On equivalence of some subcategories of modules in Morita contexts

A. I. Kashu

Algebra and discrete mathematics, no. 3 (2003), pp. 46-53.

Résumé

A Morita context $(R,\,_RV_S,\,_SW_R,\,S)$ defines the isomorphism $\mathcal L_0(R)\cong\mathcal L_0(S)$ of lattices of torsions $r\geq r_I$ of $R$-$Mod$ and torsions $s\geq r_J$ 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: \begin{equation*} R\textrm{-}Mod\supseteq\mathcal P(r) ???????????? \mathcal P(s)\subseteq S\textrm{-}Mod, \end{equation*} where $\mathcal P(r)$ and $\mathcal P(s)$ are the classes of torsion free modules. It is proved that these functors define the equivalence \begin{equation*} \mathcal P(r)\cap\mathcal J_I\approx\mathcal P(s)\cap\mathcal J_J, \end{equation*} where $\mathcal P(r)=\{_RM\mid r(M)=0\}$ and $\mathcal J_I=\{_RM\mid IM=M\}$.

Détail

Keywords: torsion (torsion theory)
Mots-clés : Morita context, torsion free module, accessible module, equivalence.