In homological algebra and algebraic geometry, 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.
A module M over a ring R is called flat if the following condition is satisfied: for any injective map of R-modules, the map
induced by is injective.
In other words, for R-modules K, L, and J, if is a short exact sequence, then M is a flat module over R if and only if is also a short exact sequence.
This definition applies also if R is not necessarily commutative, and M is a left R-module and K and L right R-modules. The only difference is that in this case and are not in general R-modules, but only abelian groups.
Characterizations of flatness
Since tensoring with M is, for any module M, a right exact functor
(between the category of R-modules and abelian groups), M is flat if and only if the preceding functor is exact.
It can also be shown in the condition defining flatness as above, it is enough to take , the ring itself, and a finitely generated ideal of R.
Flatness is also equivalent to the following equational condition, which may be paraphrased by saying that R-linear relations that hold in M stem from linear relations which hold in R: for every linear dependency, with and , there exist a matrix and an element such that and Furthermore, M 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 g to a free -module such that
Examples and relations to other notions
Flatness is related to various other conditions on a module, such as being free, projective, or torsion-free. This is partly summarized in the following graphic:
Free or projective modules vs. flat modules
Free modules are flat over any ring R. This holds since the functor
is exact. For example, vector spaces over a field are flat modules. Direct summands of flat modules are again flat. In particular, projective modules (direct summands of free modules) are flat. Conversely, for a commutative Noetherian ring R, finitely generated flat modules are projective.
Flat vs. torsion-free modules
Any flat module is torsion-free. The converse holds over the integers, and more generally over principal ideal domains. This follows from the above characterization of flatness in terms of ideals. Yet more generally, this converse holds over Dedekind rings.
An integral domain is called a Prüfer domain if every torsion-free module over it is flat.
Flatness of completions
Quotients of flat modules are not in general flat. For example, for each integer is not flat over because is injective, but tensored with it is not. Similarly, is not flat over
Further permanence properties
In general, arbitrary direct sums and filtered colimits (also known as direct limits) of flat modules are flat, a consequence of the fact that the tensor product commutes with direct sums and filtered colimits (in fact with all colimits), and that both direct sums and filtered colimits are exact functors. In particular, this shows that all filtered colimits of free modules are flat.
Lazard (1969) proved that the converse holds as well: M is flat if and only if it is a direct limit of finitely-generated free modules. As a consequence, one can deduce that every finitely-presented flat module is projective. The direct sum is flat if and only if each is flat.
Products of flat R-modules need not in general be flat. In fact, Chase (1960) showed a ring R is coherent (i.e., any finitely generated ideal is finitely presented) if and only if arbitrary products of flat R-modules are again flat.
Flat ring extensions
If is a ring homomorphism, S is called flat over R (or a flat R-algebra) if it is flat as an R-module. For example, the polynomial ring R[t] is flat over R, for any ring R. Moreover, for any multiplicatively closed subset of a commutative ring , the localization ring is flat over R. For example, is flat over (though not projective).
Let be a polynomial ring over a noetherian ring and a nonzerodivisor. Then is flat over if and only if is primitive (the coefficients generate the unit ideal). This yields an example of a flat module that is not free.
Flat ring extensions are important in algebra, algebraic geometry and related areas. A morphism of schemes is a flat morphism if, by one of several equivalent definitions, the induced map on local rings
is a flat ring homomorphism for any point x in X. Thus, the above-mentioned properties of flat (or faithfully flat) morphisms established by methods of commutative algebra translate into geometric properties of flat morphisms in algebraic geometry.
Local aspects of flatness over commutative rings
In this section, the ring R is supposed to be commutative. In this situation, flatness of R-modules is related in several ways to the notion of localization: M is flat if and only if the module is a flat -module for all prime ideals of R. In fact, it is enough to check the latter condition only for the maximal ideals, as opposed to all prime ideals. This statement reduces the question of flatness to the case of (commutative) local rings.
If R is a local (commutative) ring and either M is finitely generated or the maximal ideal of R is nilpotent (e.g., an artinian local ring) then the standard implication "free implies flat" can be reversed: in this case M is flat if and if only if its free.
The local criterion for flatness states:
- Let R be a local noetherian ring, S a local noetherian R-algebra with , and M a finitely generated S-module. Then M is flat over R if and only if
The significance of this is that S need not be finite over R and we only need to consider the maximal ideal of R instead of an arbitrary ideal of R.
Faithfully flat ring homomorphism
Let A be a ring (assumed to be commutative throughout this section) and B an A-algebra, i.e., a ring homomorphism . Then B has the structure of an A-module. Then B is said to be flat over A (resp. faithfully flat over A) if it is flat (resp. faithfully flat) as an A-module.
There is a basic characterization of a faithfully flat ring homomorphism: given a flat ring homomorphism , the following are equivalent.
- is faithfully flat.
- For each maximal ideal of ,
- If is a nonzero -module, then
- Every prime ideal of A is the inverse image under f of a prime ideal in B. In other words, the induced map is surjective.
- A is a pure subring of B (in particular, a subring); here, "pure subring" means that is injective for every -module .
Condition 2 implies a flat local homomorphism between local rings is faithfully flat. It follows from condition 5 that for every ideal (take ); in particular, if is a Noetherian ring, then is a Noetherian ring.
Condition 4 can be stated in the following strengthened form: is submersive: the topology of is the quotient topology of (this is a special case of the fact that a faithfully flat quasi-compact morphism of schemes has this property.) It compares to an integral extension of an integrally closed domain. See also flat morphism#Properties of flat morphisms for further information.
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, is faithfully flat if and only if the theorem of transition holds for it; i.e., for each -primary ideal of ,
Example. For a ring is faithfully flat. More generally, an -algebra that is free of positive rank as an -module is faithfully flat.
Example. Let be a ring and elements generating the unit ideal of Then
is faithfully flat since localizations are flat, their direct sums are then flat and
- The product of the local rings of a commutative ring is a faithfully flat module.
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.
Homological characterization using Tor functors
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 restrictiely, for any finitely generated ideal (instead of N).
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, as is shown by the above non-example . However, if A is pure in B and B is flat, then A and C are flat.
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 Fn is nonzero and Fi = 0 for i > n. 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.
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 (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. 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 (MacLane 1963) and in more recent works focussing on flat resolutions such as (Enochs & Jenda 2000).
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.
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