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

TL;DR

This paper develops an explicit construction of complete cotorsion pairs in the relative module context, providing new methods for understanding their structure via cofiltrations and coinduction functors.

Contribution

It introduces a self-dual, explicit construction of complete cotorsion pairs in the relative setting, extending classical results with new cofiltration techniques and duality principles.

Findings

Characterization of cotorsion pairs via cofiltrations coinduced from base rings

Conditions under which cotorsion pairs are preserved under coinduction functors

Application to cotorsion pairs in derived categories of curved DG-modules

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.