flat morphism

Let f:XY be a morphism of schemes. Then a sheaf of 𝒪X-modules is flat over Y at a point xX if x is a flat (http://planetmath.org/FlatModule) 𝒪Y,f(x)-module by way of the map f:𝒪Y𝒪X associated to f.

The morphismMathworldPlanetmath f itself is said to be flat if 𝒪X 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 yY, the fiber Xy 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.


Title flat morphism
Canonical name FlatMorphism
Date of creation 2013-03-22 14:11:10
Last modified on 2013-03-22 14:11:10
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 4
Author archibal (4430)
Entry type Definition
Classification msc 14A15
Synonym flat
Related topic Scheme
Related topic EtaleMorphism
Defines flat sheaf