Exercise 3, § 5

We preserve the notation of Exercise 2.

(a) Let J be a right ideal of K[F] contained in I and let u be an element of I. Show that

In particular, an element of I belongs to $I^n$ ($n \geqslant 1$) if and only if all its partial derivatives belong to $I^{n-1}$.

† For more details on the $D_x$, see R. Fox, Ann of Math., 57 (1953), pp. 547–560.

(b) Let R be a normal subgroup of F, let G = F/R and let J be the kernel of the canonical homomorphism $\gamma : K[F] \to K[G]$. Show that J is generated as a left (resp. right) ideal by the elements $r - 1$, where $r \in R$. Prove the exactness of the sequence

(*)

where $\alpha$ is derived by taking quotients from the inclusion of J in I and $\beta$ is derived by taking quotients from the restriction of $\gamma$ to I.

(c) For $u \in I, x \in X$, let $\overline{D}_x(u)$ denote the image of $D_x(u)$ in $K[G]$ under $\gamma$. Show, using (a), that the family $(\overline{D}_x)_{x \in X}$ defines on taking quotients an isomorphism of $I/(J.I)$ onto $K[G]^{\infty}$.

(d) For $r \in R$, let $\theta(r)$ be the image of $r - 1$ in $J/(J.I)$. Show that

(Use the identities

)

Deduce that $\theta$ defines a homomorphism of $R/(R, R)$ into $J/(J.I)$ compatible with the action of G and show that the image of this homomorphism generates the K-module $J/(J.I)$; when $K = \mathbf{Z}$, show that an isomorphism is thus obtained of $R/(R, R)$ onto $J/(J.I)$ (define the inverse homomorphism directly).

(e) Let $(r_\alpha)_{\alpha \in A}$ be a family of elements of R generating R as a normal subgroup of F. Show that the $\theta(r_\alpha)$ generate the $K[G]$-module $J/(J.I)$.

(f) The matrix $(\overline{D}_x(r_\alpha))_{x \in X, \alpha \in A}$ defines a homomorphism

of left $K[G]$-modules. Show that the sequence

(**)

is exact, where $\delta$ is the homomorphism $(u_x) \mapsto \sum_{x \in X} u_x(\gamma(x) - 1)$.

(Transform the exact sequence (*) using (c), (d), (e) above.)†