Flat cover

In algebra, a flat cover of a module M over a ring is a surjective 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.

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 (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

→ F₂ → F₁ → F₀ → 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

Enochs, Edgar E. (1981), "Injective and flat covers, envelopes and resolvents", Israel Journal of Mathematics, 39 (3): 189–209, doi:10.1007/BF02760849, ISSN 0021-2172, MR 0636889
Bican, L.; El Bashir, R.; Enochs, E. (2001), "All modules have flat covers", Bulletin of the London Mathematical Society, 33 (4): 385–390, doi:10.1017/S0024609301008104, ISSN 0024-6093, MR 1832549
"Flat cover", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Xu, Jinzhong (1996), Flat covers of modules, Lecture Notes in Mathematics, vol. 1634, Berlin: Springer-Verlag, doi:10.1007/BFb0094173, ISBN 3-540-61640-3, MR 1438789