The Stacks project

Lemma 34.8.3. Let $T$ be an affine scheme. Each of the following types of families of maps with target $T$ has a refinement by a standard ph covering:

  1. any Zariski open covering of $T$,

  2. $\{ W_{ji} \to T\} _{j = 1, \ldots , m, i = 1, \ldots n_ j}$ where $\{ W_{ji} \to U_ j\} _{i = 1, \ldots , n_ j}$ and $\{ U_ j \to T\} _{j = 1, \ldots , m}$ are standard ph coverings.

Proof. Part (1) follows from the fact that any Zariski open covering of $T$ can be refined by a finite affine open covering.

Proof of (3). Choose $U \to T$ proper surjective and $U = \bigcup _{j = 1, \ldots , m} U_ j$ as in Definition 34.8.1. Choose $W_ j \to U_ j$ proper surjective and $W_ j = \bigcup W_{ji}$ as in Definition 34.8.1. By Chow's lemma (Limits, Lemma 32.12.1) we can find $W'_ j \to W_ j$ proper surjective and closed immersions $W'_ j \to \mathbf{P}^{e_ j}_{U_ j}$. Thus, after replacing $W_ j$ by $W'_ j$ and $W_ j = \bigcup W_{ji}$ by a suitable affine open covering of $W'_ j$, we may assume there is a closed immersion $W_ j \subset \mathbf{P}^{e_ j}_{U_ j}$ for all $j = 1, \ldots , m$.

Let $\overline{W}_ j \subset \mathbf{P}^{e_ j}_ U$ be the scheme theoretic closure of $W_ j$. Then $W_ j \subset \overline{W}_ j$ is an open subscheme; in fact $W_ j$ is the inverse image of $U_ j \subset U$ under the morphism $\overline{W}_ j \to U$. (To see this use that $W_ j \to \mathbf{P}^{e_ j}_ U$ is quasi-compact and hence formation of the scheme theoretic image commutes with restriction to opens, see Morphisms, Section 29.6.) Let $Z_ j = U \setminus U_ j$ with reduced induced closed subscheme structure. Then

\[ V_ j = \overline{W}_ j \amalg Z_ j \to U \]

is proper surjective and the open subscheme $W_ j \subset V_ j$ is the inverse image of $U_ j$. Hence for $v \in V_ j$, $v \not\in W_ j$ we can pick an affine open neighbourhood $v \in V_{j, v} \subset V_ j$ which maps into $U_{j'}$ for some $1 \leq j' \leq m$.

To finish the proof we consider the proper surjective morphism

\[ V = V_1 \times _ U V_2 \times _ U \ldots \times _ U V_ m \longrightarrow U \longrightarrow T \]

and the covering of $V$ by the affine opens

\[ V_{1, v_1} \times _ U \ldots \times _ U V_{j - 1, v_{j - 1}} \times _ U W_{j i} \times _ U V_{j + 1, v_{j + 1}} \times _ U \ldots \times _ U V_{m, v_ m} \]

These do indeed form a covering, because each point of $U$ is in some $U_ j$ and the inverse image of $U_ j$ in $V$ is equal to $V_1 \times \ldots \times V_{j - 1} \times W_ j \times V_{j + 1} \times \ldots \times V_ m$. Observe that the morphism from the affine open displayed above to $T$ factors through $W_{ji}$ thus we obtain a refinement. Finally, we only need a finite number of these affine opens as $V$ is quasi-compact (as a scheme proper over the affine scheme $T$). $\square$

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