Lesson plan

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, 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.

Back to article

Article summary

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.

Key claims

No claims recorded.

Discussion questions

  1. Summarise the article in your own words. Which single claim would you verify first, and why?
  2. Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
  3. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

Bibliography

No sources recorded.

Key claims and bibliography are generated from the same tracked claims and sources as the article page.