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- Generic_flatness abstract "In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free. They are due to Alexander Grothendieck.Generic flatness states that if Y is an integral locally noetherian scheme, u : X → Y is a finite type morphism of schemes, and F is a coherent OX-module, then there is a non-empty open subset U of Y such that the restriction of F to u−1(U) is flat over U.Because Y is integral, U is a dense open subset of Y. This can be applied to deduce a variant of generic flatness which is true when the base is not integral. Suppose that S is a noetherian scheme, u : X → S is a finite type morphism, and F is a coherent OX module. Then there exists a partition of S into locally closed subsets S1, ..., Sn with the following property: Give each Si its reduced scheme structure, denote by Xi the fiber product X ×S Si, and denote by Fi the restriction F ⊗OS OSi; then each Fi is flat.".
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- Generic_flatness wikiPageRevisionID "649535921".
- Generic_flatness wikiPageWikiLink Alexander_Grothendieck.
- Generic_flatness wikiPageWikiLink Algebraic_geometry.
- Generic_flatness wikiPageWikiLink Category:Algebraic_geometry.
- Generic_flatness wikiPageWikiLink Category:Commutative_algebra.
- Generic_flatness wikiPageWikiLink Category:Theorems_in_abstract_algebra.
- Generic_flatness wikiPageWikiLink Commutative_algebra.
- Generic_flatness wikiPageWikiLink Dévissage.
- Generic_flatness wikiPageWikiLink Fiber_product.
- Generic_flatness wikiPageWikiLink Flat_morphism.
- Generic_flatness wikiPageWikiLink Free_module.
- Generic_flatness wikiPageWikiLink Integral_domain.
- Generic_flatness wikiPageWikiLink Module_(mathematics).
- Generic_flatness wikiPageWikiLink Noetherian_ring.
- Generic_flatness wikiPageWikiLink Pullback_(category_theory).
- Generic_flatness wikiPageWikiLink Scheme_(mathematics).
- Generic_flatness wikiPageWikiLink Sheaf_(mathematics).
- Generic_flatness wikiPageWikiLink Springer-Verlag.
- Generic_flatness wikiPageWikiLink Springer_Science+Business_Media.
- Generic_flatness wikiPageWikiLinkText "Generic flatness".
- Generic_flatness wikiPageWikiLinkText "generic flatness".
- Generic_flatness hasPhotoCollection Generic_flatness.
- Generic_flatness wikiPageUsesTemplate Template:Citation.
- Generic_flatness wikiPageUsesTemplate Template:EGA.
- Generic_flatness subject Category:Algebraic_geometry.
- Generic_flatness subject Category:Commutative_algebra.
- Generic_flatness subject Category:Theorems_in_abstract_algebra.
- Generic_flatness type Article.
- Generic_flatness type Article.
- Generic_flatness type Theorem.
- Generic_flatness comment "In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free.".
- Generic_flatness label "Generic flatness".
- Generic_flatness sameAs m.0bbvd2c.
- Generic_flatness sameAs Q5532671.
- Generic_flatness sameAs Q5532671.
- Generic_flatness wasDerivedFrom Generic_flatness?oldid=649535921.
- Generic_flatness isPrimaryTopicOf Generic_flatness.