An active field of research is that on represenation of algebras; a fundamental role in this field is played by tilting modules [1], which allow the study of various kinds of representations. The notion of tilting modules has been extended to moduels over arbitrary rings, getting a very elegant theory [2]. This theory provides with equivalences between categories of modules which generalize the classical Morita and Fuller equivalences and give examples of Colby-Fuller counter-equivalences [3].
More recently, the notion of tilting object in a Grothendieck category has been introduced [4]. This yields similar results to the classical case, providing in particular a counter-equivalence between the Grothendieck category and a category of modules; moreover it has been proved that *-modules can be regarded as tilting objects, which unifies the two theories.
In two recent works [5, 6] it was tried to generalize tilting counter-equivalences in the case both categories are Grothendieck categories. The results obtained allow to look at representation theory of infinite dimensional algebras from a new point of view.
The dualization of the concept of tilting module is that of "cotilting" module; many results have been obtained on these modules and the dualities they induce (Colby, Fuller, Colpi, Angeleri-Hügel), but it seems still unresolved the problem of giving the "correct" definition of cotilting module: see the papers [7, 8, 9, 10]. Particularly important is the difference between hereditary and non hereditary cotilting dualities and the search for examples showing the various aspects of the problem.
References
[1] D. Happel - C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399--443.
[2] R. Colpi - J. Trlifaj, Tilting modules and tilting torsion theories, J. Algebra 178 (1995), 614--634.
[3] R. R. Colby - K. R. Fuller, Tilting and torsion theory counter equivalences, Comm. Algebra 23 (1995), 4833--4849.
[4] R. Colpi, Tilting in Grothendieck categories, Forum Math. 11 (1999), 735--759
[5] E. Gregorio, Tilting equivalences for Grothendieck categories, J. Algebra (to appear)
[6] E. Gregorio, Topological tilting modules (submitted for publication)
[7] F. Mantese, Hereditary cotilting bimodules (preprint)
[8] R. Colpi, Cotilting bimodules and their dualities, in Proceedings of the Murcia 1998 Euroconference, M. Dekker, 2000
[9] R. Colpi - K. R. Fuller, Cotilting modules and bimodules, Pacific J. Math.
[10] A. Tonolo, Generalizing Morita dualities (preprint)