# 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, , Springer–Verlag, 1977 (GTM 52).
Title flat morphism FlatMorphism 2013-03-22 14:11:10 2013-03-22 14:11:10 archibal (4430) archibal (4430) 4 archibal (4430) Definition msc 14A15 flat Scheme EtaleMorphism flat sheaf