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. $\square$

 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 $A\otimes_R \kappa(\mathfrak p)$ is meant to be geometrically reduced over $\kappa(\mathfrak p)$?.

Because one can take $R=K$ a field of characteristic $p>0$, $A=L=K(\alpha)$ field extension obtaining from adjoining a $p$ root of an element in $K$ ($\alpha^p=a\in K$) and $R'=L$, so that $L\otimes_K L=L[x]/(x^p-a)=L[x]/(x-\alpha)^p$ is not reduced.

On 4 one can take $n=0$, $R=K$ a field and $R'=A=K(x)$ field of rational functions, so that $A\otimes_K R'=S^{-1}K[x,y]$, with $S$ elements of the form $f(x)g(y)$, which is a domain but not a field (so has Krull dimension $\geq 1$)

Comment #2281 by on

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

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