The Stacks project

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

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

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

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

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

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

\[ M \otimes _ R S = M \otimes _{R/I} S/IS \]

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). $\square$

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