On equivalence of some subcategories of modules in Morita contexts

Abstract

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 \({\cal J}_{I}=\{_{R}M\ |\ IM=M\}.\)