Definition 41.3.5. (See Morphisms, Definition 29.35.1 for the definition in the general case.) Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be locally of finite type. Let $x \in X$.

1. We say $f$ is unramified at $x$ if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is an unramified homomorphism of local rings.

2. The morphism $f : X \to Y$ is said to be unramified if it is unramified at all points of $X$.

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