Lemma 72.11.6 (0E04)—The Stacks project

Processing math: 100%

Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.11: Geometrically reduced algebraic spaces
Lemma 72.11.6 (cite)

numberstags

Lemma 72.11.6. Let $k$ be a field. Let $X$ be an algebraic space over $k$ . Let $k'/k$ be a field extension. Let $x \in |X|$ be a point and let $x' \in |X_{k'}|$ be a point lying over $x$ . The following are equivalent

$X$ is geometrically reduced at $x$ ,
$X_{k'}$ is geometrically reduced at $x'$ .

In particular, $X$ is geometrically reduced over $k$ if and only if $X_{k'}$ is geometrically reduced over $k'$ .

Proof. Choose an étale morphism $U \to X$ where $U$ is a scheme and a point $u \in U$ mapping to $x \in |X|$ . By Properties of Spaces, Lemma 66.4.3 we may choose a point $u' \in U_{k'} = U \times _ X X_{k'}$ mapping to both $u$ and $x'$ . By Lemma 72.11.2 the lemma follows from the lemma for $U, u, u'$ which is Varieties, Lemma 33.6.6. $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment