Remark

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 7). Then the diagram

is commutative (cf. § 4, no. 5, Corollary to Proposition 8).

The sub-$A$-module of $\bigotimes_{\lambda \in L} \Lambda(M_\lambda)$ with which $\Lambda^n(M)$ is identified by means of the isomorphism $g$ can be described more precisely. For every finite subset $J$ of $L$, we write $E_J = \bigotimes_{\lambda \in J} \Lambda(M_\lambda)$, so that $\bigotimes_{\lambda \in L} \Lambda(M_\lambda) = \lim \to E_J$ relative to the directed set $\mathscr{F}(L)$ of finite subsets of $L$, by definition ($§ 4$, no. 8, Remark 1). For every family $\nu = (n_\lambda) \in \mathbf{N}^{(L)}$ (therefore with 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 $\Lambda^{J,\nu}(M)$ for all families $\nu$ of support contained in $J$ and such that

($§ 4$, no. 7, Proposition 10 and no. 8). By way of convention we write $\Lambda^{J,\nu}(M) = \{0\}$ 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 $\bigwedge^{J,\nu}(M)$, 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 $\bigwedge^0(M_\lambda)$ 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 $\bigwedge^{J,\nu}(M) \to \bigwedge^{J',\nu}(M)$ (restriction of the canonical mapping $E_J \to E_{J'}$ to $\bigwedge^{J,\nu}(M)$) is bijective. If we write, for all $\nu \in H_n$,

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