The Stacks project

Lemma 15.21.5. Let $R \to S$ be a finite injective homomorphism of Noetherian rings. Let $M$ be an $R$-module. If $M \otimes _ R S$ is a flat $S$-module, then $M$ is a flat $R$-module.

Proof. Let $M$ be an $R$-module such that $M \otimes _ R S$ is flat over $S$. By Algebra, Lemma 10.39.8 in order to prove that $M$ is flat we may replace $R$ by any faithfully flat ring extension. By Lemma 15.21.3 we can find a finite locally free ring extension $R \subset R'$ such that $S' = S \otimes _ R R' = R'[T_1, \ldots , T_ n]/J$ for some ideal $J \subset R'[T_1, \ldots , T_ n]$ which contains the elements of the form $P_ i(T_ i)$ with $P_ i(T) = \prod _{j = 1, \ldots , d_ i} (T - \alpha _{ij})$ for some $\alpha _{ij} \in R'$. Note that $R'$ is Noetherian and that $R' \subset S'$ is a finite extension of rings. Hence we may replace $R$ by $R'$ and assume that $S$ has a presentation as in Lemma 15.21.4. Note that $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is surjective, see Algebra, Lemma 10.36.17. Thus, using Lemma 15.21.4 we conclude that $I = \bigcap J_{\underline{k}}$ is an ideal such that $V(I) = \mathop{\mathrm{Spec}}(R)$. This means that $I \subset \sqrt{(0)}$, and since $R$ is Noetherian that $I$ is nilpotent. The maps $\Phi _{\underline{k}}$ induce commutative diagrams

\[ \xymatrix{ S \ar[rr] & & R/J_{\underline{k}} \\ & R \ar[lu] \ar[ru] } \]

from which we conclude that $M/J_{\underline{k}}M$ is flat over $R/J_{\underline{k}}$. By Lemma 15.16.1 we see that $M/IM$ is flat over $R/I$. Finally, applying Algebra, Lemma 10.101.5 we conclude that $M$ is flat over $R$. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 15.21: Descent of flatness along integral maps

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0533. Beware of the difference between the letter 'O' and the digit '0'.