The Stacks project

Proof. If $R \to S$ is flat, then $R/I^ n \to S/I^ nS$ is flat for every $n$, see Algebra, Lemma 10.39.7. Hence (1) and (2) are equivalent by Algebra, Lemma 10.39.16. The equivalence of (1) with (3) follows by identifying $I$-torsion $R$-modules with $R/I$-modules, using that

for $R$-modules $M$ annihilated by $I$, and Algebra, Lemma 10.39.14. The implication (4) $\Rightarrow $ (3) is immediate. Assume (3). We have seen above that $R/I^ n \to S/I^ nS$ is flat, and by assumption it induces a surjection on spectra, as $\mathop{\mathrm{Spec}}(R/I^ n) = \mathop{\mathrm{Spec}}(R/I)$ and similarly for $S$. Hence the base change functor is faithful on modules annihilated by $I^ n$. Since any $I$-power torsion module $M$ is the union $M = \bigcup M_ n$ where $M_ n$ is annihilated by $I^ n$ we see that the base change functor is faithful on the category of all $I$-power torsion modules (as tensor product commutes with colimits). $\square$