The Stacks project

[Lemma 2.1.2, EHIK]

Lemma 37.76.4. 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. $\square$

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