Lemma 10.58.6. If $M$ is a finitely generated graded $S$-module, and if $S$ is finitely generated over $S_0$, then each $M_ n$ is a finite $S_0$-module.

Proof. Suppose the generators of $M$ are $m_ i$ and the generators of $S$ are $f_ i$. By taking homogeneous components we may assume that the $m_ i$ and the $f_ i$ are homogeneous and we may assume $f_ i \in S_{+}$. In this case it is clear that each $M_ n$ is generated over $S_0$ by the “monomials” $\prod f_ i^{e_ i} m_ j$ whose degree is $n$. $\square$

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