Grant ID489

Research Project
"Torsion Classes and Cotorsion Classes",
October 2007-September 2010.

Financed by National Authority for Scientific Research (ANCS) , through
UEFISCSU
CNCSIS

Algebra Department, Faculty of Mathematics and Computer Science "Babes-Bolyai" University, Cluj-Napoca

1. Team

2. Abstract

Team:

Simion Breaz ,
George Ciprian Modoi,
Flaviu Pop (PhD student),
Carolina Contiu (PhD student).

Abstract:

(I) Scientific objectives

THE MAIN OBJECTIVE OF THIS PROJECT IS TO STUDY THE WAY IN WHICH OBJECTS OF SOME CATEGORIES WHICH ARE GENERALIZATIONS FOR CATEGORIES OF MODULES, GROUPS OR OTHER CLASSICAL STRUCTURES MAY BE CONSTRUCTED (FROM ANOTHER OBJECTS EASIER TO UNDERSTAND) AND/OR APPROXIMATED. THE PROJECT INTENDS TO OBTAIN RESULTS WHICH WILL CONTRIBUTE TO THE STUDY OF SOME IMPORTANT PROBLEMS: THE TELESCOPE CONJECTURE, THE CLASSIFICATION OF (CO)TILTING MODULES, THE STUDY OF CELLULAR COVERS, MARGOLIS'S CONJECTURE. THE RESEARCH HAS 4 PRINCIPAL DIRECTIONS:
(1) TORSION THEORIES AND COTORSION THEORIES. WE WILL CONSIDER EQUIVALENCE RELATIONS DEFINED AS FOLLOWS: TWO OBJECTS ARE EQUIVALENT IF AND ONLY IF THEY (CO)GENERATE THE SAME (CO)TORSION THEORY, AND WE WILL SEARCH FOR NECESSARY AND SUFFICIENT CONDITIONS SUCH THAT THESE RELATIONS COINCIDE WITH ISOMORPHISMS IN SOME CATEGORIES OF FRACTIONS.
(2) SMALL OBJECTS IN ADDITIVE CATEGORIES: WE WILL STUDY SELF-SMALL ABELIAN GROUPS (WHICH REPRESENTS AN IMPORTANT SOURCE OF EXAMPLES) AND CONNECTIONS BETWEEN SMALL OBJECTS IN SOME SUBCATEGORIES AND PROPERTIES OF THESE SUBCATEGORIES. WE WILL ALSO STUDY IF WE MAY FIND SOME FUNCTORIAL CHARACTERIZATION OF SOME TRIANGULATED CATEGORIES USING THEIR SMALL OBJECTS (CALLED IN THIS CONTEXT COMPACT OBJECTS).
(3) EQUIVALENCES AND DUALITIES BETWEEN SUBCATEGORIES. WE WILL TRY TO FIND NEW METHODS TO CONSTRUCT SOME EQUIVALENCES AND DUALITIES IN MANY CONTEXTS. FOR EXAMPLE, IN THE CASE OF TRIANGULATED CATEGORIES WE WILL APPLY A FUNCTORIAL VERSION FOR BROWN REPRESENTABILITY THEOREM.
(4) RELATIVE HOMOLOGICAL ALGEBRA IN NON-STANDARD CONTEXTS. WE WILL CONSIDER NON-ABELIAN CATEGORIES, BUT HAVING SOME SIMILAR PROPERTIES (SEMI-ABELIAN CATEGORIES, BARR-EXACT CATEGORIES, TRIANGULATED CATEGORIES). WE WILL TRY TO USE MINIMAL HYPOTHESES, IN ORDER TO BE SURE THAT OUR RESULTS HOLD IN STANDARD (NON-ABELIAN) CATEGORIES, SUCH AS CATEGORY OF GROUPS OR CATEGORY OF SETS ENDOWED WITH A SEMIGROUP ACTION.

(II) Raising the international impact of the research

1. Publication of the results in journals with high international impact (ISI cited and/or reviewed by Mathematical Reviews or Zentralblatt fur Mathematik).
2. Presentation of the results on international research seminars and conferences.

(III) Improving the research conditions

1. Completion of the research infrastructure by
-- assuring the bibliographical background (books, journals, etc.)
-- assuring the material background of the research (computers, office consumables etc.)
2. Improving the level of knowledge of the team members and other young master and PhD students about the newest research directions

3. Young researchers

Pop Flaviu
Activities on directions (1), (2) and (3) with the following results:
FP1. S. Breaz, F. Pop, On some dualities induced by right adjoint contravariant functors, Studia Universitatis Babes-Bolyai -- Series Mathematica LX (2010), 75--83 pdf
FP2 S. Breaz, C. Modoi, F. Pop, Natural equivalences and dualities, in Proceedings of the International Conference on Modules and Representation Theory, Cluj University Press, 2009, 24-40 pdf
In 2010 his research is concerned on directions (2) si (3) (objective 2/2010)

Research visits:

1. Karlin University, Prague; research seminar:Algebra Seminar, November 2007, April 2008, September 2009
Berkeley University (april 2008),
Padeborn University (may 2009).

Contiu Carolina
Activities on directions (1) and (4) with the following results:
CC1. S. Breaz, C. Contiu Groups which are determined by subgroup lattices, Acta Universitatis Apulensis, Special Issue (2009) 449--463 pdf
CC2. C. Contiu Conditions under which a lattice is isomorphic to the subgroup lattice of an abelian group, pdf

In 2010 her research is concerned on direction (4) (objective 1/2010)

Research visits:
1. Montreal University (Prof. K. Benabdallah), April-May 2008
2. Renyi Institute, Budapest, September 2008
3. Napoli University, May 2009
4. Karlin University, Prague (september 2009).

Activities

Objectives and activities (tabular)

Achievements:

(A) Papers in ISI journals
1. S. Breaz, Finitistic $n$-quasi-tilting modules}, Communications in Algebra, 37:9 (2009), 3152--3170. pdf link
2. U. Albrecht, S. Breaz, W. Wickless, Self-small abelian groups, Bulletin of the Australian Mathematical Society, 80 (2009), 205--216. pdf link
3. U. Albrecht, S. Breaz, Quasi-isomorphisms and groups of quasi-homomorphisms, Journal of Algebra and its Applications 8 (2009), 617--627 pdf link
4. C. Modoi, On perfectly generating projective classes in triangulated categories, Communications in Algebra 38 (2010), 995--1011 pdf link
5. S. Breaz, Commutativity conditions using normal subgroup lattices, Rendiconti del Seminario Matematico dell'Universita di Padova, 112(2009) 161--169 pdf link
6. S. Breaz, Warfield dualities induced by self-small mixed groups, acceptata de Journal of Group Theory pdf link
7. S. Breaz, J. Trlifaj, Modules determined by their annihilator classes, Journal of the London Mathematical Society , 81 (2010), 225--240 pdf link
8. U. Albrecht, S. Breaz, A Note on Mixed $A$-reflexive groups, Journal of Algebra 323 (2010), 509--516 pdf link
9. C. Modoi, Localizations, colocalizations and non additive *-objects, acceptata de Semigroup Forum link
0. G. C. Modoi, Generelized lax epimorphism in the additive case, acceptata de Journal of Pure and Appl. Algebra,
11. U. Albrecht, S. Breaz, W. Wickless, S*-groups, acceptata de Journal of Algebra and its Applications,

(B) Papers in international journals
1. S. Breaz, C. Modoi, A reformulation of Brown Representability Theorem}, Mathematica (Cluj), 51(74) (2009), 129-133; pdf
2. S. Breaz, F. Pop, On some dualities induced by right adjoint contravariant functors, Studia Universitatis Babes-Bolyai -- Series Mathematica LX (2010), 75--83 pdf
3. S. Breaz, C. Modoi, F. Pop, Natural equivalences and dualities, in Proceedings of the International Conference on Modules and Representation Theory, Cluj University Press, 2009, 24-40 pdf
4. S. Breaz, C. Contiu Groups which are determined by subgroup lattices, Acta Universitatis Apulensis, Special Issue (2009) 449--463 pdf
5. S. Breaz, Gr. Calugareanu, P. Schultz, Modules with Dedekind-finite endomorphism rings, accepted by Mathematica (Cluj) pdf

(C) Books
S. Breaz, S. Crivei, A. Marcus (eds.) Proceedings of the International Conference on Modules and Representation Theory, Cluj University Press, 2009

(D) Preprints
1. C. Contiu Conditions under which a lattice is isomorphic to the subgroup lattice of an abelian group, pdf
b> 2. U. Albrecht, S. Breaz, W Wickless S*-groups

Abstracts 2008, href="http://math.ubbcluj.ro/~bodo/preprints/sinteza2009.pdf">2009
Presentation 2009,

Visits and conferences
Simion Breaz, 1. Visit la Algebra Department, Charles Univ., Prague:Algebra Seminar, "M" November 2007 (talk 1 hour),
2. Visit la Institutul Renyi, Budapesta, Octombrie 2008 (prezentare de 1 hour)
3. Visit la Algebra Department, Charles Univ., Prague., september 2009 (talk 1 hour)
Ciprian Modoi,:1. Visit at Algebra Department, Charles Univ., Prague:Algebra Seminar, "M" November 2007 (talk 1 hour),
2. Visits at Universitate din Padeborn, Germania; talk 1 hour, mai 2009
3. Visit at Algebra Department, Charles Univ. Prague., September 2009 (talk 1 hour)
Flaviu Pop,:1. Visits at Algebra Department, Charles Univ., Prague. Paricipare la seminarul de cercetare al departamenului:Algebra Seminar, "M" November 2007, April 2008, September 2009
Visit la Universitatea Berkeley.
Carolina Contiu: 1. Visit at Montreal University (Prof. K. Benabdallah), 2007
2. Visit la Institutul Renyi, Budapesta, 2008
3. Visit at University of Napoli, mai 2009
4. Visit la Algebra Department, Charles Univ., Prague 5. Visit at University of Padova, 2010.

7. Talks
1. S. Breaz: Abelian groups with the same orthogonal classes, Algebra Seminar, Univ. Karlin, Praga (noiembrie 2007)
2. C. Modoi: Cellular approximations in abelian categories, Algebra Seminar, Univ. Karlin, Praga (noiembrie 2007)
3. S. Breaz: Finitistic n-self-cotilting modules, International Conference on Modules and Representation Theory, ``Babe\c s-Bolyai" Cluj-Napoca 2008
4. C. Modoi: A generalization of the Rickard–Morita theory, International Conference on Modules and Representation Theory, ``Babe\c s-Bolyai" Cluj-Napoca 2008.
5. S. Breaz: Commutativity conditions using normal subgroup lattices, Inst. Renyi, Budapesta (septembrie 2008)
6. C. Modoi: Triangulated categories and singularities, Univ. Paderborn (mai 2009)
7. S. Breaz: Commutativity conditions using subgroup lattices, ICTAMI 2009 (septembrie 2009)
8. S. Breaz: Modules determined by their annihilator classes, Some Trends in Algebra, Prague (septembrie 2009).
9. C. Modoi: Brown representability via projective classes, Some Trends in Algebra, Prague (septembrie 2009).
10. C. Con\c tiu: Lattice subgroups of abelian groups, Algebra Symposium, Cluj-Napoca, 2009
11. F. Pop: Natural equivalences and dualities, Algebra Symposium, Cluj-Napoca, 2009
12. S. Breaz: Modules determined by their annihilator classes, Algebra Symposium, Cluj-Napoca, 2009
13. S. Breaz: Dualities and self--small groups of finite torsion-free rank, Algebra Seminar, Univ. Padova (septembrie 2010)
14. C. Modoi: Approximations in homotopy category and Brown representability, Algebra Seminar, Univ. Padova (septembrie 2010)

Contact

e-mail:
    user: bodo server: @math.ubbcluj.ro
    name: ssbreaz server: gmail.com
fax:
    +40 264 691906
mail:
    Simion Breaz
    Babes-Bolyai University
    Faculty of Mathematics and Computer Science
    Str. Mihail Kogalniceanu nr. 1
    RO-400084 Cluj-Napoca
    Romania