Equivalences and dualities between categories of modules and Grothendieck categories; finite and infinite dimensional algebras

Starting date
January 1, 2001
Duration (months)
24
Departments
Computer Science
Managers or local contacts
Gregorio Enrico
Keyword
CATEGORIES OF MODULES ; EQUIVALENCES ; DUALITIES ; GROTHENDIECK CATEGORIES

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)

Sponsors:

Ministero dell'Istruzione dell'Università e della Ricerca
Funds: assigned and managed by the department

Project participants

Enrico Gregorio
Associate Professor

Collaboratori esterni

Gabriella D'Este
Università di Milano Matematica Professore ordinario
Publications
Title Authors Year
Topological Tilting Modules, II E. Gregorio 2002
Weakly tilting bimodules E. Gregorio, A. Tonolo 2001

Activities

Research facilities