In
algebraic geometry 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. Promi ...
, the theorems of generic flatness and generic freeness state that under certain hypotheses, a
sheaf of
modules on a
scheme is
flat or
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 ofte ...
, 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 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 s ...
, ''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. Another version of generic freeness can be proved using
Noether's normalization lemma.
References
Bibliography
*
* {{EGA, book=IV-2
Algebraic geometry
Commutative algebra
Theorems in abstract algebra