Cotilting bimodules and their dualities

A right R-module UR is said to be cotilting if Cog(UR)=⊥UR, where ⊥UR=KerExt1R(,U). So cotilting modules generalize injective cogenerators. If U is cotilting, then T=(KerHom(−,U),Cog(U)) is a torsion theory, the so-called cotilting torsion theory. A bimodule SUR is cotilting if SUR is faithfully balanced and both UR and SU are cotilting modules. So cotilting bimodules generalize Morita bimodules. The main topic of the paper is the study of dualities induced by cotilting bimodules, as a generalization of the classical theory of Morita dualities. Let SUR be a cotilting bimodule and put Δ=Hom(,U) and Γ=Ext1(,U). Denote by Y the class of all U-reflexive modules, by C the class of all modules of the form K/L where K,L∈Y, and by X the class of all T-torsion modules from C. These classes are studied in the first part of the paper, in order to prove their closure properties, and show that they are sufficiently large. For example, by Proposition 5, C contains all finitely presented modules. The main result of the paper is the following "cotilting theorem'': If U is a cotilting bimodule, then Δ and Γ realize a duality between the classes Y and X, respectively. Moreover, there is a natural morphism γM:Γ2(M)→M such that the sequence 0→Γ2(M)@>γM>>M@>δM>>Δ2(M)@>>>0 is exact for all M∈C, where δM is the evaluation map. In the case of Morita dualities, Müller proved that U-reflexive modules coincide with the linearly compact ones. Inspired by this result, the author studies U-torsionless linearly compact (U-tl.l.c.) modules in the case when U is a cotilting module. In Proposition 10, he proves that if SUR is a cotilting bimodule then any U-tl.l.c. bimodule is U-reflexive, and asks whether the converse is true. (By a recent example of D'Este, this is not true in general.) Finally, the author introduces abstract "duality conditions'' for a pair of torsion theories. These are necessary conditions for the pair to be cogenerated by a cotilting bimodule. The conditions are not sufficient in general. Nevertheless, by Proposition 13, they ensure uniqueness of the representing bimodule.