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

Leonid Positselski

doi:10.4171/rsmup/118

Leonid Positselski

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

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

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

@article{RSMUP_2023__149__191_0,
     author = {Leonid Positselski},
     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/}
}

TY  - JOUR
AU  - Leonid Positselski
TI  - An explicit self-dual construction of complete cotorsion pairs in the relative context
JO  - Rendiconti del Seminario Matematico della Università di Padova
PY  - 2023
SP  - 191
EP  - 253
VL  - 149
UR  - http://www.numdam.org/articles/10.4171/rsmup/118/
DO  - 10.4171/rsmup/118
LA  - en
ID  - RSMUP_2023__149__191_0
ER  -

%0 Journal Article
%A Leonid Positselski
%T An explicit self-dual construction of complete cotorsion pairs in the relative context
%J Rendiconti del Seminario Matematico della Università di Padova
%D 2023
%P 191-253
%V 149
%U http://www.numdam.org/articles/10.4171/rsmup/118/
%R 10.4171/rsmup/118
%G en
%F RSMUP_2023__149__191_0

Leonid Positselski. 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/

Cité par Sources :