The Stacks project

Lemma 29.14.8. The following properties of ring maps are stable under base change.

  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. Add more here as needed1.

Proof. Omitted. $\square$

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

Comments (2)

Comment #2246 by JuanPablo on

Something seems to be wrong with 3 and 4.

On 3 maybe is meant to be geometrically reduced over ?.

Because one can take a field of characteristic , field extension obtaining from adjoining a root of an element in () and , so that is not reduced.

On 4 one can take , a field and field of rational functions, so that , with elements of the form , which is a domain but not a field (so has Krull dimension )

Comment #2281 by on

OMG!!! Yes, terrible! Thanks very much. The change is here.

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