Exercise 1, § 16

Let G be a group and $\mathbf{Z}[G]$ the algebra of G over $\mathbf{Z}$; we identify G with the canonical basis of $\mathbf{Z}[G]$. Let M be a $\mathbf{Z}$[G]-module, that is, an abelian group endowed with an action of G respecting its group structure. For any integer $n\geqslant 0$, denote by $C^n(G,M)$ the group of mappings from $G^n$ to M; set $C^n(G,M) = 0$ for $n <0$. Define a homomorphism $d^n: C^n(G,M)\rightarrow C^{n+1}(G,M)$ by the formula

For any integer $n$, set $Z^n(G,M) =$ Ker $d^n$ and $B^n(G,M) =$ Im $d^n$.

a) Prove that $B^n(G,M)$ is contained in $Z^n(G,M)$.

Set $H^n(G,M) = Z^n(G,M)/B^n(G,M)$. The group $H^0(G,M)$ is the set of elements of M fixed by the action of G. The group $Z^1(G,M)$ is the group of mappings $f: G\rightarrow M$ satisfying $f(gg') =f(g) +g f(g')$, and $B^1(G,M)$ is the subgroup of mappings $f$ for which there exists an $m\in M$ such that $f(g) =gm-m$ (cf. I, §6, p. 148, Exercise 7).

b) Let N be a $\mathbf{Z}$[G]-module and $u: M\rightarrow N$ a $\mathbf{Z}$[G]-linear mapping. For every integer $n$, the mapping $c\mapsto u\circ c$ is a homomorphism from $C^n(G,M)$ to $C^n(G,N)$ that sends $Z^n(G,M)$ into $Z^n(G,N)$ and $B^n(G,M)$ into $B^n(G,N)$ and therefore induces a homomorphism from $H^n(G,M)$ to $H^n(G,N)$ that we denote by $H^n(G, u)$ or simply $H^n(u)$. If $v: N\rightarrow P$ is a homomorphism of $\mathbf{Z}$[G]-modules, then we have $H^n(v\circ u) = H^n(v)\circ H^n(u)$.

c) Let $0\rightarrow M'\rightarrow^{\iota}M\rightarrow^{\pi}M''\rightarrow 0$ be an exact sequence of $\mathbf{Z}$[G]-modules. Let $n$ be an integer and $x''$ an element of $H^n(G,M'')$, the class of an element $c''\in$ $Z^n(G,M'')$. Let $c\in C^n(G,M)$ be such that $\pi \circ c=c''$. Prove that there exists an element $c'\in Z^{n+1}(G,M')$ such that $\iota \circ c'=d^n(c)$ and that the class of $c'$ in $H^{n+1}(G,M')$ depends only on $x''$; we denote that class by $\partial^n(x'')$. The mapping $\partial^n: H^n(G,M'')\rightarrow H^{n+1}(G,M')$ is a homomorphism (“linking homomorphism”). Prove that the sequence

$n'H^n(\iota )nH^n(\pi )n''\partial^nn+1'$

$\cdots \longrightarrow H(G,M)\longrightarrow H(G,M)\longrightarrow$ H $(G,M)\longrightarrow H(G,M)\longrightarrow \cdots$

is exact.