Corollary 2

With the hypotheses on $\Delta$ and A of Proposition 6, let M be a finitely generated graded $\mathbf{A}$-module and $\mathbf{N}$ a graded submodule of $\mathbf{M}$ such that $\mathbf{N} + \mathfrak{J}_0 \mathbf{M} = \mathbf{M}$; then $\mathbf{N} = \mathbf{M}$.

$\mathbf{M}/\mathbf{N}$ is a finitely generated graded $\mathbf{A}$-module and the hypothesis implies that $\mathfrak{J}_0 . (\mathbf{M}/\mathbf{N}) = \mathbf{M}/\mathbf{N}$; hence $\mathbf{M}/\mathbf{N} = 0$.