Proposition 11

Let $\Delta$ be a commutative monoid, $M$ a graded A-module of type $\Delta$ and $(M_\alpha)_{\alpha \in \Delta}$* its graduation. For every ordered pair $(\alpha, n) \in \Delta \times \mathbf{N}$, let $\bigwedge^{\alpha,n}(M)$* be the submodule of $\bigwedge^n(M)$* the direct sum of the submodules $\bigotimes_{\lambda \in J} \bigwedge^{n_\lambda}(M_{\alpha_\lambda})$, where $(n_\lambda)_{\lambda \in L}$* runs through the set of families of integers $\geq 0$* such that $\sum_{\lambda \in L} n_\lambda = n$, $J$* is its support and, for each $(n_\lambda)$, $(\alpha_\lambda)_{\lambda \in J}$* runs through the set of families of $\Delta^J$* such that $\sum_{\lambda \in J} \alpha_\lambda = \alpha$. Then $(\bigwedge^{\alpha,n}(M))_{(\alpha, n) \in \Delta \times \mathbf{N}}$* is the only graduation of type $\Delta \times \mathbf{N}$* compatible with the algebra structure on $\bigwedge(M)$* and which induces on $M = \bigwedge^1(M)$* the given graduation.

The fact that $\bigwedge(M)$ is the direct sum of the $\bigwedge^{\alpha,n}(M)$ follows from the Corollary to Proposition 10; the rest of the proof is identical with the end of the proof of § 5, no. 5, Proposition 7.