Exercise 10, § 7

Algebra, X, p. 201

Let $B$ be a k-bigebra (III, p. 148), which is considered as an augmented k-algebra (X, p. 196, exercise 27) by the counit $\beta : B \to k$ ; suppose that the k-module B is projective.

(a) Let M, N be two left B-modules. The homomorphism $\vee$ of exercise 7 identifies with a graded homomorphism of degree zero

H'(B, \gamma; M) \otimes_k H'(B, \gamma; N) \to H'(B \otimes_k B, \gamma \otimes \gamma; M \otimes_k N);

by composition with the homomorphism deduced from the comultiplication, deduce a graded homomorphism of degree zero $\cup : H'(B, \gamma; M) \otimes_k H'(B, \gamma; N) \to H'(B, \gamma; M \otimes_k N)$ .

(b) Let C be an associative and unital k-algebra, which is provided with the structure of B-module deduced from $\gamma$ . Prove that the homomorphism $\cup$ defines on $H'(B, \gamma; C)$ a structure of associative and unital graded k-algebra; if C is commutative and if the bigebra B is cocommutative, the algebra $H'(B, \gamma; C)$ is anticommutative.

(d) Let G be a group, M and N two $\mathbf{Z}^{(G)}$ -modules. We provide $M \otimes_Z N$ with the structure of $\mathbf{Z}^{(G)}$ -module defined by $g(x \otimes y) = gx \otimes gy$ for $g \in G, x \in M, y \in N$ . Prove that the mapping

\cup : H'(G, M) \otimes_Z H'(G, N) \to H'(G, M \otimes_Z N)

is the homomorphism induced on homology by the graded homomorphism of degree zero

u : C'(G, M) \otimes_Z C'(G, N) \to C'(G, M \otimes_Z N)

defined by $u(f \otimes h)(g_1, ..., g_{p+q}) = f(g_1, ..., g_p) \otimes g_1, ..., g_p h(g_{p+1}, ..., g_{p+q})$ for

f \in C^p(G, M), \quad h \in C^q(G, N), \quad g_1 ... g_{p+q} \in G.

(Use exercise 5, p. 185.)

Solution

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