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 Fn+1 is the flat cover of the kernel of Fn → Fn−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 J. Math., 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", Bull. London Math. Soc., 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, 1634, Berlin: Springer-Verlag, doi:10.1007/BFb0094173, ISBN 3-540-61640-3, MR 1438789
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