Let $R$ be a Noetherian ring, $I \subseteq R$ an ideal, and $S \subseteq R$ a multiplicative set.

Lemma 2.3 of Adam, Haeberly, Jackowski, and May's paper A generalisation of the Segal conjecture states that if $M = \{M_\alpha\}$ is a pro-module and $S_P^{-1}M^\wedge_P$ is pro-zero for all primes $P$ disjoint from $S$ and containing and ideal $I$, then $S^{-1}M^\wedge_I$ is pro-zero.

I'd like the following generalisation. If $M = \{M_\alpha\} \to N = \{N_\beta\}$ is a homomorphism of pro-groups such that the induced map $S_P^{-1}M^\wedge_P \to S_P^{-1}N^\wedge_P$ is an isomorphism for all primes $P$ containing $I$, then $S^{-1}M^\wedge_I \to S^{-1}N^\wedge_I$ is an isomorphism.

Is this already stated somewhere in the literature? I'm from an algebraic topology background, so I'm not really sure where I should be looking. It seems like it might be possible to generalise Adam et al's proof to this case, but if the result is already known then that would be better.

finitely generatedmodules. Would you assume the same? $\endgroup$