Let $f\colon X\to Y$ be a morphism of schemes. Then a sheaf $\mathcal{F}$ of $\mathcal{O}_X$ -modules is flat over $Y$ at a point $x\in X$ if $\mathcal{F}_x$ is a flat $\mathcal{O}_{Y,f(x)}$ -module by way of the map $f^\sharp\colon \mathcal{O}_Y\to\mathcal{O}_X$ associated to $f$ .

The morphism $f$ itself is said to be flat if $\mathcal{O}_X$ is flat over $Y$ at every point of $X$ .

This is the natural condition for $X$ to form a ``continuous family'' over $Y$ . That is, for each $y\in Y$ , the fiber $X_y$ of $f$ over $y$ is a scheme. We can consider $X$ as a family of schemes parameterized by $Y$ . If the morphism $f$ is flat, then this family should be thought of as a ``continuous family''. In particular, this means that certain cohomological invariants remain constant on the fibers of $X$ .

Bibliography

1

Robin Hartshorne, Algebraic Geometry, Springer-Verlag, 1977 (GTM 52).