Lemma 61.26.3. Let $j : U \to X$ be a weakly étale morphism of schemes. Let $i : Z \to X$ be a closed immersion such that $U \times _ X Z = \emptyset$. Let $V \to X$ be an affine object of $X_{pro\text{-}\acute{e}tale}$ such that every point of $V$ specializes to a point of $V_ Z = Z \times _ X V$. Then $j_!\mathcal{F}(V) = 0$ for all abelian sheaves on $U_{pro\text{-}\acute{e}tale}$.

Proof. Let $\{ V_ i \to V\}$ be a pro-étale covering. The lemma follows if we can refine this covering to a covering where the members have no morphisms into $U$ over $X$ (see construction of $j_!$ in Modules on Sites, Section 18.19). First refine the covering to get a finite covering with $V_ i$ affine. For each $i$ let $V_ i = \mathop{\mathrm{Spec}}(A_ i)$ and let $Z_ i \subset V_ i$ be the inverse image of $Z$. Set $W_ i = \mathop{\mathrm{Spec}}(A_{i, Z_ i}^\sim )$ with notation as in Lemma 61.5.1. Then $\coprod W_ i \to V$ is weakly étale and the image contains all points of $V_ Z$. Hence the image contains all points of $V$ by our assumption on specializations. Thus $\{ W_ i \to V\}$ is a pro-étale covering refining the given one. But each point in $W_ i$ specializes to a point lying over $Z$, hence there are no morphisms $W_ i \to U$ over $X$. $\square$

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