Exercise 2, § 16

Let $\varphi : H\rightarrow G$ be a group homomorphism, M a $\mathbf{Z}$[G]-module, N a $\mathbf{Z}$[H]-module, and $u: M\rightarrow N$ a homomorphism of abelian groups. We say that $\varphi$ and $u$ are compatible if we have $u(\varphi (h)x) =hu(x)$ for every $h\in H$ and $x\in M$.

a) For $n\geqslant 0$, let $\varphi^n: H^n\rightarrow G^n$ be the mapping deduced from $\varphi$; if $\varphi$ and $u$ are compatible, then the homomorphism $c\mapsto u\circ c\circ \varphi^n$ from $C^n(G,M)$ to $C^n(H,N)$ sends $Z^n(G,M)$ into $Z^n(H,N)$ and $B^n(G,M)$ into $B^n(H,N)$ and therefore induces a homomorphism $H^n(\varphi , u)$ from $H^n(G,M)$ to $H^n(H,N)$. b) Let $\psi : K\rightarrow H$ be another group homomorphism, P a$\mathbf{Z}$[K]-module, and $v: N\rightarrow P$ a homomorphism of abelian groups. If $\varphi$ and $u$ are compatible, and $\psi$ and $v$ also are, then we have $H^n(\varphi \circ \psi , v\circ u) = H^n(\psi , v)\circ H^n(\varphi , u)$.

c) Consider a commutative diagram

0 // $M'$ // M // $M''$ // 0

$u'uu''$

0 // $N'$ // N // $N''$ // 0 ,

where the first (resp. second) line is an exact sequence of $\mathbf{Z}$[G]-modules (resp. of $\mathbf{Z}$[H]-modules) and where $u', u, u''$ are compatible with $\varphi$. Prove that the diagram

$\cdots$ // $H_n(G,M')$ // $H_n(G,M)$ // $H_n(G,M'')$ // $H_{n+1}(G,M')$ /$/\cdots$

$H^n(\varphi ,u')H^n(\varphi ,u)H^n(\varphi ,u'')H^{n+1}(\varphi ,u)$

$\cdots$ // $H_n(H,N')$ // $H_n(H,N)$ // $H_n(H,N'')$ // $H_{n+1}(H,N')$ /$/\cdots$

(whose horizontal lines are the long exact sequences defined in Exercise 1, c)) is commutative.

d) Examples: If H = G, then the mapping $u$ is compatible with Id$_G$ if and and only if it is $\mathbf{Z}$[G]-linear, and we have $H^n$(Id$_G, u$) $= H^n(G, u)$. If N is equal to M endowed with the $\mathbf{Z}$[G]-module structure deduced from $\varphi$, then the homomorphisms $\varphi$ and $1_M$ are compatible; we denote by Res$^q: H^q(G,M)\rightarrow H^q(H,M)$ the homomorphism $H^q(\varphi ,1_M)$ (“restriction homomorphism”).

e) Let H be a normal subgroup of G, and let $M^H$ be the subgroup of M consisting of the elements invariant under H, endowed with the action of $G/H$ deduced from that of G by passing to the quotient. The canonical surjection $p: G\rightarrow$ $G/H$ and the canonical injection $j: M^H\rightarrow M$ are compatible; we denote by Inf$^q: H^q(G/H,M^H)\rightarrow H^q(G,M)$ the homomorphism $H^q(p, j)$ (“inflation homomorphism”). Prove that the sequence

is exact. If, moreover, $H^1(H,M)$ is zero, then the sequence

is exact.