COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

09. Cohomology of groups Let G be a group. Let us consider a functor

F^{G} : M ↦ M^{G} = {m ∈ M; g.m = m(∀g ∈ G)}

The functor is left-exact. The derived functor of this functor

H^{j}(G, M) = R^{j}F^{G}(M)

is called the j-th cohomology of G with coefficients in M. Let us consider ℤ as a G-module with trivial G-action. Then we may easily verify that

F^{G}(M) = M^{G} ≅ Hom_{G}(ℤ, M).

Thus we have

H^{j}(G, M) = Ext_{G}^{j}(ℤ, M).

To compute cohomologies of G, it is useful to use ℤ[G]-resolution of ℤ. For any tuples g_{0}, g_{1}, g_{2}, …, g_{t} of G, we introduce a symbol

[g_{0}, g_{1}, g_{2}, …, g_{t}]

and we consider the following sequence (∗_{G})

0 ← ℤ ←^{d} ⨁_{g_{0}∈G} ℤ·[g_{0}] ←^{d} ⨁_{g_{0},g_{1}∈G} ℤ·[g_{0}, g_{1}] ←^{d} ⨁_{g_{0},g_{1},g_{2}∈G} ℤ·[g_{0}, g_{1}, g_{2}] ←^{d} …

where ϵ, d are determined by the following rules.

d([g_{0}]) = 1

d([g_{0}, g_{1}]) = [g_{1}] − [g_{0}]

d([g_{0}, g_{1}, g_{2}]) = [g_{1}, g_{2}] − [g_{0}, g_{2}] + [g_{0}, g_{1}]

d([g_{0}, g_{1}, g_{2}, g_{3}]) = [g_{1}, g_{2}, g_{3}] − [g_{0}, g_{2}, g_{3}] + [g_{0}, g_{1}, g_{3}] − [g_{0}, g_{1}, g_{2}] …

To see that the sequence ∗_{G} is acyclic, we consider a homotopy

h([g_{0}, g_{1}, …, g_{t}]) = [1, g_{0}, g_{1}, …, g_{t}]

Exercise 9.1. Show that h ∘ d + d ∘ h = id

Lemma 9.1. (1) Each of the modules that appears in the sequence ∗_{G} admits an action of G determined by

g.[g_{0}, g_{1}, g_{2}, …, g_{t}] = [g · g_{0}, g · g_{1}, g · g_{2}, …, g · g_{t}]

(2)

C_{t} = ⨁_{g_{0},g_{1},g_{2},...g_{t}∈G} ℤ · [g_{0}, g_{1}, g_{2}, …, g_{t}]

is ℤ[G]-free

YOSHIFUMI TSUCHIMOTO

There are several choices for the ℤ[G]-basis of C_{t}. One such is clearly

{[1, g_{1}, g_{2}, g_{3}, …, g_{t}]; g_{1}, g_{2}, …, g_{t} ∈ G}.

It is traditional (and probably useful) to use another basis

{⟨g_{1}, g_{2}, g_{3}, …, g_{t}⟩; g_{1}, g_{2}, …, g_{t} ∈ G}.

where

⟨g_{1}, g_{2}, g_{3} … g_{t}⟩ = [1, g_{1}, g_{1}g_{2}, g_{1}g_{2}g_{3}, …, g_{1}g_{2}g_{3} … g_{t}].

Conversely we have

[1, a_{1}, a_{2}, …, a_{t}] = ⟨a_{1}, a_{1}^{−1}a_{2}, a_{2}^{−1}a_{3}, …, a_{t−1}^{−1}a_{t}⟩. Definition 9.2. For any group G, the derived functor of a functor

F_{G} : (G − modules) → (modules)

defined by

M ↦ M_{G} = M/(ℤ − span{g.m − m; g ∈ G, M ∈ M}) is called the homology of G with coefficients in M. We denote the homology group L_{j}F_{G}(M) by H_{j}(G; M).

Lemma 9.3.

H_{j}(G; M) ≅ Tor_{j}^{ℤ[G]}(ℤ, M)