Corollary 3

With the hypotheses on $\Delta$ and $\mathbf{A}$ of Proposition 6, let $u : \mathbf{M} \to \mathbf{N}$ be a graded homomorphism of graded right $\mathbf{A}$-modules, where $\mathbf{N}$ is assumed to be finitely generated. If the homomorphism

is surjective, then $u$ is surjective.

$u(\mathbf{M})$ is a graded submodule of $\mathbf{N}$ and the $(\mathbf{A}/\mathfrak{J}_0)$-module

is isomorphic to $(N \otimes_{\mathbf{A}} (\mathbf{A}/\mathfrak{J}_0))/\mathrm{Im}(u \otimes 1)$ (§ 3, no. 6, Proposition 6). The hypothesis therefore implies $(N/u(\mathbf{M})) \otimes_{\mathbf{A}} (\mathbf{A}/\mathfrak{J}_0) = 0$ and hence $\mathbf{N} = u(\mathbf{M})$ by Corollary 1.