The Stacks project

Lemma 34.10.14. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be a V covering. Then

\[ \coprod \nolimits _{i \in I} f_ i : \coprod \nolimits _{i \in I} X_ i \longrightarrow X \]

is a universally submersive morphism of schemes (Morphisms, Definition 29.24.1).

Proof. We will use without further mention that the base change of a V covering is a V covering (Lemma 34.10.9). In particular it suffices to show that the morphism is submersive. Being submersive is clearly Zariski local on the base. Thus we may assume $X$ is affine. Then $\{ X_ i \to X\} $ can be refined by a standard V covering $\{ Y_ j \to X\} $. If we can show that $\coprod Y_ j \to X$ is submersive, then since there is a factorization $\coprod Y_ j \to \coprod X_ i \to X$ we conclude that $\coprod X_ i \to X$ is submersive. Set $Y = \coprod Y_ j$ and consider the morphism of affines $f : Y \to X$. By Lemma 34.10.13 we know that we can lift any specialization $x' \leadsto x$ in $X$ to some specialization $y' \leadsto y$ in $Y$. Thus if $T \subset X$ is a subset such that $f^{-1}(T)$ is closed in $Y$, then $T \subset X$ is closed under specialization. Since $f^{-1}(T) \subset Y$ with the reduced induced closed subscheme structure is an affine scheme, we conclude that $T \subset X$ is closed by Algebra, Lemma 10.41.5. Hence $f$ is submersive. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 34.10: The V topology

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Your email address will not be published.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