The Stacks project

Slight generalization of [Lemme 1, Beauville-Laszlo].

Lemma 15.90.2. Let $R$ be a ring, let $f \in R$ be an element, and let $R \to R'$ be a ring map which induces isomorphisms $R/f^ nR \to R'/f^ nR'$ for $n > 0$. For any $f$-power torsion $R$-module $M$ the map $M \to M \otimes _ R R'$ is an isomorphism. For example, we have $M \cong M \otimes _ R R^\wedge $.

Proof. If $M$ is annihilated by $f^ n$, then

\[ M \otimes _ R R' \cong M \otimes _{R/f^ nR} R'/f^ n R' \cong M \otimes _{R/f^ nR} R/f^ n R \cong M. \]

Since $M = \bigcup M[f^ n]$ and since tensor products commute with direct limits (Algebra, Lemma 10.12.9), we obtain the desired isomorphism. The last statement is a special case of the first statement by Lemma 15.90.1. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 15.90: The Beauville-Laszlo theorem

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Your email address will not be published.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 0BNK. Beware of the difference between the letter 'O' and the digit '0'.