The Stacks project

Lemma 62.5.8. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $r \geq 0$ be an integer. Let $g : S' \to S$ be a surjective morphism of schemes. Set $S'' = S' \times _ S S'$ and let $f' : X' \to S'$ and $f'' : X'' \to S''$ be the base changes of $f$. Let $x \in X$ with $\text{trdeg}_{\kappa (f(x))}(\kappa (x)) = r$.

  1. There exists an $x' \in X'$ mapping to $x$ with $\text{trdeg}_{\kappa (f'(x'))}(\kappa (x')) = r$.

  2. If $x'_1, x'_2 \in X'$ are both as in (1), then there exists an $x'' \in X''$ with $\text{trdeg}_{\kappa (f''(x''))}(\kappa (x'')) = r$ and $\text{pr}_ i(x'') = x'_ i$.

Proof. Part (1) is Morphisms, Lemma 29.28.3. Let $x'_1, x'_2$ be as in (2). Then since $X'' = X' \times _ X X'$ we see that there exists a $x'' \in X''$ mapping to both $x'_1$ and $x'_2$ (see for example Descent, Lemma 35.13.1). Denote $s'' \in S''$, $s'_ i \in S'$, and $s \in S$ the images of $x''$, $x'_ i$, and $x$. Denote $k = \kappa (s)$ and let $Z \subset X_ k$ be the integral closed subscheme whose generic point is $x$. Then $x'_ i$ is a generic point of an irreducible component of $Z_{\kappa (s'_ i)}$. Let $Z'' \subset Z_{\kappa (s'')}$ be an irreducible component containing $x''$. Denote $\xi '' \in Z''$ the generic point. Since $\xi '' \leadsto x''$ we see that $\xi ''$ must also map to $x'_ i$ under the two projections. On the other hand, we see that $\text{trdeg}_{\kappa (s'')}(\kappa (\xi '')) = r$ because it is a generic point of an irreducible component of the base change of $Z$. $\square$

Comments (0)

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Your email address will not be published.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 0H4W. Beware of the difference between the letter 'O' and the digit '0'.