Algebraic Number Theory (Spring 2013)

107 Pages·2014·0.808 MB·English

by  Joe Rabinoff, J. Bland

Algebraic Number Theory
Spring 2013, taught by Joe Rabinoff.
This consists of only the second semester of Math 223.
Contents
1 Statements of Class Field Theory 2
2 G-Modules 4
3 Group Cohomology 6
4 Behavior with Respect to Induction 8
5 The Standard Resolution of Z as a G-Module 9
6 Change of Group 12
7 Tate Cohomology 16
8 Complete Resolutions 19
9 Cup Products 21
9.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
10 Cohomology of Finite Groups 25
10.1 The Cyclic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
10.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
11 Cohomologically Trivial Modules 32
11.1 Over p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
11.2 Over General Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
12 The Theorems of Tate and Nakayama 38
13 Galois Cohomology 41
14 Structure of Local Fields 46
15 Class Formations 53
1
16 Construction of a Class Formation 59
16.1 Cohomology of Z(cid:98) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
16.2 Quasi-Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
17 Lubin-Tate Theory 65
17.1 The Case of K = Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
p
17.2 The General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
18 Higher Ramification Groups 74
19 Global Proofs 77
20 Cohomology of Idèles 78
21 The First Inequality 81
22 The Second Inequality 82
22.1 The Case of Kummer Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
22.2 p-extensions in characteristic p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
23 Global Class Formations 90
24 Local-Global Compatibility 99
25 Universal Norms 100
26 Elliptic Curves With Complex Multiplication 104
1 Statements of Class Field Theory
Here K will be a local or global field. Let
(cid:40)
K× local
A = (1.1)
K A×/K× global
K
Theorem 1.1 (CFT for K). There exists a canonical continuous homomorphism with dense image
ψ : A → Gab called the Artin map.
K K K
(Functoriality) For L/K a finite separable extension, the following diagrams commute:
2
A ψL Gab
L L
(1.2)
N
L/K
A ψK Gab
K K
and
A ψL Gab
L L
(1.3)
Ver
A ψK Gab
K K
(Existence) Every finite index open subgroup of A exists as the kernel of
K
ψ
L/K
A Gab Gal(L/K) (1.4)
K K
for L/K a unique finite abelian extension of K (so kerψ = N (A )).
L/K L/K L
If K is global of characteristic 0, then ψ is surjective, with kernel
K
D = {x ∈ A× : infinitely divisible}. (1.5)
K K
If K is global of characteristic p with k the constant field and A1 = (A×)1/K×, or K is local
K K
with k the residue field and A1 = O×, then we have the commutative diagram
K K
0 A1 A val Z 0
K K
∼ ψK ∼
0 I Wab Z 0 (1.6)
K
0 I Gab Z(cid:98) 0
K
with Z(cid:98) = Gal(k/k). In particular, if L/K is a finite unramified extension over a local field K,
then Gab → Gal(L/K) kills I, so ψ kills O× and sends π to Frob. More generally, ψ (O×)
K L/K K K L/K K
is the inertia in G = Gal(L/K), and ψ (1+mi) = Gi, the higher ramification groups.
L/K
If K is global and v ∈ V , we have a map K× (cid:44)→ A×/K× by x (cid:55)→ (1,..., x ,...,1). Kab (cid:44)→
K v K
(cid:124)(cid:123)(cid:122)(cid:125)
v
Kab (cid:32) Gab → Gab.
v Kv K
3
Theorem 1.2 (Local-global compatibility). The following diagram commutes:
K× A×/K×
v K
(1.7)
ψKv ψK
Gab Gab
Kv K
Hence, if L/K is a finite abelian extension, then ψ kills O× if and only if v is unramified, in
L/K v
which case π (cid:55)→ Frob . This determines ψ , along with continuity.
v v K
2 G-Modules
Let G be a group. A left (or right) G-module is an abelian group A equipped with a left (right)
action of G; that is, a homomorphism G → Aut(A) or Gop → Aut(A), Examples include:
• If G = GL (C), then Cn is a G-module.
n
• If G = Gal(L/K), then (L,+) and (L×,∗) are G-modules.
• Z[G], the free abelian group on elements of G, is a left G-module and a right G-module by
multiplication. In fact, Z[G] is a ring, called the group ring.
There is an equivalence between G-modules and Z[G]-modules; a G-module extends uniquely to
a Z[G]-action by linearity.
Remark. For R any commutative ring, set R[G] = R⊗ZZ[G]. Then there is an equivalence between
R-modules with G-linear action and R[G]-modules.
AG-modulehomomorphismA → A(cid:48) isahomomorphismofZ[G]-modules.Thisgroupisdenoted
by Hom (A,A(cid:48)). Kernels and quotients of G-modules are G-modules. The category Mod of G-
G G
modules is abelian.
A ∈ Mod is injective if the functor A(cid:48) (cid:55)→ Hom (A(cid:48),A) is exact, and A ∈ Mod is projective if
G G G
the functor A(cid:48) (cid:55)→ Hom (A,A(cid:48)) is exact. (Both functors are always left exact.)
G
Fact. Mod has enough injectives. That is, any G-module can be embedded in an injective one.
G
That Mod has enough projectives is easier to see, since free modules are projective.
G
ThecategoryModG admitsproductsanddirectsums.Also,ifA,A(cid:48) ∈ ModG,thenA⊗ZA(cid:48) isalso
aG-moduleintheexpectedway.Thetensorproductsatisfiesthefollowinguniversalproperty:giving
amapA⊗ZA(cid:48) → B isequivalenttogivingabilinearF : A×A(cid:48) → B suchthatF(ga,ga(cid:48)) = gF(a,a(cid:48)).
Let ϕ : H → G be a homomorphism. Then ϕ induces a map Z[H] → Z[G], which we’ll also call
ϕ. We define the following:
• The restriction ResG : Mod → Mod by mapping A to A viewed as an H-module (h·a =
H G H
ϕ(h)a).
• (Compact) induction indG : Mod → Mod by A (cid:55)→ Z[G]⊗ A.
H H G Z[H]
4
• (Co-)induction IndG : Mod → Mod by A (cid:55)→ Hom (Z[G],A).
H H G H
For compact induction, if R is a non-commutative ring, A is a right R-module, and B is a left
R-module, then A⊗ B is an abelian group. So Z[G] is viewed as a right Z[H]-module. It is viewed
R
as a left Z[G]-module by the action of left multiplication g·(x⊗y) = (gx)⊗y.
If H ≤ G, let {g } be coset representatives for H\G. Then {g−1} are representatives for G/H,
i i
so
(cid:77)
Z[G] = (g−1)Z[H] (2.1)
i
i
=⇒ indGHA ∼= (cid:77)gi−1Z[H]⊗Z[H]A (2.2)
i
(cid:77)
= g−1A. (2.3)
i
The group action is then
(cid:88) (cid:88)
g· g−1a = g−1(g gg−1)a (2.4)
i i j j i i
where for each i, gg−1 ∈ g−1H, so g g ∈ Hg . As a result, IndG(A) is in bijection with finitely
i j j i H
supported maps from H\G to A.
As an example, when H is the trivial subgroup {1}, an H-module is simply an abelian group.
indG1 = Z[G]⊗ZA = A[G]. If A(cid:48) ∼= A[G] as G-modules, then we say that A(cid:48) is induced.
Proposition 2.1 (Frobenius Reciprocity). Let A ∈ Mod and A(cid:48) ∈ Mod . Then
H G
Hom (A,ResGA(cid:48)) = Hom (indGA,A(cid:48)). (2.5)
H H G H
Proof. Hom (A,ResGA(cid:48)) = Hom (Z[G]⊗ A,A(cid:48)).
Z[H] H Z[G] Z[H]
If A = ResGA(cid:48), we have the identity element 1 ∈ Hom (A,ResGA(cid:48)); then Z[G]⊗ A (cid:16) A(cid:48)
H H H Z[H]
by g ⊗a (cid:55)→ ga. On the other hand, if A(cid:48) = indGA, we have 1 ∈ Hom (indGA,A(cid:48)). We then have
H G H
A(cid:48) → indGResGA(cid:48) by a(cid:48) (cid:55)→ 1⊗a(cid:48).
H H
Remark. Any A ∈ ModG is canonically a quotient of the induced module Z[G]⊗Z A → A. And
Z[G]⊗ZA ∼= Z[G]⊗ZResG1A by g⊗ga →(cid:55) g⊗a, so there is no ambiguity here.
A direct summand of an induced module is called relatively projective.
For co-induction: the G-module structure on Hom (Z[G],A) is given by (gf)(x) = f(xg).
H
If H ≤ G, then IndGA ↔ Maps(H\G,A) with an appropriate choice of coset representatives.
H
In fact,
IndGA = {f : G → A : f(hg) = hf(g)∀h ∈ H}. (2.6)
H
We have an identification
5
(cid:89)
IndGA = A·g (2.7)
H i
gi∈H\G
by (f : Z[G] → A) ↔ (f(g )·g ). Check that
i i
g·(a·g ) = ((g gg−1)a ·g ) (2.8)
i j i i i
where g g ∈ Hg .
j i
If H ≤ G, then
indGA = Maps (H\G,A) (cid:44)→ Maps(H\G,A) = IndGA. (2.9)
H f H
If [G : H] < ∞, then indGA = IndGA.
H H
If H = {1}, then IndGA = Maps(G,A). A G-module of this form is called co-induced, and a
1
direct summand of a co-induced module is called relatively injective.
Proposition 2.2 (Frobenius Reciprocity). For A ∈ Mod and A(cid:48) ∈ Mod , we have
H G
Hom (ResGA(cid:48),A) = Hom (A(cid:48),IndGA). (2.10)
H H G H
Proof. We have ϕ (cid:55)→ F by F(a(cid:48))(x) = ϕ(xa(cid:48)), and ϕ →(cid:55) F by ϕ(a) = F(a)(1).
Remark. By taking H = {1}, any A is canonically embedded in a co-induced one.
Remark. If [G : H] < ∞, then Ind and Res are both left and right adjoint.
Remark. If G is finite, then induced is the same as co-induced, and relatively projective is the same
as relatively injective.
The map indGA → IndGA as described earlier is in fact canonical. By Frobenius Reciprocity,
H H
Hom (indGA,IndGA) = Hom (ResGindGA,A). (2.11)
G H H H H H
Now consider the map Z[G]⊗ A → A by
Z[H]
(cid:40)
ga g ∈ H
g⊗a = (2.12)
0 g ∈/ H
3 Group Cohomology
Given a G-module A, consider the abelian group AG of G-invariants. We have AG = Hom (Z,A)
G
whereZhasthetrivialG-action,sothefunctor(cid:3)G isleftexact.Similarly,wehavetheabeliangroup
of coinvariants A , the “largest quotient with trivial G-action”, given by A/(cid:104)ga−a : g ∈ G,a ∈ A(cid:105).
G
We have A = Z⊗ A, where Z is a right G-module with trivial action, and ⊗ means ⊗ . So
G G G Z[G]
(cid:3) is right exact.
G
We have the exact sequence
6
0 → I → Z[G] −“−de−g→” Z → 0 (3.1)
G
(cid:80) (cid:80)
where I is the augmentation ideal { a g : a = 0}. It is generated by g−e for g ∈ G, so
G g g
A = A/I A.
G G
We define Hi(G,A) and H (G,A) such that Hi(G,(cid:3)) is the right derived functor of (cid:3)G and
i
H (G,(cid:3))istheleftderivedfunctorof(cid:3) .Recallhowthesefunctorsareconstructed:forcohomology,
i G
choose an injective resolution
0 → A → I0 → I1 → I2 → ··· (3.2)
and then Hi(G,A) = Hi((I•)G). Similarly, for homology choose a projective resolution
··· → P → P → P → A → 0 (3.3)
2 1 0
and then H (G,A) = H ((P ) ).
i i • G
Properties:
1. Hi(G,A) and H (G,A) are independent of choice of resolution up to canonical isomorphism.
i
2. These are covariant functors in A.
3. H0(G,A) = AG and H (G,A) = A .
0 G
4. Given 0 → A → B → C → 0 exact, there are maps
Hi(G,C) −→δi Hi+1(G,A) (3.4)
and
H (G,C) −d→i H (G,A) (3.5)
i+1 i
which produce long exact sequences of homology and cohomology.
5. A map between two exact sequences yields a map on long exact sequences of homology and
cohomology.
Remark. If A is injective, then 0 → A → A → 0 is an injective resolution, so Hi(G,A) = 0 for
i > 0. Similarly, H (G,A) = 0 if A is projective.
i
Remark. A product of injectives is injective, so Hi(G,(cid:81)A ) = (cid:81)Hi(G,A ). Similarly, homology
j j
commutes with direct sums.
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4 Behavior with Respect to Induction
Suppose H → G is a group homomorphism. If 0 → A → B → C → 0 is exact in Mod , then
G
0 → ResGA → ResGB → ResGC → 0 is exact. Now for I ∈ Mod , we have the exact sequence
H H H H
0 → Hom (ResGC,I) → Hom (ResGB,I) → Hom(ResGA,I) → 0 (4.1)
H H H H H
which by Frobenius Reciprocity, gives the exact
0 → Hom (C,IndGI) → Hom (B,IndGI) → Hom (A,IndGI) → 0 (4.2)
G H C H G H
So IndGI is injective in Mod . More generally, a functor with an exact left adjoint preserves
H G
injectives.
Now assume H ≤ G. Then IndGA is exact in A.
H
Remark. If G = {1}, then Ind1 A = Hom (Z[G],A) = Hom (Z,A) = AH.
H H H
We have (IndGA)G = Hom (Z,IndGA) = Hom (Z,A) = AH. Now if 0 → A → I• is an
H G H H
injectiveresolutioninMod ,then0 → IndA → IndI• isaninjectiveresolutioninMod .Therefore:
H G
Lemma 4.1 (Shapiro).
Hi(G,IndA) = Hi((IndI•)G) = Hi((I•)H) = Hi(H,A). (4.3)
In a similar way, we can show that ind preserves projectives, and if H ≤ G, then
H (G,indA) = H (H,A) (4.4)
i i
Corollary 4.2. If A is relatively injective, then Hi(G,A) = 0 for i > 0, and if A is relatively
projective, then H (G,A) = 0 for i > 0.
i
Fact. We can calculate derived functors (such as cohomology and homology) using acyclic resolu-
tions; that is, resolutions by objects whose higher cohomology or homology is zero.
Observe that since (cid:3)G = Hom (Z,(cid:3)), we have that
G
Hi(G,A) = Exti (Z,A) (4.5)
Z[G]
and similarly
H (G,A) = TorZ[G](Z,A). (4.6)
i i
To calculate these groups, we can resolve the first variable, Z, instead. In both cases, we need a
projective resolution.
8
5 The Standard Resolution of Z as a G-Module
For i > 0, let P = Z[Gi+1]. The diagonal morphism G → Gi+1 makes P into a Z[G]-algebra, in
i i
particular a G-module. Define the boundary map d : P → P by
i+1 i
i+1
(cid:88)
d(g ,...,g ) = (−1)j(g ,...,g ,...,g ) (5.1)
0 i+1 0 (cid:98)j i+1
j=0
Also d : P = Z[G] → Z is the degree map.
0
Claim. The sequence
··· → P → P → P → Z → 0 (5.2)
2 1 0
is exact.
(This is left as an exercise.)
We have found a projective (in fact free) resolution of Z, called the standard resolution.
From this resolution, we have
Hi(G,A) = Exti (Z,A) = Hi(Hom (P ,A)). (5.3)
Z[G] G •
Explicitly, we know that
Hom (P ,A) = {f : Gi+1 → A : f(gg ,...,gg ) = gf(g ,...,g )} = IndGi+1A. (5.4)
G i 0 i 0 i G
Such an f is called a homogeneous cochain. With this description,
(δf)(g ,...,g ) = f(d(g ,...,g )) (5.5)
0 i+1 0 i+1
 
i+1
(cid:88)
= f (−1)j(g0,...,g(cid:98)j,...,gi+1) (5.6)
j=0
i+1
(cid:88)
= (−1)jf(g ,...,g ,...,g ). (5.7)
0 (cid:98)j i+1
j=0
A cocycle is an f such that δf = 0, and a coboundary is an f of the form δf . We have
0
Hi(G,A) = {i-cocycles}/{i-coboundaries}.
Observe that a homogeneous cochain f is uniquely determined by the values for which the first
argument is 1. Define
f(cid:101)(g1,...,gi) = f(1,g1,g1g2,...,g1···gi) (5.8)
so we can recover f from f(cid:98)by
9
f(g,0,...,gi) = g0f(cid:98)(g0−1g1,g1−1g2,...,gi−−11gi). (5.9)
We have a bijection
Hom (P ,A) = Maps(Gi,A) = Ci(G,A), (5.10)
G i
the set of inhomogeneous cochains, via f(cid:98). We have
(δf)(g ,...,g ) = (δf)(1,g ,g g ,...,g ···g ) (5.11)
1 i+1 1 1 2 1 i+1
= f(g ,g g ,...,g ···g ) (5.12)
1 1 2 1 i+1
i+1
(cid:88)
+ (−1)jf(1,g ,g g ,...,g(cid:92)···g ,...,g ···g ) (5.13)
1 1 2 1 j i i+1
j=1
= g1f(cid:98)(g2,g3,...,gi+1) (5.14)
i
(cid:88)
+ (−1)jf(cid:98)(g1,g2,...,gjgj+1,...,gi+1) (5.15)
j=1
+(−1)i+1f(cid:98)(g1,...,gi). (5.16)
Write Zi(G,A) = {f ∈ Ci(G,A) : δf = 0} the set of inhomogeneous cocycles and Bi(G,A) =
{f ∈ Zi(G,A) : f = δf(cid:48)} the set of inhomogeneous coboundaries. We have
Hi(G,A) = Zi(G,A)/Bi(G,A). (5.17)
As an example, C0(G,A) = Maps({1},A) = A and δ(a) = ga−a, so
H0(G,A) = Z0(G,A) = {a ∈ A : ga = a∀g ∈ G} = AG. (5.18)
In degree 1, C1(G,A) = Maps(G,A) and
(δf)(g ,g ) = g f(g )−f(g g )+f(g ). (5.19)
1 2 1 2 1 2 1
In particular, the cocycles are those f for which f(g g ) = f(g )+g f(g ), called crossed homo-
1 2 1 1 2
morphisms.InthecasethatAhasatrivialaction,crossedhomomorphismsarejusthomomorphisms
and the principal crossed homomorphisms are all zero, so H1(G,A) = Hom(G,A).
For group homology, we still use the standard resolution of Z. The right action of G on P is
i
taken to be
(g ,...,g )g = (g−1g ,...,g−1g ). (5.20)
0 i 0 i
Choose coset representatives {x } for G\Gi+1. Now we have
j
10