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

Positselski, Leonid

doi:10.4171/rsmup/118

Positselski, Leonid

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

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Résumé

Let $R \to A$ 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.

Publié le : 2023-02-16

MR Zbl

DOI : 10.4171/rsmup/118

Classification : 16, 17

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     author = {Positselski, Leonid},
     title = {An explicit self-dual construction of complete cotorsion pairs in the relative context},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {191--253},
     volume = {149},
     year = {2023},
     doi = {10.4171/rsmup/118},
     mrnumber = {4575369},
     zbl = {07688322},
     language = {en},
     url = {http://www.numdam.org/articles/10.4171/rsmup/118/}
}

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Positselski, Leonid. An explicit self-dual construction of complete cotorsion pairs in the relative context. Rendiconti del Seminario Matematico della Università di Padova, Tome 149 (2023), pp. 191-253. doi : 10.4171/rsmup/118. http://www.numdam.org/articles/10.4171/rsmup/118/

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