flat morphism
Let $f:X\to Y$ be a morphism of schemes. Then a sheaf $\mathcal{F}$ of ${\mathcal{O}}_{X}$-modules is flat over $Y$ at a point $x\mathrm{\in}X$ if ${\mathcal{F}}_{x}$ is a flat (http://planetmath.org/FlatModule) ${\mathcal{O}}_{Y,f(x)}$-module by way of the map ${f}^{\mathrm{\u266f}}:{\mathcal{O}}_{Y}\to {\mathcal{O}}_{X}$ 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 |