Flat module

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 formula_1 of -modules, the map
is also injective, where formula_3 is the map
induced by formula_4
For this definition, it is enough to restrict the injections formula_5 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 formula_6 the sequence formula_7 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 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 formula_9 and formula_10, there exist elements formula_11 and formula_12 such that
It is equivalent to define elements of a module, and a linear map from formula_14 to this module, which maps the standard basis of formula_14 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 formula_16 where formula_17 is a finitely generated free -module, and for every finitely generated -submodule formula_18 of formula_19 the map formula_20 factors through a map to a free -module formula_21 such that formula_22
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 free, and every free module is projective. 
There are finitely generated modules that are flat and not projective. However, finitely generated 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 formula_23 and formula_24 such that formula_25 In particular, every free module is projective (take formula_26 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 formula_27 and formula_28
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 formula_29 in the above characterization of flatness in terms of linear maps. The condition formula_30 implies the existence of a linear map formula_23 such that formula_32 and thus formula_33 As is surjective, one has thus formula_34 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 formula_35 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 formula_36 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.
Direct sums, limits and products.
A direct sum formula_44 of modules is flat is and only if each formula_45 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 formula_46 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 formula_47 of a commutative ring formula_48, the localization ring formula_49 is flat over (it is projective only in exceptional cases). For example, formula_43 is flat and not projective over formula_51
If formula_39 is an ideal of a Noetherian commutative ring formula_53 the completion formula_54 of formula_48 with respect to formula_39 is flat. It is faithfully flat if and only if formula_39 is contained in the Jacobson radical of formula_58 (See also Zariski ring.)
Localization.
In this section, denotes a commutative ring. If formula_59 is a prime ideal of , the localization at formula_59 is, as usual, denoted with formula_59 as an index. That is, formula_62 and, if is an -module, formula_63
If an -module if flat, then formula_64 is a flat formula_65-module for every prime ideal formula_66
Conversely, if formula_67 is a flat formula_68-module for every maximal ideal formula_69, then is a flat -module (and formula_64 is a flat formula_65-module for every prime ideal formula_59).
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 formula_73 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 formula_75. The inclusion formula_76 then determines the flat morphism
Each (geometric) fiber formula_78 is the curve of equation formula_79 See also flat degeneration and deformation to normal cone.
Let formula_80 be a polynomial ring over a commutative Noetherian ring formula_48 and formula_82 a nonzerodivisor. Then formula_83 is flat over formula_48 if and only if formula_20 is primitive (the coefficients generate the unit ideal). An example ispg 3 formula_86 which is flat (and even free) over formula_87 (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 only this case that is considered here, even if some results can be generalizes to the case of modules over a non-commutaive ring.
In this section, formula_88 is a ring homomorphism of commutative rings, which gives to formula_47 the structures of an formula_48-algebra and an formula_48-module. If formula_47 is a formula_48-module flat (or faithfully flat), one says commonly that formula_47 is flat (or faithfully flat) over formula_95 and that formula_20 is flat (or faithfully flat).
If formula_47 is flat over formula_53 the following conditions are equivalent.
The second condition implies implies that a flat homomorphism of local rings is faithfully flat. It follows from the last condition that formula_119 for every ideal formula_39 of formula_48 (take formula_122). In particular, if formula_47 is a Noetherian ring, then formula_48 is also Noetherian.
The last but one condition can be stated in the following strengthened form: formula_125 is "submersive", which means that the Zariski topology of formula_126 is the quotient topology of that of formula_127 (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.
The two last examples are implictly behind the wide use of localization in commutative algebra and algebraic geometry.
Faithfully flat local homomorphisms.
Here is one characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism formula_147 such that formula_148 is an formula_149-primary ideal, the homomorphism formula_150 is faithfully flat if and only if the theorem of transition holds for it; that is, for each formula_69-primary ideal formula_152 of formula_48, formula_154
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
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 formula_158 and formula_159 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 "F""i" 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 formula_162 is nonzero and formula_163 for formula_164. 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 → "F"0 → "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 automoprhism. 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.