The Stacks project

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

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 $.

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. $\square$


Comments (4)

Comment #8329 by Rankeya on

Lemma 0BNK works when you replace f by a finitely generated ideal I and take M to be an I-power torsion R-module, right? Can it be stated in this generality?

The consequence also continues to hold because of Tag 05GG.

Comment #8333 by on

Indeed, this is true. We could say that the first part works if is killed by a power of and is an isomorphism modulo any power of without assuming is finitely generated. The conclusion about the -adic completion only (as far as I know) works when is finitely generated by the reference you pointed out. I will make these changes the next time I go through all the comments and I will move this lemma into Section 15.88 because it is more suitable.

Comment #8339 by Rankeya on

Thank you, this will be very helpful! I needed the fact that if is the injective hull of the residue field of a noetherian local ring (or for that matter any Artinian -module), then , but could not find a suitable reference for the last isomorphism.


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