Abstract
Let \(\Lambda \) be an artin algebra of finite global dimension. We study when the composition of three irreducible morphisms between indecomposable complexes in \({{\mathbf {K}}^{b}(\mathrm {proj}\,\Lambda )}\) is a non-zero morphism in the fourth power of the radical. We apply such results to prove that the composition of three irreducible morphisms between indecomposable complexes in the bounded derived category of a gentle Nakayama algebra, not selfinjective, whose ordinary quiver is an oriented cycle, belongs to the fourth power of the radical if and only if it vanishes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auslander, M., Reiten, I., Smalø, S.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)
Bautista, R.: The category of morphisms between projectives modules. Commun. Algebra 32(11), 4303–4331 (2004)
Bautista, R., Souto Salorio, M.J.: Irreducible morphisms in the bounded derived category. J. Pure Appl. 215, 866–884 (2011)
Bautista, R., Souto Salorio, M.J., Zuazua, R.: Almost split sequences for complexes of fixed size. J. Algebra 287, 140–168 (2005)
Bekkert, V., Merklen, H.: Indecomposables in derived categories of gentle algebras. Algebras Represent. Theory 6(4), 285–302 (2003)
Bobiński, G., Geiss, C., Skowroński, A.: Classification of discrete derived categories. Cent. Eur. J. Math. 1, 1–31 (2004)
Chaio, C.: Degrees and composition of irreducible morphisms in almost pre-sectional paths. Algebras Represent. Theory 17(2), 407–432 (2014)
Chaio, C., Coelho, F.U., Trepode, S.: On the composite of two irreducible morphisms in radical cube. J. Algebra 312, 650–667 (2007)
Chaio, C., Coelho, F.U., Trepode, S.: On the composite of irreducible morphisms in almost sectional paths. J. Pure Appl. Algebra 212, 244–261 (2008)
Chaio, C., Coelho, F.U., Trepode, S.: On the composite of three irreducible morphisms in the fourth power of the radical. Commun. Algebra 39(2), 555–559 (2011)
Chaio, C., González Chaio, A., Pratti, I.: On Non-homogeneous Tubes and Components of Type\({\mathbb{Z}}A_{\infty }\) in the Bounded Derived Category. To appear in Algebras and Representation theory (2020)
Chaio, C., Le Meur, P., Trepode, S.: Degrees of irreducible morphisms and finite-representation type. J. Lond. Math. Soc. II 84(1), 35–57 (2011)
Chaio, C., Pratti, I., Souto Salorio, M.J.: On sectional paths in a category of complexes of fixed size. Algebras Represent. Theory 20, 289–311 (2017)
Chaio, C., Souto Salorio, M.J., Trepode, S.: Composite of irreducible morphisms in the bounded derived category. J. Pure Appl. Algebra 215(11), 2957–2968 (2011)
Geiss, C., de la Peña, J.A.: Auslander–Reiten components for clans. Bol. Soc. Mat. Mex. 5, 307–326 (1999)
Happel, D.: Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Mathematical Society Lecture Notes Series 119, Cambridge University Press, Cambridge (1988)
Happel, D.: On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62(3), 339–389 (1987)
Happel, D.: Auslander–Reiten triangles in derived categories of finite-dimensional algebras. Proc. Am. Math. Soc. 112(3), 641–648 (1991)
Igusa, K., Todorov, G.: A characterization of finite Auslander–Reiten quivers. J. Algebra 89, 148–177 (1984)
Liu, S.: Degree of irreducible maps and the shapes of Auslander–Reiten quivers. J. Lond. Math. Soc. 2(45), 32–54 (1992)
Liu, S.: Auslander–Reiten theory in a Krull–Schmidt category. Sao Paulo J. Math. Sci. 4, 425–472 (2010)
Madsen, D.: Projective dimension and Nakayama algebras. In: Fields Institute Communications, vol 45 (2005)
Scherotzke, S.: Examples of Auslander–Reiten components in the bounded derived category (2009). arXiv:0906.4987
Wheeler, W.: The triangle structure of a stable derived category. J. Algebra 165, 23–40 (1994)
Yuefei, Z., Zhaoyong, H.: Auslander–Reiten triangles in homotopy categories. Commun. Algebra 44(11), 4995–5003 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Bernhard Keller.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chaio, C., González Chaio, A. & Pratti, I. On the Composition of Three Irreducible Morphisms in the Bounded Homotopy Category. Appl Categor Struct 30, 1043–1074 (2022). https://doi.org/10.1007/s10485-022-09682-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-022-09682-0