flat morphism

Let $f:X\to Y$ be a morphism of schemes. Then a sheaf $ℱ$ of ${𝒪}_X$-modules is flat over $Y$ at a point $x\mathrm{\in }X$ if ${ℱ}_x$ is a flat (http://planetmath.org/FlatModule) ${𝒪}_{Y,f(x)}$-module by way of the map $f^{\mathrm{♯}}:{𝒪}_Y\to {𝒪}_X$ associated to $f$.

The morphism $f$ itself is said to be flat if ${𝒪}_X$ 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 $X$.