Flat morphism
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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′ of Y, such that f restricted to Y′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of fiber product, applied to f and the inclusion map of Y′ 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.
[edit] Properties of flat morphisms
- Flat morphisms, which are locally of finite type are open.
- The dimension of fibers f − 1(y) of a flat map
is given by 
. (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.
[edit] References
- Eisenbud, David (1995), Commutative algebra, vol. 150, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1322960, ISBN 978-0-387-94268-1; 978-0-387-94269-8, section 6.
