Lemma 33.16.8. Let $f : X \to Y$ be a morphism of schemes locally of finite type over a base scheme $S$. Let $x \in X$ be a point. Set $y = f(x)$ and assume that $\kappa (y) = \kappa (x)$. Then the following are equivalent

1. $\text{d}f : T_{X/S, x} \longrightarrow T_{Y/S, y}$ is injective, and

2. $f$ is unramified at $x$.

Proof. The morphism $f$ is locally of finite type by Morphisms, Lemma 29.15.8. The map $\text{d}f$ is injective, if and only if $\Omega _{Y/S, y} \otimes \kappa (y) \to \Omega _{X/S, x} \otimes \kappa (x)$ is surjective (Lemma 33.16.6). The exact sequence $f^*\Omega _{Y/S} \to \Omega _{X/S} \to \Omega _{X/Y} \to 0$ (Morphisms, Lemma 29.32.9) then shows that this happens if and only if $\Omega _{X/Y, x} \otimes \kappa (x) = 0$. Hence the result follows from Morphisms, Lemma 29.35.14. $\square$

Comment #2159 by Ariyan on

Typo: The last sentence "Hence the result by Morphisms" should be "Hence the result follows from.."

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