The Stacks project

Lemma 61.13.1. Let $T = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. The following are equivalent

  1. $A$ is w-contractible, and

  2. every pro-étale covering of $T$ can be refined by a Zariski covering of the form $T = \coprod _{i = 1, \ldots , n} U_ i$.

Proof. Assume $A$ is w-contractible. By Lemma 61.12.5 it suffices to prove we can refine every standard pro-étale covering $\{ f_ i : T_ i \to T\} _{i = 1, \ldots , n}$ by a Zariski covering of $T$. The morphism $\coprod T_ i \to T$ is a surjective weakly étale morphism of affine schemes. Hence by Definition 61.11.1 there exists a morphism $\sigma : T \to \coprod T_ i$ over $T$. Then the Zariski covering $T = \coprod \sigma ^{-1}(T_ i)$ refines $\{ f_ i : T_ i \to T\} $.

Conversely, assume (2). If $A \to B$ is faithfully flat and weakly étale, then $\{ \mathop{\mathrm{Spec}}(B) \to T\} $ is a pro-étale covering. Hence there exists a Zariski covering $T = \coprod U_ i$ and morphisms $U_ i \to \mathop{\mathrm{Spec}}(B)$ over $T$. Since $T = \coprod U_ i$ we obtain $T \to \mathop{\mathrm{Spec}}(B)$, i.e., an $A$-algebra map $B \to A$. This means $A$ is w-contractible. $\square$

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