Flat cover

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title: "Flat cover" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["module-theory"] topic_path: "general/module-theory" source: "https://en.wikipedia.org/wiki/Flat_cover" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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 . The conjecture turned out to be true, resolved positively and proved simultaneously by . 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

  • {{citation |last=Enochs | first= Edgar E. |title=Injective and flat covers, envelopes and resolvents |journal=Israel Journal of Mathematics |volume=39 |year=1981 |number=3 |pages=189–209 |issn=0021-2172 |mr=636889 |doi=10.1007/BF02760849 | doi-access=}}
  • {{citation |last1=Bican |first1=L. |last2=El Bashir |first2=R. |last3=Enochs |first3=E. |title=All modules have flat covers |journal=Bulletin of the London Mathematical Society |volume=33 |year=2001 |number=4 |pages=385–390 |issn=0024-6093 |mr=1832549 |doi=10.1017/S0024609301008104 }}

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module-theory