In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion free modules. Formally, a module M over a ring R is flat if  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.

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 ideals 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 allows 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 is flat, and every free module 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 domains 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  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. 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  of modules is flat if and only if each  is flat.

A direct limit of flat is flat. In particular, a direct limit of free modules is flat. Conversely, every flat module can be written as a direct limit of finitely-generated free modules.

Direct products 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. If  is a prime ideal 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 and deformation to normal cone.

Let  be a polynomial ring over a commutative Noetherian ring  and  a nonzerodivisor. Then  is flat over  if and only if  is primitive (the coefficients generate the unit ideal). An example ispg 3  which is flat (and even free) over  (see also below for the geometric meaning). Such flat extensions can be used to yield examples of flat modules that are not free and do not result from a localization.

Faithful flatness 
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. Although the concept is defined for modules over a non-necessary commutative ring, it is used mainly for commutative algebras. So, this is the only case that is considered here, even if some results can be generalized to the case of modules over a non-commutaive ring.

In this section,  is a ring homomorphism of commutative rings, which gives to  the structures of an -algebra and an -module. If  is a -module flat (or faithfully flat), one says commonly that  is flat (or faithfully flat) over  and that  is flat (or faithfully flat).

If  is flat over  the following conditions are equivalent.
  is faithfully flat.
 For each maximal ideal  of , one has 
 If  is a nonzero -module, then 
 For every prime ideal  of  there is a prime ideal  of  such that  In other words, the map  induced by  on the spectra is surjective.
  is injective, and  is a pure subring of  that is,  is injective for every -module .

The second condition implies that a flat local homomorphism of local rings is faithfully flat. It follows from the last condition that  for every ideal  of  (take ). In particular, if  is a Noetherian ring, then  is also Noetherian.

The last but one condition can be stated in the following strengthened form:  is submersive, which means that the Zariski topology of  is the quotient topology of  that of  (this is a special case of the fact that a faithfully flat quasi-compact morphism of schemes has this property.). See also Flat morphism#Properties of flat morphisms.

Examples
A ring homomorphism  such that  is a nonzero free -module is faithfully flat. For example:
Every field extension is faithfully flat. This property is implicitly behind the use of complexification for proving results on real vector spaces.
A polynomial ring is a faithfully flat extension of its ring of coefficients.
If  is a monic polynomial, the inclusion  is faithfully flat.
Let  The direct product  of the localizations at the  is faithfully flat over  if and only if  generate the unit ideal of  (that is, if  is a linear combination of the ).
The direct sum of the localizations  of  at all its prime ideals is a faithfully flat module that is not an algebra, except if there are finitely many prime ideals.

The two last examples are implicitly behind the wide use of localization in commutative algebra and algebraic geometry.

For a given ring homomorphism  there is an associated complex called the Amitsur complex:where the coboundary operators  are the alternating sums of the maps obtained by inserting 1 in each spot; e.g., . Then (Grothendieck) this complex is exact if  is faithfully flat.

Faithfully flat local homomorphisms
Here is one characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism  such that  is an -primary ideal, the homomorphism  is faithfully flat if and only if the theorem of transition holds for it; that is, for each -primary ideal  of ,

Homological characterization using Tor functors
Flatness may also be expressed using the Tor functors, the left derived functors of the tensor product. A left R-module M is flat if and only if

 for all  and all right R-modules X).
In fact, it is enough to check that the first Tor term vanishes, i.e., M is flat if and only if

for any R-module N or, even more restrictively, when  and  is any finitely generated ideal.

Using the Tor functor's long exact sequences, one can then easily prove facts about a short exact sequence

If A and C are flat, then so is B. Also, if B and C are flat, then so is A. If A and B are flat, C need not be flat in general. However, if A is pure in B and B is flat, then A and C are flat.

Flat resolutions
A flat resolution of a module M is a resolution of the form

where the Fi are all flat modules. Any free or projective resolution is necessarily a flat resolution. Flat resolutions can be used to compute the Tor functor.

The length of a finite flat resolution is the first subscript n such that  is nonzero and  for . If a module M admits a finite flat resolution, the minimal length among all finite flat resolutions of M is called its flat dimension and denoted fd(M). If M does not admit a finite flat resolution, then by convention the flat dimension is said to be infinite. As an example, consider a module M such that fd(M) = 0. In this situation, the exactness of the sequence 0 → F0 → M → 0 indicates that the arrow in the center is an isomorphism, and hence M itself is flat.

In some areas of module theory, a flat resolution must satisfy the additional requirement that each map is a flat pre-cover of the kernel of the map to the right. For projective resolutions, this condition is almost invisible: a projective pre-cover is simply an epimorphism from a projective module. These ideas are inspired from Auslander's work in approximations. These ideas are also familiar from the more common notion of minimal projective resolutions, where each map is required to be a projective cover of the kernel of the map to the right. However, projective covers need not exist in general, so minimal projective resolutions are only of limited use over rings like the integers.

Flat covers

While projective covers for modules do not always exist, it was speculated that for general rings, every module would have a flat cover, that is, every module M would be the epimorphic image of a flat module F such that every map from a flat module onto M factors through F, and any endomorphism of F over M is an automorphism. This flat cover conjecture was explicitly first stated in . The conjecture turned out to be true, resolved positively and proved simultaneously by L. Bican, R. El Bashir and E. Enochs. This was preceded by important contributions by P. Eklof, J. Trlifaj and J. Xu.

Since flat covers exist for all modules over all rings, minimal flat resolutions can take the place of minimal projective resolutions in many circumstances. The measurement of the departure of flat resolutions from projective resolutions is called relative homological algebra, and is covered in classics such as  and in more recent works focussing on flat resolutions such as .

In constructive mathematics 

Flat modules have increased importance in constructive mathematics, where projective modules are less useful. For example, that all free modules are projective is equivalent to the full axiom of choice, so theorems about projective modules, even if proved constructively, do not necessarily apply to free modules. In contrast, no choice is needed to prove that free modules are flat, so theorems about flat modules can still apply.

See also
Generic flatness
Flat morphism
von Neumann regular ring – those rings over which all modules are flat.
Normally flat ring

References 

 
 

 

 

 - page 33

 

Homological algebra
Algebraic geometry
Module theory