In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, a flat module over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'' is an ''R''-
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 ...
''M'' such that taking the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
over ''R'' with ''M'' preserves
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
s. 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
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,
:f_P\colon \mathcal_ \t ...
.
Definition
A module over a ring is ''flat'' if the following condition is satisfied: for every injective
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
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 ideals into .
Equivalently, an -module is flat if the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
with is an
exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much o ...
; that is if, for every
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
of -modules
the sequence
is also exact. (This is an equivalent definition since the tensor product is a
right exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much o ...
.)
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
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
.
Characterizations
Flatness can also be characterized by the following equational condition, which means that -
linear relation
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.
More precisely, if e_1,\dots,e_n are elements of a (left) module over a ring ( ...
s 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 module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type.
Related concepts inclu ...
s 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 ring
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
s.
An integral domain over which every torsion-free module is flat is called a
Prüfer domain In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely gene ...
.
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
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 ...
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
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type.
Related concepts inclu ...
(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
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
, every finitely generated flat module is projective, since every finitely generated module is finitely presented. The same result is true over an
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 ...
, even if it is not Noetherian.
On a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
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 sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
s whose terms belong to a fixed field . It is a commutative ring with addition and multiplication defined componentwise. This ring is
absolutely flat
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the element ...
(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
Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence ...
(that is an element equal to its square). In particular, if is an
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 ...
,
is flat only if
equals or is the
zero ideal
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive identi ...
.
* 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
In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented.
Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over cohe ...
(that is, every finitely generated ideal is finitely presented).
Flat ring extensions
A
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preservi ...
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 In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold:
* 1 \in S,
* xy \in S for all x, y \in S.
In other words, ''S'' is closed under taking finite ...
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 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 ...
commutative ring
the
completion of
with respect to
is flat. It is faithfully flat if and only if
is contained in the
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yie ...
of
(See also
Zariski ring In commutative algebra, a Zariski ring is a commutative Noetherian topological ring ''A'' whose topology is defined by an ideal \mathfrak a contained in the Jacobson radical, the intersection of all maximal ideals. They were introduced by under the ...
.)
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
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
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
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
, 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 ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
s. They are often expressed by saying that flatness is a
local property In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points).
Pr ...
.
Flat morphisms of schemes
A morphism
of
schemes is a
flat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,
:f_P\colon \mathcal_ \t ...
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 and
deformation to normal cone.
Let