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Semicontinuous maps on module varieties

  • Christof Geiß , Daniel Labardini-Fragoso and Jan Schröer EMAIL logo
Published/Copyright: August 23, 2024

Abstract

We study semicontinuous maps on varieties of modules over finite-dimensional algebras. We prove that truncated Euler maps are upper or lower semicontinuous. This implies that 𝑔-vectors and 𝐸-invariants of modules are upper semicontinuous. We also discuss inequalities of generic values of some upper semicontinuous maps.

References

[1] T. Adachi, O. Iyama and I. Reiten, 𝜏-tilting theory, Compos. Math. 150 (2014), no. 3, 415–452. 10.1112/S0010437X13007422Search in Google Scholar

[2] I. Assem, D. Simson and A. Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Math. Soc. Stud. Texts 65, Cambridge University, Cambridge 2006. 10.1017/CBO9780511614309Search in Google Scholar

[3] M. Auslander, I. Reiten and S. O. Smalø, Representation theory of Artin algebras, Cambridge Stud. Adv. Math. 36, Cambridge University, Cambridge 1997. Search in Google Scholar

[4] K. Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), no. 2, 245–287. 10.1006/aima.1996.0053Search in Google Scholar

[5] A. T. Carroll and C. Chindris, On the invariant theory for acyclic gentle algebras, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3481–3508. 10.1090/S0002-9947-2014-06191-6Search in Google Scholar

[6] G. Cerulli Irelli, D. Labardini-Fragoso and J. Schröer, Caldero–Chapoton algebras, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2787–2822. 10.1090/S0002-9947-2014-06175-8Search in Google Scholar

[7] W. Crawley-Boevey and J. Schröer, Irreducible components of varieties of modules, J. reine angew. Math. 553 (2002), 201–220. 10.1515/crll.2002.100Search in Google Scholar

[8] H. Derksen and J. Fei, General presentations of algebras, Adv. Math. 278 (2015), 210–237. 10.1016/j.aim.2015.03.012Search in Google Scholar

[9] H. Derksen, J. Weyman and A. Zelevinsky, Quivers with potentials and their representations. I. Mutations, Selecta Math. (N. S.) 14 (2008), no. 1, 59–119. 10.1007/s00029-008-0057-9Search in Google Scholar

[10] H. Derksen, J. Weyman and A. Zelevinsky, Quivers with potentials and their representations II. Applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), no. 3, 749–790. 10.1090/S0894-0347-10-00662-4Search in Google Scholar

[11] P. Gabriel, Finite representation type is open, Proceedings of the International Conference on Representations of Algebras (Ottawa 1974), Carleton Math. Lecture Notes 9, Carleton University, Ottawa (1974), Paper No. 10. Search in Google Scholar

[12] C. Geiß, D. Labardini-Fragoso and J. Schröer, The representation type of Jacobian algebras, Adv. Math. 290 (2016), 364–452. 10.1016/j.aim.2015.09.038Search in Google Scholar

[13] C. Geiß, D. Labardini-Fragoso and J. Schröer, Generic Caldero–Chapoton functions with coefficients and applications to surface cluster algebras, preprint (2020), https://arxiv.org/abs/2007.05483. Search in Google Scholar

[14] C. Geiß, D. Labardini-Fragoso and J. Schröer, Schemes of modules over gentle algebras and laminations of surfaces, Selecta Math. (N. S.) 28 (2022), no. 1, Paper No. 8. 10.1007/s00029-021-00710-wSearch in Google Scholar

[15] C. Geiß, D. Labardini-Fragoso and J. Schröer, Geometrization of DWZ-mutations, preprint (2024), in preparation. Search in Google Scholar

[16] C. Geiss, B. Leclerc and J. Schröer, Generic bases for cluster algebras and the Chamber ansatz, J. Amer. Math. Soc. 25 (2012), no. 1, 21–76. 10.1090/S0894-0347-2011-00715-7Search in Google Scholar

[17] C. Geiss, B. Leclerc and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), no. 1, 61–158. 10.1007/s00222-016-0705-1Search in Google Scholar

[18] V. G. Kac, Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78 (1982), no. 1, 141–162. 10.1016/0021-8693(82)90105-3Search in Google Scholar

[19] C. Pfeifer, Email communication. Search in Google Scholar

[20] P.-G. Plamondon, Generic bases for cluster algebras from the cluster category, Int. Math. Res. Not. IMRN 2013 (2013), no. 10, 2368–2420. 10.1093/imrn/rns102Search in Google Scholar

[21] P.-G. Plamondon, T. Yurikusa and B. Keller, Tame algebras have dense g-vector fans, Int. Math. Res. Not. IMRN 2023 (2023), no. 4, 2701–2747. 10.1093/imrn/rnab105Search in Google Scholar

[22] F. Qin, Bases for upper cluster algebras and tropical points, J. Eur. Math. Soc. (JEMS) 26 (2024), no. 4, 1255–1312. 10.4171/jems/1308Search in Google Scholar

[23] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math. 1099, Springer, Berlin 1984. 10.1007/BFb0072870Search in Google Scholar

[24] A. Schofield, General representations of quivers, Proc. Lond. Math. Soc. (3) 65 (1992), no. 1, 46–64. 10.1112/plms/s3-65.1.46Search in Google Scholar

Received: 2023-02-04
Revised: 2024-06-09
Published Online: 2024-08-23
Published in Print: 2024-11-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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