Lemma 10.55.1. Let $S$ be a graded ring. Let $M$ be a graded $S$-module.

1. If $S_+M = M$ and $M$ is finite, then $M = 0$.

2. If $N, N' \subset M$ are graded submodules, $M = N + S_+N'$, and $N'$ is finite, then $M = N$.

3. If $N \to M$ is a map of graded modules, $N/S_+N \to M/S_+M$ is surjective, and $M$ is finite, then $N \to M$ is surjective.

4. If $x_1, \ldots , x_ n \in M$ are homogeneous and generate $M/S_+M$ and $M$ is finite, then $x_1, \ldots , x_ n$ generate $M$.

Proof. Proof of (1). Choose generators $y_1, \ldots , y_ r$ of $M$ over $S$. We may assume that $x_ i$ is homogeneous of degree $d_ i$. After renumbering we may assume $d_ r = \min (d_ i)$. Then the condition that $S_+M = M$ implies $y_ r = 0$. Hence $M = 0$ by induction on $r$. Part (2) follows by applying (1) to $M/N$. Part (3) follows by applying (2) to the submodules $\mathop{\mathrm{Im}}(N \to M)$ and $M$. Part (4) follows by applying (3) to the module map $\bigoplus S(-d_ i) \to M$, $(s_1, \ldots , s_ n) \mapsto \sum s_ i x_ i$. $\square$

Comment #5356 by Peter Bruin on

In the proof of (1), shouldn't $x_i$ be $y_i$?

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