The Stacks project

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

as in More on Morphisms, Lemma 37.54.2 for the finitely presented morphism

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

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 37.48.7), 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. $\square$