Theorem 76.24.1. Let S be a scheme. Let f : X \to Y and Y \to Z be morphisms of algebraic spaces over S. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Assume
X, Y, Z locally Noetherian, and
\mathcal{F} a coherent \mathcal{O}_ X-module.
Let x \in |X| and let y \in |Y| and z \in |Z| be the images of x. If \mathcal{F}_{\overline{x}} \not= 0, then the following are equivalent:
\mathcal{F} is flat over Z at x and the restriction of \mathcal{F} to its fibre over z is flat at x over the fibre of Y over z, and
Y is flat over Z at y and \mathcal{F} is flat over Y at x.
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