Lemma 10.56.1. Let $S$ be a graded ring. Let $M$ be a graded $S$-module.

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

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

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.

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 $M$.

## Comments (2)

Comment #5356 by Peter Bruin on

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