flat morphism

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 (http://planetmath.org/FlatModule) $\mathcal{O}_{Y,f(x)}$ -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 $X$ .

References

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

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