The Stacks project

Proof. Since $M$ is flat we have $F' \otimes _ R M \subset F \otimes _ R M$ if $F' \subset F$ is a submodule, hence the statement makes sense. Let $I = \{ F' \subset F \mid x \in F' \otimes _ R M\} $ and for $i \in I$ denote $F_ i \subset F$ the corresponding submodule. Then $x$ maps to zero under the map

whence by Proposition 10.89.5 $x$ maps to zero under the map

Since $M$ is flat the kernel of this arrow is $(\bigcap F_ i) \otimes _ R M$ which proves that $F' = \bigcap F_ i$. To see that $F'$ is a finite module, suppose that $x = \sum _{j = 1, \ldots , m} f_ j \otimes m_ j$ with $f_ j \in F'$ and $m_ j \in M$. Then $x \in F'' \otimes _ R M$ where $F'' \subset F'$ is the submodule generated by $f_1, \ldots , f_ m$. Of course then $F'' = F'$ and we conclude the final statement holds. $\square$