Exercise 1, § 5

Let the algebra $K[F]$ of $F$ be given the filtration $(I^n)$ consisting of the powers of the augmentation ideal $I$ (§ 4, Exercise 6). (a) Let $\tilde{g}: K[F] \to \hat{A}(X)$ be the unique algebra homomorphism extending $g: F \to \hat{A}(X)^*$. Show that $\tilde{g}$ maps $I^n$ to the ideal $\hat{A}_n(X)$ (cf. no. 1) and defines when taking quotients an isomorphism $\tilde{g}_n$ of $K[F]/I^n$ onto $\hat{A}(X)/\hat{A}_n(X)$. (Define an inverse homomorphism of $\tilde{g}_n$ by means of the homomorphism of $A(X)$ into $K[F]$ which maps $x$ to $x - 1$ for all $x \in X$.) Deduce that $\hat{A}(X)$ is isomorphic to the Hausdorff completion of $K[F]$ under the topology defined by $(I^n)$. (b) Suppose that $K = \mathbf{Z}$. Show that the filtration $(I^n)$ is separated (use Proposition 3 and Exercise 6 (e) of § 4) and that the filtration of $F$ defined by it (Exercise 6 (a) of § 4) coincides with the filtration $(C^nF)$. Deduce that $\tilde{g}: \mathbf{Z}[F] \to \hat{A}_\mathbf{Z}(X)$ is injective.