The Stacks project

Lemma 61.31.2. Let $S$ be a scheme. Let $S_{pro\text{-}\acute{e}tale}\subset S_{pro\text{-}\acute{e}tale}'$ be two small pro-├ętale sites of $S$ as constructed in Definition 61.12.8. Then the inclusion functor satisfies the assumptions of Sites, Lemma 7.21.8. Hence there exist morphisms of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) \ar[r]^ g & \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}') \ar[r]^ f & \mathop{\mathit{Sh}}\nolimits (S_{pro\text{-}\acute{e}tale}) } \]

whose composition is isomorphic to the identity and with $f_* = g^{-1}$. Moreover,

  1. for $\mathcal{F}' \in \textit{Ab}(S_{pro\text{-}\acute{e}tale}')$ we have $H^ p(S_{pro\text{-}\acute{e}tale}', \mathcal{F}') = H^ p(S_{pro\text{-}\acute{e}tale}, g^{-1}\mathcal{F}')$,

  2. for $\mathcal{F} \in \textit{Ab}(S_{pro\text{-}\acute{e}tale})$ we have

    \[ H^ p(S_{pro\text{-}\acute{e}tale}, \mathcal{F}) = H^ p(S_{pro\text{-}\acute{e}tale}', g_*\mathcal{F}) = H^ p(S_{pro\text{-}\acute{e}tale}', f^{-1}\mathcal{F}). \]

Proof. The inclusion functor is fully faithful and continuous. We have seen that $S_{pro\text{-}\acute{e}tale}$ and $S_{pro\text{-}\acute{e}tale}'$ have fibre products and final objects and that our functor commutes with these (Lemma 61.12.10). It follows from Lemma 61.31.1 that the inclusion functor is cocontinuous. Hence the existence of $f$ and $g$ follows from Sites, Lemma 7.21.8. The equality in (1) is Cohomology on Sites, Lemma 21.7.2. Part (2) follows from (1) as $\mathcal{F} = g^{-1}g_*\mathcal{F} = g^{-1}f^{-1}\mathcal{F}$. $\square$


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