Lemma 110.58.1. There exists a sheaf of abelian groups $G$ on $\mathit{Sch}_{\acute{e}tale}$ with the following properties

$G(\mathop{\mathrm{Spec}}(k)) = 0$ whenever $k$ is a field,

$G$ is limit preserving,

if $X \subset X'$ is a thickening, then $G(X) = G(X')$, and

$G$ is not zero.

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