The Stacks project

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

  1. $f$ is formally étale,

  2. for every $x \in X$ there exist opens $x \in U \subset X$ and $f(x) \in V \subset Y$ with $f(U) \subset V$ such that $f|_ U : U \to V$ is formally étale,

  3. for every pair of affine opens $U \subset X$ and $V \subset Y$ with $f(U) \subset V$ the ring map $\mathcal{O}_ Y(V) \to \mathcal{O}_ X(U)$ is formally étale, and

  4. there exists an affine open covering $Y = \bigcup V_ j$ and for each $j$ an affine open covering $f^{-1}(V_ j) = \bigcup U_{ji}$ such that $\mathcal{O}_ Y(V) \to \mathcal{O}_ X(U)$ is a formally étale ring map for all $j$ and $i$.

Proof. Combining Lemmas 37.8.5 and 37.8.8 we see that (1) $\Rightarrow $ (3). It is immediate that (3) $\Rightarrow $ (4). We have (4) $\Rightarrow $ (2) by Lemma 37.8.8. If (2) holds, then $\Omega _{X/S}$ and $\mathcal{C}_{X/S}$ vanish in a neighbourhood of every point by Lemma 37.8.6 (this also uses Morphisms, Lemma 29.33.3 and Lemma 37.7.8) whereupon Lemma 37.8.6 tells us that $f$ is formally unramified. $\square$

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