Proposition 9

The canonical homomorphism $g$ (formula (10)) is a graded algebra isomorphism (cf. § 4, no. 8, Remark 1).

To prove that $g$ is bijective, we define an algebra homomorphism

such that $g \circ h$ and $h \circ g$ are respectively the identity mappings on $S(M)$ and $\bigotimes_{\lambda \in L} S(M_\lambda)$. For each $\lambda \in L$, let $u_\lambda$ be the composite linear mapping

There exists one and only one $A$-linear mapping $u : M \to \bigotimes_{\lambda \in L} S(M_\lambda)$ such that $u \circ j_\lambda = u_\lambda$ for all $\lambda \in L$. As the $S(M_\lambda)$ are commutative, so is their tensor product (§ 4, no. 5) and hence (no. 1, Proposition 2) there exists a unique algebra homomorphism $h : S(M) \to \bigotimes_{\lambda \in L} S(M_\lambda)$ such that $h \circ \phi'_M = u$; on the other hand, it is immediate that $u(M)$ is contained in the submodule of elements of degree 1 of the graded algebra $\bigotimes_{\lambda \in L} S(M_\lambda)$ and hence $h$ is a graded algebra homomorphism. For $x_\lambda \in M_\lambda$, as the $u_\lambda(x_\lambda)$ generate the algebra $\bigotimes_{\lambda \in L} S(M_\lambda)$ (§ 4, no. 5, Proposition 8), $h \circ g$ is certainly the identity mapping. Similarly, and, as the elements $\phi'_M(j_\lambda(x_\lambda))$ generate the algebra $S(M)$, $g \circ h$ is certainly the identity mapping.

Remark (1) Let $N = \bigoplus_{\lambda \in L} N_\lambda$ be the direct sum of another family of $A$-modules with $L$ as indexing set and, for all $\lambda \in L$, let $v_\lambda : M_\lambda \to N_\lambda$ be an $A$-linear mapping, whence there is an $A$-linear mapping $v = \bigoplus_\lambda v_\lambda : M \to N$ (II, § 1, no. 6, Proposition 6). Then the diagram

is commutative, as follows from the definitions (§ 4, no. 5, Corollary to Proposition 8).

The sub-$A$-module of $\bigotimes_{\lambda \in L} S(M_\lambda)$ with which $S^n(M)$ is identified by means of the isomorphism $g$ can be described more precisely. For every finite subset $J$ of $L$, we write $E_J = \bigotimes_{\lambda \in J} S(M_\lambda)$, so that $\bigotimes_{\lambda \in L} S(M_\lambda) = \varprojlim E_J$ relative to the directed set $\mathscr{F}(L)$ of finite subsets of $L$, by definition (§ 4, no. 5). For every family $\nu = (n_\lambda) \in \mathbf{N}^{(L)}$ (thus having finite support) such that $\sum_{\lambda \in L} n_\lambda = n$ and every finite subset $J$ of $L$ containing the support of the family $\nu$, we write

so that the submodule $E_{J,n}$ of elements of degree $n$ in $E_J$ is the direct sum of the $S^{J,\nu}(M)$ over all the families $\nu$ of support contained in $J$ and such that $\sum_{\lambda \in L} n_\lambda = n$ (§ 4, no. 7, Proposition 10 and § 4, no. 8). As a convention we write $S^{J,\nu}(M) = \{0\}$ for the families $\nu$ whose support is not contained in $J$; then $E_{J,n}$ can also be called the direct sum of all the $S^{J,\nu}(M)$, where $\nu$ runs through the set $H_n$ of all families $\nu = (n_\lambda)_{\lambda \in L}$ such that $\sum_{\lambda \in L} n_\lambda = n$. Since $S^0(M_\lambda)$ is identified with $A$, clearly also, for two finite subsets $J \subset J'$ of $L$ and a family $\nu$ of support contained in $J$, the canonical mapping $S^{J,\nu}(M) \to S^{J',\nu}(M)$ (restriction of the canonical mapping $E_J \to E_J$ to $S^{J,\nu}(M)$) is bijective. If we write, for all $\nu \in H_n$,

it is seen that, taking account of II, § 6, no. 2, Proposition 5: