Lemma 15.88.8. Let R \to R' be a ring map. Let I \subset R be an ideal such that R/I^ n \to R'/I^ nR' is an isomorphism for n > 0. For any I-power torsion R-module M the map M \to M \otimes _ R R' is an isomorphism. For example, if I is finitely generated and R^\wedge is the completion of R with respect to I, then we have M \cong M \otimes _ R R^\wedge .
Slight generalization of [Lemme 1, Beauville-Laszlo].
Proof. If M is annihilated by I^ n, then
M \otimes _ R R' \cong M \otimes _{R/I^ n} R'/I^ n R' \cong M \otimes _{R/I^ n} R/I^ n \cong M.
If M is I-power torsion, then M = \bigcup M[I^ n]. 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 Algebra, Lemma 10.96.3. \square
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