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. $\square$

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