Lemma 15.89.1. Let $\varphi : R \to S$ be a ring map. Let $I \subset R$ be an ideal. The following are equivalent

$\varphi $ is flat and $R/I \to S/IS$ is faithfully flat,

$\varphi $ is flat, and the map $\mathop{\mathrm{Spec}}(S/IS) \to \mathop{\mathrm{Spec}}(R/I)$ is surjective.

$\varphi $ is flat, and the base change functor $M \mapsto M \otimes _ R S$ is faithful on modules annihilated by $I$, and

$\varphi $ is flat, and the base change functor $M \mapsto M \otimes _ R S$ is faithful on $I$-power torsion modules.

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