Lemma 29.14.8 (01SY)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.14: Types of morphisms defined by properties of ring maps
Lemma 29.14.8 (cite)

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

(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 $R_{\mathfrak p} \to A_{\mathfrak q}$.
(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 $R_ f \to A_ f$.
Add more here as needed ¹.

Proof. Omitted. $\square$

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

Comments (2)

Comment #2246 by JuanPablo on September 30, 2016 at 01:25

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 Johan on November 03, 2016 at 18:30

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

Comments (2)

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