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
Comments (0)