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Grant PN2-IDEI-PCE-2008-2 ID_2271, January 2009-December 2011
"Locally finitely presented categories, submodule categories and applications"  

Financed by CNCSIS
Faculty of Mathematics and Computer Science,  "Babes-Bolyai" University, Cluj-Napoca
Research team
Assoc. Prof. Dr. Septimiu Crivei (project manager), Assoc. Prof. Dr. Csaba Szanto, researcher Dan Salajan, researcher Stefan Suteu Szollosi.

Brief presentation of the project

In this project we deal with problems of present interest on locally finitely presented categories and submodule categories. We are also interested in applying our  theoretical results to other fields, such as matrix pencils theory, and possible algorithmic implementations. The scientific objectives of the project are the following ones:

2009

1. Development of new techniques for the study of special objects in locally finitely presented categories.
2. Embedding Kronecker modules via Ringel-Hall algebras. Analysing Kronecker invariants associated to pencils.
3. Study of injective and flat objects in locally finitely presented categories.
4. Embedding Kronecker modules via Ringel-Hall algebras and solving the subpencil problem.
5. Development of GAP algorithms.

2010

1. Find new classes of locally finitely presented Grothendieck categories symmetric to each other.
2. Solving the subpencil problem in certain cases.
3. Development of GAP algorithms.

2011

1. Establishing further results on approximations, equivalences and dualities in locally finitely presented categories.
2. Describing bounded submodule categories in certain cases.
3. Development of GAP algorithms.

Results
12 articles in ISI journals, 3 articles in IDB (International Data Bases) journals, 1 preprint.

Articles in ISI journals

1. S. Crivei, M. Prest, B. Torrecillas, Covers in finitely accessible categories, Proc. Amer. Math. Soc. 138 (2010), 1213-1221. doi: 10.1090/S0002-9939-09-10178-8
2. S. Crivei, C. Nastasescu, B. Torrecillas, On the Osofsky-Smith theorem, Glasgow Math. J. 52A (2010), 61-67. doi: 10.1017/S0017089510000169
3. S. Crivei, On Krull-Schmidt finitely accessible categories, Bull. Austral. Math. Soc. 84 (2011), 90-97. doi:10.1017/S0004972711002085
4. S. Crivei, Maximal exact structures on additive categories revisited, Math. Nachr. 285 (2012), 440-446. doi:10.1002/mana.201000065
5. S. Crivei, M.C. Iovanov, Symmetry for comodule categories, J. Algebra Appl. 11 (2012), 1250009 (16 pages). doi:10.1142/S0219498811005324
6. S. Crivei, C. Nastasescu, L. Nastasescu, A generalization of the Mitchell Lemma: The Ulmer Theorem and the Gabriel-Popescu Theorem revisited, J. Pure Appl. Algebra 216 (2012), 2126-2129. doi:10.1016/j.jpaa.2012.01.014
7. S. Bazzoni, S. Crivei, One-sided exact categories, J. Pure Appl. Algebra 217 (2013), 377-391. doi:10.1016/j.jpaa.2012.06.019  
8. S. Crivei, S. Sahinkaya, Modules whose closed submodules with essential socle are direct summands, Taiwanese J. Math. 18 (2014), 989-1002. doi:10.11650/tjm.18.2014.3388 
9. C. Szanto, A. Horvath, Formulas for Kronecker invariants using a representation theoretical approach, Linear Alg. Appl. 430 (2009), 664-673. doi:10.1016/j.laa.2008.09.024
10. C. Szanto, On the cardinalities of Kronecker quiver Grassmannians, Math. Z. 269 (2011), 833-846. doi:10.1007/s00209-010-0762-x
11. C. Szanto, I. Szollosi, The terms in the Ringel-Hall product of preinjective Kronecker modules, Period. Math. Hung. 63 (2011), 227-244. doi:10.1007/s10998-011-8227-5
12. I. Szollosi, Computing the extensions of preinjective and preprojective Kronecker modules, J. Algebra 408 (2014), 205-221. doi:10.1016/j.jalgebra.2013.09.003 

Articles in IDB journals

1. S. Crivei, Epic envelopes by generalized flat modules, Mathematica (Cluj) 51 (74) (2009), No. 1, 47-53.
2. I. Crivei, S. Crivei, Associated classes of modules, Studia Univ. Babes-Bolyai, Mathematica 54 (2009), No. 2, 23-32.
3. S. Crivei, Finitely accessible categories, generalized module categories and approximations, Proceedings of the International Conference on Modules and Representation Theory (Cluj-Napoca, Romania, July 7-12, 2008), Cluj University Press, Cluj-Napoca, 2009, pp. 53-60.

Preprint

1. C. Szanto, On extensions of Kronecker modules.

Computer packages

Development of GAP algorithms related to the topics of the project has been achieved.

Talks at conferences and research seminars

1. Algebra Seminar, Vrije Universiteit, Bruxelles, Belgium, April 30, 2009; S. Crivei, invited talk with the title: On the Osofsky-Smith theorem.
2. Algebra Symposium, Cluj-Napoca, Romania, May 15-16, 2009; S. Crivei, talk with the title: On an open problem of Mark Teply.
3. Algebra Symposium, Cluj-Napoca, Romania, May 15-16, 2009; S. Suteu Szollosi, talk with the title: Subgroup lattice algorithms related to extending and lifting abelian groups.
4. Some Trends in Algebra, Prague, Czech Republic, September 7-11, 2009; S. Crivei, talk with the title: Monads, comonads and approximations.
5. Algebra Seminar, Renyi Institute of Mathematics, Budapest, Hungary, October 26, 2009: C. Szanto, invited talk with the title: On the cardinalities of Kronecker quiver Grassmannians.
6. Algebra Seminar, Universita di Padova, Italy, February 22, 2010: S. Crivei, invited talk with the title: Finitely accessible categories and approximations.
7. The First Conference of PhD Students in Mathematics, Szeged, Hungary, June 29 - July 2, 2010: I.Szollosi, talk with the title: On the Hall product of preinjective Kronecker modules.
8. Algebra Seminar, Gebze Institute of Technology, Turkey, November 1, 2010: S. Crivei, invited talk with the title: Frobenius functors and generalizations.
9. Algebra Seminar, Universita di Padova, Italy, May 17, 2011: S. Crivei, invited talk with the title: Krull-Schmidt finitely accessible categories.
10. 13th Postgraduate Group Theory Conference, University of Aberdeen, Aberdeen, Scotland, June 23-25, 2011; I. Szollosi, talk with the title: Kronecker modules and matrix pencils.
11. The 7th Congress of Romanian Mathematicians, Brasov, Romania, June 29 – July 5, 2011; S. Crivei, talk with the title: Krull-Schmidt finitely accessible categories.
12. A^3 Abstract Algebra and Algorithms Conference, Eger, Hungary, August 14-17, 2011; I. Szollosi, talk with the title: Computational methods in the theory of Kronecker modules.
13. Some Trends in Algebra, Prague, Czech Republic, September 6-10, 2011; S. Crivei, talk with the title: One-sided exact categories.
14. Algebra Seminar, Cluj-Napoca, Romania, September 28, 2011; I. Szollosi, talk with the title: Kronecker modules and matrix pencils.
15. Algebra Seminar, Universita di Padova, Italy, November 8, 2011: S. Crivei and C. Nastasescu, invited talk with the title: A generalization of the Mitchell Lemma and applications.
16. Algebra Seminar, Renyi Institute of Mathematics, Budapest, Hungary, November 21, 2011: S. Crivei, invited talk with the title: One-sided exact categories.

Other participations in conferences
1. Algebra Symposium, Cluj-Napoca, Romania, May 15-16, 2009; C. Szanto.
2. Advanced School and Conference on Homological and Geometrical Methods in Representation Theory, Trieste, Italy, January 18 - February 5, 2010; S. Suteu Szollosi.
3. Advanced School on Lie and Representation Theories, Sevilla, Spain, February 8-12, 2010; S. Suteu Szollosi.

Contact data
Address: Septimiu Crivei, Faculty of Mathematics and Computer Science, Babes-Bolyai University, Str. M. Kogalniceanu 1, 400084 Cluj-Napoca, Romania
E-mail: crivei at math dot ubbcluj dot ro,  scrivei at gmail dot com
Tel.: ++40-264-405327
Fax: ++40-264-591906