In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphismf 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.[1] A map of rings 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.[2]
Two basic intuitions regarding flat morphisms are:
flatness is a generic property; and
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 of Y, such that f restricted to Y′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of the fiber product of schemes, applied to f and the inclusion map of 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.
algebraic geometry, a flatmorphism 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.,...
that a faithfully flat quasi-compact morphism of schemes has this property.). See also Flatmorphism § Properties of flatmorphisms. A ring homomorphism...
differential geometry, flat map is a mapping that converts vectors into corresponding 1-forms, given a non-degenerate (0,2)-tensor. Flatmorphism Sharp map, the...
situations Flatness (systems theory), a property of nonlinear dynamic systems Spectral flatnessFlat intonation Flat module in abstract algebra Flatmorphism in...
space Flat (matroids), a further generalization of flats from linear algebra to the context of matroids Flat module in ring theory Flatmorphism in algebraic...
Faithfully flat may refer to: Faithfully flatmorphism, in the theory of schemes in algebraic geometry Faithfully flat module, for sequences in algebra...
Faithfully flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flatmorphism. Such...
présentation finie, and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation. fpqc stands...
morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism...
definitions of an fpqc morphism, both variations of faithfully flatmorphisms. Sometimes an fpqc morphism means one that is faithfully flat and quasicompact...
naturally the structure of a locally ringed space; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces. If X...
geometry, a morphism f : X → S {\displaystyle f:X\to S} between schemes is said to be smooth if (i) it is locally of finite presentation (ii) it is flat, and...
abelian. Specifically: AB1) Every morphism has a kernel and a cokernel. AB2) For every morphism f, the canonical morphism from coim f to im f is an isomorphism...
associated to the proper morphism Z → X {\displaystyle Z\to X} , and the second homomorphism is pullback with respect to the flatmorphism X − Z → X {\displaystyle...
spectrum of a field) and f : X → S {\displaystyle f:X\to S} a faithfully flatmorphism, locally of finite type. Assume f {\displaystyle f} has a section x...
a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism. Contents: !$@ A B C D E F G H I J K L M N O P Q R S T U V W XYZ...
: X → B {\displaystyle f:X\to B} of X {\displaystyle X} is a proper flatmorphism f {\displaystyle f} to a smooth curve such that f ∗ O X ≅ O B {\displaystyle...
pulled-back morphism loses all information about the original morphism. But if the morphism Z → Y is flat and surjective (also called faithfully flat) and quasi-compact...
In algebraic geometry, an unramified morphism is a morphism f : X → Y {\displaystyle f:X\to Y} of schemes such that (a) it is locally of finite presentation...
two morphs of the same clone and Tetramorium ants. Aphids of the round morph cause the ants to farm them, as with many other aphids. The flatmorph aphids...
reduction theorems state that, given a proper flatmorphism X → S {\displaystyle X\to S} , there exists a morphism S ′ → S {\displaystyle S'\to S} (called base...
over a scheme S and if i is an S-morphism, then i is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. If Spec...
constructed as a fiber of a morphism; namely, viewing L as the total space of it, the section s is a X-morphism of L: a morphism s : X → L {\displaystyle...
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