Lemma 61.24.1. Let f : Z \to X be a finite morphism of schemes which is locally of finite presentation. Then f_{{pro\text{-}\acute{e}tale}, *} : \textit{Ab}(Z_{pro\text{-}\acute{e}tale}) \to \textit{Ab}(X_{pro\text{-}\acute{e}tale}) is exact.
61.24 Finite morphisms and pro-étale sites
It is not clear that a finite morphism of schemes determines an exact pushforward on abelian pro-étale sheaves.
Proof. The prove this we may work (Zariski) locally on X and assume that X is affine, say X = \mathop{\mathrm{Spec}}(A). Then Z = \mathop{\mathrm{Spec}}(B) for some finite A-algebra B of finite presentation. The construction in the proof of Proposition 61.11.3 produces a faithfully flat, ind-étale ring map A \to D with D w-contractible. We may check exactness of a sequence of sheaves by evaluating on U = \mathop{\mathrm{Spec}}(D) be such an object. Then f_{{pro\text{-}\acute{e}tale}, *}\mathcal{F} evaluated at U is equal to \mathcal{F} evaluated at V = \mathop{\mathrm{Spec}}(D \otimes _ A B). Since D \otimes _ A B is w-contractible by Lemma 61.11.6 evaluation at V is exact. \square
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