The Stacks project

Lemma 34.10.9. Let $T$ be a scheme.

  1. If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is a V covering of $T$.

  2. If $\{ T_ i \to T\} _{i\in I}$ is a V covering and for each $i$ we have a V covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a V covering.

  3. If $\{ T_ i \to T\} _{i\in I}$ is a V covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is a V covering.

Proof. Assertion (1) is clear.

Proof of (3). Let $U' \subset T'$ be an affine open subscheme. Since $U'$ is quasi-compact we can find a finite affine open covering $U' = U'_1 \cup \ldots \cup U'$ such that $U'_ j \to T$ maps into an affine open $U_ j \subset T$. Choose a standard V covering $\{ U_{jl} \to U_ j\} _{l = 1, \ldots , n_ j}$ refining $\{ T_ i \times _ T U_ j \to U_ j\} $. By Lemma 34.10.4 the base change $\{ U_{jl} \times _{U_ j} U'_ j \to U'_ j\} $ is a standard V covering. Note that $\{ U'_ j \to U'\} $ is a standard V covering (for example by Lemma 34.10.2). By Lemma 34.10.5 the family $\{ U_{jl} \times _{U_ j} U'_ j \to U'\} $ is a standard V covering. Since $\{ U_{jl} \times _{U_ j} U'_ j \to U'\} $ refines $\{ T_ i \times _ T U' \to U'\} $ we conclude.

Proof of (2). Let $U \subset T$ be affine open. First we pick a standard V covering $\{ U_ k \to U\} _{k = 1, \ldots , m}$ refining $\{ T_ i \times _ T U \to U\} $. Say the refinement is given by morphisms $U_ k \to T_{i_ k}$ over $T$. Then

\[ \{ T_{i_ kj} \times _{T_{i_ k}} U_ k \to U_ k\} _{j \in J_{i_ k}} \]

is a V covering by part (3). As $U_ k$ is affine, we can find a standard V covering $\{ U_{ka} \to U_ k\} _{a = 1, \ldots , b_ k}$ refining this family. Then we apply Lemma 34.10.5 to see that $\{ U_{ka} \to U\} $ is a standard V covering which refines $\{ T_{ij} \times _ T U \to U\} $. This finishes the proof. $\square$

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