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 OX-module, then there is a non-empty open subset U of Y such that the restriction of F to u1(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

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 an 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.

References

  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

Bibliography

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