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

→ 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 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

This article is issued from Wikipedia - version of the Saturday, January 23, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.