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′ 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 aren't detected, for example, simply by requiring a smooth morphism. 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 etale morphism (and so etale cohomology) depends on the flat morphism concept: an etale morphism being flat, finite, and unramified.