Corollary 1

With the hypotheses on $\Delta$ and A of Proposition 6, if M is a finitely generated graded A-module such that $\mathfrak{I}_0 M = M$, then $M = \{0\}$.

Suppose $M \neq \{0\}$. Let $\lambda_0$ be a minimal element of the set of degrees of a finite generating system of M consisting of homogeneous elements $\neq 0$; then the hypotheses of Proposition 6 would be fulfilled, which implies a contradiction.