The Stacks project

[Lemma 2.1.2, EHIK]

Lemma 37.75.4. Let $f : X \to Y$ be a morphism of schemes. Assume $f$ is completely decomposed, $f$ is locally of finite presentation, and $Y$ is quasi-compact and quasi-separated. Then there exist $n \geq 0$ and morphisms $Z_ i \to Y$, $i = 1, \ldots , n$ with the following properties

  1. $\coprod Z_ i \to Y$ is surjective,

  2. $Z_ i \to Y$ is an immersion for all $i$,

  3. $Z_ i \to Y$ is of finite presentation for all $i$, and

  4. the base change $X \times _ Y Z_ i \to Z_ i$ has a section for all $i$.

Proof. Let $y \in Y$. By assumption there is a morphism $\sigma : \mathop{\mathrm{Spec}}(\kappa (y)) \to X$ over $Y$. We can write $\mathop{\mathrm{Spec}}(\kappa (y))$ as a directed limit of affine schemes $Z$ over $Y$ such that $Z \to Y$ is an immersion of finite presentation. Namely, choose an affine open $y \in \mathop{\mathrm{Spec}}(A) \subset Y$ and say $y$ corresponds to the prime ideal $\mathfrak p$ of $A$. Then $\kappa (\mathfrak p)$ is the filtered colimit of the rings $(A/I)_ f$ where $I \subset \mathfrak p$ is a finitely generated ideal and $f \in A$, $f \not\in \mathfrak p$. The morphisms $Z = \mathop{\mathrm{Spec}}((A/I)_ f) \to Y$ are immersions of finite presentation; quasi-compactness of $Z \to Y$ follows as $Y$ is quasi-separated, see Schemes, Lemma 26.21.14. By Limits, Proposition 32.6.1 for some such $Z$ there is a morphism $\sigma ' : Z \to X$ over $Y$ agreeing with $\sigma $ on the spectrum of $\kappa (\mathfrak p)$. Since $\sigma '$ is a morphism over $Y$, we obtain a section of the projection $X \times _ Y Z \to Z$

We conclude that $Y$ is the union of the images of immersions $Z \to Y$ of finite presentation such that $X \times _ Y Z \to Z$ has a section. Since the image of $Z \to Y$ is constructible (Morphisms, Lemma 29.22.2) and since $Y$ is compact in the constructible topology (Properties, Lemma 28.2.4 and Topology, Lemma 5.23.2), we see that a finite number of these suffice. $\square$


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Your email address will not be published.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