generic freeness

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, is a finite type morphism of schemes, and ''F'' is a coherent ''O''_''X''-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, is a finite type morphism, and ''F'' is a coherent ''O''_''X''-module. Then there exists a partition of ''S'' into locally closed subsets ''S''₁, ..., ''S''_''n'' with the following property: Give each ''S''_''i'' its reduced scheme structure, denote by ''X''_''i'' the fiber product , and denote by ''F''_''i'' the restriction ; then each ''F''_''i'' 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''_''f'' is a free ''A''_''f''-module. 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''_''f'' is free.Eisenbud, Theorem 14.4

Generic freeness is proved using Grothendieck's technique of dévissage. Another version of generic freeness can be proved using Noether's normalization lemma.

References

Bibliography

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* {{EGA, book=IV-2

Algebraic geometry
Commutative algebra
Theorems in abstract algebra