Definition 17.20.3. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules.

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

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

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