Definition 29.25.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules.

We say $f$ is

*flat at a point $x \in X$*if the local ring $\mathcal{O}_{X, x}$ is flat over the local ring $\mathcal{O}_{S, f(x)}$.We say that $\mathcal{F}$ is

*flat over $S$ at a point $x \in X$*if the stalk $\mathcal{F}_ x$ is a flat $\mathcal{O}_{S, f(x)}$-module.We say $f$ is

*flat*if $f$ is flat at every point of $X$.We say that $\mathcal{F}$ is

*flat over $S$*if $\mathcal{F}$ is flat over $S$ at every point $x$ of $X$.

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