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flat morphism
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(Definition)
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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 $\mathcal{O}_{Y,f(x)}$ -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 $X$ .
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- Robin Hartshorne, Algebraic Geometry, Springer-Verlag, 1977 (GTM 52).
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"flat morphism" is owned by archibal.
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Cross-references: invariants, scheme, fiber, morphism, map, point, sheaf, morphism of schemes
There are 6 references to this entry.
This is version 1 of flat morphism, born on 2004-02-23.
Object id is 5614, canonical name is FlatMorphism.
Accessed 6515 times total.
Classification:
| AMS MSC: | 14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms) |
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Pending Errata and Addenda
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