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, {{nowrap|''u'' : ''X'' → ''Y''}} is a finite type morphism of schemes, and ''F'' is a coherent ''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>−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 {{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 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 ''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 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. Another version of generic freeness can be proved using 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 | 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