The Stacks project

Lemma 37.61.4. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. The morphism $f$ is weakly étale.

  2. For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is weakly étale.

  3. There exists an open covering $S = \bigcup _{j \in J} V_ j$ and open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that each of the morphisms $U_ i \to V_ j$, $j\in J, i\in I_ j$ is weakly étale.

  4. There exists an affine open covering $S = \bigcup _{j \in J} V_ j$ and affine open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that the ring map $\mathcal{O}_ S(V_ j) \to \mathcal{O}_ X(U_ i)$ is of weakly étale, for all $j\in J, i\in I_ j$.

Moreover, if $f$ is weakly étale then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_ U : U \to V$ is weakly-étale.

Proof. Suppose given open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$. Then $U \times _ V U \subset X \times _ Y X$ is open (Schemes, Lemma 26.17.3) and the diagonal $\Delta _{U/V}$ of $f|_ U : U \to V$ is the restriction $\Delta _{X/Y}|_ U : U \to U \times _ V U$. Since flatness is a local property of morphisms of schemes (Morphisms, Lemma 29.25.3) the final statement of the lemma is follows as well as the equivalence of (1) and (3). If $X$ and $Y$ are affine, then $X \to Y$ is weakly étale if and only if $\mathcal{O}_ Y(Y) \to \mathcal{O}_ X(X)$ is weakly étale (use again Morphisms, Lemma 29.25.3). Thus (1) and (3) are also equivalent to (2) and (4). $\square$


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