The morphism itself is said to be flat if
is flat over at every point of .
This is the natural condition for to form a “continuous family” over . That is, for each , the fiber of over is a scheme. We can consider
as a family of schemes parameterized by . If the morphism 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 .