# Noetherian assumptions in basic properties of coherent sheaves of modules

Using Hartshorne's definition of 'coherent sheaf':

Proposition 5.11c Let $S$ be a graded ring, $M$ a graded $S$-module, $X=\operatorname{Proj} S$. Then $\tilde M$ is a quasi-coherent $\mathscr O_X$ module. If $S$ is noetherian and $M$ is finitely generated, then $\tilde M$ is coherent.

Why is $S$ required to be noetherian for the second statement to hold?

## 1 Answer

Assume $S$ is a noetherian graded ring and $M$ a f.g. graded $S$-module. Let $X=\operatorname{Proj} S$ and $\tilde M$ be the quasi-coherent $\mathscr O_X$ module associated to $M$. Let $f\in S_+$; then $M_f$ is f.g. over $S_f$ noeth. hence $M_f$ is a noetherian module. Suppose $M_{(f)}$ were not a noetherian $S_{(f)}$-module. Then there would be a strictly ascending chain of $S_{(f)}$-submodules of $M_{(f)}$ which did not terminate:

$$M_1\subsetneqq M_2\subsetneqq\cdots.$$ Then $$S_fM_1\subsetneqq S_fM_2\subsetneqq\cdots$$ would be a strictly ascending chain of $S_f$-submodules of $M_f$ which did not terminate (strictly ascending since $S_fM_i\cap M_{(f)}=S_{(f)}M_i=M_i$).