algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

and

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...

, the theorems of generic flatness and generic freeness state that under certain hypotheses, a

sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper ser ...

module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...

s on a

scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...

flat Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...

free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...

. 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 In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often w ...

, 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 In mathematics, the adjective Noetherian is used to describe Category_theory#Categories.2C_objects.2C_and_morphisms, objects that satisfy an ascending chain condition, ascending or descending chain condition on certain kinds of subobjects, meaning t ...

integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...

, ''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 In algebraic geometry, dévissage is a technique introduced by Alexander Grothendieck for proving statements about coherent sheaves on noetherian schemes. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applicati ...

. Another version of generic freeness can be proved using

Noether's normalization lemma In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field ''k'', and any finitely generated commutative ''k''-algebra ''A'', there exists a non-negativ ...

References

Bibliography

* * {{EGA, book=IV-2 Algebraic geometry Commutative algebra Theorems in abstract algebra