The Stacks project

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 $M \cong M \otimes_R \widehat{R}^I$ also continues to hold because of Tag 05GG.

Indeed, this is true. We could say that the first part works if $M$ is killed by a power of $I$ and $R \to S$ is an isomorphism modulo any power of $I$ without assuming $I$ is finitely generated. The conclusion about the $I$-adic completion only (as far as I know) works when $I$ 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.

Thank you, this will be very helpful! I needed the fact that if $E$ is the injective hull of the residue field of a noetherian local ring $(R,\mathfrak m)$ (or for that matter any Artinian $R$-module), then $E \otimes_R \widehat{R}^{\mathfrak m} \cong E$, but could not find a suitable reference for the last isomorphism.

This has now been done. See this change.