In
algebra, a flat module over a
ring ''R'' is an ''R''-
module ''M'' such that taking the
tensor product over ''R'' with ''M'' preserves
exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact.
Flatness was introduced by in his paper ''
Géometrie Algébrique et Géométrie Analytique''. See also
flat morphism.
Definition
A module over a ring is ''flat'' if the following condition is satisfied: for every injective
linear map of -modules, the map
:
is also injective, where
is the map
induced by
For this definition, it is enough to restrict the injections
to the inclusions of
finitely generated ideal
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pre ...
s into .
Equivalently, an -module is flat if the
tensor product with is an
exact functor; that is if, for every
short exact sequence of -modules
the sequence
is also exact. (This is an equivalent definition since the tensor product is a
right exact functor.)
These definitions apply also if is a non-commutative ring, and is a left -module; in this case, , and must be right -modules, and the tensor products are not -modules in general, but only
abelian groups.
Characterizations
Flatness can also be characterized by the following equational condition, which means that -
linear relations in stem from linear relations in . An -module is flat if and only if, for every linear relation
:
with
and
, there exist elements
and
such that
:
It is equivalent to define elements of a module, and a linear map from
to this module, which maps the standard basis of
to the elements. This allow rewriting the previous characterization in terms of homomorphisms, as follows.
An -module is flat if and only if the following condition holds: for every map
where
is a finitely generated free -module, and for every finitely generated -submodule
of
the map
factors through a map to a free -module
such that
Relations to other module properties
Flatness is related to various other module properties, such as being free, projective, or torsion-free. In particular, every flat module is
torsion-free, every
projective module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...
is flat, and every
free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
is projective.
There are
finitely generated modules that are flat and not projective. However, finitely generated flat modules are all projective over the rings that are most commonly considered.
This is partly summarized in the following graphic.
Torsion-free modules
Every flat module is
torsion-free. This results from the above characterization in terms of relations by taking
The converse holds over the integers, and more generally over
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
s and
Dedekind rings.
An integral domain over which every torsion-free module is flat is called a
Prüfer domain.
Free and projective modules
A module is
projective if and only if there is a
free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
and two linear maps
and
such that
In particular, every free module is projective (take
and
Every projective module is flat. This can be proven from the above characterizations of flatness and projectivity in terms of linear maps by taking
and
Conversely,
finitely generated flat modules are projective under mild conditions that are generally satisfied in
commutative algebra and
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 ...
. This makes the concept of flatness useful mainly for modules that are not finitely generated.
A
finitely presented module (that is the quotient of a finitely generated free module by a finitely generated submodule) that is flat is always projective. This can be proven by taking surjective and
in the above characterization of flatness in terms of linear maps. The condition
implies the existence of a linear map
such that
and thus
As is surjective, one has thus
and is projective.
Over a
Noetherian ring, every finitely generated flat module is projective, since every finitely generated module is finitely presented. The same result is true over an
integral domain, even if it is not Noetherian.
On a
local ring every finitely generated flat module is free.
A finitely generated flat module that is not projective can be built as follows. Let
be the set of the
infinite sequences whose terms belong to a fixed field . It is a commutative ring with addition and multiplication defined componentwise. This ring is
absolutely flat (that is, every module is flat). The module
where is the ideal of the sequences with a finite number of nonzero terms, is thus flat and finitely generated (only one generator), but it is not projective.
Non-examples
* If is an ideal in a Noetherian commutative ring , then
is not a flat module, except if is generated by an
idempotent (that is an element equal to its square). In particular, if is an
integral domain,
is flat only if
equals or is the
zero ideal.
* Over an integral domain, a flat module is
torsion free. Thus a module that contains nonzero torsion elements is not flat. In particular
and all fields of positive characteristics are non-flat
-modules, where
is the ring of integers, and
is the field of the rational numbers.
Direct sums, limits and products
A
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of modules is flat if and only if each
is flat.
A
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
of flat is flat. In particular, a direct limit of
free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
s is flat. Conversely, every flat module can be written as a direct limit of
finitely-generated free modules.
Direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
s of flat modules need not in general be flat. In fact, given a ring , every direct product of flat -modules is flat if and only if is a
coherent ring (that is, every finitely generated ideal is finitely presented).
Flat ring extensions
A
ring homomorphism is ''flat'' if is a flat -module for the module structure induced by the homomorphism. For example, the polynomial ring is flat over , for any ring .
For any
multiplicative subset of a commutative ring
, the
localization ring is flat over (it is
projective only in exceptional cases). For example,
is flat and not projective over
If
is an ideal of a
Noetherian commutative ring
the
completion of
with respect to
is flat. It is faithfully flat if and only if
is contained in the
Jacobson radical of
(See also
Zariski ring.)
Localization
In this section, denotes a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
. If
is a
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
of , the
localization at
is, as usual, denoted with
as an index. That is,
and, if is an -module,
If an -module is flat, then
is a flat
-module for every prime ideal
Conversely, if
is a flat
-module for every
maximal ideal , then is a flat -module (and
is a flat
-module for every prime ideal
).
These properties are fundamental in commutative algebra, since they reduce the question of flatness to the case of
local rings. They are often expressed by saying that flatness is a
local property.
Flat morphisms of schemes
A morphism
of
schemes is a
flat morphism if the induced map on local rings
:
is a flat ring homomorphism for any point in . Thus, properties of flat (or faithfully flat) ring homomorphisms extends naturally to geometric properties of flat morphisms in algebraic geometry. For example, consider the previous example of
. The inclusion
then determines the flat morphism
:
Each (geometric) fiber
is the curve of equation
See also
flat degeneration In algebraic geometry, a degeneration (or specialization) is the act of taking a limit of a family of varieties. Precisely, given a morphism
:\pi: \mathcal \to C,
of a variety (or a scheme) to a curve ''C'' with origin 0 (e.g., affine or projective ...
and
deformation to normal cone.
Let