Definition 66.30.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

We say $f$ is

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

*flat at $x$*if the equivalent conditions of Lemma 66.22.5 hold with $\mathcal{Q} =$“induced map local rings is flat”.

Note that the second part makes sense by Descent, Lemma 35.33.4.

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