The Stacks project

Lemma 35.10.1. In Lemma 35.8.5 the morphism of ringed sites $\text{id}_{small, {\acute{e}tale}, Zar} : S_{\acute{e}tale}\to S_{Zar}$ is flat.

Proof. Let us denote $\epsilon = \text{id}_{small, {\acute{e}tale}, Zar}$ and $\mathcal{O}_{\acute{e}tale}$ and $\mathcal{O}_{Zar}$ the structure sheaves on $S_{\acute{e}tale}$ and $S_{Zar}$. We have to show that $\mathcal{O}_{\acute{e}tale}$ is a flat $\epsilon ^{-1}\mathcal{O}_{Zar}$-module. Recall that étale morphisms are open, see Morphisms, Lemma 29.36.13. It follows (from the construction of pullback on sheaves) that $\epsilon ^{-1}\mathcal{O}_{Zar}$ is the sheafification of the presheaf $\mathcal{O}'$ on $S_{\acute{e}tale}$ which sends an étale morphism $f : V \to S$ to $\mathcal{O}_ S(f(V))$. If both $V$ and $U = f(V) \subset S$ are affine, then $V \to U$ is an étale morphism of affines, hence corresponds to an étale ring map. Since étale ring maps are flat, we see that $\mathcal{O}_ S(U) = \mathcal{O}'(V) \to \mathcal{O}_{\acute{e}tale}(V) = \mathcal{O}_ V(V)$ is flat. Finally, for every étale morphism $f : V \to S$, i.e., object of $S_{\acute{e}tale}$, there is an affine open covering $V = \bigcup V_ i$ such that $f(V_ i)$ is an affine open in $S$ for all $i$1. Thus the result by Modules on Sites, Lemma 18.28.4. $\square$

[1] Namely, for $y \in V$, we pick an affine open $y \in V' \subset V$ with $f(V')$ contained in an affine open $U \subset S$. Then we pick an affine open $f(y) \in U' \subset f(V')$. Then $V'' = f^{-1}(U') \subset V'$ is affine as it is equal to $U' \times _ U V'$ and $f(V'') = U'$ is affine too.

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