Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-18T09:14:05.470Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinite Dimensional Representations of Canonical Algebras

Published online by Cambridge University Press:  20 November 2018

Idun Reiten  and
Claus Michael Ringel
Idun Reiten
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, 7490 Trondheim, Norway e-mail: idunr@math.ntnu.no
Claus Michael Ringel
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, 7490 Trondheim, Norway e-mail: idunr@math.ntnu.no
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to the more general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with canonical algebras. The investigation is centered around the generic and the Prüfer modules, and how other modules are determined by these modules.

Keywords

16D7016D9016G2016G6016G70
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 1 , 01 February 2006 , pp. 180 - 224
Copyright
Copyright © Canadian Mathematical Society 2006

References

[AB] Auslander, M. and Buchweitz, R. O., The homological theory of maximal Cohen-Macaulay approximations. Mém.s Soc. Math. France. 38(1989), 5–37.Google Scholar
[AC] Angeleri-Hügel, L., and Coelho, F. U., Infinitely generated tilting modules of finite projective dimension. Forum Math. 13(2001), no. 2, 239–250.Google Scholar
[ATT] Angeleri-Hügel, L., Tonolo, A., and Trlifaj, J., Tilting preenvelopes and cotilting precovers. Algebr. Represent. Theory 4(2001), no. 2, 155–170.Google Scholar
[BK] Buan, A., and Krause, H., Cotilting modules over tame hereditary algebras. Pacific J. Math 211(2003), no. 1, 41–59.Google Scholar
[BS] Buan, A. and Solberg, Ø., Limits of pure-injective cotilting modules. Algebr. Represent. Theory. (To appear).Google Scholar
[CF] Colby, R. R. and Fuller, K. R., Tilting, cotilting and serially tilted rings. Comm. Algebra 18(1990), no. 5, 1585–1615.Google Scholar
[CET] Colpi, R., D’Este, G., and Tonolo, A., Quasi-tilting modules and counter equivalences. J. Algebra 191(1997), no. 2, 461–494. Corrigendum in J. Algebra 206(1998), 370.Google Scholar
[CT] Colpi, R. and Trlifaj, J., Tilting modules and tilting torsion theories. J. Algebra 178(1995), no. 2, 614–634.Google Scholar
[DR] Dlab, V., and Ringel, C. M., Indecomposable Representations of Graphs and Algebras. Mem. Amer. Math. Soc 6(1976), no. 173.Google Scholar
[ET] Eklof, P., and Trlifaj, J., Covers induced by Ext. J. Algebra 231(2000), no. 2, 640–651.Google Scholar
[E] Enochs, E. E., Injective and flat covers, envelopes and resolvents. Israel J. Math. 39(1981), no. 3, 189–209.Google Scholar
[EJ] Enochs, E. E. and Jenda, O. M. G., Relative Homological Algebra. De Gruyter. Berlin, 2000.Google Scholar
[F] Fuchs, L., Infinite abelian groups. II. Academic Press, New York 1973.Google Scholar
[HR] Happel, D. and Reiten, I., An introduction to quasitilted algebras. An. Şiinţ. Univ. Ovidius Constanţa Ser. Mat. 4(1996), no. 2, 137–149.Google Scholar
[HRS] Happel, D., Reiten, I., and Smalø, S. O., Tilting in abelian categories and quasitilted algebras. Mem. Amer.Math. Soc. 120(1996), no. 575.Google Scholar
[K] Krause, H., Generic modules over Artin algebras. Proc. London Math. Soc. (3) 76(1998), no. 2, 276–306.Google Scholar
[L] Lenzing, H., Generic modules over tubular algebras. In: Advances in Algebra and Model Theory. Algebra Logic Appl. 9, Gordon and Breach, Amsterdam, 1997, pp. 375–385.Google Scholar
[LM] 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, pp. 455–473.Google Scholar
[LP] Lenzing, H. and de la Peña, J., Concealed-canonical algebras and separating tubular families. Proc. London Math. Soc. (3) 78(1999), 513–540.Google Scholar
[RS] Reiten, I. and Skowroñski, A., Sincere stable tubes. J. Algebra. 232(2000) no. 1, 64–75.Google Scholar
[R1] Ringel, C. M., Infinite-dimensional representations of finite- dimensional hereditary algebras. In: Symposia Mathematica 23, Academic Press, London, 1979, pp. 321–412.Google Scholar
[R2] Ringel, C. M., Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics 1099, Springer-Verlag, Berlin, 1984.Google Scholar
[R3] Ringel, C. M., Representation theory of finite-dimensional algebras. In: Representations of Algebras, London Math. Soc. Lecture Note Series 116, Cambridge University Press, Cambridge, 1986, 7–79.Google Scholar
[R4] Ringel, C. M., The canonical algebras. In: Topics in Algebra. Banach Center Publications 26, Part 1, Warsaw, 1990 pp. 407–432.Google Scholar
[R5] Ringel, C. M., The Ziegler spectrum of a tame hereditary algebra. Colloq. Math. 76(1998), 105–115.Google Scholar
[R6] Ringel, C. M., A Construction of endofinite modules. In: Advances in Algebra andModel Theory. Gordon and Breach, Amsterdam. 1997, pp. 387–399.Google Scholar
[R7] Ringel, C. M., Tame algebras are wild. Algebra Colloq. 6(1999), 473–480.Google Scholar
[R8] Ringel, C. M., Infinite length modules. Some examples as introduction. In: Infinite Length Modules, Birkhäuser, Basel, 2000, 1–73.Google Scholar
[R9] Ringel, C. M., Algebraically compact modules arising from tubular families. A Survey. Algebra Colloq. 11(2004), no. 1, 155–172.Google Scholar