Generic flatness

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 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.^[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 O_X module. Then there exists a partition of S into locally closed subsets S₁, ..., S_n with the following property: Give each S_i its reduced scheme structure, denote by X_i the fiber product X ×_S S_i, and denote by F_i the restriction F ⊗_{O_S} O_{S_i}; 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.^[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 M_f 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.

References

^ EGA IV₂, Théorème 6.9.1
^ EGA IV₂, Corollaire 6.9.3
^ EGA IV₂, Lemme 6.9.2
^ Eisenbud, Theorem 14.4

Bibliography

Eisenbud, 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; Dieudonné, Jean (1965). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie". Publications Mathématiques de l'IHÉS. 24. doi:10.1007/bf02684322. MR 0199181.

[1] EGA IV₂, Théorème 6.9.1

[2] EGA IV₂, Corollaire 6.9.3

[3] EGA IV₂, Lemme 6.9.2

[4] Eisenbud, Theorem 14.4

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