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Complete cotorsion pairs in the category of complexes. (English) Zbl 1285.18016
A cotorsion pair in an abelian category is a pair of families of objects \(({\mathcal A},{\mathcal B})\) such that \(\text{Ext}^1(A,B)=0\) for all \(A\) in \({\mathcal A}\) if and only if \(B\in{\mathcal B}\) and such that \(\text{Ext}^1(A,B)=0\) for all \(B\) in \({\mathcal B}\) if and only if \(A\in{\mathcal A}\). In this paper the authors consider certain cotorsion pairs in the the category of chain complexes of \(R\)-modules obtained from a cotorsion pair \(({\mathcal A},{\mathcal B})\) in the category of \(R\)-modules, where \(R\) is a ring. They show that the cotorsion pairs in the category of chain complexes are complete and perfect if \({\mathcal A}\) has certain closure properties. As a consequence they show that every complex over a right coherent ring has a Gorenstein flat cover.

MSC:
18E10 Abelian categories, Grothendieck categories
18G35 Chain complexes (category-theoretic aspects), dg categories
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
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