Exercise 12, § 6

Let $G$ be a group. A $\mathbf Z^{(G)}$-module $M$ is said to be co-induced (resp. induced) if there exists a $\mathbf Z$-module $A$ such that $M$ is isomorphic to $\operatorname{Hom}_{\mathbf Z}(\mathbf Z^{(G)},A)$ (resp. $\mathbf Z^{(G)}\otimes_{\mathbf Z}A$).

a) Every $\mathbf Z^{(G)}$-module is isomorphic to a submodule of a co-induced module and to a quotient of an induced module.

b) Let $M$ be a $\mathbf Z^{(G)}$-module. Now prove that the following conditions are equivalent:

α) $M$ is a direct factor of a co-induced module.

β) $M$ is injective relative to $\mathbf Z$ (exercise 19, p. 170).

γ) There exists a $G$-average on $M$ (I, p. 136, exercise 8), that is to say a $\mathbf Z^{(G)}$-homomorphism

(the $G$-module structure on $C^1(G,M)$ being defined by

for $g\in G$, $x\in M$) such that if $f_x$ is the constant function with value $x\in M$, one has $m(f_x)=x$.

c) Now prove that the following conditions are equivalent:

α) $M$ is a direct factor of an induced module.

β) $M$ is projective relative to $\mathbf Z$ (exercise 19, p. 170).

γ) There exists a $\mathbf Z$-endomorphism $\rho$ of $M$ such that for every $x\in M$, the family $(\rho(g^{-1}x))_{g\in G}$ has finite support and that one has

d) Let $M$ be a $\mathbf Z^{(G)}$-module injective (resp. projective) relative to $\mathbf Z$. Now prove that $H^q(G,M)=0$ (resp. $H_q(G,M)=0$) for every $q\geq 1$.

e) If $G$ is finite, now prove that the notions of induced and co-induced module (resp. projective and injective relative to $\mathbf Z$) coincide.

EXERCISES