The Stacks project

Example 61.19.12. Let $X$ be a scheme. Let $A$ be an abelian group. Denote $fun(-, A)$ the sheaf on $X_{pro\text{-}\acute{e}tale}$ which maps $U$ to the set of all maps $U \to A$ (of sets of points). Consider the sequence of sheaves

\[ 0 \to \underline{A} \to fun(-, A) \to \mathcal{F} \to 0 \]

on $X_{pro\text{-}\acute{e}tale}$. Since the constant sheaf is the pullback from the final topos we see that $\underline{A} = \epsilon ^{-1}\underline{A}$. However, if $A$ has more than one element, then neither $fun(-, A)$ nor $\mathcal{F}$ are pulled back from the étale site of $X$. To work out the values of $\mathcal{F}$ in some cases, assume that all points of $X$ are closed with separably closed residue fields and $U$ is affine. Then all points of $U$ are closed with separably closed residue fields and we have

\[ H^1_{pro\text{-}\acute{e}tale}(U, \underline{A}) = H^1_{\acute{e}tale}(U, \underline{A}) = 0 \]

by Lemma 61.19.6 and Étale Cohomology, Lemma 59.80.3. Hence in this case we have

\[ \mathcal{F}(U) = fun(U, A)/\underline{A}(U) \]

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