An explicit self-dual construction of complete cotorsion pairs in the relative context

Abstract

Let $R\to A$ be a homomorphism of associative rings, and let $(\mathcal F,\mathcal C)$ be a hereditary complete cotorsion pair in $R\mathsf{-Mod}$. Let $(\mathcal F_A,\mathcal C_A)$ be the cotorsion pair in $A\mathsf{-Mod}$ in which $\mathcal F_A$ is the class of all left $A$-modules whose underlying $R$-modules belong to $\mathcal F$. Assuming that the $\mathcal F$-resolution dimension of every left $R$-module is finite and the class $\mathcal F$ is preserved by the coinduction functor $\operatorname{Hom}_R(A,-)$, we show that $\mathcal C_A$ is the class of all direct summands of left $A$-modules finitely (co)filtered by $A$-modules coinduced from $R$-modules from $\mathcal C$. Assuming that the class $\mathcal F$ is closed under countable products and preserved by the functor $\operatorname{Hom}_R(A,-)$, we prove that $\mathcal C_A$ is the class of all direct summands of left $A$-modules cofiltered by $A$-modules coinduced from $R$-modules from $\mathcal C$, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from $\mathcal F$ have finite $\mathcal F$-resolution dimension bounded by $k$, involves cofiltrations indexed by the ordinal $\omega+k$. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra arXiv:0708.3398. In addition, we discuss the $n$-cotilting and $n$-tilting cotorsion pairs, for which we obtain better results using a suitable version of a classical Bongartz-Ringel lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.