Introduction to Group Cohomology [thesis]

66 Pages·2009·0.618 MB·English

by  Maxim Stykow

Introduction to
Group Cohomology
Maxim Stykow
SID: 19718683
Supervisor: Dr. Daniel Delbourgo
This thesis is submitted in partial fulfillment of
the requirements for the degree of
Bachelor of Science (Honours)
Department of Mathematical Sciences
Monash University
Australia
October 17, 2008
Contents
Introduction 1
1 Modules 2
1.1 Λ-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Direct Sums and Products . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Free and Projective Modules . . . . . . . . . . . . . . . . . . . . . . . 6
2 Categories and Functors 8
2.1 Basic Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Ext and Tor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Derived Functors 18
3.1 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 The Cohomology of Groups 25
4.1 G-modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Definition of (Co)Homology . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 The Zeroth (Co)Homology Groups . . . . . . . . . . . . . . . . . . . . 27
4.4 The First (Co)Homology Groups with Trivial Coefficient Modules. . . 28
4.5 Shapiro’s Lemma, Cyclic and Free Groups and the Tate Cohomology . 29
4.6 Derivations, Semidirect Products and Hilbert’s Theorem 90 . . . . . . 36
4.7 The Standard Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.8 H2 and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.9 Change of Groups and the Restriction-Inflation Sequence . . . . . . . 46
A Commutative Algebra 49
A.1 More Module Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.2 Modules of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
A.3 Integral Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
A.4 Chain Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Acknowledgements 63
Bibliography 64
Introduction
This work is the author’s Honours thesis as an undergraduate student at Monash
University in Australia. Its purpose is to introduce cohomology groups assuming
as background little more than group, ring and field theory. The first three chap-
ters are devoted to building up the necessary machinery from homological algebra.
This includes module theory, basic category theory and the theory of derived func-
tors. In chapter four these concepts are then applied to group cohomology. Whilst
group cohomology did not historically arise out of the theory of derived functors, the
author favored this approach nonetheless as the extra work required to understand
homological algebra seemed a fair price to pay for a deeper understanding.
Thecohomologytheoryofgroupsarosefrombothtopologicalandalgebraicsources.
As an example from algebra, suppose that the product of two matrices f and g is
zero, i.e. g◦f =0. Suppose also that g·v =0 for some n-dimensional column vector
v. It is then not always true that v =f·u for some vector u. However, the failure to
be able to conclude so can be measured by the defect
d=nullityg−rankf =n−rankf −rankg.
Inmodernlanguage,wewouldcalldthehomology module andwriteH =kerg/imf.
On the topological side of things, it was Hurewicz who first introduced higher
homotopy groups π (X) for n ≥ 2 of a topological space X. Homology groups arise
n
quite naturally in algebraic topology. We recall that a triangulation of a space X is
a simplicial complex K together with a homeomorphism φ : K → X. Within such
a complex, we can pass from higher dimensional simplices to lower dimensional ones
through a so-called boundary operator ∂ . Quite intuitively, the boundary operator
n
∂ would send a 1-simplex or line segment AB of a simplicial complex to the point
1
B−A. Similarly, we send a 2-simplex or solid triangle ABC to its boundary, a line
segment, BC −AC +AB via ∂ . If we denote the subspace of all n-simplices of Rn
2
by S , we thus have a natural way of constructing a chain from any n-dimensional
n
simplex to the empty set:
C:···→S →∂n S →···→S →∂1 S →∂0 0.
n n−1 1 0
In fact, with the boundary operator defined as we hint to above, we could show even
more: ∂ ∂ = 0. This is important because it tells us that the image of ∂ lies
n n+1 n+1
in the kernel of ∂ which enables us to form the nth homology group (in the sense of
n
a topological group) H (C) = ker∂ /im∂ . Note that this quotient again really
n n n+1
measures a defect, namely by how much the boundary operator fails to be “exact”,
that is, by how much the image and the kernel of two successive boundary operators
differ. The study of these defects is at the heart of homological algebra which is said
to have begun with the cohomology of groups.
1
Chapter 1
Modules
In this chapter we review some background material from module theory needed in
all subsequent chapters. For the entire article, we will let Λ denote a ring with unity
element 1 6=0. We begin with the definition of a Λ-module.
λ
1.1 Λ-modules
A left Λ-module is an abelian group A together with a ring homomorphism ω : Λ →
EndA where EndA is the ring of endomorphisms of A. We will talk of Λ operating
on A from the left and denote this action by simply writing λa. The following rules
are then satisfied for a,a ,a ∈A and λ,λ ,λ ∈Λ:
1 2 1 2
M1: (λ +λ )a=λ a+λ a
1 2 1 2
M2: (λ λ )a=λ (λ a)
1 2 1 2
M3: 1 a=a
λ
M4: λ(a +a )=λa +λa .
1 2 1 2
A Λ-module homomorphism φ : M → N is defined as a homomorphism of abelian
groups such that φ(λm) = λφ(m) for m ∈ M and λ ∈ Λ. We also define a sequence
of Λ-modules and Λ-module homomorphisms
···−→M −f→i M −fi→+1 M −→···
i−1 i i+1
to be exact at M if imf =kerf . The sequence is said to be exact if it is exact at
i i i+1
each M . We have in particular:
i
0→M0 →f M →g M00 →0
is exact if and only if f is injective, g is surjective and g induces an isomorphism of
cokerf =M/f(M0) onto M00. A sequence of the preceding type is called short exact
and we will also often write M0 ,→f M (cid:16)g M00.
2
Let A,B,C,D be Λ-modules and let α,β,γ,δ be Λ-module homomorphisms. We
say that the diagram
α //
A B
γ β
(cid:15)(cid:15) (cid:15)(cid:15)
δ //
C D
is commutative if βα=δγ :A→D. We shall need the following
Lemma 1.1.1. Let A0 ,→µ A(cid:16)ε A00 and B0 ,→µ0 B (cid:16)ε0 B00 be two short exact sequences.
Suppose that in the commutative diagram
0 //A0 µ //A ε //A00 //0
α0 α α00
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 //B0 µ0 //B ε0 //B00 //0
any two of the three homomorphisms α0,α,α00 are isomorphisms. Then the third is
an isomorphism also.
Proof. Theproofisanexerciseindiagramchasingusingexactnessandcommutativity.
Details can be found in [HS, p. 15].
LetHom (A,B)denotethesetofallΛ-modulehomomorphismsfromAtoB. We
Λ
givethissetanabeliangroupstructurebydefiningforanyΛ-modulehomomorphisms
f,g :A→B
(f +g)(a)=f(a)+g(a) (1.1)
for all a ∈ A. Given Λ-module homomorphisms β : B → B and α : A → A we
1 2 2 1
obtain induced mappings
β :Hom (A,B )→Hom (A,B ), β (f)=βf :A→B
∗ Λ 1 Λ 2 ∗ 2
and
α∗ :Hom (A ,B)→Hom (A ,B), α∗(f)=fα:A →B. (1.2)
Λ 1 Λ 2 2
Proposition 1.1.2. Let A0 →µ A →ε A00 → 0 and 0 → B0 →µ B →ε B00 be exact
sequences of Λ-modules. For any Λ-modules A and B the induced sequences
0→Hom (A00,B)→ε∗ Hom (A,B)→µ∗ Hom (A0,B)
Λ Λ Λ
0→Hom (A,B0)→µ∗ Hom (A,B)→ε∗ Hom (A,B00)
Λ Λ Λ
are exact.
Proof. This is again straightforward and can be found in [HS, p. 17].
Example 1.1.3. Let us calculate some Hom-groups. Let φ be an element of the
respective Hom-group then
• HomZ(Z,Zn)’Zn since φ is determined by where it sends 1∈Z and there are
n choices;
3
• HomZ(Zm,Zn) ’ Zgcd(m,n) since mφ(1) = 0 mod n iff mdφ(1) = 0 mod nd iff
φ(1)∈{0,n,...,(d−1)n};
d d
• HomZ(Z,Z)’Z;
• HomZ(Zn,Z)=0 since nφ(1)=0 iff φ(1)=0;
• HomZ(Q,Z)=0becauselettingpbeaprimenotdividingf(x)forsomenonzero
homomorphismf :Q→Zandsomex∈Q, wegetf(x)=f(x/p+...+x/p)=
pf(x/p) which is a contradiction;
• HomZ(Q,Q)’Q.
We now show how to give Hom a Λ-module structure. Let A be a right Λ-module
andletGbeanabeliangroup. RegardingAasanabeliangroupalso,wecanformthe
abelian group HomZ(A,G) as described above. Using the right Λ-module structure
of A we define
(λφ)(a)=φ(aλ), a∈A,λ∈Λ,φ∈HomZ(A,G)
givingHomZ(G,A)aleftΛ-modulestructure. Wecanofcourseinterchange“left”and
“right” in the above construction. We obtain the following important proposition:
Proposition 1.1.4. Let A be a left Λ-module and let G be an abelian group. Regard
HomZ(Λ,G) as a left Λ-module via the right Λ-module structure of Λ. Then there
exists an isomorphism of abelian groups
η =ηA :HomΛ(A,HomZ(Λ,G)) →˜ HomZ(A,G).
Moreover, for every Λ-module homomorphism α:A→B the diagram
HomΛ(B,HomZ(Λ,G)) ηB //HomZ(B,G)
α∗ α∗
(cid:15)(cid:15) (cid:15)(cid:15)
HomΛ(A,HomZ(Λ,G)) ηA //HomZ(A,G)
is commutative.
Proof. Given a Λ-module homomorphism φ : A → HomZ(Λ,G), define a homomor-
phism of abelian groups φ0 : A → G by φ0(a) = (φ(a))(1),a ∈ A. Conversely, given
a homomorphism of abelian groups φ : A → G, define a Λ-module homomorphism
ψ0 :A→HomZ(Λ,G) by (ψ0(a))(λ)=ψ(λa),a∈A,λ∈Λ. The rest is just definition
checking.
1.2 Direct Sums and Products
Let {A },j ∈J be a family of Λ-modules indexed by J. We define a Λ-module called
j
L L
the direct sum A of the modules A as follows: an element of A is a
j∈J j j j∈J j
sequence (a ) with a ∈ A and a = 0 for all but a finite number of subscripts.
j j∈J j j j
Addition is defined by (a )+(b )=(a +b ) and the Λ-module operation by λ(a )=
j j j j j
L
(λa ). We also define injections ι : A → A by ι (a ) = (b ) with b = 0
j k k j∈J j k k j j
when j 6=k and b =a for a ∈A .
k k k k
4
Proposition 1.2.1. Let M be a Λ-module and let {ψ :A →M},j ∈J be a family
j j
of Λ-module homomorphisms. Then there exists a unique homomorphism ψ =hψ i:
j
L
A →M, such that the diagram
j∈J j
ιjA(cid:15)(cid:15)jHHHHHHψHjHHH$$
L _ψ_ _//
j∈JAj M
is commutative for all j ∈J.
P
Proof. We define ψ((a )) = ψ (a ). This is well-defined since all but finitely
j j∈J j j
many terms in this sum are zero.
We remark that the direct sum together with its injections is unique up to iso-
morphism but omit the proof which just uses the above property also known as a
Q
universal property. Similarly, we define the direct product A of a family of
j∈J j
Λ-modules {A },j ∈J indexed by J where the elements are again sequences (a )
j j j∈J
butwithnorestrictionsimposedontheterms. Wecouldthushaveaninfinitenumber
of nonzero terms in such a sequence. Rather than injections, we define projections
Q
π : A → A by π ((a )) = a and we have the following proposition very
k j∈J j k k j k
similarly to (1.2.1):
Proposition 1.2.2. Let M be a Λ-module and let {φ :M →A },j ∈J be a family
j j
of Λ-module homomorphisms. Then there exists a unique homomorphism φ={φ }:
j
Q
M → A such that for every j ∈J the diagram
j∈J j
vvvφvjvvvvvv;;AOOjπj
_ _φ_//Q
M j∈JAj
is commutative.
Proof. We define φ(m)=(φ (m)) .
j j∈J
Finally we prove that Hom (−,−) preserves sums and products.
Λ
Proposition 1.2.3. Let A,B be Λ-modules and {A },{B },j ∈ J be families of
j j
Λ-modules. Then there are isomorphisms
 
M Y
η :HomΛ Aj,B →˜ HomΛ(Aj,B)
j∈J j∈J
 
Y Y
ζ :HomΛA, Bj →˜ HomΛ(A,Bj).
j∈J j∈J
L
Proof. Givenψ : A →B,defineη(ψ)=(ψι ) . Converselyweusetheuniversal
j j j∈J
property of the direct sum since a given family {ψ : A → B} then gives rise to a
j j
L
mapψ : A →B. Theprooffortheζ-isomorphismissimilarbutusestheuniversal
j
property of the direct product.
5
1.3 Free and Projective Modules
Let A be a Λ-module and let S be a subset of A. If the set
( )
X
A = λ s:λ ∈Λ,λ =0 for all but finitely many s∈S ⊆A
0 s s s
s∈S
is all of A, we say that S is a set of generators of A. If A admits a finite set of
generators, we say it is finitely generated. A set S of generators of A is called a basis
ifeverya∈AisuniquelyexpressibleasanelementofA . Thisisequivalenttosaying
0
that S must be linearly independent. Finally, if S is a basis of the Λ-module P, then
P is said to be free on S or just free if we do not specify S. Alternatively, we could
L
say that P is free if and only if P ’ Λ by the following proposition:
s∈S
L
Proposition 1.3.1. Suppose the Λ-module P is free on S. Then P ’ Λ
s∈S s
L
where Λ = Λ as a left module for s ∈ S. Conversely, Λ is free on the set
s s∈S s
{1 ,s∈S}.
Λs
L P
Proof. Define φ : P → Λ by φ(a) = (λ ) where a = λ s is the unique
s s s∈S s∈S s
expression for a in terms of the given basis. Given s∈S we also define ψ :Λ →P
s s
by ψ (λ )=λ s. By the universal property of the direct sum the family {ψ },s∈S
s s s s
L
givesrisetoamapψ : Λ →P. Itisclearthatφandψaremutualinverses.
s∈S s
We will also need the following property:
Proposition 1.3.2. Let P be free on S. To every Λ-module M and to every function
f from S into the set underlying M, there is a unique Λ-module homomorphism
φ:P →M extending f.
P P
Proof. Let f(s)=m . Define φ(a)=φ( λ s)= λ m .
s s s s
This yields two very important results:
Proposition 1.3.3. Every Λ-module A is a quotient of a free module P.
L
Proof. Let S be a set of generators of A and let P = Λ with Λ = Λ. As P
s∈S s s
is free on W = {1 ,s ∈ S}, we define f : W → A by f(1 ) = s and construct an
Λs P P Λs
extension φ:P →A of f by φ( λ 1 )= λ s which shows that φ is indeed
s∈S s s s∈S s
surjective and thus completes the proof.
Proposition 1.3.4. Let P be a free Λ-module. To every surjective homomorphism
ε : B (cid:16) C of Λ-modules and to every homomorphism γ : P → C there exists a
homomorphism β :P →B such that εβ =γ.
Proof. LetP befreeonS. Sinceεissurjective,wecanfindb ∈B suchthatforevery
s
s∈S,ε(b )=γ(s). Then simply define β as the extension of the function f(s)=b .
s s
By the uniqueness of (1.3.2), we conclude εβ =γ.
Note that the last proposition states that Hom (P,−) gives rise to a short exact
Λ
sequence when P is free. To see this, let A ,→µ B (cid:16)ε C be a short exact sequence of
Λ-modules. Then given a homomorphism γ : P → C, we can find a homomorphism
β :P →B such that εβ =γ. But this is exactly what we need to conclude that
0→Hom (P,A)→µ∗ Hom (P,B)→ε∗ Hom (P,C)→0 (1.3)
Λ Λ Λ
6
is exact. Conversely, the exactness of (1.3) immediately implies (1.3.4). Modules
possessing this property have a special name: a Λ-module P is projective if to any
homomorphisms ε,γ with ε surjective there exists β making the below diagram com-
mute.
P
~
β ~
~ γ
(cid:127)(cid:127)~ (cid:15)(cid:15)
ε //
B C
For example, we see by (1.3.4) that every free module is projective.
Proposition 1.3.5. For a Λ-module P the following are equivalent:
i) P is projective;
ii) for every short exact sequence, the corresponding Hom (P,−)-sequence is short
Λ
exact also;
iii) P is a direct summand in a free module.
Proof. We already proved i) ⇔ ii). So suppose P is projective. (1.3.3) constructs a
surjectivemapφ:F →P fromafreemoduleF toP. SinceP isprojective,therethus
exists a map ι:P →F such that φι=1 . But this shows that ι is injective whence
P
L
P is a direct summand in F ’ Λ. Conversely, suppose P is a direct summand in
a free module. Then P0 ’P ⊕Q for some free Λ-module P0 and some Λ-module Q.
Hence P is projective by proposition (1.3.6) below. This completes i)⇔iii).
Dualizing the concept of a projective module, we obtain the following definition:
a Λ-module I is called injective if to any homomorphisms α,µ with µ injective there
exists β making the following diagram commutative:
µ //
A B
~
~
α ~
(cid:15)(cid:15) (cid:127)(cid:127)~ β
I
We also state:
Proposition 1.3.6. A direct sum (direct product) of modules is projective (injective)
if and only if each summand (factor) is projective (injective).
Proof. See [HS, pp. 24, 30].
7
Chapter 2
Categories and Functors
This chapter is very heavy in definitions and many proofs of results will be omitted.
We need, however, to go through these concepts to introduce the theory of derived
functors upon which much of homological algebra is built.
2.1 Basic Category Theory
To define a category C we must first be given three definitions of their own:
(1) a class of objects A,B,C,...;
(2) to each pair of objects A,B of C, a set C(A,B) of morphisms from A to B;
(3) to each triple of objects A,B,C, a law of composition C(A,B) × C(B,C) →
C(A,C).
Suppose we are given a morphism f ∈ C(A,B) and a morphism g ∈ C(B,C), then
we will denote the composition law by writing gf for the morphism in C(A,C). The
rationaleforthisisthatveryoftenfunctioncompositionwillbethiscompositionlaw.
For f ∈ C(A,B) we call A the domain and B the codomain of f. To be a category,
C must then obey the following axioms:
A1: The sets C(A ,B ),C(A ,B ) are disjoint unless A =A ,B =B .
1 1 2 2 1 2 1 2
A2: Given f :A→B,g :B →C,h:C →D, then h(gf)=(hg)f.
A3: To each object A there is a morphism 1 :A→A such that, for any
A
f :A→B,g :C →A,f1 =f,1 g =g.
A A
We call A2 the associativity of composition and A3 the existence of identities. 1
A
is called the identity morphism of A which is unique by A3. We also define a zero
object 0 in C to be an object with the property that, for any object X ∈ C, the sets
C(X,0)andC(0,X)bothconsistofexactlyoneelement. However,noteverycategory
hasazeroobject. Takeforinstancethecategoryofringswithunityelementandring
homomorphisms.
An example of a category with zero object is the category of abelian groups with
group homomorphisms: any one-element group is a zero object.
8