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

Positselski, Leonid

Rendiconti del Seminario Matematico della Università di Padova, Tome 149 (2023), pp. 191-253.

Let $R\to A$ be a homomorphism of associative rings, and let $(ℱ,𝒞)$ be a hereditary complete cotorsion pair in $R-\mathrm{𝖬𝗈𝖽}$. Let $({ℱ}_A,{𝒞}_A)$ be the cotorsion pair in $A-\mathrm{𝖬𝗈𝖽}$ in which ${ℱ}_A$ is the class of all left $A$-modules whose underlying $R$-modules belong to $ℱ$. Assuming that the $ℱ$-resolution dimension of every left $R$-module is finite and the class $ℱ$ is preserved by the coinduction functor ${Hom}_R(A,-)$, we show that ${𝒞}_A$ is the class of all direct summands of left $A$-modules finitely (co)filtered by $A$-modules coinduced from $R$-modules from $𝒞$. Assuming that the class $ℱ$ is closed under countable products and preserved by the functor ${Hom}_R(A,-)$, we prove that ${𝒞}_A$ is the class of all direct summands of left $A$-modules cofiltered by $A$-modules coinduced from $R$-modules from $𝒞$, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from $ℱ$ have finite $ℱ$-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 (2010). 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.

Classification : 16, 17