The Stacks project

Lemma 67.30.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is flat if and only if $f$ is flat at all points of $|X|$.

Proof. Choose a commutative diagram

\[ \xymatrix{ U \ar[d]_ a \ar[r]_ h & V \ar[d]^ b \\ X \ar[r]^ f & Y } \]

where $U$ and $V$ are schemes, the vertical arrows are ├ętale, and $a$ is surjective. By definition $f$ is flat if and only if $h$ is flat (Definition 67.22.2). By definition $f$ is flat at $x \in |X|$ if and only if $h$ is flat at some (equivalently any) $u \in U$ which maps to $x$ (Definition 67.22.6). Thus the lemma follows from the fact that a morphism of schemes is flat if and only if it is flat at all points of the source (Morphisms, Definition 29.25.1). $\square$

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