# 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.

## Definitions

The homomorphism *F*→*M* 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*.

## 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

- →
*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

- 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 636889 (83a:16031) - 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 - Hazewinkel, Michiel, ed. (2001), "Flat cover",
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - 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