If MH denotes the subgroup of M fixed by each element of a subgroup, H, then

denote the subset of continuous cochains in Cn(G; M). It is clear that d in (1.3) preserves continuous (i.e., locally constant) cochains, and we may define continuous cohomology by

When G is the profinite group given by the absolute Galois group of a field, K:

The resulting continuous cohomology is called Galois cohomology. In (1.24) the groups, G(N/K), are the Galois groups of finite Galois extensions N of K, and the inverse limit is taken over such N/K.

In this example we have

where U(N) = ker(ΩK → G(N/K)).

2. BASIC PROPERTIES

(2.1) Suppose that f : H → G is a homomorphism of groups (or of topological groups in the context of §1.21). Clearly,f induces, by composition, a homomorphism

such that d(fn · h) = fn + ¹(dh). Hence (2.2) induces a restriction map on cohomology

such that 1* = 1 and f*·(f′)* = (f′f)*.

(2.4) PRODUCTS

Let H and G be groups, and let

module structure given by

Define (d d)n = ∂n:En + 1 → En by

and set

Each En [H × G]-module, and

[H × G. One sees that (2.6) is exact by means of (S l)n:En → En + l defined, using the chain homotopy of (1.9).

The diagonal homomorphism ∆: G → G × G makes ([G[G]-module, (here H = G):

An explicit ∆i, is given by the following formula [Mac, p. 296]:

If

then we may project from En + m onto the factor BnBmG and compose with f f′ to yield

From (2.5) and (2.7) we see easily that if f and f′ are cocycles, so is f f′, and its cohomology class depends only on that of f and f′. In this manner we obtain the external cup-product.

given by [f]∪[f′], where [ – ] denotes a cohomology class. The cup-product is associative.

The internal product is

If T*:Ha(G;M M′) → Ha(G;M′ M) is induced by interchanging M and M′, then the product is commutative in the sense that

(2.13) By virtue of the explicit formula (2.8), one finds that the products of (2.10) and (2.11) make sense for continuous cochains and induce analogous products on continuous cohomology groups.

(2.14) Example

Suppose that R is a ring with trivial G action. We have a product m:R × R → R so that we may form

Explicitly, by (2.8), this product is given by

0≤nHn(G;R) into a graded ring, H*(G;Ris a graded ring.

(2.15) HOMOLOGY

The nth homology of a discrete group, G[G]-module, M, is denoted by Hn(G; M). It is defined by

where {BnG,dn} is the bar resolution of (1.7). As in §1.13, one can show that(up to canonical isomorphism) {BnG,dn[G]-module .

[G]-module. Then we are entitled to two long exact sequences:

In the upper sequence α*[f] = [α·f], β*[h] = [β·h],and in the lower sequence α*[aα)(a)].

To construct each of the sequences in (2.18), we first produce a commutative diagram of the form

in which the vertical columns are chain complexes (i.e., dd = 0) and the horizontal rows are short exact. The cohomology sequence arises from the case

whereas the homology sequence comes from the case

If x∈ker d ⊂ Cn + l, we define

This is a homology class for the chain complex (A*,d) which depends only on the homology class [x]∈Hn + 1(C*, d).

It is straightforward to prove the next lemma, from which (2.18) follows, being special cases (2.20) and (2.21).

(2.23) Lemma

In the situation of (2.19),

is an exact sequence.

(2.24) Application

is a chain complex of free modules over a principal ideal domain, R. Suppose also that M is a left R-module. We can form the cohomology groups Hn(C*; M) given by the homology of the chain complex {HomR(Cn, M), d*} or we can form the R-module given by Hn(C*, d), the homology of {C*, d}. These are related by the universal coefficient exact sequence:

Briefly, I will recall the derivation of (2.25). Firstly Zn = ker d and Bn = im d are free sub-R-modules of Cn so that the exact sequence

is a free R-module resolution of Hn(C*). Hence, by definition, the map

satisfies

Now take the exact sequence

and form a diagram like (2.19) by applying HomR(—, M) to (2.29), where d: Zn → Zn – 1 and d:Bn → Bn – 1 are defined to be zero. By §2.23, there results a long exact sequence

and (2.25)follows from(2.28)and the verification that ∂ in (2.30) equals i* of (2.27).

a similar argument yields the following exact sequence:

(2.33) Particularly useful special cases of (2.25) and (2.32) are the following exact sequence and isomorphism: Let G be a discrete group, and let p /p having trivial G-action)

(2.34) TRANSFER OR CORESTRICTION

Suppose that i: H ⊂ G are discrete groups with index [G:H] = m. Let {xi: 1 ≤ i ≤ m} be a set of coset representatives for G/H[H]-resolution

-module,

[G. Hence we have an isomorphism, in which M [G]-module,

[G. For, if then g ∈ G, then gxi ∈ xσ(i)H for some permutation,σ ∈ ∑m, so that g(τ(z) = τ(z) = τ(g · z). The induced map

composes with (2.33) to yield the transfer homomorphism

In an entirely analogous manner we obtain the homology transfer map:

(2.39) Lemma

In §2.24 each composite homomorphism and is multiplication by m = [G : H].

Proof[G]-resolution

then the homology of {HomG(Pi, M), d*} is H*(G; M[G[H]-module, the homology of {HomH (Pi, M), d*} is H*(H; M). The natural map

is induced by

. Therefore, in cohomology, i*iinduces multiplication by m

(2.40) Remark

Returning to the definition of the transfer in cohomology, observe that if

is a [G]-module resolution, then there is an adjunction isomorphism

given by . This induces an isomorphism

The transfer is induced by the [G]-module homomorphism [H-S, p. 266]:

Here G acts on the left of HomH [G], M) by (gΦ)(x) = Φ(xg) so that

(2.43) Theorem

Let H and K be subgroups of finite index in G, and let M be a [G]-module. Then the composite

is equal to the sum

where ψg is the composite,

Proof. H*(H; M) is the homology of the chain complex, HomH(Qi, M), where {Qi, d[GHomH (Qi, M) → HomG (Qi, M),

However, as a K-map in HomK(Qi, Mis the sum of maps such as

one for each double coset of K\G/H.

However, the bijection

equals the sum as g runs over coset representatives of K/K ∩ (xiHxi. However, this sum is the representative of the composition, ψ g

(2.44) Remark

I leave to the interested reader the task of proving the analogous double coset formula in homology.

(2.45) Corollary

If H G has finite index and M is a [G]-module, then

equals ∑g(g – g–1)* where g runs through a set of representatives of G/H.

Proof. When H = K G, then H = K ∩ (xiHxiso that ψg becomes (g – g

(2.46) I will close this section with the explicit description of the transfer on H¹ (G; M) when M is a trivial G-module. We use the notation of §2.34.

From the exact sequence associated to

we find [H-S, p. 193] that, in general,

When G acts trivially on M, (2.47) yields

where the second isomorphism sends f : I(G) → M to the homomorphism μ(f)(g) = f(g – 1).

For g ∈ G, we have a permutation π ∈ ∑m and h(i, g)∈H (1 ≤ i ≤ m), satisfying

We will derive the following formula:

(2.50) Proposition

Let M be a trivial G-module. Then the following diagram commutes

where I(α)(gα(h(i, g)).

Proof. By (2.42), we must unravel the isomorphism between H¹ (H; M) and

On the other hand, the diagram of exact sequences

gives isomorphisms

where the second isomorphism is induced by restriction to I(H). The isomorphism connecting (2.51) and (2.52) is given by

given by λ(f)(a ⊗ b) = f(b)(a).

Now suppose f ∈ Hom (H, M∈ HomH (I(G), M) by

corresponds to f in (2.52). If λ(Fin (2.53), then

and i*(f) is represented, by (2.42), by the map J : I(G) → M,

since xi– 1 acts trivially on M. However,

3. EXAMPLES OF COHOMOLOGY RINGS

In this section I will evaluate some rings, H*(G; Λ) for suitable finite groups, G, and rings, Λ.

(3.1) THE CYCLIC GROUPS

Let G /n, the cyclic group of order n[G]-modules:

given by ε(g) = 1 for g∈ G, d(z) = (x – 1)z, where x generates G and Δ(z. At once one sees H/n) is the cohomology of the complex

so that

From the universal coefficient theorem and Künneth formula (2.25), and (2.32), we can evaluate H/n: M) and H/n; M) for any trivial module, M.

Suppose that the i[G], in the preceding resolution, has generator ei (i ≥ 0). Then the differential is given by

If {Bi/n), di} is the bar resolution (see §1.6), define, for i /n-module homomorphism Φ: Pi → Bi/n) by

(3.5) Lemma

Φ is a chain map in (3.4).

Proof. We must show that Φ∂ = dΦ. However,

whereas

(3.6) If G /n, we have a chain map

where Δ is defined in (2.8).

If f ∈ HomG (B2 G) and h ∈ HomG(Bn G) are cocycles, then the product, [f][h], is represented by (f × h)· ϕn + 2. If n = 2s, we obtain

The generator of H/n) is given by [f], where f(e2) = 1 so that we have deduced the following