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The Stacks project

Lemma 61.12.19. Let S be a scheme contained in a big pro-étale site \mathit{Sch}_{pro\text{-}\acute{e}tale}. A sheaf \mathcal{F} on the big pro-étale site (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale} is given by the following data:

  1. for every T/S \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}) a sheaf \mathcal{F}_ T on T_{pro\text{-}\acute{e}tale},

  2. for every f : T' \to T in (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale} a map c_ f : f_{small}^{-1}\mathcal{F}_ T \to \mathcal{F}_{T'}.

These data are subject to the following conditions:

  1. given any f : T' \to T and g : T'' \to T' in (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale} the composition c_ g \circ g_{small}^{-1}c_ f is equal to c_{f \circ g}, and

  2. if f : T' \to T in (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale} is weakly étale then c_ f is an isomorphism.

Proof. Identical to the proof of Topologies, Lemma 34.4.20. \square


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