The Stacks project

Definition 76.23.2. Let $S$ be a scheme. Let $X \to Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in |X|$ be a point and denote $z \in |Z|$ its image.

  1. We say the restriction of $\mathcal{F}$ to its fibre over $z$ is flat at $x$ over the fibre of $Y$ over $z$ if the equivalent conditions of Lemma 76.23.1 are satisfied.

  2. We say the fibre of $X$ over $z$ is flat at $x$ over the fibre of $Y$ over $z$ if the equivalent conditions of Lemma 76.23.1 hold with $\mathcal{F} = \mathcal{O}_ X$.

  3. We say the fibre of $X$ over $z$ is flat over the fibre of $Y$ over $z$ if for all $x \in |X|$ lying over $z$ the fibre of $X$ over $z$ is flat at $x$ over the fibre of $Y$ over $z$

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View Definition 76.23.2 as pdf