On equivalence of some subcategories of modules in Morita contexts

• Datum: 2018
• 1. Verfasser: Kashu, A. I.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Lugansk National Taras Shevchenko University 2018
• Schlagworte: torsion (torsion theory); Morita context; torsion free module; accessible module; equivalence; 16S90; 16D90
• Online Zugang: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/963
• Назва журналу: Algebra and Discrete Mathematics

A Morita context \((R,\,_{R}\!V_{S},\,_{S}\!W_{R},S)\) defines the isomorphism \({\cal L}_{0}(R) \cong {\cal L}_{0}(S)\) of lattices of torsions \(r\geq r_{\scriptscriptstyle I}\) of \(R\)-\(Mod\) and torsions \(s\geq r_{\scriptscriptstyle J}\) of \(S\)-\(Mod\), where \(I\) and \(J\) are the trace ideals of the given context. For every pair \((r,s)\) of corresponding torsions the modifications of functors \(T^{W}=W\otimes _{R}\)- and \(T^{V}=V\otimes _{S}\)- are considered:\[R\textrm{-}Mod\supseteq \mathcal{P}(r)\begin{array}{c}\begin{array}{c}\underrightarrow{\quad\bar{T}^W=(1/s)\cdot T^W\quad}\\\overleftarrow{\quad\bar{T}^V=(1/r)\cdot T^{V}\quad}\end{array}\end{array}\mathcal{P}(s)\subseteq S\textrm{-}Mod,\]where \({\cal P}(r)\) and \({\cal P}(s)\) are the classes of torsion free modules. It is proved that these functors define the equivalence \begin{equation*} {\cal P}(r)\cap {\cal J}_{I}\approx {\cal P}(s)\cap {\cal J}_{J}, \end{equation*} where \({\cal P}(r)=\{_{R}M\ |\ r(M)=0\}\) and \({\cal J}_{I}=\{_{R}M\ |\ IM=M\}.\)