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The Stacks project

Lemma 61.12.2. Let T be a scheme. Let \{ f_ i : T_ i \to T\} _{i \in I} be a family of morphisms of schemes with target T. The following are equivalent

  1. \{ f_ i : T_ i \to T\} _{i \in I} is a pro-étale covering,

  2. each f_ i is weakly étale and \{ f_ i : T_ i \to T\} _{i \in I} is an fpqc covering,

  3. each f_ i is weakly étale and for every affine open U \subset T there exist quasi-compact opens U_ i \subset T_ i which are almost all empty, such that U = \bigcup f_ i(U_ i),

  4. each f_ i is weakly étale and there exists an affine open covering T = \bigcup _{\alpha \in A} U_\alpha and for each \alpha \in A there exist i_{\alpha , 1}, \ldots , i_{\alpha , n(\alpha )} \in I and quasi-compact opens U_{\alpha , j} \subset T_{i_{\alpha , j}} such that U_\alpha = \bigcup _{j = 1, \ldots , n(\alpha )} f_{i_{\alpha , j}}(U_{\alpha , j}).

If T is quasi-separated, these are also equivalent to

  1. each f_ i is weakly étale, and for every t \in T there exist i_1, \ldots , i_ n \in I and quasi-compact opens U_ j \subset T_{i_ j} such that \bigcup _{j = 1, \ldots , n} f_{i_ j}(U_ j) is a (not necessarily open) neighbourhood of t in T.

Proof. The equivalence of (1) and (2) is immediate from the definitions. Hence the lemma follows from Topologies, Lemma 34.9.2. \square


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