The Stacks project

Lemma 37.28.2. Let $f : X \to Y$ be a morphism of schemes. Let $g : Y' \to Y$ be any morphism, and denote $f' : X' \to Y'$ the base change of $f$. Then

\begin{align*} \{ y' \in Y' \mid X'_{y'}\text{ is geometrically connected}\} \\ = g^{-1}(\{ y \in Y \mid X_ y\text{ is geometrically connected}\} ). \end{align*}

Proof. This comes down to the statement that for $y' \in Y'$ with image $y \in Y$ the fibre $X'_{y'} = X_ y \times _ y y'$ is geometrically connected over $\kappa (y')$ if and only if $X_ y$ is geometrically connected over $\kappa (y)$. This follows from Varieties, Lemma 33.7.3. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 37.28: Connected components of fibres

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 055E. Beware of the difference between the letter 'O' and the digit '0'.