The Stacks project

Lemma 34.9.11. Let $T$ be an affine scheme.

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

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

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

Proof. This follows formally from the fact that compositions and base changes of flat morphisms are flat (Morphisms, Lemmas 29.25.8 and 29.25.6) and that fibre products of affine schemes are affine (Schemes, Lemma 26.17.2). $\square$

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