Lemma 66.31.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in |X|$. The following are equivalent

for some 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, $a, b$ are étale, and $u \in U$ mapping to $x$ the module $a^*\mathcal{F}$ is flat at $u$ over $V$,

the stalk $\mathcal{F}_{\overline{x}}$ is flat over the étale local ring $\mathcal{O}_{Y, \overline{y}}$ where $\overline{x}$ is any geometric point lying over $x$ and $\overline{y} = f \circ \overline{x}$.

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