# Generic flatness

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In [algebraic geometry](/source/algebraic_geometry) and [commutative algebra](/source/commutative_algebra), the theorems of '''generic flatness''' and '''generic freeness''' state that under certain hypotheses, a [sheaf](/source/sheaf_(mathematics)) of [module](/source/module_(mathematics))s on a [scheme](/source/scheme_(mathematics)) is [flat](/source/flat_morphism) or [free](/source/free_module). They are due to [Alexander Grothendieck](/source/Alexander_Grothendieck).

Generic flatness states that if ''Y'' is an integral [locally noetherian scheme](/source/locally_noetherian_scheme), {{nowrap|''u'' : ''X'' → ''Y''}} is a [finite type](/source/morphism_of_finite_type) [morphism of schemes](/source/morphism_of_schemes), and ''F'' is a [coherent](/source/coherent_sheaf) ''O''<sub>''X''</sub>-module, then there is a non-empty open subset ''U'' of ''Y'' such that the restriction of ''F'' to ''u''<sup>&minus;1</sup>(''U'') is flat over ''U''.<ref>EGA IV<sub>2</sub>, Théorème 6.9.1</ref>

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.<ref>EGA IV<sub>2</sub>, Corollaire 6.9.3</ref> Suppose that ''S'' is a noetherian scheme, {{nowrap|''u'' : ''X'' → ''S''}} is a finite type morphism, and ''F'' is a coherent ''O''<sub>''X''</sub>-module. Then there exists a partition of ''S'' into locally closed subsets ''S''<sub>1</sub>, ..., ''S''<sub>''n''</sub> with the following property: Give each ''S''<sub>''i''</sub> its reduced scheme structure, denote by ''X''<sub>''i''</sub> the [fiber product](/source/fiber_product_of_schemes) {{nowrap|''X'' ×<sub>''S''</sub> ''S''<sub>''i''</sub>}}, and denote by ''F''<sub>''i''</sub> the restriction {{nowrap|''F'' ⊗<sub>''O''<sub>''S''</sub></sub> ''O''<sub>''S''<sub>''i''</sub></sub>}}; then each ''F''<sub>''i''</sub> is flat.

== Generic freeness ==
Generic flatness is a consequence of the generic freeness lemma. Generic freeness states that if ''A'' is a [noetherian](/source/noetherian_ring) [integral domain](/source/integral_domain), ''B'' is a [finite type](/source/algebra_of_finite_type) ''A''-algebra, and ''M'' is a finite type ''B''-module, then there exists a non-zero element ''f'' of ''A'' such that ''M''<sub>''f''</sub> is a free ''A''<sub>''f''</sub>-module.<ref>EGA IV<sub>2</sub>, Lemme 6.9.2</ref> Generic freeness can be extended to the [graded](/source/graded_ring) 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 ''M''<sub>''f''</sub> is free.<ref>Eisenbud, Theorem 14.4</ref>

Generic freeness is proved using Grothendieck's technique of [dévissage](/source/d%C3%A9vissage). Another version of generic freeness can be proved using [Noether's normalization lemma](/source/Noether's_normalization_lemma).

== References ==
<references/>

== Bibliography ==
* {{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra with a view toward algebraic geometry | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1 |mr=1322960 | year=1995 | volume=150}}
* {{EGA|book=IV-2}}

Category:Algebraic geometry
Category:Commutative algebra
Category:Theorems in abstract algebra

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