Editing Flat module (section)

== Flat covers ==

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 {{harvs|txt|last=Enochs|year=1981|loc=p. 196}}. The conjecture turned out to be true, resolved positively and proved simultaneously by L. Bican, R. El Bashir and E. Enochs.{{sfn|Bican|El Bashir|Enochs|2001|ps=none}} 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 {{harvs|txt|last=Mac Lane|year=1963}} and in more recent works focussing on flat resolutions such as {{harvs|txt|last1=Enochs|last2=Jenda|year=2000}}.