Flat cover

In algebra, a flat cover of a module M over a ring is a homomorphism from a flat module 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 hulls, and are related to projective covers and torsion-free covers.


The homomorphism FM is defined to be a flat cover of M if 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.


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 (Enochs 1981, p 196). The conjecture turned out to be true, resolved positively and proved simultaneously by Bican, El Bashir & Enochs (2001). 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

F2F1F0M → 0

such that each Fn+1 is the flat cover of the kernel of FnFn−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.


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