Flat module

In abstract 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.

Vector spaces over a field are flat modules. Free modules, or more generally projective modules, are also flat, over any "R". Over a Noetherian local ring, flatness, projectivity, and freeness are all equivalent.

In commutative algebra, and more generally in algebraic geometry, flatness has come to play a major role since Serre's paper "Géometrie Algébrique et Géométrie Analytique". See also flat morphism.

Case of commutative rings

In the case when "R" is a commutative ring, one can say that flatness for an "R"-module "M" is equivalent to tensor product with "M" being an exact functor from the category of "R"-modules to itself.

For any multiplicatively closed subset "S" of "R", the localization ring $S^\{-1\}R$ is flat as an "R"-module.

When R is Noetherian and M is a finitely-generated R-module, being flat is the same as being locally free in the following sense: M is a flat R-module if and only if for every prime ideal (or even just for every maximal ideal) P of R, the localization $M\_P$ is free as a module over the localization $R\_P$.

General rings

When "R" isn't commutative one needs the more careful statement, that (if "M" is a left "R"-module) the tensor product with "M" maps exact sequences of right "R"-modules to exact sequences of abelian groups.

Taking tensor products (over arbitrary rings) is always a right exact functor. Therefore, the "R"-module "M" is flat if and only if for any injective homomorphism "K" → "L" of "R"-modules, the induced homomorphism "K"$otimes$"M" → "L"$otimes$"M" is also injective.

Categorical colimits

In general, arbitrary direct sums and direct limits of flat modules are flat, a consequence of the fact that the tensor product commutes with direct sums and direct limits (in fact with all colimits), and that both direct sums and direct limits are exact functors. Submodules and factor modules of flat modules need not be flat in general. However we have the following result: the homomorphic image of a flat module "M" is flat if and only if the kernel is a pure submodule of "M".

D.Lazard proved in 1969 that a module "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.

An abelian group is flat (viewed as a Z-module) if and only if it is torsion-free.

Homological algebra

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 Tor_n^"R"(–, "M") = 0 for all $n\; ge\; 1$ (i.e., if and only if Tor_n^"R"("X", "M") = 0 for all $n\; ge\; 1$ and all right "R"-modules "X"). Similarly, a right "R"-module "M" is flat if and only if Tor_n^"R"("M", "X") = 0 for all $n\; ge\; 1$ and all left "R"-modules "X". 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"
* If "B" and "C" are flat, then so is "A"If "A" and "B" are flat, "C" need not be flat in general. However, it can be shown that
* If "A" is pure in "B" and "B" is flat, then "A" and "C" are flat.

Flat resolutions

A flat resolution of a module is a resolution by flat modules. Any projective resolution is therefore a flat resolution. These flat resolutions can also be used to compute the Tor functor.

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 exist for all modules over all rings, so minimal flat resolutions can take their place 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 harv|MacLane|1963 and in more recent works focussing on flat resolutions such as harv|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, harv|Richman|1997.

References

* - page 33
* | year=2000 | volume=30
* | year=1995 | volume=150
* | year=1963
* | year=1997 | journal=New Zealand Journal of Mathematics | issn=1171-6096 | volume=26 | issue=2 | pages=263–273

See also

*localization of a module
*flat morphism
*von Neumann regular ring: those rings over which "all" modules are flat.