The Stacks project

Lemma 37.8.9. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

  1. The morphism $f$ is étale, and

  2. the morphism $f$ is locally of finite presentation and formally étale.

Proof. Assume $f$ is étale. An étale morphism is locally of finite presentation, flat and unramified, see Morphisms, Section 29.36. Hence $f$ is locally of finite presentation and formally étale, see Lemma 37.8.7.

Conversely, suppose that $f$ is locally of finite presentation and formally étale. Being étale is local in the Zariski topology on $X$ and $S$, see Morphisms, Lemma 29.36.2. By Lemma 37.8.5 we can cover $X$ by affine opens $U$ which map into affine opens $V$ such that $U \to V$ is formally étale (and of finite presentation, see Morphisms, Lemma 29.21.2). By Lemma 37.8.8 we see that the ring maps $\mathcal{O}(V) \to \mathcal{O}(U)$ are formally étale (and of finite presentation). We win by Algebra, Lemma 10.150.2. (We will give another proof of this implication when we discuss formally smooth morphisms.) $\square$

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