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- Flat_morphism abstract "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.,fP: OY,f(P) → OX,Pis 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.".
- Flat_morphism wikiPageExternalLink item?id=AIF_1956__6__1_0.
- Flat_morphism wikiPageID "542520".
- Flat_morphism wikiPageLength "16540".
- Flat_morphism wikiPageOutDegree "27".
- Flat_morphism wikiPageRevisionID "690237111".
- Flat_morphism wikiPageWikiLink Algebraic_geometry.
- Flat_morphism wikiPageWikiLink Algebraic_surface.
- Flat_morphism wikiPageWikiLink Annales_de_lInstitut_Fourier.
- Flat_morphism wikiPageWikiLink Birational_geometry.
- Flat_morphism wikiPageWikiLink Blowing_down.
- Flat_morphism wikiPageWikiLink Category:Morphisms_of_schemes.
- Flat_morphism wikiPageWikiLink Commutative_algebra.
- Flat_morphism wikiPageWikiLink Fiber_of_a_morphism_of_schemes.
- Flat_morphism wikiPageWikiLink Finiteness_condition_on_a_morphism_of_schemes.
- Flat_morphism wikiPageWikiLink Flat_module.
- Flat_morphism wikiPageWikiLink Flat_topology.
- Flat_morphism wikiPageWikiLink Fpqc_morphism.
- Flat_morphism wikiPageWikiLink Generic_flatness.
- Flat_morphism wikiPageWikiLink Generic_property.
- Flat_morphism wikiPageWikiLink Glossary_of_algebraic_geometry.
- Flat_morphism wikiPageWikiLink Inclusion_map.
- Flat_morphism wikiPageWikiLink Mathematics.
- Flat_morphism wikiPageWikiLink Pullback_(category_theory).
- Flat_morphism wikiPageWikiLink Ramification_(mathematics).
- Flat_morphism wikiPageWikiLink Scheme_(mathematics).
- Flat_morphism wikiPageWikiLink Semi-continuity.
- Flat_morphism wikiPageWikiLink Springer_Science+Business_Media.
- Flat_morphism wikiPageWikiLink Stalk_(sheaf).
- Flat_morphism wikiPageWikiLink Étale_cohomology.
- Flat_morphism wikiPageWikiLink Étale_morphism.
- Flat_morphism wikiPageWikiLinkText "Flat morphism".
- Flat_morphism wikiPageWikiLinkText "flat morphism".
- Flat_morphism wikiPageWikiLinkText "flat".
- Flat_morphism wikiPageWikiLinkText "flatness".
- Flat_morphism wikiPageUsesTemplate Template:=.
- Flat_morphism wikiPageUsesTemplate Template:Citation.
- Flat_morphism wikiPageUsesTemplate Template:EGA.
- Flat_morphism wikiPageUsesTemplate Template:Hartshorne_AG.
- Flat_morphism wikiPageUsesTemplate Template:Reflist.
- Flat_morphism subject Category:Morphisms_of_schemes.
- Flat_morphism hypernym Morphism.
- Flat_morphism type Morphism.
- Flat_morphism comment "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.,fP: OY,f(P) → OX,Pis a flat map for all P in X.".
- Flat_morphism label "Flat morphism".
- Flat_morphism sameAs Q3324325.
- Flat_morphism sameAs Morphisme_plat.
- Flat_morphism sameAs 평탄_사상.
- Flat_morphism sameAs m.02nb3k.
- Flat_morphism sameAs Q3324325.
- Flat_morphism wasDerivedFrom Flat_morphism?oldid=690237111.
- Flat_morphism isPrimaryTopicOf Flat_morphism.