Flat morphism

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Flat morphism

In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism "f" from a scheme "X" to a scheme "Y" is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,

:"fP":"OY,f(P)"→"OX,P"

is a flat map for all "P" in "X".

The definition here has its roots in homological algebra, rather than geometric considerations. Two of the basic intuitions are that "flatness is a generic property", and that "the failure of flatness occurs on the jumping set of the morphism".

The first of these comes from commutative algebra: subject to some finiteness conditions on "f", it can be shown that there is a non-empty open subscheme "Y"&prime; of "Y", such that "f" restricted to "Y"&prime; is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of fiber product, applied to "f" and the inclusion map of "Y"&prime; into "Y".

For the second, the idea is that morphisms in algebraic geometry can exhibit discontinuities of a kind that are detected by flatness. For instance, the operation of blowing down in the birational geometry of an algebraic surface, can give a single fiber that is of dimension 1 when all the others have dimension 0. It turns out (retrospectively) that flatness in morphisms is directly related to controlling this sort of semicontinuity, or one-sided jumping.

Flat morphisms are used to define (more than one version of) the flat topos, and flat cohomology of sheaves from it. This is a deep-lying theory, and has not been found easy to handle. The concept of étale morphism (and so étale cohomology) depends on the flat morphism concept: an étale morphism being flat, of finite type, and unramified.

Properties of flat morphisms

* Flat morphisms, which are locally of finite type are open.
* The dimension of fibers $f^\left\{-1\right\}\left(y\right)$ of a flat map $f: X ightarrow Y$ is given by $mathrm\left\{dim\right\}, X - mathrm\left\{dim\right\}, Y$. (In general, the dimension of the fibers is greater or equal than this difference).
* If the local rings of X are Cohen-Macaulay, then the converse statement holds, too.

References

* | year=1995 | volume=150, section 6.

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