Cohomology of the idèles I: the "First Inequality"

Section 7.1 Cohomology of the idèles I: the “First Inequality”

Reference.

[37] VII.2-VII.4; [38] VI.3, but see below. See also this blog post by Akhil Mathew

amathew.wordpress.com/2010/05/30/the-first-inequality-cohomology-of-the-idele-classes/

By analogy with local class field theory, we want to prove that for

L / K

a cyclic extension of number fields and

C_{K}, C_{L}

the respective idèle class groups of

K

and

L,

🔗

H^{1} (Gal (L / K), C_{L}) = 1, H^{2} (Gal (L / K), C_{L}) = Z / [L : K] Z .

Our first step is to calculate the Herbrand quotient (in analogy with Proposition 4.2.11).

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Theorem 7.1.1.

For

L / K

a cyclic extension of number fields,

🔗

h (C_{L}) = [L : K] .

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Proof.

This will follow by combining Corollary 7.1.6, Definition 7.1.8, and Lemma 7.1.9.

This will end up reducing to a study of lattices in a real vector space, much as in the proof of Dirichlet’s units theorem (Corollary 6.2.11).

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From Theorem 7.1.1, we will deduce the so-called “First Inequality”.

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Theorem 7.1.2. First Inequality.

For

L / K

a cyclic extension of number fields,

🔗

# H_{T}^{0} (Gal (L / K), C_{L}) \geq [L : K] .

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Proof.

Theorem 7.1.1

# H_{T}^{1} (Gal (L / K), C_{L}) \geq 1 .

The “Second Inequality” will be the reverse, which will be a bit more subtle (see Theorem 7.2.9).

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Subsection Some basic observations

Definition 7.1.3. Notational shorthand.

Let

L / K

be a Galois extension of number fields with Galois group

G .

(We do not yet need

G

to be cyclic.) For any finite set

S

of places of

K

containing all infinite places, write

I_{L, S}

to mean the group

I_{L, T}

where

T

denotes the set of places of

L

lying over some place in

S .

Similarly, write

o_{L, S}

to mean

o_{L, T} .

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Note that each

I_{L, S}

is stable under the action of

G

and that

I_{L}

is the direct limit of the

I_{L, S}

over all

S .

Moreover, by Corollary 6.2.10, for

S

sufficiently large we have

🔗

I_{L} = I_{L, S} L^{*} .

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Proposition 7.1.4.

Let

L / K

be a Galois extension of number fields with Galois group

G .

Let

S

be a finite set of places of

K

containing all infinite places and all places that ramify in

K .

For each

i > 0,

🔗

H^{i} (G, I_{L, S}) = ⨁_{v \in S} H^{i} (G_{w}, L_{w}^{*})

where

w

denotes a single place of

L

above

v .

Similarly, for each

i \in Z,

🔗

H_{T}^{i} (G, I_{L, S}) = ⨁_{v \in S} H_{T}^{i} (G_{w}, L_{w}^{*}) .

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Proof.

The group

H^{i} (G, I_{L, S})

is the product of

H^{i} (G, \prod_{w | v} L_{w}^{*})

over all

v \in S

and

H^{i} (G, \prod_{w | v} o_{L_{w}}^{*})

over all

v \notin S .

By Shapiro’s lemma (Lemma 3.2.3), we may reinterpret

H^{i} (G, I_{L, S})

as the product of

H^{i} (G_{w}, L_{w}^{*})

over all

v \in S

and

H^{i} (G_{w}, o_{L_{w}}^{*})

over all

v \notin S .

The latter vanish for all

i > 0

by Corollary 4.2.6, so we get what we want. The argument for Tate groups is analogous.

Corollary 7.1.5.

Let

L / K

be a Galois extension of number fields with Galois group

G .

For each

i > 0,

🔗

H^{i} (G, I_{L}) = ⨁_{v} H^{i} (G_{w}, L_{w}^{*}),

where

v

runs over places of

K

and

w

denotes a single place of

L

above

v .

Similarly, for each

i \in Z,

🔗

H_{T}^{i} (G, I_{L}) = ⨁_{v} H_{T}^{i} (G_{w}, L_{w}^{*}) .

🔗

Proof.

View

I_{L}

as the direct limit of the

I_{L, S}

over all finite sets

S

of places of

K

containing all infinite places and all ramified places; then

H^{i} (G, I_{L})

is the direct limit of the

H^{i} (G, I_{L, S}) .

We may thus apply Proposition 7.1.4 to deduce the claims.

Corollary 7.1.6.

Let

L / K

be a Galois extension of number fields with Galois group

G .

For any finite set of places

S

K

containing all infinite places and all places which ramify in

L,

🔗

H^{1} (G, I_{L, S}) = 0, H^{2} (G, I_{L, S}) = ⨁_{v \in S} \frac{1}{[L_{w} : K_{v}]} Z / Z .

In addition,

🔗

H^{1} (G, I_{L}) = 0, H^{2} (G, I_{L}) = ⨁_{v} \frac{1}{[L_{w} : K_{v}]} Z / Z .

🔗

Proof.

This follows by combining Proposition 7.1.4, the computation of cohomology of local fields (Lemma 1.2.3 and Proposition 4.2.1), and the equality

H^{2} (Gal (C / R), C^{*}) ≅ H_{T}^{0} (Gal (C / R), C^{*}) = R^{*} / R^{+} ≅ Z / 2 Z

which already arose in the proof of Lemma 6.4.1.

Remark 7.1.7. Sanity check.

The case of

H_{T}^{0}

in Corollary 7.1.5 asserts something that is evidently true and already featured in the proof of Lemma 6.4.1: an idèle in

I_{K}

is a norm from

I_{L}

if and only if each component is a norm.

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Subsection Cohomology of the units: first steps

Definition 7.1.8.

Let

L / K

be a cyclic extension of number fields with Galois group

G .

Apply Corollary 6.2.10 to choose a finite set

S

of places of

K,

containing all infinite places and all places which ramify in

L,

such that

I_{L} = I_{L, S} L^{*} .

From the exact sequence

🔗

1 \to o_{L, S}^{*} \to I_{L, S} \to I_{L, S} / o_{L, S}^{*} = C_{L} \to 1

we have an equality of Herbrand quotients

🔗

h (C_{L}) = h (I_{L, S}) / h (o_{L, S}^{*}) .

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By Corollary 7.1.6,

🔗

h (I_{L, S}) = \prod_{v \in S} [L_{w} : K_{v}] .

To get

h (C_{L}) = [L : K],

it will thus suffice to establish Lemma 7.1.9 below.

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Lemma 7.1.9.

Let

L / K

be a cyclic extension of number fields. Let

S

be a finite set of places of

K

containing all infinite places. Then

🔗

h (o_{L, S}^{*}) = \frac{1}{[L : K]} \prod_{v \in S} [L_{w} : K_{v}] .

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Proof.

See the calculations in Definition 7.1.10 and Definition 7.1.12, plus Lemma 7.1.13.

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Subsection Cohomology of the units: a computation with $S$ -units

At this point, we have reduced the computation of the Herbrand quotient

h (I_{L}),

and by extension the First Inequality, to the computation of

h (o_{L, S}^{*})

for a suitable set

S

of places of

K .

We treat this point next, using similar ideas to the proof of Dirichlet’s units theorem (Corollary 6.2.11).

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Definition 7.1.10.

Let

L / K

be a cyclic extension of number fields with Galois group

G .

Let

S

be a finite set of places of

K

containing all infinite places, and let

T

be the set of places of

L

lying above places of

S .

Let

V

be the real vector space consisting of one copy of

R

for each place in

T .

Define the map

o_{L, S}^{*} \to V

🔗

α \to \prod_{w \in T} \log | α |_{w}

with normalizations as in Definition 6.1.11. By the product formula (Proposition 6.1.12) and Dirichlet’s units theorem (Corollary 6.2.11), the kernel of this map consists of roots of unity, and the image

M

is a lattice in the trace-zero hyperplane

H

V .

Since

G

acts compatibly on

o_{L, S}^{*}

and

V

(the latter by permuting the factors), it also acts on

M .

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Remark 7.1.11. Caveat.

At this point, we deviate from [38] due to an error therein. Namely, Lemma VI.3.4 is only proved assuming that

G

acts transitively on the coordinates of

V,

but in Definition 7.1.10 this is not the case:

G

permutes the places above any given place

v

K

but those are separate orbits. So we’ll follow [37] instead.

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Definition 7.1.12.

Continuing from Definition 7.1.10, we can write down two natural lattices in

V .

One of them is the lattice generated by

M

together with the all-ones vector, on which

G

acts trivially. As a

G

-module, the Herbrand quotient of that lattice is

h (M) h (Z) = [L : K] h (M) .

The other is the lattice

M^{'}

in which, in the given coordinate system, each element has integral coordinates. To compute the Herbrand quotient of

M^{'},

notice that the projection of this lattice onto the coordinates corresponding to the places

w \in T

above some

v \in S

form a copy of

{Ind}_{G_{v}}^{G} Z .

Writing

h (G, ∙)

for the Herbrand quotient to emphasize the dependence on the group

G,

we have

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\begin{aligned} h (G, M^{'}) & = \prod_{v \in S} h (G, {Ind}_{G_{v}}^{G} Z) \\ = \prod_{v \in S} h (G_{v}, Z) \\ = \prod_{v \in S} # G_{v} = \prod_{v \in S} [L_{w} : K_{v}] . \end{aligned}

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To sum up, the calculations from Definition 7.1.10 and Definition 7.1.12 reduce Lemma 7.1.9 to a comparison of Herbrand quotients of two lattices in the same real vector space. See Lemma 7.1.13.

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Subsection Herbrand quotients of real lattices

We conclude the proof of the First Inequality with the following statement.

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Lemma 7.1.13.

Let

V

be a real vector space on which a finite cyclic group

G

acts linearly, and let

L_{1}

and

L_{2}

G

-stable lattices in

V .

Then

h (L_{1}) = h (L_{2}) .

(Note that both Herbrand quotients are automatically well-defined because

L_{1}

and

L_{2}

are finitely generated abelian groups; see Example 3.2.24.)

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Proof.

Note that

L_{1} \otimes_{Z} Q

and

L_{2} \otimes_{Z} Q

are

Q [G]

-modules which become isomorphic to

V,

and hence to each other, after tensoring over

Q

with

R .

By Lemma 7.1.14, this implies that

L_{1} \otimes_{Z} Q

and

L_{2} \otimes_{Z} Q

are isomorphic as

Q [G]

-modules.

From this isomorphism, we see that as a

Z [G]

-module,

L_{1}

is isomorphic to some sublattice of

L_{2} .

Since a lattice has the same Herbrand quotient as any sublattice (the quotient is finite, so its Herbrand quotient is 1 by Remark 3.4.8), that means

h (L_{1}) = h (L_{2}) .

Lemma 7.1.14.

Let

F / E

be an extension of infinite fields. Let

G

be a finite group. Let

V_{1}

and

V_{2}

be two right

E [G]

-modules which are finite-dimensional as

E

-vector spaces. If

V_{1} \otimes_{E} F

and

V_{2} \otimes_{E} F

are isomorphic as

F [G]

-modules, then

V_{1}

and

V_{2}

are isomorphic.

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Proof.

By hypothesis, the

F

-vector space

W_{F} := {Hom}_{F} (V_{1} \otimes_{E} F, V_{2} \otimes_{E} F),

on which

G

acts by the formula

T^{g} (x) = T (x^{g^{- 1}})^{g},

contains an invariant vector which, as a linear transformation, is invertible. Now

W_{F}

can also be written as

W \otimes_{E} F, W := {Hom}_{E} (V_{1}, V_{2}) .

The fact that

W_{F}

has an invariant vector says that a certain set of linear equations has a nonzero solution over

F,

namely the equations that express the fact that the action of

G

leaves the vector invariant. But those equations have coefficients in

E,

W^{G} \otimes_{E} F = W_{F}^{G};

in particular, the space of invariant vectors in

W

is also nonzero.

It remains to check that some element of

W^{G}

corresponds to a map

V_{1} \to V_{2}

which is actually an isomorphism; for this, we argue as in Exercise 3. Fix an isomorphism of vector spaces between

V_{2} \otimes_{E} F

and

V_{1} \otimes_{E} F

(which need not respect the

G

-action). By composing each element of

W

with this isomorphism and taking the determinant, we get a well-defined polynomial function on

W,

which we can restrict to

W^{G} .

By hypothesis, this function is not identically zero on

W_{F}^{G},

so (because

E

is infinite) it cannot be identically zero on

W^{G}

either.

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Subsection Splitting of primes

As a consequence of the First Inequality, we record the following fact which was previously stated as a consequence of the Chebotaryov density theorem (Theorem 2.4.11), but which will be needed logically earlier in the arguments. (See [38], Corollary VI.3.8 for more details.) Alternatively, we will get something stronger out of the analytic proof of the Second Inequality (Proposition 7.2.3).

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Corollary 7.1.15.

For any nontrivial extension

L / K

of number fields, there are infinitely many primes of

K

which do not split completely in

L .

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Proof.

Suppose first that

L / K

is cyclic. Suppose that all but finitely many primes split completely; we can then take a finite set

S

of places which contains all of them as well as all of the infinite places and all of the ramified places. For each

v \in S,

the group

U_{v} := {Norm}_{L_{w} / K_{v}} L_{w}^{*}

is open of finite index in

K_{v}^{*} .

For any

α \in I_{K},

using the approximation theorem (Proposition 6.1.18) we can then find

β \in K^{*}

such that

(α / β)_{v} \in U_{v}

for all

v \in S .

For each place

v \notin S,

we have

L_{w} = K_{v},

α / β \in {Norm}_{L / K} I_{L} .

We deduce that the class of

α

C_{L}

is a norm; that is,

C_{K} = {Norm}_{L / K} C_{L},

whereas Theorem 7.1.2 asserts that

H_{T}^{0} (Gal (L / K), C_{L}) \geq [L : K],

contradiction.

In the general case, let

M

be the Galois closure of

L / K;

then a prime of

K

splits completely in

L

if and only if it splits completely in

M .

Since

Gal (M / K)

is a nontrivial finite group, it contains a cyclic subgroup; let

N

be the fixed field of this subgroup. By the previous paragraph, there are infinitely many prime ideals of

N

which do not split completely in

M,

proving the original result.

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Exercises Exercises

1.

Let

K

be a number field. Let

L_{1}, \dots, L_{r}

be cyclic extensions of

K

of the same prime degree

p

such that

L_{i} \cap L_{j} = K

for

i \neq j .

Using the First Inequality (Theorem 7.1.2), prove that there are infinitely many primes of

K

which split completely in

L_{2}, \dots, L_{r}

but not in

L_{1} .

🔗

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Section 7.1 Cohomology of the idèles I: the “First Inequality”

Reference.

Theorem 7.1.1.

Proof.

Theorem 7.1.2. First Inequality.

Proof.

Subsection Some basic observations

Definition 7.1.3. Notational shorthand.

Proposition 7.1.4.

Proof.

Corollary 7.1.5.

Proof.

Corollary 7.1.6.

Proof.

Remark 7.1.7. Sanity check.

Subsection Cohomology of the units: first steps

Definition 7.1.8.

Lemma 7.1.9.

Proof.

Subsection Cohomology of the units: a computation with S-units

Definition 7.1.10.

Remark 7.1.11. Caveat.

Definition 7.1.12.

Subsection Herbrand quotients of real lattices

Lemma 7.1.13.

Proof.

Lemma 7.1.14.

Proof.

Subsection Splitting of primes

Corollary 7.1.15.

Proof.

Exercises Exercises

1.

Subsection Cohomology of the units: a computation with $S$ -units