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On Extensions for Gentle Algebras

Part of: Algebraic combinatorics Representation theory of rings and algebras Homological methods Abelian categories

Published online by Cambridge University Press:  28 January 2020

İlke Çanakçı ,
David Pauksztello  and
Sibylle Schroll
İlke Çanakçı
Affiliation:
Department of Mathematics, VU Amsterdam, Amsterdam 1081 HV, The Netherlands Email: i.canakci@vu.nl
David Pauksztello
Affiliation:
Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, United Kingdom Email: d.pauksztello@lancaster.ac.uk
Sibylle Schroll
Affiliation:
Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom Email: schroll@leicester.ac.uk
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Abstract

We give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.

Keywords

gentle algebraextensionsbounded derived categoryhomotopy string and bandsstring combinatorics

MSC classification

Primary: 16G10: Representations of Artinian rings 16E35: Derived categories 18E30: Derived categories, triangulated categories 05E10: Combinatorial aspects of representation theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 1 , February 2021 , pp. 249 - 292
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

This work has been supported by the EPSRC through the grants EP/K026364/1, EP/K022490/1 and EP/N005457/1. The third author is supported by the EPSRC through an Early Career Fellowship EP/P016294/1.

References

Arnesen, K. K., Laking, R., and Pauksztello, D., Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra. J. Algebra 467(2016), 1–46. https://doi.org/10.1016/j.jalgebra.2016.07.019CrossRefGoogle Scholar
Assem, I., Brüstle, T., Charbonneau-Jodoin, G., and Plamondon, P.-G., Gentle algebras arising from surface triangulations. Algebra Number Theory 4(2010), 2, 201–229. https://doi.org/10.2140/ant.2010.4.201CrossRefGoogle Scholar
Assem, I. and Skowroński, A., Iterated tilted algebras of type ˜A. Math. Z. 195(1987), 269–290. https://doi.org/10.1007/BF01166463CrossRefGoogle Scholar
Baur, K. and Coelho Simões, R., A geometric model for the module category of a gentle algebra. IMRN, to appear. https://doi.org/10.1093/imrn/rnz150CrossRefGoogle Scholar
Bekkert, V. and Merklen, H., Indecomposables in derived categories of gentle algebras. Algebr. Represent. Theory 6(2003), 285–302. https://doi.org/10.1023/A:1025142023594CrossRefGoogle Scholar
Bocklandt, R., Noncommutative mirror symmetry for punctured surfaces. Trans. Amer. Math. Soc. 368(2016), 1, 429–469. https://doi.org/10.1090/tran/6375CrossRefGoogle Scholar
Brüstle, T., Douville, G., Mousavand, K., Thomas, H., and Yıldırım, E., On the Combinatorics of Gentle Algebras. Canad. J. Math., to appear. https://doi.org/10.4153/S0008414X19000397CrossRefGoogle Scholar
Butler, M. C. R. and Ringel, C. M., Auslander–Reiten sequences with few middle terms and applications to string algebras. Comm. Algebra 15(1987), 145–179. https://doi.org/10.1080/00927878708823416CrossRefGoogle Scholar
Çanakçı, İ., Pauksztello, D., and Schroll, S., Mapping cones in the bounded derived category of a gentle algebra. J. Algebra 530(2019), 163–194. https://doi.org/10.1016/j.jalgebra.2019.04.005CrossRefGoogle Scholar
Çanakçı, İ., Pauksztello, D., and Schroll, S., Addendum and corrigendum: mapping cones for morphisms involving a band complex in the bounded derived category of a gentle algebra. arxiv:2001.06435Google Scholar
Çanakçı, İ. and Schroll, S., Extensions in Jacobian algebras and cluster categories of marked surfaces. Adv. Math. 313(2017), 1–49. https://doi.org/10.1016/j.aim.2017.03.016CrossRefGoogle Scholar
Crawley-Boevey, W. W., Maps between representations of zero-relation algebras. J. Algebra 126(1989), 259–263. https://doi.org/10.1016/0021-8693(89)90304-9CrossRefGoogle Scholar
Garcia Elsener, A., Gentle m-Calabi-Yau tilted algebras. arxiv:1701.07968Google Scholar
Haiden, F., Katzarkov, L., and Kontsevich, M., Flat surfaces and stability structures. Publ. Math. Inst. Hautes Études Sci. 126(2017), 247–318. https://doi.org/10.1007/s10240-017-0095-yCrossRefGoogle Scholar
Happel, D., Triangulated categories in the representation theory of finite dimensional algebras. In: London Mathematical Society Lecture Notes Series, 119. Cambridge University Press, Cambridge, 1988. https://doi.org/10.1017/CBO9780511629228Google Scholar
Huerfano, R. S. and Khovanov, M., A category for the adjoint representation. J. Algebra 246(2001), 2, 514–542. https://doi.org/10.1006/jabr.2001.8962CrossRefGoogle Scholar
Huisgen-Zimmermann, B. and Smalø, S. O., The homology of string algebras. I. J. Reine Angew. Math. 580(2005), 1–37. https://doi.org/10.1515/crll.2005.2005.580.1CrossRefGoogle Scholar
Kalck, M., Singularity categories of gentle algebras. Bull. Lond. Math. Soc. 47(2015), 1, 65–74. https://doi.org/10.1112/blms/bdu093CrossRefGoogle Scholar
Krause, H., Maps between tree and band modules. J. Algebra 137(1991), 186–194. https://doi.org/10.1016/0021-8693(91)90088-PCrossRefGoogle Scholar
Labardini-Fragoso, D., Quivers with potentials associated to triangulated surfaces. Proc. Lond. Math. Soc. (3) 98(2009), 3, 797–839. https://doi.org/10.1112/plms/pdn051CrossRefGoogle Scholar
Lekili, Y. and Polishchuk, A., Derived equivalences of gentle algebras via Fukaya categories. Math. Ann. 376(2020), 1–2, 187–225. https://doi.org/10.1007/s00208-019-01894-5CrossRefGoogle Scholar
McConville, T., Lattice structure of Grid-Tamari orders. J. Combin. Theory Ser. A 148(2017), 27–56. https://doi.org/10.1016/j.jcta.2016.12.001CrossRefGoogle Scholar
Opper, S., Plamondon, P.-G., and Schroll, S., A geometric model for the derived category of gentle algebras. arxiv:1801.09659Google Scholar
Palu, Y., Pilaud, V., and Plamondon, P.-G., Non-kissing complexes and 𝜏 tilting for gentle algebras. Mem. Amer. Math. Soc., to appear. arxiv:1707.07574Google Scholar
Schröer, J., Modules without self-extensions over gentle algebras. J. Algebra 216(1999), 1, 178–189. https://doi.org/10.1006/jabr.1998.7696CrossRefGoogle Scholar
Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras. Vol. 3. Representation-infinite tilted algebras. London Mathematical Society Student Texts, 72, Cambridge University Press, Cambridge, 2007.Google Scholar
Vossieck, D., The algebras with discrete derived category. J. Algebra 243(2001), 168–176. https://doi.org/10.1006/jabr.2001.8783CrossRefGoogle Scholar
Wald, B. and Waschbüsch, J., Tame biserial algebras. J. Algebra 95(1985), 480–500. https://doi.org/10.1016/0021-8693(85)90119-XCrossRefGoogle Scholar
Zhang, J., On the indecomposable exceptional modules over gentle algebras. Comm. Alg. 42(2014), 3096–3199. https://doi.org/10.1080/00927872.2013.781369CrossRefGoogle Scholar