Lemma 15.22.4. Let $R \to R'$ be a flat homomorphism of domains. If $M$ is a torsion free $R$-module, then $M \otimes _ R R'$ is a torsion free $R'$-module.
Proof. If $M$ is torsion free, then $M \subset M \otimes _ R K$ is injective where $K$ is the fraction field of $R$. Since $R'$ is flat over $R$ we see that $M \otimes _ R R' \to (M \otimes _ R K) \otimes _ R R'$ is injective. Since $M \otimes _ R K$ is isomorphic to a direct sum of copies of $K$, it suffices to see that $K \otimes _ R R'$ is torsion free. This is true because it is a localization of $R'$. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like
$\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.