flat morphismFlatMorphism2004-02-23 01:46:252004-02-23 01:46:25Definitionflat sheaf% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{cor}{Corollary}Let $f\colon X\to Y$ be a morphism of schemes. Then a sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules is \emph{flat over $Y$ at a point $x\in X$} if $\mathcal{F}_x$ is a \PMlinkname{flat}{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 \emph{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 $X$.
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\bibitem{hartshorne} Robin Hartshorne, {\em Algebraic
Geometry}, Springer--Verlag, 1977 (GTM {\bf 52}).
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