The Stacks project

Lemma 29.35.2. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. The morphism $f$ is 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 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 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 étale, for all $j\in J, i\in I_ j$.

Moreover, if $f$ is 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 étale.

Proof. This follows from Lemma 29.14.3 if we show that the property “$R \to A$ is étale” is local. We check conditions (a), (b) and (c) of Definition 29.14.1. These all follow from Algebra, Lemma 10.142.3. $\square$


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