Proposition 4

Let S be a subset of X. If $\mathbf{N}^{(S)}$ is identified with its canonical image in $\mathbf{N}^{(X)}$ (Algebra, Chapter I, § 7, no. 7), then $L(S) = \sum_{\alpha \in \mathbf{N}^{(S)}} L^\alpha(X)$. Further, for all $\alpha \in \mathbf{N}^{(S)}$, the homogeneous component of degree $\alpha$ under the multigraduation on $L(S)$ is equal to $L^\alpha(X)$.

Let $\alpha \in \mathbf{N}^{(S)}$. The module $L^\alpha(S)$ is generated by the images in $L(X)$ of the elements $w$ in $M(S)$ such that $\phi(w) = \alpha$, that is (Algebra, § 7, no. 9, formulae (23) and (24)) the set of $w$ in $M(X)$ such that $\phi(w) = \alpha$. Hence $L^\alpha(S) = L^\alpha(X)$.

The proposition follows from this and the relation $L(S) = \sum_{\alpha \in \mathbf{N}^{(S)}} L^\alpha(S)$.