Exercise 3, § 16

Let G be a group and M a $\mathbf{Z}$[G]-module. Let G act on $C^1(G,M)$ by sending $(g, f)\in G\times C^1(G,M)$ to the function $h\mapsto gf(g^{-1}h)$ on G. Recall (I, §6, p. 149, Exercise 8) that a G-mean on M is a $\mathbf{Z}$[G]-linear mapping $\mu: C^1(G,M)\rightarrow M$ satisfying $\mu(c_x) =x$, where $c_x$ denotes the constant function with value $x\in M$.

a) Prove that if M admits a G-mean, then $H^n(G,M)$ is zero for $n\geqslant 1$. (For $q\geqslant 1$, let $h^q: C^q(G,M)\rightarrow C^{q-1}(G,M)$ be the following homomorphism: for $g_1, . . . , g_{q-1}$ in G and $c\in C^q(G,M)$, define $h^q(c)(g_1, . . . , g_{q-1})$ as the image by $\mu$ of the function $g\mapsto c(g_1, . . . , g_{q-1},(g_1\cdots g_{q-1})^{-1}g)$. Prove the equality $h^{q+1}\circ d^q+d^{q-1}\circ h^q=$ $1_{C^q(G,M)}.$)

b) Let A be an abelian group; denote by $\mathscr{F}(G,A)$ the group of mappings from G to A, endowed with the action of G defined by $^gf(h) =f(hg)$ for $g, h$ in G and $f$ in $\mathscr{F}(G,A)$. For any mapping $c$ from G to $\mathscr{F}(G,A)$, denote by $\mu(c)$ the mapping $g\mapsto c(g^{-1})(g)$ from G to A. Show that $\mu$ is a G-mean on $\mathscr{F}(G,A)$; we therefore have $H^q(G,\mathscr{F}(G,A)) = 0$ for $q\geqslant 1$.

c) Let $j: M\rightarrow \mathscr{F}(G,M)$ be the mapping that sends an element $m$ of M to the function $g\mapsto gm$, and let $M'$ be its cokernel. Prove that $j$ is injective and $\mathbf{Z}[G]$-linear. Deduce that the homomorphism $\partial^q: H^q(G,M')\rightarrow H^{q+1}(G,M)$ is bijective for $q\geqslant 1$.

d) Let L be a finite Galois extension of the commutative field K and G be its Galois group. Deduce from b) and the normal basis theorem (V, §10, No. 9, p. 73) that the groups $H^n(G,L)$ are zero for $n\geqslant 1$.