Homology and Tate groups

Section 3.3 Homology and Tate groups

Reference.

[37], II.2.

You may not be surprised to learn that there is a “dual” theory to the theory of group cohomology, namely group homology. What you may be surprised to learn is that one can actually fit the two together, so that in a sense the homology groups become cohomology groups with negative indices. (Since the arguments are similar to those for cohomology, I’m going to skip some details.)

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Subsection Homology

Definition 3.3.1.

Let

M_{G}

denote the maximal quotient of

M

on which

G

acts trivially. In other words,

M_{G}

is the quotient of

M

by the submodule spanned by

m^{g} - m

for all

m \in M

and

g \in G .

In yet other words,

M_{G} = M / M I_{G},

where

I_{G}

is the augmentation ideal of the group algebra

Z [G] :

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I_{G} := {\sum_{g \in G} z_{g} [g] : \sum_{g} z_{g} = 0} .

Or if you like,

M_{G} = M \otimes_{Z [G]} Z .

Since

M^{G}

is the group of

G

-invariants, we call

M_{G}

the group of

G

-coinvariants.

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The functor

M \to M^{G}

is right exact but not left exact: if

0 \to M^{'} \to M \to M^{″} \to 0

is exact, then

M_{G}^{'} \to M_{G} \to M_{G}^{″} \to 0

is exact but the map on the left is not necessarily injective. Again, we can fill in the exact sequence by defining homology groups.

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Definition 3.3.2.

G

-module

M

is projective if for any surjection

N \to N^{'}

G

-modules and any map

ϕ : M \to N^{'},

there exists a map

ψ : M \to N

lifting

ϕ .

This definition is dual to the definition of an injective

G

-module, but this symmetry is a bit misleading: it is much easier to describe projective

G

-modules than injective

G

-modules. For example, any

G

-module which is a free module over the ring

Z [G]

is projective, such as

Z [G]

itself!

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Definition 3.3.3.

A projective resolution of

M

is an exact sequence

\dots \to P_{1} \to P_{0} \to M \to 0

G

-modules in which the

P_{i}

are projective. Given such a resolution, take coinvariants to get a complex

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\dots \overset{d_{2}}{\to} P_{2} \overset{d_{1}}{\to} P_{1} \overset{d_{0}}{\to} P_{0} \to 0,

then put

H_{i} (G, M) = \ker (d_{i - 1}) / im (d_{i}) .

Again, this is canonically independent of the resolution and functorial, and there is a long exact sequence which starts out

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\dots \to H_{1} (G, M^{″}) \overset{δ}{\to} H_{0} (G, M^{'}) \to H_{0} (G, M) \to H_{0} (G, M^{″}) \to 0 .

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Definition 3.3.4.

We say that

M

is acyclic (for homology) if

H_{i} (G, M) = 0

for

i > 0 .

As with group cohomology, we can replace a projective resolution with an acyclic resolution and get the same homology groups. For example, induced modules are again acyclic and the analogue of Shapiro’s lemma holds (key point: any free

Z [H]

-module induces to a free

Z [G]

-module).

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Remark 3.3.5.

One can give a concrete description of homology as well, but we won’t need it for our purposes. Even without one, though, we can calculate

H_{1} (G, Z)

using the exact sequence

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0 \to I_{G} \to Z [G] \to Z \to 0 .

By the long exact sequence in homology,

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0 = H_{1} (G, Z [G]) \to H_{1} (G, Z) \to H_{0} (G, I_{G}) \to H_{0} (G, Z [G])

is exact, i.e.

0 \to H_{1} (G, Z) \to I_{G} / I_{G}^{2} \to Z [G] / I_{G}

is exact. The last map is induced by

I_{G} ↪ Z [G]

and so is the zero map. Thus

H_{1} (G, Z) ≅ I_{G} / I_{G}^{2};

recall that in Exercise 5, it was shown that the map

g \mapsto [g] - 1

defines an isomorphism

G^{ab} \to I_{G} / I_{G}^{2} .

This can be thought of as an algebraic analogue of the fact that the first homology group of a (reasonable) topological space equals the abelianization of the fundamental group.

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Subsection The Tate groups

We now stitch together the long exact sequences of cohomology and homology to get a doubly infinite exact sequence. The value of doing this may be unclear at first, but will become apparent when we compute the Tate cohomology of cyclic groups (Theorem 3.4.1).

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Definition 3.3.6.

Let

M

be a

G

-module. Define the map

{Norm}_{G} : M \to M

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{Norm}_{G} (m) = \sum_{g \in G} m^{g} .

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Then

{Norm}_{G}

induces a homomorphism

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{Norm}_{G} : H_{0} (G, M) = M_{G} \to M^{G} = H^{0} (G, M) .

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Remark 3.3.7.

You might be wondering why

{Norm}_{G}

is called a “norm” rather than a “trace”. The reason is that in practice, our modules

M

will most often be groups which are most naturally written multiplicatively, e.g., the nonzero elements of a field.

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Definition 3.3.8.

We now define the Tate cohomology groups (or Tate homology groups if you prefer) as follows:

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H_{T}^{i} (G, M) := {\begin{cases} H^{i} (G, M) & i > 0 \\ M^{G} / {Norm}_{G} M & i = 0 \\ \ker ({Norm}_{G}) / M I_{G} & i = - 1 \\ H_{- i - 1} (G, M) & i < - 1. \end{cases}

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Lemma 3.3.9.

For any short exact sequence

0 \to M^{'} \to M \to M^{″} \to 0

G

-modules, we have a canonical exact sequence

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\dots \to H_{T}^{i - 1} (G, M^{″}) \to H_{T}^{i} (G, M^{'}) \to H_{T}^{i} (G, M) \to H_{T}^{i} (G, M^{″}) \to H_{T}^{i + 1} (G, M^{'}) \to \dots

which extends infinitely in both directions.

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Proof.

Since we already have long exact sequences for homology and cohomology, the only remaining issue is exactness between

H_{T}^{- 2} (G, M^{″})

and

H_{T}^{1} (G, M^{'})

inclusive. This follows by diagram-chasing, as in the proof of the snake lemma (Lemma 3.1.16) on the commutative diagram Figure 3.3.10 with exact rows, noting that the diagram remains commutative with the dashed arrows added.

Remark 3.3.11.

M

is an induced

G

-module, then

H_{T}^{i} (G, M) = 0

for all

i

(see Exercise 1). That is, induced modules are acyclic for all of cohomology, homology, and Tate (co)homology.

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Subsection Extended functoriality revisited

The extended functoriality for cohomology groups (Definition 3.2.22) has analogues for homology groups and Tate cohomology groups, but under more restrictive conditions.

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Definition 3.3.12.

Again, let

M

be a

G

-module and

M^{'}

G^{'}

-module, and consider a homomorphism

α : G^{'} \to G

of groups and a homomorphism

β : M \to M^{'}

of abelian groups which are compatible in the sense of Definition 3.2.22. We would like to obtain canonical homomorphisms

H_{i} (G, M) \to H_{i} (G^{'}, M^{'}),

but for this we need to add an additional condition to ensure that

M \to M^{'}

induces a well-defined map

M_{G} \to M_{G^{'}}^{'} .

For instance, this holds if

α

is surjective.

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For Tate cohomology groups, there is a further complication that the map

M^{G} \to (M^{'})^{G^{'}}

does not necessarily induce a map

Norm (M) \to Norm (M^{'}) .

However, this does occur if

α

is injective, so for instance we have well-defined restriction maps

Res : H_{T}^{0} (G, M) \to H_{T}^{0} (H, M)

whenever

H

is a subgroup of

G .

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Remark 3.3.13.

In Example 3.2.24, we used the restriction and corestriction maps to show that for

G

a finite group and

M

G

-module, the groups

H^{i} (G, M)

are torsion groups killed by

# G

for all

i > 0 .

While we cannot extend the corestriction map to Tate cohomology, we may still argue directly that

H_{T}^{0} (G, M)

is killed by

# G .

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Exercises Exercises

1.

Prove that if

M

is an induced

G

-module, then

H_{T}^{i} (G, M) = 0

for all

i \in Z .

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Hint.

Use the fact that induced

G

-modules are acyclic for both cohomology and homology to reduce to checking the cases

i = - 1, 0 .

Another option is to extend Shapiro’s lemma to Tate cohomology groups.

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2.

Let

G \subseteq H

be an inclusion of finite groups. Show that via the identification from Remark 3.3.5, the map

Res : H_{T}^{- 2} (G, Z) \to H_{T}^{- 2} (H, Z)

corresponds to the transfer (Verlagerung) map

V : G^{ab} \to H^{ab} .

This provides another way to derive the existence of the latter.

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