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.,

:"f_P":"O_Y,f(P)"→"O_X,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.

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 $f:\; X\; ightarrow\; Y$ is given by $mathrm\{dim\},\; X\; -\; mathrm\{dim\},\; Y$. (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.