Cotilting modules and bimodules
A module \(U_R\) is cotilting if \(\text{Cogen}(U_R)=\text{KerExt}^1_R(-,U_R)\) (this notion is dual to that of a tilting module). A bimodule \(_SU_R\) is cotilting if \(U_R\) and \(_SU\) are cotilting modules. For a bimodule \(_SU_R\) by \(\Delta\) are denoted both the functors \(\Hom_R(-,U)\) and \(\Hom_S(-,U)\), and by \(\Gamma\) both the functors \(\text{Ext}^1_R(-,U)\) and \(\text{Ext}^1_S(-,U)\). If \(U_R\) is a cotilting module, then \((\text{Ker }\Delta,\text{Ker }\Gamma)\) is a torsion theory in Mod-\(R\). Linear compactness with respect to a torsion theory \((\mathcal{T,F})\) in Mod-\(R\) is defined: \(M_R\) is \(\mathcal F\)-linearly compact if \(M\in{\mathcal F}\) and for any inverse system of morphisms \(\{p_\lambda\colon M\to M_\lambda\}\) with \(M_\lambda\in{\mathcal F}\) and \(\text{Coker}(p_\lambda)\in{\mathcal F}\) for all \(\lambda\)'s, it happens that \(\text{Coker}(\varprojlim p_\lambda)\in{\mathcal T}\). If \(U_R\) is a cotilting module, a module \(M_R\) is called \(U\)-torsionless linearly compact (\(U\)-tl.l.c.) if \(M_R\) is \(\text{Ker }\Gamma\)-linearly compact. A module \(M_R\) is \(U\)-tl.l.c. if and only if \(M_R\) is \(\Delta\)-reflexive and for all \(L\overset {i}\hookrightarrow\Delta(M)\) we have \(\text{Coker }\Delta(i)\in\text{Ker }\Delta\). Let \(_SU_R\) be a cotilting bimodule. For any module \(M_R\in\text{Ker }\Gamma\) the following conditions are equivalent: (1) \(M_R\) is \(U\)-tl.l.c.; (2) \(M_R\) is \(\Delta\)-reflexive and for all \(_SL\subseteq\Delta(M)\) we have \(\Delta\Gamma(\Delta(M)/L)=0\); (3) Any \(S\)-submodule of \(\Delta(M)\) is \(\Delta\)-reflexive. A method for obtaining a cotilting bimodule as Morita dual of a tilting bimodule is shown. The cotilting bimodules over Noetherian serial rings are studied. If such a ring \(R\) possesses a self-duality induced by a bimodule \(_RW_R\) and if \(_RV\) is a tilting module with \(S=\text{End}(_RV)\), then \(_SU_R=\Hom_R(V,W)\) is a cotilting bimodule.
scientific article; zbMATH DE number 1463761