Covers, precovers, and purity

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Covers in finitely accessible categories (2010, methodology): “We first extend the above result to modules over a ring with enough idempotents, giving a proof based on an idea, different from that of [27], employed in [30] for showing that the class of absolutely pure modules over a right coherent ring is covering.”
Finiteness conditions and cotorsion pairs (2015, methodology): “esis R ∈ FP∞-Inj and [BGH14, Proposition 2.7]. We conclude this section by showing that the class FP n-Flat is also the left half of a perfect cotorsion pair, for every ring R. We use the result from [Hr08] again for that purpose. Observe that in Theorem 2.14 from [BGH14], the authors prove that the”
Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves (2013, methodology): “It is fair to remark that there is an alternative approach to approximation theory, namely Bican’s and El Bashir’s proof of the Flat Cover Conjecture in [8] and its follow ups [16, 20, 40, 63], which does not seem to fit in our framework.”