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 {{harv|Enochs|1981|loc=p 196}}. The conjecture turned out to be true, resolved positively and proved simultaneously by {{harvtxt|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
- {{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 }}
- {{eom|id=flat_cover|title=Flat cover}}
- {{citation|mr= 1438789|last= Xu|first= Jinzhong|title= Flat covers of modules|series= Lecture Notes in Mathematics|volume= 1634|publisher= Springer-Verlag|place= Berlin|year= 1996|isbn= 3-540-61640-3|doi= 10.1007/BFb0094173|url-access= registration|url= https://archive.org/details/flatcoversofmodu1634xuji|doi-access= free}}