The Stacks project

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.

1. 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 $\mathcal{O}_{S, f(x)}$.

2. 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.

3. We say $f$ is flat if $f$ is flat at every point of $X$.

4. We say that $\mathcal{F}$ is flat over $S$ if $\mathcal{F}$ is flat over $S$ at every point $x$ of $X$.