In this paper we give two different proofs that the flat cover conjecture is true: that is, every module has a flat cover. The two proofs are of completely different nature, and, we hope, will have different applications. The first of the two proofs (due to the third author) is essentially an application of the work of P. Eklof and J. Trlifaj (work which is more set‐theoretic). The second proof (due to the first two authors) is more direct, and has a model‐theoretic flavour.

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We find conditions on F , a class of objects of a Grothendieck category, sufficient for the existence of F -covers. The theory includes the existence of flat covers of modules. Mathematics Subject Classifications (2000)Primary 16D40 · Secondary 16D90

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