Cotilting modules and bimodules

Cotilting theory (for arbitrary modules over arbitrary unital rings) extends Morita duality in analogy to the way tilting theory extends Morita equivalence. In particular, cotilting modules generalize injective cogenerators similarly as tilting modules generalize progenerators. Here, right R-module UR is cotilting if UR has injective dimension ≤1, ExtR(Uα,U)=0 for any cardinal α, and KerHomR(−,U)∩KerExtR(−,U)=0. A bimodule SUR is cotilting if SUR is faithfully balanced and both UR and SU are cotilting modules. Let UR be a cotilting module and CR the class of all modules cogenerated by UR. Then CR is a torsion-free class and every module has a special CR-precover. The key problem of the cotilting theory is to characterize the subclasses of CR and CS formed by all U-reflexive modules (= the modules for which HomR(−,U) and HomS(−,U) induce a duality) in the case when SUR is a cotilting bimodule. In the classical works of Müller, the problem was solved in the Morita case, that is, in the case when SUR is a Morita bimodule: the U-reflexive modules are exactly the linearly compact ones. In Section 1, the authors define the notion of a U-torsionless linearly compact (U-tl.l.c.) module. Colpi [in Interactions between ring theory and representations of algebras (Murcia), 81--93, Dekker, New York, 2000; MR1758403 (2001f:16015); see the following review] proved that if SUR is a cotilting bimodule then any U-tl.l.c. bimodule is U-reflexive. The main result of Section 1---Theorem 1.8---then gives a characterization of the U-tl.l.c. modules among the U-reflexive ones. Corollary 1.9 shows that the two classes coincide iff the class of all reflexive S-modules is closed under submodules. Note that applying this corollary, D'Este recently proved that the two classes may be different in general. Section 2 deals with constructing cotilting bimodules as Morita duals of tilting bimodules. Assume A and R are Morita dual rings via a Morita bimodule AWR, and AV is a tilting module with S=End(AV). Put SXR=HomA(V,W). By Proposition 2.6, SXR is a cotilting bimodule iff (∗) ExtR(Xα,X)=0 for any cardinal α. In Section 3, the results are applied to the case when R is a Noetherian serial ring with a self-duality (so A=R). By Theorem 3.4, (∗) always holds, so SXR is a cotilting bimodule. By Proposition 3.7, if R is, moreover, hereditary, then any tilting module RX is a finitistic cotilting module in the sense of Colby, and RX satisfies condition (∗).