Exercise 2, § 5

Let $G$ be a group, let $K[G]$ be its algebra over $K$ and let $I$ be its augmentation ideal (§ 4, Exercise 6). Let $\varepsilon$ denote the canonical homomorphism of $K[G]$ onto $K$; then $\operatorname{Ker}(\varepsilon) = I$. (a) Let $M$ be a left $K[G]$-module and let $Z(G, M)$ be the group of crossed homomorphisms of $G$ into $M$ (Algebra, Chapter I, § 6, Exercise 7). If $f \in \mathrm{Hom}_{K[G]}(I, M)$, let $f_1$ be the mapping $g \mapsto f(g - 1)$ of G into M. Show that $f_1$ is a crossed homomorphism (use the identity

to show that $f_1(gg') = g . f_1(g') + f_1(g)$) and that $f \mapsto f_1$ is an isomorphism of $\mathrm{Hom}_{K[G]}(I, M)$ onto $Z(G, M)$.

(b) Take G to be the free group $F = F(X)$. Show that, for every mapping $\theta : X \to M$, there exists one and only one element $f_\theta$ of $Z(M, F)$ which extends $\theta$. (Use the interpretation of $Z(F, M)$ in terms of the semi-direct product of F by M, cf. Algebra, Chapter I, loc. cit.) Deduce, using (a), that the family $(x - 1)_{x \in X}$ is a basis of I as a left K[F]-module.

(c) By (b) every element $u \in K[F]$ can be written uniquely in the form

where $D_x(u) \in K[F]$ and the $D_x(u)$ are zero except for a finite number. The mapping $u \mapsto D_x(u)$ is called the partial derivative with respect to x. It is characterized by the following properties: (i) $D_x$ is a K-linear mapping of K[F] into K[F]. (ii) $D_x(uv) = u . D_x(v) + \varepsilon(v) . D_x(u)$ for u, v in K[F]. (iii) $D_x(x) = 1$ and $D_x(y) = 0$ if $y \in X - \{x\}$. If $n \geqslant 1$, then

(d) Let $\varepsilon_A$ be the canonical homomorphism of A(X) onto K. Show that every element u of A(X) can be written uniquely in the form

where $d_x(u) \in A(X)$ and the $d_x(u)$ are zero except for a finite number. Show that $d_x : u \mapsto d_x(u)$ is an endomorphism of A(X) which can be extended by continuity to $\hat{A}(X)$ and enjoys properties analogous to (i), (ii), (iii) above. Show $d_x \circ \tilde{g} = \tilde{g} \circ D_x$, where $\tilde{g}$ is the homomorphism of K[F] into $\hat{A}(X)$ which extends g (cf. Exercise 1).†