Lemma 10.58.2. A graded ring $S$ is Noetherian if and only if $S_0$ is Noetherian and $S_{+}$ is finitely generated as an ideal of $S$.

Proof. It is clear that if $S$ is Noetherian then $S_0 = S/S_{+}$ is Noetherian and $S_{+}$ is finitely generated. Conversely, assume $S_0$ is Noetherian and $S_{+}$ finitely generated as an ideal of $S$. Pick generators $S_{+} = (f_1, \ldots , f_ n)$. By decomposing the $f_ i$ into homogeneous pieces we may assume each $f_ i$ is homogeneous. By Lemma 10.58.1 we see that $S_0[X_1, \ldots X_ n] \to S$ sending $X_ i$ to $f_ i$ is surjective. Thus $S$ is Noetherian by Lemma 10.31.1. $\square$

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