Corollary

Let $A = \bigoplus_{i \in \mathbf{Z}} A_i$, be a graded commutative ring with positive degrees and $m$ the graded ideal, $\bigoplus_{i \geq 1} A_i$.

(i) The following conditions are equivalent: (a) The ideal $m$ is a finitely generated A-module. (b) The ring $A$ is a finitely generated A,-module.

(ii) Suppose that the conditions in (i) hold and let $M = \bigoplus_{i \in \mathbf{Z}} M_i$ be a finitely generated graded A-module. Then, for all $i \in \mathbf{Z}$, $M_i$ is a finitely A,-module and there exists $i_0$ such that $M_i = \{0\}$ for $i < i_0$.

(i) If a family $(y_\mu)$ of elements of A is a system of generators of the A-module m (resp. of the A,-module A), so is the family consisting of the homogeneous components of the $y_\mu$; the equivalence of conditions (a) and (b) then follows from Proposition 1.

(ii) We may suppose that A is generated (as an A,-algebra) by homogeneous elements $a_i$ ($1 < i < r$) of degree $> 1$ and M is generated (as an A-module) by homogeneous elements $x_j$ ($1 < j < s$); let $h_i = \deg(a_i), k_j = \deg(x_j)$. Clearly M, consists of the linear combinations with coefficients in $A_0$ of the elements $a_1^{\alpha_1} a_2^{\alpha_2} \ldots a_r^{\alpha_r} x_j$ such that the $\alpha_i$ are integers $\geq 0$ satisfying the relation $k_j + \sum_{i=1}^r \alpha_i h_i = n$; for each $n$ there is only a finite number of families $(\alpha_i)_{1 \leq i \leq r}$ satisfying these conditions, since $h_i \geq 1$ for all $i$; we conclude that $M_n$ is a finitely generated A,-module and moreover clearly $M_n = \{0\}$ when $n < \inf(k_j)$.