In algebra, a flat cover of a module ''M'' over a ring is a surjective homomorphism from a

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 seq ...

''F'' to ''M'' that is in some sense minimal. Any module over a ring has a flat cover that is unique up to (non-unique) isomorphism. Flat covers are in some sense dual to

injective hull In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in . Definition ...

s, and are related to projective covers and

torsion-free cover In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule is reduced to its ...

Definitions

The homomorphism ''F''→''M'' is defined to be a flat cover of ''M'' if it is surjective, ''F'' is flat, every homomorphism from flat module to ''M'' factors through ''F'', and any map from ''F'' to ''F'' commuting with the map to ''M'' is an automorphism of ''F''.

History

While projective covers for modules do not always exist, it was speculated that for general rings, every module would have a flat cover. This flat cover conjecture was explicitly first stated in . The conjecture turned out to be true, resolved positively and proved simultaneously by . This was preceded by important contributions by P. Eklof, J. Trlifaj and J. Xu.

Minimal flat resolutions

Any module ''M'' over a ring has a resolution by flat modules :→ ''F''₂ → ''F''₁ → ''F''₀ → ''M'' → 0 such that each ''F''_''n''+1 is the flat cover of the kernel of ''F''_''n'' → ''F''_''n''−1. Such a resolution is unique up to isomorphism, and is a minimal flat resolution in the sense that any flat resolution of ''M'' factors through it. Any homomorphism of modules extends to a homomorphism between the corresponding flat resolutions, though this extension is in general not unique.

References

* * * *{{citation, mr= 1438789, last= Xu, first= Jinzhong, title= Flat covers of modules, series= Lecture Notes in Mathematics, volume= 1634, publisher= Springer-Verlag, place= Berlin, year= 1996, isbn= 3-540-61640-3, doi= 10.1007/BFb0094173, url-access= registration, url= https://archive.org/details/flatcoversofmodu1634xuji, doi-access= free Module theory