Definition 66.37.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

We say $f$ is

*smooth*if the equivalent conditions of Lemma 66.22.1 hold with $\mathcal{P} =$“smooth”.Let $x \in |X|$. We say $f$ is

*smooth at $x$*if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is smooth.

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