- Flat morphism
In

mathematics , in particular in the theory of schemes inalgebraic 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 aflat 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 ageneric 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 offiber product , applied to "f" and theinclusion 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 thebirational geometry of analgebraic 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 ofsemicontinuity , or one-sided jumping.Flat morphisms are used to define (more than one version of) the

flat topos , andflat cohomology of sheaves from it. This is a deep-lying theory, and has not been found easy to handle. The concept ofétale morphism (and soétale 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.

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