# zbMATH — the first resource for mathematics

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.)
Full Text: