Lemma 29.14.9. The following properties of ring maps are stable under composition.

1. (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$.

2. (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi (f) \not\in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_ f \to A_ f$.

3. (Locally Noetherian on the target.) The ring map $R \to A$ has the property that $A$ is Noetherian.

4. Add more here as needed1.

Proof. Omitted. $\square$

[1] But only those properties that are not already dealt with separately elsewhere.

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