Preservation of finite presentation along restriction of scalars

Question

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

• $R$ and $S$ are coherent.
• $S$ is finitely presented 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.

Answer 1

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, $M$ is a finitely presented $R$-module.

I don't think coherence is needed.