Generic flatness
title: "Generic flatness" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["algebraic-geometry", "commutative-algebra", "theorems-in-abstract-algebra"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Generic_flatness" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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.
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, 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 OSi; 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.
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
References
- EGA IV2, Théorème 6.9.1
- EGA IV2, Corollaire 6.9.3
- EGA IV2, Lemme 6.9.2
- Eisenbud, Theorem 14.4
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