The Stacks project

Lemma 37.25.7. Let $f : X \to Y$ be a flat proper morphism of finite presentation. Then the set $\{ y \in Y \mid X_ y\text{ is geometrically reduced}\} $ is open in $Y$.

Proof. We may assume $Y$ is affine. Then $Y$ is a cofiltered limit of affine schemes of finite type over $\mathbf{Z}$. Hence we can assume $X \to Y$ is the base change of $X_0 \to Y_0$ where $Y_0$ is the spectrum of a finite type $\mathbf{Z}$-algebra and $X_0 \to Y_0$ is flat and proper. See Limits, Lemma 32.10.1, 32.8.7, and 32.13.1. Since the formation of the set of points where the fibres are geometrically reduced commutes with base change (Lemma 37.25.2), we may assume the base is Noetherian.

Assume $Y$ is Noetherian. The set is constructible by Lemma 37.25.5. Hence it suffices to show the set is stable under generalization (Topology, Lemma 5.19.10). By Properties, Lemma 28.5.10 we reduce to the case where $Y = \mathop{\mathrm{Spec}}(R)$, $R$ is a discrete valuation ring, and the closed fibre $X_ y$ is geometrically reduced. To show: the generic fibre $X_\eta $ is geometrically reduced.

If not then there exists a finite extension $L$ of the fraction field of $R$ such that $X_ L$ is not reduced, see Varieties, Lemma 33.6.4. There exists a discrete valuation ring $R' \subset L$ with fraction field $L$ dominating $R$, see Algebra, Lemma 10.120.18. After replacing $R$ by $R'$ we reduce to Lemma 37.25.6. $\square$


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Your email address will not be published.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 0C0E. Beware of the difference between the letter 'O' and the digit '0'.