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.[1]
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.[2] 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 productX ×SSi, and denote by Fi the restriction F ⊗OSOSi; then each Fi is flat.
Generic freenessedit
Generic flatness is a consequence of the generic freeness lemma. Generic freeness states that if A is a noetherianintegral domain, B is a finite type A-algebra, and M is a finite type B-module, then there exists a non-zero element f of A such that Mf is a free Af-module.[3] Generic freeness can be extended to the graded situation: If B is graded by the natural numbers, A acts in degree zero, and M is a graded B-module, then f may be chosen such that each graded component of Mf is free.[4]
Generic freeness is proved using Grothendieck's technique of dévissage. Another version of generic freeness can be proved using Noether's normalization lemma.
generic, flatness, algebraic, geometry, commutative, algebra, theorems, generic, flatness, generic, freeness, state, that, under, certain, hypotheses, sheaf, modules, scheme, flat, free, they, alexander, grothendieck, states, that, integral, locally, noetheria. 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 1 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 2 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 Generic freeness editGeneric flatness is a consequence of the generic freeness lemma Generic freeness states that if A is a noetherian integral domain B is a finite type A algebra and M is a finite type B module then there exists a non zero element f of A such that Mf is a free Af module 3 Generic freeness can be extended to the graded situation If B is graded by the natural numbers A acts in degree zero and M is a graded B module then f may be chosen such that each graded component of Mf is free 4 Generic freeness is proved using Grothendieck s technique of devissage Another version of generic freeness can be proved using Noether s normalization lemma References edit EGA IV2 Theoreme 6 9 1 EGA IV2 Corollaire 6 9 3 EGA IV2 Lemme 6 9 2 Eisenbud Theorem 14 4Bibliography editEisenbud David 1995 Commutative algebra with a view toward algebraic geometry Graduate Texts in Mathematics vol 150 Berlin New York Springer Verlag ISBN 978 0 387 94268 1 MR 1322960 Grothendieck Alexandre Dieudonne Jean 1965 Elements de geometrie algebrique IV Etude locale des schemas et des morphismes de schemas Seconde partie Publications Mathematiques de l IHES 24 doi 10 1007 bf02684322 MR 0199181 Retrieved from https en wikipedia org w index php title Generic flatness amp oldid 1034733101, wikipedia, wiki, book, books, library,
