Cotilting Modules and Bimodules
by
Kent R. Fuller
University of Iowa
Coauthors: Riccardo Colpi

A cotilting module U_R over a ring R is a right R-module such that Cogen(U_R) = KerExt_R¹(-, U_R). Any injective cogenerator is cotilting module, and any bimodule inducing a Morita duality is a cotilting bimodule. According to [5] (inspired by R.R. Colby's definition in [1]) U_R is a cotilting module if and only if the following three conditions are satisfied: (i) inj dim(U) < 2; (ii) Ext_R¹(U^c, U) = 0 for any cardinal c; (iii) Hom_R(M, U) = 0 = Ext_R¹(M, U) only if M=. A cotilting bimodule is a faithfully balanced bimodule _SU_R that is a cotilting module on both sides. Earlier, Colpi [4] had shown that any cotilting bimodule _SU_R induces a pair of dualities between large subcategories of torsion-free and torsion modules in Mod-R and S-Mod. Here we point out a connection between a notion of U-torsionless linear compactness and the U-reflexivity of modules for a cotilting module U_R, and show that a cotilting bimodule U induces a generalized Morita duality in the sense of Colby [2] if and only if the classes of U-reflexive modules coincide with those of the U-torsionless linearly compact modules. Also, employing the classification of tilting modules over hereditary noetherian serial rings in [3], we provide concrete examples of cotilting bimodules, constructed by applying self-duality to the tilting modules over linearly compact noetherian serial rings.

REFERENCES

1. R.R. Colby, A generalization of Morita duality and the tilting theorem, Comm. in Algebra 17 (1989), 1709-1722.
2. R.R. Colby, A cotilting theorem for rings, Methods in Module Theory, M. Dekker, New York, 1993, pp.33-37.
3. R.R. Colby and K.R. Fuller, Tilting, cotilting, and serially tilted rings, Comm. in Algebra 18 (1990), 1585-1615.
4. R. Colpi, Cotilting bimodules and their dualities, (Preprint 1998).
5. R. Colpi, G. D'Este and A. Tonolo, Quasi-tilting modules and counter equivalences, J. Algabra 191 (1997), 461-494.

Date received: January 8, 1999