Let be a morphism of schemes. Then a sheaf of -modules is flat over at a point if is a flat -module by way of the map associated to .

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 .

Bibliography

1

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