Does the $I$-torsion functor commute with inverse limit?

Question

Let $I$ be an ideal of a commutative ring with unit. Is $\Gamma_I(\varprojlim M_j)\cong \varprojlim(\Gamma_I M_j)$?

Any reference of the proof or a counterexample is appreciated. It seems this should not be true, but i don't have a counterexample.

Answer 1

A very simple counterexample: let $R=\mathbb Z_p$ the $p$-adics, and $m=p\mathbb Z_p$ its maximal ideal. Then $\Gamma_m(\mathbb Z/p^n\mathbb Z)=\mathbb Z/p\mathbb Z$ for all $n\ge 1$, while $\Gamma_m(\mathbb Z_p)=0$.