Exercise 11, § 5

(a) Let $A$ be a filtered algebra satisfying the conditions of § 4, no. 5 and let $\Gamma = A^* \cap (1 + A_0^+)$. $\Gamma$ is given the filtration induced by that on $A$ (loc. cit., Proposition 2). Show that, if $K$ is a field of characteristic $p > 0$, $(\Gamma_\alpha)$ is a restricted filtration (Exercise 10) and the embedding of $\mathrm{gr}(\Gamma)$ in $\mathrm{gr}(A)$ defined in loc. cit., Proposition 3 is compatible with the Lie $p$*-algebra structures on $\mathrm{gr}(\Gamma)$ and $\mathrm{gr}(A)$.

(b) Suppose that $K = \mathbf{F}_p$. We apply the above to the filtered algebra $\hat{A}(K)$, in which $F = F(X)$ is embedded by means of the homomorphism $g$. The filtration $(\Gamma_n)$ of $\Gamma$ induces a filtration $(F_n)$ on $F$, which is a restricted filtration. The Lie $p$*-algebra $\mathrm{gr}(F)$ is identified with a Lie $p$*-subalgebra of $A(X)$ containing $X$. Deduce that $\mathrm{gr}(F)$ contains the free Lie $p$*-algebra $L(X, p)$, cf. § 3, Exercise 4 (e).

(c) Let $H$ be a Hall set relative to $X$; for every integer $i$, let $H_i$ be the set of elements of $H$ of length $i$. Let $w \in F$ and let $\alpha_i \in \mathbf{Z}^{(H_i)}$ be such that

for all $n$, cf. no. 4. For all $m \in H$, let $l(m)$ denote the length of $m$ and $q(m, w)$ the largest power of $p$ which divides $\alpha_i(m)$, where $i = l(w)$ (if $\alpha_i(m) = 0$, we make the convention $q(m, w) = +\infty$). Let $N = \inf_{m \in H} (l(m) \cdot q(m, w))$. Show that the $\phi(m)^{\alpha_i(m)}$ belong to $F_N$ and even to $F_{N+1}$ if $l(w) \cdot q(m, w) > N$. Show that, if $l(w) \cdot q(m, w) = N$, the image of $\phi(m)^{\alpha_i(m)}$ in $\mathrm{gr}_N(\hat{A}(X)) = A^N(X)$ is a non-zero multiple of $\overline{m}^{q(m, w)}$, where $\overline{m}$ is the element of $L(X)$ defined by $m$ (§ 2, no. 11). Now these elements are linearly independent (§ 3, Exercise 4); deduce that the image of $w$ in $\mathrm{gr}_N(F)$ is $\neq 0$ and belongs to $L(X, p)$, whence the fact that $\mathrm{gr}(F) = L(X, p)$.