BIREP:
Representation Theory of Algebras:
Striking New Results.
(Compiled by
C.M.Ringel)
Nr.3:
Enochs: Proof of the flat cover conjecture.
Reference: Enochs, Edgar E. / Overtoun M. G. Jenda
Relative Homological Algebra. De Gruyter (2000), Theorem 7.4.4:
Theorem. Any module over any ring has a flat cover.
This means the following: Let R be a ring and M an R-module. Then there
exists a map p : F → M with F flat and the following properties:
- For any map p' : F' → M with F' flat, there is a map f : F' → F
with p' = pf. (precover property)
- Any endomorphism e : F → F with p = pe is an automorphism
(minimality).
(A flat cover is also called a minimal right F-approximation,
where F is the class of flat modules; and the existence of a
minimal right F-approximation asserts that F
is a contravariantly
finite subcategory of the category of all R-modules.)
Begin: 14.2.1998. Updated: 22.11.2000
Fakultät für Mathematik, C.M.Ringel
E-Mail:
ringel@mathematik.uni-bielefeld.de