Exercise 15, § 6

Let G be a group and k a commutative field, which are endowed with the structures of $\mathbf{Z}^{(G)}$-module and of $k^{(G)}$-module deduced from the trivial action of G. a) Now prove that the group $H^q(G, k)$ is isomorphic to $\mathrm{Ext}^q_{k^{(G)}}(k, k)$ for all $q \geqslant 0$, and therefore admits a structure of k-vector space; we denote by $h^q(G, k)$ its dimension over k.

b) Suppose that the group G admits a presentation (I, p. 86) defined by d generators and r relations. Prove the inequalities

(use exercise 3, d), p. 179).