Proposition 10

The canonical homomorphism g (formula (14)) is a graded algebra isomorphism.

To prove that g is bijective, we define a graded algebra homomorphism

such that $g \circ h$ and $h \circ g$ are respectively the identity mappings on $\Lambda(M)$ and $\bigotimes_{\lambda \in L}^{\mathfrak{g}} \Lambda(M_\lambda)$. For each $\lambda \in L$, consider the composite linear mapping

There exists one and only one A-linear mapping $u : M \to \bigotimes_{\lambda \in L}^{\mathfrak{g}} \Lambda(M_\lambda)$ such that $u \circ j_\lambda = u_\lambda$ for all $\lambda \in L$. The skew tensor product $\bigotimes_{\lambda \in L}^{\mathfrak{g}} \Lambda(M_\lambda)$ is an alternating algebra: for, for L finite, this follows from § 4, no. 9, Proposition 14 and, for L arbitrary, this follows from the definition of this product given in § 4, no. 8, Remark 1 and the fact that a direct limit of alternating graded algebras is alternating. As also $u(M)$ is contained in the submodule of elements of degree 1 in the graded algebra $\bigotimes_{\lambda \in L} \Lambda(M_\lambda)$, it follows from no. 1, Proposition 1 and Remark 1 that there exists a unique graded algebra homomorphism.

such that $h \circ \phi''_M = u$. The verification of the fact that $g \circ h$ and $h \circ g$ are the identity mappings is then performed as in § 6, no. 6, Proposition 9 replacing $S$ by $\Lambda$ and $\phi'$ by $\phi''$.