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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