Representations of the generalized Kronecker quiver with countably many arrows

Mahrt, Nils

doi:10.1090/S0002-9939-08-09552-X

Representations of the generalized Kronecker quiver with countably many arrows
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by Nils Mahrt

Proc. Amer. Math. Soc. 137 (2009), 815-824

DOI: https://doi.org/10.1090/S0002-9939-08-09552-X

Published electronically: September 10, 2008

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Abstract:

Let $Q$ be the generalized Kronecker quiver with countably many arrows and let $k$ be a field. We prove that the category of representations of $Q$ over $k$ has no right almost split morphism whose domain is projective. More precisely, we show that any indecomposable non-projective representation is the image of an epimorphism whose domain has no non-zero projective direct summand. This result does not hold for any finite subquiver of $Q$.

References

Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. Techniques of representation theory. MR 2197389, DOI 10.1017/CBO9780511614309
M. Auslander and Sverre O. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61–122. MR 591246, DOI 10.1016/0021-8693(80)90113-1
D. J. Benson, Representations and cohomology. I, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. Basic representation theory of finite groups and associative algebras. MR 1644252
Riccardo Colpi, Gabriella D’Este, and Alberto Tonolo, Quasi-tilting modules and counter equivalences, J. Algebra 191 (1997), no. 2, 461–494. MR 1448804, DOI 10.1006/jabr.1997.6873
Gabriella D’Este, Free modules obtained by means of infinite direct products, Algebra and its applications (Athens, OH, 1999) Contemp. Math., vol. 259, Amer. Math. Soc., Providence, RI, 2000, pp. 161–173. MR 1778499, DOI 10.1090/conm/259/04092
Edgar Enochs and Sergio Estrada, Projective representations of quivers, Comm. Algebra 33 (2005), no. 10, 3467–3478. MR 2175445, DOI 10.1081/AGB-200058181
Edgar Enochs, Luis Oyonarte, and Blas Torrecillas, Flat covers and flat representations of quivers, Comm. Algebra 32 (2004), no. 4, 1319–1338. MR 2100360, DOI 10.1081/AGB-120028784
Dieter Happel and Luise Unger, A family of infinite-dimensional non-self-extending bricks for wild hereditary algebras, Representations of finite-dimensional algebras (Tsukuba, 1990) CMS Conf. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 1991, pp. 181–189. MR 1143851, DOI 10.1016/0096-3003(91)90033-J
Mark Hovey, Classifying subcategories of modules, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3181–3191. MR 1828603, DOI 10.1090/S0002-9947-01-02747-7
Nils Mahrt, Darstellungen des verallgemeinerten Kronecker-Köchers mit abzählbar vielen Pfeilen, Diplomarbeit, Fakultät für Mathematik, Universität Bielefeld, July 2006.
Claus Michael Ringel, Representations of $K$-species and bimodules, J. Algebra 41 (1976), no. 2, 269–302. MR 422350, DOI 10.1016/0021-8693(76)90184-8