The Stacks project

Lemma 29.14.7. The following properties of a ring map $R \to A$ are local.

  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. (Reduced fibres.) For every prime $\mathfrak p$ of $R$ the fibre ring $A \otimes _ R \kappa (\mathfrak p)$ is reduced.

  4. (Fibres of dimension at most $n$.) For every prime $\mathfrak p$ of $R$ the fibre ring $A \otimes _ R \kappa (\mathfrak p)$ has Krull dimension at most $n$.

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

  6. 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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