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

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