Flat module
Formally, a module M over a ring R is flat if taking the tensor product over R with M preserves exact sequences.
Flatness was introduced by Jean-Pierre Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique.
A left module M over a ring R is flat if the following condition is satisfied: for every injective linear map
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
(This is an equivalent definition since the tensor product is a right exact functor.)
such that[1] and It is equivalent to define n elements of a module, and a linear map from
An R-module M is flat if and only if the following condition holds: for every map
Flatness is related to various other module properties, such as being free, projective, or torsion-free.
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.
The converse holds over the integers, and more generally over principal ideal domains and Dedekind rings.
This can be proven from the above characterizations of flatness and projectivity in terms of linear maps by taking
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.
[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.
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.
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).
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.
(See also flat degeneration and deformation to normal cone.)
(this is a special case of the fact that a faithfully flat quasi-compact morphism of schemes has this property.[10]).
are the alternating sums of the maps obtained by inserting 1 in each spot; e.g.,
is faithfully flat if and only if the theorem of transition holds for it; that is, for each
[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.
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.
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.
[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).
