Lemma 33.7.14 (04KV)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.7: Geometrically connected schemes
Lemma 33.7.14 (cite)

numberstags

Lemma 33.7.14. Let $k$ be a field. Let $X$ be a scheme over $k$ . Assume $X$ is connected and has a point $x$ such that $k$ is algebraically closed in $\kappa (x)$ . Then $X$ is geometrically connected. In particular, if $X$ has a $k$ -rational point and $X$ is connected, then $X$ is geometrically connected.

Proof. Set $T = \mathop{\mathrm{Spec}}(\kappa (x))$ . Let $\overline{k}$ be a separable algebraic closure of $k$ . The assumption on $\kappa (x)/k$ implies that $T_{\overline{k}}$ is irreducible, see Algebra, Lemma 10.47.8. Hence by Lemma 33.7.13 we see that $X_{\overline{k}}$ is connected. By Lemma 33.7.7 we conclude that $X$ is geometrically connected. $\square$

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