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

Leonid Positselski

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

Let R→A be a homomorphism of associative rings, and let (ℱ,𝒞) be a hereditary complete cotorsion pair in R-𝖬𝗈𝖽. Let (ℱ A ,𝒞 A ) be the cotorsion pair in A-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 ω+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