Proyecciones
Vol. 24, No 1, pp. 79-87, May 2005.
Universidad Católica del Norte
Antofagasta - Chile

A NOTE ON THE FUNDAMENTAL GROUP OF A ONE-POINT EXTENSION

I. ASSEM
Universite de Sherbrooke, Canada.

J. C. BUSTAMANTE
Universite de Sherbrooke, Canada.

D. CASTONGUAY
Universidade Federal de Goias, Goiania.

and

C. NOVOA
Universidade Católica de Goias, Brasil.

Correspondencia a :

ABSTRACT

In this note, we consider an algebra A which is a one-point extension of another algebra B and we study the morphism of fundamental groups induced by the inclusion of (the bound quiver of ) B into (that of ) A. Our main result says that the cokernel of this morphism is a free group and we prove some consequences from this fact.

Keywords and phrases : Fundamental groups, bound quivers, presentations of algebras.

Subject classification : 16G20.

References

[1] I. Assem, D. Castonguay, E. N. Marcos, and S. Trepode. Schurian strongly simply connected algebras and multiplicative bases. J. Algebra, 283 : pp. 161—189, (2005).        [ Links ]

[2] I. Assem and J.A. de la Peña. The fundamental groups of a triangular algebra. Comm. Algebra, 24 (1) : pp. 187—208, (1996).        [ Links ]

[3] K. Bongartz and P. Gabriel. Covering spaces in representation theory.Invent. Math, 65 (3) : pp. 331—378, (1981)-(1982).        [ Links ]

[4] M. J. Bardzell and E. N. Marcos. H1(Ë) and presentations of finite dimensional algebras. Number 224 in Lecture Notes in Pure and Applied Mathematics, pages 31—38. Marcel Dekker, (2001).        [ Links ]

[5] J. C. Bustamante. On the fundamental group of a schurian algebra. Comm. Algebra, 30 (11) : pp. 5305—5327, (2002).        [ Links ]

[6] J. C. Bustamante. The classifying space of a bound quiver. J. Algebra, 277 : pp. 431—455, (2004).        [ Links ]

[7] R. Martínez-Villa and J.A. de la Peña. The universal cover of a quiver with relations. J. Pure Appl. Algebra, 30 : pp. 873—887, (1983).        [ Links ]

[8] A. Skowronski. Simply connected algebras and Hochschild cohomologies.In Proceedings of the sixth international conference on representation of algebras, number 14 in Ottawa-Carleton Math. Lecture Notes Ser., pp. 431—448, Ottawa, ON, (1992).        [ Links ]

Received : March 2005. Accepted : April 2005

Ibrahim Assem
Département de Mathematiques
Universit´e de Sherbrooke.
2500 Boulevard de l’Universite
Sherbrooke J1K 2R1, Quebec
Canada.
Ibrahim.Assem@USherbrooke.ca

Juan Carlos Bustamante
Département de Mathematiques
Universite de Sherbrooke.
2500 Boulevard de l’Universite
Sherbrooke J1K 2R1, Québec
Canada.
jc.bustamante@Usherbrooke.ca

Diane Castonguay
Instituto de Informatica
Universidade Federal de Goias
Bloco IMF I, Campus II, Samambaia
Caixa Postal 131, CEP 74001-970, Goiania, Goias
Brasil.
diane@inf.ufg.br

and

Cristian Novoa
Departameto de Matematica e Física Universidade Católica de GoiasAv. Universitária 1440, Setor Universitario CEP 74610-010 Goiania, Goias,
Brasil
cristian.cristiannovoa@gmail.com