Let $\theta \colon R \to S$ be a morphism of commutative rings (with unit). We assume that:


* $R$ and $S$ are [coherent](https://en.wikipedia.org/wiki/Coherent_ring).
* $S$ is [finitely presented](https://en.wikipedia.org/wiki/Finitely_generated_module#Finitely_presented_module) as an $R$-module.


Now, let $M$ be an $S$-module, and assume that it is finitely presented (over $S$). How can I prove (as elementarily as possible) that its restriction along $\theta \colon R \to S$ is finitely presented as an $R$-module? The "slogan" would be: *finite presentation is preserved by restriction of scalars*.



Preservation of finite presentation along restriction of scalars
There is an exact sequence of $S$-modules,
$$ 0 \to K \to S^n \to M \to 0 $$
where $K$ is a finite $S$-module. Restricting to the category of $R$-modules, $S^n$ is finitely presented and $K$ is finite (because $S$ is a finitely presented $R$-module). Therefore, by [Tag 0519](https://stacks.math.columbia.edu/tag/0519), $M$ is a finitely presented $R$-module.


I don't think coherence is needed.


