Lemma 1

Let $A$ be a graded commutative ring such that $A = A_0[A_1]$, $M$ a graded $A$-module and $(y_\lambda)_{\lambda \in L}$ a system of homogeneous generators of $M$ such that $\deg(y_\lambda) \leqslant n_0$ for all $\lambda \in L$. Then, for all $n \geqslant n_0$ and all $k \geqslant 0$, $M_{n+k} = A_k \cdot M_n$.

Let $n \geqslant n_0$ and $x \in M_n$. Since the $y_\lambda$ generate $M$, there exists a family $(a_\lambda)_{\lambda \in L}$ of elements of $A$ of finite support such that $x = \sum_\lambda a_\lambda y_\lambda$; we may further suppose that each $a_\lambda$ is homogeneous and of degree $n + 1 - \deg(y_\lambda)$ (replacing it if need be by its homogeneous component of that degree). As

$A = A_0[A_1]$ and $\deg(a_\lambda) > 0$, each $a_\lambda$ is a sum of elements of the form $bb'$ where $b \in A_0, b' \in A$, whence $x \in A_1 M_n$. Then $M_{n+1} = A_1 M_n$, whence $M_{n+k} = A_k M_n$ by induction on $k$.