Lemma 10.18.4 (07BJ)—The Stacks project

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Lemma 10.18.4. Let $\varphi : R \to S$ be a ring map. Assume $R$ and $S$ are local rings. The following are equivalent:

$\varphi$ is a local ring map,
$\varphi (\mathfrak m_ R) \subset \mathfrak m_ S$ , and
$\varphi ^{-1}(\mathfrak m_ S) = \mathfrak m_ R$ .
For any $x \in R$ , if $\varphi (x)$ is invertible in $S$ , then $x$ is invertible in $R$ .

Proof. Conditions (1) and (2) are equivalent by definition. If (3) holds then (2) holds. Conversely, if (2) holds, then $\varphi ^{-1}(\mathfrak m_ S)$ is a prime ideal containing the maximal ideal $\mathfrak m_ R$ , hence $\varphi ^{-1}(\mathfrak m_ S) = \mathfrak m_ R$ . Finally, (4) is the contrapositive of (2) by Lemma 10.18.3. $\square$

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