The Stacks project

Lemma 38.34.1. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be a family of morphisms of schemes with fixed target with $f_ i$ locally of finite presentation for all $i$. The following are equivalent

  1. $\{ X_ i \to X\} $ is a ph covering, and

  2. $\{ X_ i \to X\} $ is a V covering.

Proof. Let $U \subset X$ be affine open. Looking at Topologies, Definitions 34.8.4 and 34.10.7 it suffices to show that the base change $\{ X_ i \times _ X U \to U\} $ can be refined by a standard ph covering if and only if it can be refined by a standard V covering. Thus we may assume $X$ is affine and we have to show $\{ X_ i \to X\} $ can be refined by a standard ph covering if and only if it can be refined by a standard V covering. Since a standard ph covering is a standard V covering, see Topologies, Lemma 34.10.3 it suffices to prove the other implication.

Assume $X$ is affine and assume $\{ f_ i : X_ i \to X\} _{i \in I}$ can be refined by a standard V covering $\{ g_ j : Y_ j \to X\} _{j = 1, \ldots , m}$. For each $j$ choose an $i_ j$ and a morphism $h_ j : Y_ j \to X_{i_ j}$ such that $g_ j = f_{i_ j} \circ h_ j$. Since $Y_ j$ is affine hence quasi-compact, for each $j$ we can find finitely many affine opens $U_{j, k} \subset X_{i_ j}$ such that $\mathop{\mathrm{Im}}(h_ j) \subset \bigcup U_{j, k}$. Then $\{ U_{j, k} \to X\} _{j, k}$ refines $\{ X_ i \to X\} $ and is a standard V covering (as it is a finite family of morphisms of affines and it inherits the lifting property for valuation rings from the corresponding property of $\{ Y_ j \to X\} $). Thus we reduce to the case discussed in the next paragraph.

Assume $\{ f_ i : X_ i \to X\} _{i = 1, \ldots , n}$ is a standard V covering with $f_ i$ of finite presentation. We have to show that $\{ X_ i \to X\} $ can be refined by a standard ph covering. 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

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

of affines. 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 (see Properties, Lemma 28.24.1), 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 standard ph covering and hence a standard V covering (see above). By the transitivity property of standard V coverings (Topologies, Lemma 34.10.5) it suffices to show that the pullback of the covering $\{ X_ i \to X\} $ to each $Y_ k$ can be refined by a standard V covering. This reduces us to the case described in the next paragraph.

Assume $\{ f_ i : X_ i \to X\} _{i = 1, \ldots , n}$ is a standard V covering with $f_ i$ of finite presentation and there is a dense quasi-compact open $U \subset X$ such that $X_ i \times _ X U \to U$ is flat. 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. 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 scheme theoretically dense (see Remark 38.30.1). Hence for every point $x \in X'$ there is a valuation ring $V$ and a morphism $g : \mathop{\mathrm{Spec}}(V) \to X'$ such that the generic point of $\mathop{\mathrm{Spec}}(V)$ maps into $U$ and the closed point of $\mathop{\mathrm{Spec}}(V)$ maps to $x$, see Morphisms, Lemma 29.6.5. Since $\{ X_ i \to X\} $ is a standard V covering, we can choose an extension of valuation rings $V \subset W$, an index $i$, and a morphism $\mathop{\mathrm{Spec}}(W) \to X_ i$ such that the diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[d] \ar[rr] & & X_ i \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r] & X' \ar[r] & X } \]

is commutative. Since $X'_ i \subset X' \times _ X X_ i$ is a closed subscheme containing the open $U \times _ X X_ i$, since $\mathop{\mathrm{Spec}}(W)$ is an integral scheme, and since the induced morphism $h : \mathop{\mathrm{Spec}}(W) \to X' \times _ X X_ i$ maps the generic point of $\mathop{\mathrm{Spec}}(W)$ into $U \times _ X X_ i$, we conclude that $h$ factors through the closed subscheme $X'_ i \subset X' \times _ X X_ i$. We conclude that $\{ f'_ i : X'_ i \to X'\} $ is a V covering. In particular, $\coprod f'_ i$ is surjective. In particular $\{ X'_ i \to X'\} $ is an fppf covering. Since an fppf covering is a ph covering (More on Morphisms, Lemma 37.47.7), we can find a standard ph covering $\{ Y_ j \to X'\} $ refining $\{ X'_ i \to X\} $. Say this covering is given by a proper surjective morphism $Y \to X'$ and a finite affine open covering $Y = \bigcup Y_ j$. Then the composition $Y \to X$ is proper surjective and we conclude that $\{ Y_ j \to X\} $ is a standard ph covering. This finishes the proof. $\square$


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