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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.,
is a flat map for all P in X. A map of rings A → B {\displaystyle A\to B} is called flat if it is a homomorphism that makes B a flat A-module. A morphism of schemes is called faithfully flat if it is both surjective and flat.
Two basic intuitions regarding flat morphisms are:
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 ′ {\displaystyle Y'} of Y, such that f restricted to Y′ is a flat morphism. Here 'restriction' is interpreted by means of the fiber product of schemes, applied to f and the inclusion map of Y ′ {\displaystyle Y'} into Y.
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