Noetherian fibrewise flatness for a module finite over the target

Set $I=\mathfrak{m}S⊊S$. For any $S$-module $M$, there is a natural surjection $\mathfrak{m}{\otimes }_{R}M⟶I{\otimes }_{S}M\mathrm{.}$ Indeed $I$ is generated as an $S$-module by images of elements of $\mathfrak{m}$; a pure tensor $({\sum }_a{r}_a{s}_a)\otimes m$ with $r_a\in \mathfrak{m}$, $s_a\in S$ is the image of ${\sum }_a{r}_a\otimes s_{a}m$. This works whether or not $M$ is finite over either smaller ring.

The composite $\mathfrak{m}{\otimes }_{R}M⟶I{\otimes }_{S}M⟶M$ is the multiplication map of [F1], hence injective. Since the first arrow is surjective by step 1.1, the second arrow $I{\otimes }_{S}M\to M$ is injective. The quotient $M\mathrm{/}IM$ is exactly $M\mathrm{/}\mathfrak{m}M$, flat over $S\mathrm{/}I=S\mathrm{/}\mathfrak{m}S$ by hypothesis. Apply [F2] to $S\to S^{\mathrm{\prime }}$ and $I$ to conclude that $M$ is flat over $S$.

If $M=0$, both hypotheses and the conclusion hold, and the same argument applies. The Axiom of Choice is inherited from [F2]; no additional infinite selection is used.