The Stacks project

Definition 41.9.3. (See Morphisms, Definition 29.25.1). Let $f : X \to Y$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

1. Let $x \in X$. We say $\mathcal{F}$ is flat over $Y$ at $x \in X$ if $\mathcal{F}_ x$ is a flat $\mathcal{O}_{Y, f(x)}$-module. This uses the map $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ to think of $\mathcal{F}_ x$ as a $\mathcal{O}_{Y, f(x)}$-module.

2. Let $x \in X$. We say $f$ is flat at $x \in X$ if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is flat.

3. We say $f$ is flat if it is flat at all points of $X$.

4. A morphism $f : X \to Y$ that is flat and surjective is sometimes said to be faithfully flat.