Abstract
We give a combinatorial description of a family of indecomposable objects in the bounded derived categories of two new classes of algebras: string almost gentle (SAG) algebras and SUMP algebras. These indecomposable objects are, up to isomorphism, the string and band complexes introduced by Bekkert and Merklen (Algebras Rep Theory 6:285–302, 2003). With this description, we give a necessary and sufficient condition for a given string complex to have infinite minimal projective resolution and we extend this condition for the case of string algebras. Using this characterization we establish a sufficient condition for a string almost gentle algebra (or a string algebra) to have infinite global dimension.
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Acknowledgements
We acknowledge the important collaboration and very helpful comments and suggestions made by Viktor Bekkert, which were given during the visit of the first author to the Departamento de Matemática of the Universidade Federal de Minas Gerais, and also for his hospitality. We would like to thank the referee for the painstaking effort, and attention to detail put into the manuscript we have submitted. We are deeply grateful with the referee who provided many suggestions and corrections that improved the quality and the readability of this paper. The second and the third authors are grateful to Germán Benítez Monsalve and José A. Vélez-Marulanda for several useful discussions about this work.
Funding
This research was supported by Beca Doctorado Nacional Colciencias (Convocatoria 647 de 2014), CODI (Universidad de Antioquia, U de A), and Colciencias-Ecopetrol (No. 0266-2013).
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Communicated by Henning Krause.
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Franco, A., Giraldo, H. & Rizzo, P. String and Band Complexes Over String Almost Gentle Algebras. Appl Categor Struct 30, 417–452 (2022). https://doi.org/10.1007/s10485-021-09661-x
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DOI: https://doi.org/10.1007/s10485-021-09661-x