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*. A map of rings A → B is called **flat**, if it is a homomorphism that makes B a flat A-module.

A morphism of schemes *f* is a **faithfully flat morphism** if *f* is a surjective flat morphism.

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.

Read more about Flat Morphism: Properties of Flat Morphisms

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**flat**at x in X ... In particular, if f is faithfully

**flat**, then X reduced or normal implies that Y is reduced or normal, respectively ... If f is faithfully

**flat**and of finite presentation, then all the fibers of f reduced or normal implies that X is reduced or normal, respectively ...

### Famous quotes containing the word flat:

“Castaway, your time is a *flat* sea that doesn’t stop,

with no new land to make for and no new stories to swap.”

—Anne Sexton (1928–1974)