Lemma 5

Let A be a ring, r its radical, $R = \bigoplus_{i \in \mathbf{Z}} R_i$ a graded A-algebra, $M = \bigoplus_{i \in \mathbf{Z}} M_i$ a graded R-module. Suppose that each $M_i$ is a finitely generated A-module and that $M/rM$ is a finitely generated $R/rR$-module. Then M is a finitely generated R-module.

Let $m_1, ..., m_n$ be homogeneous elements of M whose images in $M/rM$ generate the $R/rR$-module $M/rM$. Let N be the (graded) sub-$R$-module of M generated by $\{ m_1, ..., m_n \}$. For every $i \in \mathbf{Z}$, we have $M_i = N_i + rM_i$, hence $M_i = N_i$ (II, § 3, No. 2, Prop. 4); consequently we have $M = N$.