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Normality of Maximal Orbit Closures for Euclidean Quivers

Published online by Cambridge University Press:  20 November 2018

Grzegorz Bobiński
Grzegorz Bobiński*
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland email: [email protected]
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Abstract

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Let $\Delta $ be a Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of $\Delta $ are normal and Cohen–Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

Keywords

16G2014L30normal varietycomplete intersectionEuclidean quiverconcealed-canonical algebra
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 6 , 01 December 2012 , pp. 1222 - 1247
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Abeasis, S., Del Fra, A., and Kraft, H., The geometry of representations of Am. Math. Ann. 256(1981), 401–418. http://dx.doi.org/10.1007/BF01679706 Google Scholar
[2] Assem, I., Simson, D., and Skowronśki, A., Elements of the Representation Theory of Associative Algebras. I. London Math. Soc. Stud. Texts 65, Cambridge University Press, Cambridge, 2006.Google Scholar
[3] Auslander, M., Reiten, I., and Smalø, S. O., Representation Theory of Artin Algebras. Cambridge Stud. Adv. Math. 36, Cambridge University Press, Cambridge, 1995.Google Scholar
[4] Bender, J. and Bongartz, K., Minimal singularities in orbit closures of matrix pencils. Linear Algebra Appl. 365(2003), 13–24. http://dx.doi.org/10.1016/S0024-3795(01)00356-1 Google Scholar
[5] Bobinśki, G., Geometry of regular modules over canonical algebras. Trans. Amer. Math. Soc. 360(2008), 717–742. http://dx.doi.org/10.1090/S0002-9947-07-04174-8 Google Scholar
[6] Bobinśki, G., Orbit closures of directing modules are regular in codimension one. J. London Math. Soc. (2) 79(2009), 211–224. http://dx.doi.org/10.1112/jlms/jdn067 Google Scholar
[7] Bobinśki, G., Riedtmann, Ch., and Skowronśki, A., Semi-invariants of quivers and their zero sets. In: Trends in Representation Theory of Algebras and Related Topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, 49–99.Google Scholar
[8] Bobinśki, G. and Skowronśki, A., Geometry of modules over tame quasi-tilted algebras. Colloq. Math. 79(1999), 85–118.Google Scholar
[9] Bobinśki, G. and Skowronśki, A., Geometry of directing modules over tame algebras. J. Algebra 215(1999), 603–643. http://dx.doi.org/10.1006/jabr.1998.7748 Google Scholar
[10] Bobinśki, G. and Skowronśki, A., Geometry of periodic modules over tame concealed and tubular algebras. Algebr. Represent. Theory 5, (2002), 187–200. http://dx.doi.org/10.1023/A:1015606729502 Google Scholar
[11] Bobinśki, G. and Zwara, G., Schubert varieties and representations of Dynkin quivers. Colloq. Math. 94(2002), 285–309. http://dx.doi.org/10.4064/cm94-2-10 Google Scholar
[12] Bobinśki, G. and Zwara, G., Normality of orbit closures for directing modules over tame algebras. J. Algebra 298(2006), 120–133. http://dx.doi.org/10.1016/j.jalgebra.2005.06.023 Google Scholar
[13] Bongartz, K., Algebras and quadratic forms. J. London Math. Soc. 28(1983), 461–469. http://dx.doi.org/10.1112/jlms/s2-28.3.461 Google Scholar
[14] Bongartz, K., A geometric version of the Morita equivalence. J. Algebra 139(1991), 159–171. http://dx.doi.org/10.1016/0021-8693(91)90288-J Google Scholar
[15] Bongartz, K., Minimal singularities for representations of Dynkin quivers. Comment. Math. Helv. 69(1994), 575–611. http://dx.doi.org/10.1007/BF02564505 Google Scholar
[16] Bongartz, K., Degenerations for representations of tame quivers. Ann. Sci. École Norm. Sup. (4) 28(1995), 647–668.Google Scholar
[17] Bongartz, K., On degenerations and extensions of finite-dimensional modules. Adv. Math. 121(1996), 245–287. http://dx.doi.org/10.1006/aima.1996.0053 Google Scholar
[18] Bongartz, K., Frank, G. and I.Wolters, On minimal disjoint degenerations of modules over tame path algebras. Adv. Math. 226(2011), 1875–1910. http://dx.doi.org/10.1016/j.aim.2010.09.002 Google Scholar
[19] Chindris, C., On orbit closures for infinite type quivers. arxiv:0709.3613.Google Scholar
[20] Chindris, C., Orbit semigroups and the representation type of quivers. J. Pure Appl. Algebra 213(2009), 1418–1429. http://dx.doi.org/10.1016/j.jpaa.2008.12.003 Google Scholar
[21] Crawley-Boevey, W. W., On tame algebras and bocses. Proc. London Math. Soc. (3) 56(1988), 451–483. http://dx.doi.org/10.1112/plms/s3-56.3.451 Google Scholar
[22] Crawley-Boevey, W. W. and Schröer, J., Irreducible components of varieties of modules. J. Reine Angew. Math. 553(2002), 201–220. http://dx.doi.org/10.1515/crll.2002.100 Google Scholar
[23] Derksen, H. and Weyman, J., Semi-invariants for quivers with relations. J. Algebra 258(2002), 216–227. http://dx.doi.org/10.1016/S0021-8693(02)00501-X Google Scholar
[24] Domokos, M., Relative invariants for representations of finite-dimensional algebras. Manuscripta Math. 108(2002), 123–133. http://dx.doi.org/10.1007/s002290200258 Google Scholar
[25] Domokos, M. and Lenzing, H., Moduli spaces for representations of concealed-canonical algebras. J. Algebra 251(2002), 371–394. http://dx.doi.org/10.1006/jabr.2001.9117 Google Scholar
[26] Drozd, Yu. A., Tame and wild matrix problems. In: Representation Theory II, Lecture Notes in Math. 832, Springer, Berlin, 1980, 242–258.Google Scholar
[27] Eisenbud, D., Commutative Algebra, Grad. Texts in Math. 150, Springer, New York, 1995.Google Scholar
[28] Gabriel, P., Unzerlegbare Darstellungen. I. Manuscripta Math. 6(1972), 71–103. http://dx.doi.org/10.1007/BF01298413 Google Scholar
[29] King, A. D., Moduli of representations of finite-dimensional algebras. Quart. J. Math. Oxford Ser. (2) 45(1994), 515–530. http://dx.doi.org/10.1093/qmath/45.4.515 Google Scholar
[30] Kunz, E., Introduction to Commutative Algebra and Algebraic Geometry. Birkhäuser, Boston, MA, 1985.Google Scholar
[31] Lenzing, H. and Meltzer, H., Tilting sheaves and concealed-canonical algebras. In: Representation Theory of Algebras, CMS Conf. Proc. 18, Amer. Math. Soc., Providence, RI, 1996, 455–473.Google Scholar
[32] Lenzing, H. and de la, J. A. Pe˜na, Concealed-canonical algebras and separating tubular families. Proc. London Math. Soc. (3) 78(1999), 513–540. http://dx.doi.org/10.1112/S0024611599001872 Google Scholar
[33] Loc, N. Q. and Zwara, G., Regular orbit closures in module varieties. Osaka J. Math. 44(2007), 945–954.Google Scholar
[34] Riedtmann, Ch. and Zwara, G., Orbit closures and rank schemes. Comment. Math. Helv., to appear.Google Scholar
[35] Ringel, C. M., The rational invariants of the tame quivers. Invent. Math. 58(1980), 217–239. http://dx.doi.org/10.1007/BF01390253 Google Scholar
[36] Ringel, C. M., Tame Algebras and Integral Quadratic Forms. Lecture Notes in Math. 1099, Springer, Berlin, 1984.Google Scholar
[37] Simson, D. and Skowronśki, A., Elements of the Representation Theory of Associative Algebras. Vol. 2. London Math. Soc. Stud. Texts 71, Cambridge University Press, Cambridge, 2007.Google Scholar
[38] Simson, D. and Skowronśki, A., Elements of the Representation Theory of Associative Algebras. Vol. 3. London Math. Soc. Stud. Texts 71, Cambridge University Press, Cambridge, 2007.Google Scholar
[39] Skowronśki, A., On omnipresent tubular families of modules. In: Representation Theory of Algebras, CMS Conf. Proc. 18, Amer. Math. Soc., Providence, RI, 1996, 641–657.Google Scholar
[40] Skowronśki, A. and J.Weyman, The algebras of semi-invariants of quivers. Transform. Groups 5(2000), 361–402. http://dx.doi.org/10.1007/BF01234798 Google Scholar
[41] Skowronśki, A. and Zwara, G., Degenerations for indecomposable modules and tame algebras. Ann. Sci. École Norm. Sup. (4) 31(1998), 153–180.Google Scholar
[42] Skowronśki, A. and Zwara, G., Derived equivalences of selfinjective algebras preserve singularities. Manuscripta Math. 112(2003), 221–230. http://dx.doi.org/10.1007/s00229-003-0396-y Google Scholar
[43] Voigt, D., Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen. Lecture Notes in Math. 592, Springer, Berlin, 1977.Google Scholar
[44] Zwara, G., Degenerations for modules over representation-finite algebras. Proc. Amer. Math. Soc. 127(1999), 1313–1322. http://dx.doi.org/10.1090/S0002-9939-99-04714-0 Google Scholar
[45] Zwara, G., Degenerations of finite-dimensional modules are given by extensions. Compositio Math. 121(2000), 205–218. http://dx.doi.org/10.1023/A:1001778532124 Google Scholar
[46] Zwara, G., Unibranch orbit closures in module varieties. Ann. Sci. École Norm. Sup. (4) 35(2002), 877–895.Google Scholar
[47] Zwara, G., An orbit closure for a representation of the Kronecker quiver with bad singularities. Colloq. Math. 97(2003), 81–86. http://dx.doi.org/10.4064/cm97-1-8 Google Scholar
[48] Zwara, G., Regularity in codimension one of orbit closures in module varieties. J. Algebra 283(2005), 821–848. http://dx.doi.org/10.1016/j.jalgebra.2004.09.021 Google Scholar
[49] Zwara, G., Orbit closures for representations of Dynkin quivers are regular in codimension two. J. Math. Soc. Japan 57(2005), 859–880. http://dx.doi.org/10.2969/jmsj/1158241938 Google Scholar
[50] Zwara, G., Singularities of orbit closures in module varieties and cones over rational normal curves. J. London Math. Soc. (2) 74(2006), 623–638. http://dx.doi.org/10.1112/S0024610706023271 Google Scholar
[51] Zwara, G., Codimension two singularities for representations of extended Dynkin quivers. Manuscripta Math. 123(2007), 237–249. http://dx.doi.org/10.1007/s00229-007-0093-3 Google Scholar
[52] Zwara, G., Singularities of orbit closures in module varieties. In: Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 661–725.Google Scholar