Geometry of regular modules over canonical algebras

Bobiński, Grzegorz

doi:10.1090/S0002-9947-07-04174-8

Geometry of regular modules over canonical algebras
HTML articles powered by AMS MathViewer

by Grzegorz Bobiński PDF

Trans. Amer. Math. Soc. 360 (2008), 717-742 Request permission

Abstract:

We classify canonical algebras such that for every dimension vector of a regular module the corresponding module variety is normal (respectively, a complete intersection). We also prove that for the dimension vectors of regular modules normality is equivalent to irreducibility.

References

M. Barot and Jan Schröer, Module varieties over canonical algebras, J. Algebra 246 (2001), no. 1, 175–192. MR 1872616, DOI 10.1006/jabr.2001.8933
Grzegorz Bobiński and Andrzej Skowroński, Geometry of modules over tame quasi-tilted algebras, Colloq. Math. 79 (1999), no. 1, 85–118. MR 1671811, DOI 10.4064/cm-79-1-85-118
Grzegorz Bobiński and Andrzej Skowroński, Geometry of periodic modules over tame concealed and tubular algebras, Algebr. Represent. Theory 5 (2002), no. 2, 187–200. MR 1909550, DOI 10.1023/A:1015606729502
Klaus Bongartz, Algebras and quadratic forms, J. London Math. Soc. (2) 28 (1983), no. 3, 461–469. MR 724715, DOI 10.1112/jlms/s2-28.3.461
Klaus Bongartz, Minimal singularities for representations of Dynkin quivers, Comment. Math. Helv. 69 (1994), no. 4, 575–611. MR 1303228, DOI 10.1007/BF02564505
Klaus Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), no. 2, 245–287. MR 1402728, DOI 10.1006/aima.1996.0053
Klaus Bongartz, Some geometric aspects of representation theory, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 1–27. MR 1648601
William Crawley-Boevey and Jan Schröer, Irreducible components of varieties of modules, J. Reine Angew. Math. 553 (2002), 201–220. MR 1944812, DOI 10.1515/crll.2002.100
Mátyás Domokos and Helmut Lenzing, Invariant theory of canonical algebras, J. Algebra 228 (2000), no. 2, 738–762. MR 1764590, DOI 10.1006/jabr.2000.8298
M. Domokos and H. Lenzing, Moduli spaces for representations of concealed-canonical algebras, J. Algebra 251 (2002), no. 1, 371–394. MR 1900290, DOI 10.1006/jabr.2001.9117
Werner Geigle and Helmut Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 265–297. MR 915180, DOI 10.1007/BFb0078849
Christof Geiß, Geometric methods in representation theory of finite-dimensional algebras, Representation Theory of Algebras and Related Topics (Mexico City, 1994), 1996, pp. 53–63.
Dieter Happel, A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), no. 2, 381–398. MR 1827736, DOI 10.1007/s002220100135
Dieter Happel, Idun Reiten, and Sverre O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88. MR 1327209, DOI 10.1090/memo/0575
Hanspeter Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984 (German). MR 768181, DOI 10.1007/978-3-322-83813-1
Hanspeter Kraft, Geometric methods in representation theory, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 944, Springer, Berlin-New York, 1982, pp. 180–258. MR 672117
Helmut Lenzing and José Antonio de la Peña, Concealed-canonical algebras and separating tubular families, Proc. London Math. Soc. (3) 78 (1999), no. 3, 513–540. MR 1674837, DOI 10.1112/S0024611599001872
Markus Reineke, The monoid of families of quiver representations, Proc. London Math. Soc. (3) 84 (2002), no. 3, 663–685. MR 1888427, DOI 10.1112/S0024611502013497
Claus Michael Ringel, The rational invariants of the tame quivers, Invent. Math. 58 (1980), no. 3, 217–239. MR 571574, DOI 10.1007/BF01390253
Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
Andrzej Skowroński, On omnipresent tubular families of modules, Representation theory of algebras (Cocoyoc, 1994) CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 641–657. MR 1388078
Andrzej Skowroński and Jerzy Weyman, Semi-invariants of canonical algebras, Manuscripta Math. 100 (1999), no. 3, 391–403. MR 1726226, DOI 10.1007/s002290050208
Detlef Voigt, Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen, Lecture Notes in Mathematics, Vol. 592, Springer-Verlag, Berlin-New York, 1977 (German). Mit einer englischen Einführung. MR 0486168, DOI 10.1007/BFb0086128