Geodesic
Mutation graph of support $\tau $-tilting modules over a skew-gentle algebra
He, Ping orcid ; Zhou, Yu orcid ; Zhu, Bin orcid
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e113

Voir la notice de l'article provenant de la source Cambridge University Press

Let $\mathcal {D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object R. Let $\Lambda =\operatorname {End}_{\mathcal {D}}R$ be the endomorphism algebra of R. We introduce the notion of mutation of maximal rigid objects in the two-term subcategory $R\ast R[1]$ via exchange triangles, which is shown to be compatible with the mutation of support $\tau $-tilting $\Lambda $-modules. In the case that $\mathcal {D}$ is the cluster category arising from a punctured marked surface, it is shown that the graph of mutations of support $\tau $-tilting $\Lambda $-modules is isomorphic to the graph of flips of certain collections of tagged arcs on the surface, which is moreover proved to be connected. Consequently, the mutation graph of support $\tau $-tilting modules over a skew-gentle algebra is connected. This generalizes one main result in [49]. 
He, Ping; Zhou, Yu; Zhu, Bin. Mutation graph of support $\tau $-tilting modules over a skew-gentle algebra. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e113. doi: 10.1017/fms.2025.49
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[1] Adachi, T., Iyama, O. and Reiten, I., ‘-tilting theory’, Compos. Math. 150(2014), 415–452. http://dx.doi.org/10.1112/S0010437X13007422 Google Scholar | DOI

[2] Amiot, C., ‘Cluster categories for algebras of global dimension 2 and quiver with potentials’, Ann. Inst. Fourier (Grenoble) 59(2009), 2525–2590. http://dx.doi.org/10.5802/aif.2499 Google Scholar | DOI

[3] Amiot, C., ‘Indecomposable objects in the derived category of skew-gentle algebra using orbifolds’ in Representations of Algebras and Related Structures (EMS Ser. Congr. Rep., 2023), 1–24. http://dx.doi.org/10.4171/ECR/19 Google Scholar

[4] Amiot, C. and Brüstle, T., ‘Derived equivalences between skew-gentle algebras using orbifolds’, Doc. Math. 27(2022), 933–982. http://dx.doi.org/10.4171/DM/889 Google Scholar | DOI

[5] Amiot, C. and Plamondon, P-G., ‘The cluster category of a surface with punctures via group actions’, Adv. Math. 389(2021), 107884. http://dx.doi.org/10.1016/j.aim.2021.107884 Google Scholar | DOI

[6] Amiot, C., Plamondon, P-G. and Schroll, S., ‘A complete derived invariant for gentle algebras via winding numbers and Arf invariants’, Sel. Math. New Ser. 29(2023) Paper No.30, 36pp. https://doi.org/10.1007/s00029-022-00822-x Google Scholar | DOI

[7] Asai, S., ‘Non-rigid regions of real Grothendieck groups of gentle and special biserial algebras’, Preprint, 2022, . Google Scholar | arXiv

[8] Assem, I., Brüstle, T., Charbonneau-Jodoin, G. and Plamondon, P-G., ‘Gentle algebras arising from surface triangulations’, Algebra Number Theory, 4(2010), 201–229. http://dx.doi.org/10.2140/ant.2010.4.201 Google Scholar | DOI

[9] Baur, K., and Coelho Simões, R., ‘A geometric model for the module category of a gentle algebra’, Int. Math. Res. Not. IMRN 2021(2019), 11357–11392. http://dx.doi.org/10.1093/imrn/rnz150 Google Scholar | DOI

[10] Bondarenko, V. M., ‘Representations of bundles of semichained sets and their applications’, St. Petersburg Math. J. 3(1992), 937–996. Google Scholar

[11] Breaz, S., Marcus, A. and Modoi, G. C., ‘Support -tilting modules and semibricks over group graded algebras’, J. Algebra 637(2024), 90–111. http://dx.doi.org/10.1016/j.jalgebra.2023.08.030 Google Scholar | DOI

[12] Brüstle, T., and Qiu, Y., ‘Tagged mapping class groups: Auslander-Reiten translation’, Math. Z. 279(2015), 1103–1120. http://dx.doi.org/10.1007/s00209-015-1405-z Google Scholar | DOI

[13] Brüstle, T., and Zhang, J., ‘On the cluster category of a marked surface without punctures’, Algebra Number Theory 5(2011), 529–566. http://dx.doi.org/10.2140/ant.2011.5.529 Google Scholar | DOI

[14] Buan, A. B., Marsh, R., Reineke, M., Reiten, I. and Todorov, G., ‘Tilting theory and cluster combinatorics’, Adv. Math. 204(2006), 572–618. http://dx.doi.org/10.1016/j.aim.2005.06.003 Google Scholar | DOI

[15] Buan, A. B., Marsh, R. and Vatne, D. F., ‘Cluster structures from 2-Calabi-Yau categories with loops’, Math. Z. 265(2010), 951–970. http://dx.doi.org/10.1007/s00209-009-0549-0 Google Scholar | DOI

[16] Burban, I., Iyama, O., Keller, B. and Reiten, I., ‘Cluster tilting for one-dimensional hypersurface singularities’, Adv. Math. 217(2008), 2443–2484. http://dx.doi.org/10.1016/j.aim.2007.10.007 Google Scholar | DOI

[17] Canakci, I. and Schroll, S., ‘Extensions in Jacobian algebras and cluster categories of marked surfaces’, Adv. Math. 313(2017), 1–49. http://dx.doi.org/10.1016/j.aim.2017.03.016 Google Scholar | DOI

[18] Chang, W., Zhang, J. and Zhu, B., ‘On support -tilting modules over endomorphism rigid algebras of rigid objects’, Acta Math. Sin. (Engl. Ser.) 31(2015), 1508–1516. https://doi.org/10.1007/s10114-015-4161-4 Google Scholar | DOI

[19] Crawley-Boevey, W., ‘Functorial filtrations II: Clans and the Gelfand problem’, J. London Math. Soc. 40(1989), 9–30. http://dx.doi.org/10.1112/jlms/s2-40.1.9 Google Scholar | DOI

[20] David-Roesler, L. and Schiffler, R., ‘Algebras from surfaces without punctures’, J. Algebra 350(2012), 218–244. http://dx.doi.org/10.1016/j.jalgebra.2011.10.034 Google Scholar | DOI

[21] Dehy, R., and Keller, B., ‘On the combinatorics of rigid objects in 2-Calabi-Yau categories’, Int. Math. Res. Not. IMRN 2008(2008), rnn029. http://dx.doi.org/10.1093/imrn/rnn029 Google Scholar

[22] Deng, B., ‘On a problem of Nazarova and Roiter’, Comment. Math. Helv. 75(2000), 368–409. http://dx.doi.org/10.1007/S000140050132 Google Scholar | DOI

[23] Derksen, H., Weyman, J. and Zelevinski, A., ‘Quivers with potentials and their representations I: Mutations’, Sel. Math. New Ser. 14(2008), 59–119. http://dx.doi.org/10.1007/s00029-008-0057-9 Google Scholar | DOI

[24] Disarlo, V., and Palier, H., ‘The geometry of flip graphs and mapping class groups’, Trans. Amer. Math. Soc. 372(2019), 3809–3844. http://dx.doi.org/10.1090/tran/7356 Google Scholar | DOI

[25] Fomin, S., Shapiro, M. and Thurston, D., ‘Cluster algebras and triangulated surfaces. Part I: Cluster complexes’, Acta Math. 201(2008), 83–146. http://dx.doi.org/10.1007/s11511-008-0030-7 Google Scholar | DOI

[26] Fomin, S., and Thurston, D., ‘Cluster algebras and triangulated surfaces. Part II: Lambda lengths’, Mem. Amer. Math. Soc. 1223(2018), 255–295. http://dx.doi.org/10.1090/memo/1223 Google Scholar

[27] Fomin, S., and Zelevinsky, A., ‘Cluster algebras I: Foundations’, J. Amer. Math. Soc. 15(2002), 497–529. http://dx.doi.org/10.1090/s0894-0347-01-00385-x Google Scholar | DOI

[28] Fu, C., Geng, S., Liu, P. and Zhou, Y., ‘On support -tilting graphs of gentle algebras’, J. Algebra 628(2023), 189–211. http://dx.doi.org/10.1016/j.jalgebra.2023.03.013 Google Scholar | DOI

[29] Geiß, C., ‘Maps between representations of clans’, J. Algebra 218(1999), 131–164. http://dx.doi.org/10.1006/jabr.1998.7829 Google Scholar | DOI

[30] Geiß, C., Labardini-Fragoso, D. and Schröer, J., ‘The representation type of Jacobian algebras’, Adv. Math. 290(2016), 364–452. http://dx.doi.org/10.1016/j.aim.2015.09.038 Google Scholar | DOI

[31] Geiß, C., and De La Peña, J. A., ‘Auslander-Reiten components for clans’, Boll. Soc. Mat. Mexicana 5(1999), 307–326. Google Scholar

[32] Haiden, F., Katzarkov, L. and Kontsevich, M., ‘Flat surfaces and stability structures’, Publ. Math. Inst. Hautes Études Sci. 126(2017), 247–318. http://dx.doi.org/10.1007/s10240-017-0095-y Google Scholar | DOI

[33] He, P., Zhou, Y. and Zhu, B., ‘A geometric model for the module category of a skew-gentle algebra’, Math. Z. (2023) Paper No.18. https://doi.org/10.1007/s00209-023-03275-w Google Scholar | DOI

[34] Irelli, G. C., and Labardini-Fragoso, D., ‘Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials’, Compos. Math. 148(2012), 1833–1866. http://dx.doi.org/10.1112/S0010437X12000528 Google Scholar | DOI

[35] Iyama, O., and Yoshino, Y., ‘Mutations in triangulated categories and rigid Cohen-Macaulay modules’, Invent. Math. 172(2008), 117–168. http://dx.doi.org/10.1007/s00222-007-0096-4 Google Scholar | DOI

[36] Keller, B., ‘Deformed Calabi-Yau completions’, J. Reine. Angew. Math. 654(2011), 125–180. http://dx.doi.org/10.1515/crelle.2011.031 Google Scholar

[37] Keller, B. and Yang, D., ‘Derived equivalences from mutations of quivers with potential’, Adv. Math. 226(2011), 2118–2168. http://dx.doi.org/10.1016/j.aim.2010.09.019 Google Scholar | DOI

[38] Kimura, Y., Koshio, R., Kozakai, Y., Minamoto, H. and Mizuno, Y., ‘-tilting theory and silting theory of skew group algebra extensions’, Preprint, 2024, . Google Scholar | arXiv

[39] Koenig, S., and Zhu, B., ‘From triangulated categories to abelian categories: cluster-tilting in a general framework’, Math. Z. 258(2008), 143–160. http://dx.doi.org/10.1007/s00209-007-0165-9 Google Scholar | DOI

[40] Labardini-Fragoso, D., ‘Quivers with potentials associated to triangulated surfaces’, Proc. Lond. Math. Soc. 98(2009), 797–839. http://dx.doi.org/10.1112/plms/pdn051 Google Scholar | DOI

[41] Labardini-Fragoso, D., ‘Quivers with potentials associated to triangulated surfaces, Part II: Arc representations’, Preprint, 2009, . Google Scholar | arXiv

[42] Labardini-Fragoso, D., Schroll, S. and Valdivieso, Y., ‘Derived categories of skew-gentle algebras and orbifolds’, Glasg. Math. J. 63(2022), 649–674. http://dx.doi.org/10.1017/S0017089521000422 Google Scholar | DOI

[43] Lekili, Y. and Polishchuk, A., ‘Derived equivalences of gentle algebras via Fukaya categories’, Math. Ann. 376(2020), 187–225. http://dx.doi.org/10.1007/s00208-019-01894-5 Google Scholar | DOI

[44] Marsh, R., and Palu, Y., ‘Coloured quivers for rigid objects and partial triangulations: the unpunctured case’, Proc. Lond. Math. Soc. 108(2014), 411–440. http://dx.doi.org/10.1112/plms/pdt032 Google Scholar | DOI

[45] Opper, S., ‘On auto-equivalences and complete derived invariants of gentle algebras’, Preprint, 2019, . Google Scholar | arXiv

[46] Opper, S., Plamondon, P. G. and Schroll, S., ‘A geometric model for the derived category of gentle algebras’, Preprint, 2018, . Google Scholar | arXiv

[47] Palu, Y., ‘Cluster characters for 2-Calabi-Yau triangulated categories’, Ann. Inst. Fourier (Grenoble) 58(2008), 2221–2248. http://dx.doi.org/10.5802/aif.2412 Google Scholar | DOI

[48] Plamondon, P-G., ‘Cluster algebras via cluster categories with infinite-dimensional morphism spaces’, Compos. Math. 147(2011), 1921–1954. https://doi.org/10.1112/S0010437X11005483 Google Scholar | DOI

[49] Qiu, Y., and Zhou, Y., ‘Cluster categories for marked surfaces: punctured case’, Compos. Math. 153(2017), 1779–1819. http://dx.doi.org/10.1112/S0010437X17007229 Google Scholar | DOI

[50] Reading, N., ‘Universal geometric cluster algebras from surfaces’, Trans. Amer. Math. Soc. 366(2014), 6647–6685. http://dx.doi.org/10.1090/s0002-9947-2014-06156-4 Google Scholar | DOI

[51] Yurikusa, T., ‘Density of g-vector cones from triangulated categories’, Int. Math. Res. Not. IMRN 2020(2020), 8081–8119. http://dx.doi.org/10.1093/imrn/rnaa008 Google Scholar | DOI

[52] Zhang, Y., and Huang, Z., ‘G-stable support -tilting modules’, Front. Math. China 11(2016), 1057–1077. http://dx.doi.org/10.1007/s11464-016-0560-9 Google Scholar | DOI

[53] Zhang, J., Zhou, Y. and Zhu, B., ‘Cotorsion pairs in the cluster category of a marked surface’, J. Algebra 391(2013), 209–226. http://dx.doi.org/10.1016/j.jalgebra.2013.06.014 Google Scholar | DOI

[54] Zhou, Y., and Zhu, B., ‘Maximal rigid subcategories in 2-Calabi-Yau triangulated categories’, J. Algebra 348(2011), 49–60. http://dx.doi.org/10.1016/j.jalgebra.2011.09.027 Google Scholar | DOI

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