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 Jean-Pierre Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique.
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A left module M over a ring R is flat if the following condition is satisfied: for every injective linear map of right R-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 R.
Equivalently, an R-module M is flat if the tensor product with M is an exact functor; that is if, for every short exact sequence of R-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 R is a non-commutative ring, and M is a left R-module; in this case, K, L and J must be right R-modules, and the tensor products are not R-modules in general, but only abelian groups.
Flatness can also be characterized by the following equational condition, which means that R-linear relations in M stem from linear relations in R.
A left R-module M is flat if and only if, for every linear relation
with and , there exist elements and such that1
and
It is equivalent to define n elements of a module, and a linear map from to this module, which maps the standard basis of to the n elements. This allows rewriting the previous characterization in terms of homomorphisms, as follows.
An R-module M is flat if and only if the following condition holds: for every map where is a finitely generated free R-module, and for every finitely generated R-submodule of the map factors through a map g to a free R-module such that
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. Moreover, a finitely generated module is flat if and only it is locally free, meaning all the localizations at prime ideals are free modules.
This is partly summarized in the following graphic.
Every flat module is torsion-free. This results from the above characterization in terms of relations by taking m = 1.
The converse holds over the integers, and more generally over principal ideal domains and .
A ring homomorphism is flat if S is a flat R-module for the module structure induced by the homomorphism. For example, the polynomial ring R[t] is flat over R, for any ring R.
For any multiplicative subset of a commutative ring , the localization is a flat R-algebra (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.6 It is faithfully flat if and only if is contained in the Jacobson radical of (See also Zariski ring.)7
In this section, R denotes a commutative ring. If is a prime ideal of R, the localization at is, as usual, denoted with as an index. That is, and, if M is an R-module,
If M is an R-module the three following conditions are equivalent:
This property is fundamental in commutative algebra and algebraic geometry, since it reduces the study of flatness to the case of local rings. They are often expressed by saying that flatness is a local property.
The definition of a flat morphism of schemes results immediately from the local property of flatness.
A morphism of schemes is a flat morphism if the induced map on local rings
is a flat ring homomorphism for any point x in X.
Thus, properties of flat (or faithfully flat) ring homomorphisms extend naturally to geometric properties of flat morphisms in algebraic geometry.
For example, consider the flat -algebra (see below). The inclusion induces the flat morphism
The (geometric) fiber is the curve given by the equation (See also flat degeneration and .)
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-commutative 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.
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 second-last 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 .
Flatness may also be expressed using the Tor functors, the left derived functors of the tensor product. A left -module 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 -module 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 and are flat, then so is . Also, if and are flat, then so is . If and are flat, need not be flat in general. However, if is pure in and is flat, then and are flat.
A flat resolution of a module is a resolution of the form
where the 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 admits a finite flat resolution, the minimal length among all finite flat resolutions of is called its flat dimension14 and denoted . If does not admit a finite flat resolution, then by convention the flat dimension is said to be infinite. As an example, consider a module such that . In this situation, the exactness of the sequence indicates that the arrow in the center is an isomorphism, and hence itself is flat.c
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.
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 Enochs (1981, p. 196). The conjecture turned out to be true, resolved positively and proved simultaneously by L. Bican, R. El Bashir and E. Enochs.15 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 Mac Lane (1963) and in more recent works focussing on flat resolutions such as Enochs and Jenda (2000).
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.16
An integral domain over which every torsion-free module is flat is called a Prüfer domain.
A module M is projective if and only if there is a free module G 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 f 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 f is surjective, one has thus and M 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.2
On a local ring every finitely generated flat module is free.3
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 F. 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 I 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.
The direct sum of modules is flat if and only if each is flat.
The direct limit of flat modules is flat. In particular, the direct limit of free modules is flat. Conversely, every flat module can be written as the direct limit of finitely-generated free modules.4
Direct products of flat modules need not in general be flat. In fact, given a ring R, every direct product of flat R-modules is flat if and only if R is a coherent ring (that is, every finitely generated ideal is finitely presented).5
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).8 An example is9 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.
The two last examples are implicitly behind the wide use of localization in commutative algebra and algebraic geometry.
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.
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 , 13
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