The Stacks project

Lemma 38.34.3. Let $X$ be a Noetherian scheme. Let $\{ X_ i \to X\} _{i \in I}$ be a finite family of finite type morphisms. The following are equivalent

  1. $\coprod _{i \in I} X_ i \to X$ is universally submersive (Morphisms, Definition 29.24.1), and

  2. $\{ X_ i \to X\} _{i \in I}$ is an h covering.

Proof. The implication (2) $\Rightarrow $ (1) follows from the more general Topologies, Lemma 34.10.14 and our definition of h covers. Assume $\coprod X_ i \to X$ is universally submersive. We will show that $\{ X_ i \to X\} $ can be refined by a ph covering; this will suffice by Topologies, Lemma 34.8.7 and our definition of h coverings. The argument will be the same as the one used in the proof of Lemma 38.34.1.

Choose a generic flatness stratification

\[ X = S \supset S_0 \supset S_1 \supset \ldots \supset S_ t = \emptyset \]

as in Lemma 38.21.4 for the finitely presented morphism

\[ \coprod \nolimits _{i = 1, \ldots , n} f_ i : \coprod \nolimits _{i = 1, \ldots , n} X_ i \longrightarrow X \]

We are going to use all the properties of the stratification without further mention. By construction the base change of each $f_ i$ to $U_ k = S_ k \setminus S_{k + 1}$ is flat. Denote $Y_ k$ the scheme theoretic closure of $U_ k$ in $S_ k$. Since $U_ k \to S_ k$ is a quasi-compact open immersion (all schemes in this paragraph are Noetherian), we see that $U_ k \subset Y_ k$ is a quasi-compact dense (and scheme theoretically dense) open immersion, see Morphisms, Lemma 29.6.3. The morphism $\coprod _{k = 0, \ldots , t - 1} Y_ k \to X$ is finite surjective, hence $\{ Y_ k \to X\} $ is a ph covering. By the transitivity property of ph coverings (Topologies, Lemma 34.8.8) it suffices to show that the pullback of the covering $\{ X_ i \to X\} $ to each $Y_ k$ can be refined by a ph covering. This reduces us to the case described in the next paragraph.

Assume $\coprod X_ i \to X$ is universally submersive and there is a dense open $U \subset X$ such that $X_ i \times _ X U \to U$ is flat for all $i$. By Theorem 38.30.7 there is a $U$-admissible blowup $X' \to X$ such that the strict transform $f'_ i : X'_ i \to X'$ of $f_ i$ is flat for all $i$. Observe that the projective (hence closed) morphism $X' \to X$ is surjective as $U \subset X$ is dense and as $U$ is identified with an open of $X'$. After replacing $X'$ by a further $U$-admissible blowup if necessary, we may also assume $U \subset X'$ is dense (see Remark 38.30.1). Hence for every point $x \in X'$ there is a discrete valuation ring $A$ and a morphism $g : \mathop{\mathrm{Spec}}(A) \to X'$ such that the generic point of $\mathop{\mathrm{Spec}}(A)$ maps into $U$ and the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $x$, see Limits, Lemma 32.15.1. Set

\[ W = \mathop{\mathrm{Spec}}(A) \times _ X \coprod X_ i = \coprod \mathop{\mathrm{Spec}}(A) \times _ X X_ i \]

Since $\coprod X_ i \to X$ is universally submersive, there is a specialization $w' \leadsto w$ in $W$ such that $w'$ maps to the generic point of $\mathop{\mathrm{Spec}}(A)$ and $w$ maps to the closed point of $\mathop{\mathrm{Spec}}(A)$. (If not, then the closed fibre of $W \to \mathop{\mathrm{Spec}}(A)$ is stable under generalizations, hence open, which contradicts the fact that $W \to \mathop{\mathrm{Spec}}(A)$ is submersive.) Say $w' \in \mathop{\mathrm{Spec}}(A) \times _ X X_ i$ so of course $w \in \mathop{\mathrm{Spec}}(A) \times _ X X_ i$ as well. Let $x'_ i \leadsto x_ i$ be the image of $w' \leadsto w$ in $X' \times _ X X_ i$. Since $x'_ i \in X'_ i$ and since $X'_ i \subset X' \times _ X X_ i$ is a closed subscheme we see that $x_ i \in X'_ i$. Since $x_ i$ maps to $x \in X'$ we conclude that $\coprod X'_ i \to X'$ is surjective! In particular $\{ X'_ i \to X'\} $ is an fppf covering. But an fppf covering is a ph covering (More on Morphisms, Lemma 37.48.7). Since $X' \to X$ is proper surjective, we conclude that $\{ X'_ i \to X\} $ is a ph covering and the proof is complete. $\square$


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