Exercise 11, § 7

Assume that the bigebra $B$ admits an inversion $i$ (cf. III, p. 198, exercise 4). Let $M$ and $N$ be $B$-modules. The $k$-module $M \otimes_k N$ (resp. $\mathrm{Hom}_k(M, N)$) has a natural structure of $B \otimes_k B$-module (resp. $B \otimes_k B^\circ$-module); it is provided with the structure of $B$-module deduced from the homomorphism c (resp. $(1_B \otimes i) \circ c$).

a) Now prove that the homomorphism $u$ (exercise 10) defines a graded homomorphism of degree zero

b) Let $0 \to M \to R_n \to ... \to R_1 \to k \to 0$ be an exact sequence of $B$-modules, with $R_i$ projective $(1 \leq i \leq n)$; let $\theta$ be its class in $\mathrm{Ext}_B^n(k, M) = H^n(B, \gamma; M)$. Now prove that, for $p \geq 1$, the mapping $x \mapsto x \cup \theta$ is a $k$-isomorphism from $H^p(B, \gamma; \mathrm{Hom}_k(M, N))$ onto $H^{p+n}(B, \gamma; N)$.

(Prove that this mapping factors through

c) Let $G$ be a group, $N$ a $\mathbf{Z}^{(G)}$-module. Let $F$ be a free group, $p : F \to G$ a surjective homomorphism, $R = \mathrm{Ker}\,(p)$, $\overline{R} = R/(R, R)$, $\overline{F} = F/(R, R)$. The extension $\overline{R} \to \overline{F} \to G$ defines a class $\varepsilon \in H^2(G, \overline{R})$; prove that the mapping $x \mapsto x \cup \varepsilon$ from $H^p(G, \mathrm{Hom}_\mathbf{Z}(\overline{R}, N))$ into $H^{p+2}(G, N)$ is an isomorphism for $p \geqslant 1$ (use exercise 3, d), p. 179).