Lemma 2

Let V be a vector space, E a cogebra and $f : S(V) \to E$ a cogebra morphism. If the restriction of f to $S^0(V) + S^1(V)$ is injective, then f is injective. Let $n \geq 0$; we write $S_n = \sum_{i \geq n} S^i(V)$ and $c_S$ for the coproduct of $S(V)$ and show by induction on n that $f | S_n$ is injective. Since the assertion is trivial for $n = 0$ and $n = 1$, we assume that $n \geq 2$ and let $u \in S_n$ be such that $f(u) = 0$. Then

As $c_S^+(u) \in S_{n-1} \otimes S_{n-1}$, by (11) the induction hypothesis shows that u is a primitive element of $S(V)$, hence is of degree 1 (no. 5, Remark 4) and hence is zero, since $f | S^1(V)$ is injective. It follows in particular that the family $(X_\alpha)$ is free. We show that $f_E$ is surjective if E has a filtration compatible with its bigebra structure. Let $(E_n)_{n \geq 0}$ be such a filtration and write $E_n^+ = E_n \cap \mathrm{Ker}(\varepsilon_E)$. We show by induction on n that $E_n^+$ is contained in the image of $f_E$. As $E = K.1 + \bigcup_{n \geq 0} E_n^+$, this will imply the surjectivity of $f_E$. The assertion is trivial for $n = 0$ and follows from the Corollary to Proposition 6 of no. 3 for $n = 1$; suppose henceforth that $n \geq 2$ and let $x \in E_n^+$. By Proposition 6 of no. 3,

and by the induction hypothesis there exist scalars $\lambda_{\alpha, \beta}$, where $\alpha, \beta$ are in $\mathbf{N}^{(\Omega)}$, which are zero except for a finite number, such that

(19)

Hence by formula (18)

By Proposition 3 of no. 1 and the linear independence of the $X_\alpha$ it follows that

(20)

On the other hand, the coproduct $c_E$ is cocommutative; the same argument as above implies

(21)

Suppose that there exists a family of scalars $(\mu_\alpha)$ with $|\alpha| \geq 2$, such that

(22)

Then

by formula (18), hence $y = x - \sum_{|\gamma| \geq 2} \mu_\gamma X_\gamma$ is primitive and hence belongs to $P(E) \subset \mathrm{Im}(f_E)$. Hence

The proof will therefore be complete when we have proved the following lemma: