Representation theory of two-dimensionalbrauer graph rings

Wolfgang Rump

Colloquium Mathematicae (2000)

  • Volume: 86, Issue: 2, page 239-251
  • ISSN: 0010-1354

Abstract

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We consider a class of two-dimensional non-commutative Cohen-Macaulay rings to which a Brauer graph, that is, a finite graph endowed with a cyclic ordering of edges at any vertex, can be associated in a natural way. Some orders Λ over a two-dimensional regular local ring are of this type. They arise, e.g., as certain blocks of Hecke algebras over the completion of ℤ [ q , q - 1 ] at (p,q-1) for some rational prime p . For such orders Λ, a class of indecomposable maximal Cohen-Macaulay modules (see introduction) has been determined by K. W. Roggenkamp. We prove that this list of indecomposables of Λ is complete.

How to cite

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Rump, Wolfgang. "Representation theory of two-dimensionalbrauer graph rings." Colloquium Mathematicae 86.2 (2000): 239-251. <http://eudml.org/doc/210853>.

@article{Rump2000,
abstract = {We consider a class of two-dimensional non-commutative Cohen-Macaulay rings to which a Brauer graph, that is, a finite graph endowed with a cyclic ordering of edges at any vertex, can be associated in a natural way. Some orders Λ over a two-dimensional regular local ring are of this type. They arise, e.g., as certain blocks of Hecke algebras over the completion of $ℤ[q,q^\{-1\}]$ at (p,q-1) for some rational prime $p$. For such orders Λ, a class of indecomposable maximal Cohen-Macaulay modules (see introduction) has been determined by K. W. Roggenkamp. We prove that this list of indecomposables of Λ is complete.},
author = {Rump, Wolfgang},
journal = {Colloquium Mathematicae},
keywords = {Brauer graph; order; Cohen-Macaulay; Auslander-Reiten quiver; orders; Cohen-Macaulay modules; Hecke algebras; lattices; Brauer graph orders},
language = {eng},
number = {2},
pages = {239-251},
title = {Representation theory of two-dimensionalbrauer graph rings},
url = {http://eudml.org/doc/210853},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Rump, Wolfgang
TI - Representation theory of two-dimensionalbrauer graph rings
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 2
SP - 239
EP - 251
AB - We consider a class of two-dimensional non-commutative Cohen-Macaulay rings to which a Brauer graph, that is, a finite graph endowed with a cyclic ordering of edges at any vertex, can be associated in a natural way. Some orders Λ over a two-dimensional regular local ring are of this type. They arise, e.g., as certain blocks of Hecke algebras over the completion of $ℤ[q,q^{-1}]$ at (p,q-1) for some rational prime $p$. For such orders Λ, a class of indecomposable maximal Cohen-Macaulay modules (see introduction) has been determined by K. W. Roggenkamp. We prove that this list of indecomposables of Λ is complete.
LA - eng
KW - Brauer graph; order; Cohen-Macaulay; Auslander-Reiten quiver; orders; Cohen-Macaulay modules; Hecke algebras; lattices; Brauer graph orders
UR - http://eudml.org/doc/210853
ER -

References

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  1. [1] M. Auslander, Isolated singularities and existence of almost split sequences, in: Representation Theory II, Groups and Orders, Lecture Notes in Math. 1178, Springer, 1986, 194-242. 
  2. [2] M. Auslander, Functors and morphisms determined by objects, in: Proc. Conf. Representation Theory (Philadelphia, 1976), Dekker, 1978, 1-244. 
  3. [3] M. Auslander and I. Reiten, Almost Split Sequences for Cohen-Macaulay Modules, Math. Ann. 277 (1987), 345-349. Zbl0611.13009
  4. [4] M. Auslander, I. Reiten and S. Smalο, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math. 36, Cambridge Univ. Press, 1995. 
  5. [5] C. W. Curtis and I. Reiner, Methods of Representation Theory, I, II, Wiley,1987. 
  6. [6] P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, in: Lecture Notes in Math. 831, Springer, 1980, 1-71. 
  7. [7] J. A. Green, Walking around the Brauer tree, J. Austral. Math. Soc. 17 (1974), 197-213. Zbl0299.20006
  8. [8] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. Zbl0499.20035
  9. [9] G. Lusztig, On quantum groups, J. Algebra 131 (1990), 466-475. Zbl0698.16007
  10. [10] J. C. McConnell, Localization in enveloping rings, J. London Math. Soc. 43 (1968), 421-428. Zbl0162.05203
  11. [11] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley, 1987. 
  12. [12] I. Reiner, Maximal Orders, London, 1975. 
  13. [13] I. Reiten and M. Van den Bergh, Two-dimensional tame and maximal orders of finite representation type, Mem. Amer. Math. Soc. 408 (1989). Zbl0677.16002
  14. [14] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, 1984. 
  15. [15] K. W. Roggenkamp, Indecomposable Representations of Orders, in: Topics in Algebra, Part I: Rings and Representations of Algebras, Banach Center Publ. 26, PWN, Warszawa, 1990, 449-491. 
  16. [16] K. W. Roggenkamp, Cohen-Macaulay modules over two-dimensional graph orders, Colloq. Math. 82 (2000), 25-48. Zbl0945.16013
  17. [17] K. W. Roggenkamp, Blocks with cyclic defect of Hecke orders of Coxeter groups, preprint. Zbl0960.16015
  18. [18] K. W. Roggenkamp and W. Rump, Orders in non-semisimple algebras, Comm. Algebra 27 (1999), 5267-5301. Zbl0939.16009
  19. [19] W. Rump, Green walks in a hypergraph, Colloq. Math. 78 (1998), 133-147. Zbl0938.16008
  20. [20] W. Rump, Non-commutative Cohen-Macaulay rings, manuscript, Stuttgart, 1999. 
  21. [21] W. Rump, Non-commutative regular rings, manuscript, Stuttgart, 1999. 
  22. [22] W. Rump, *-modules, tilting, and almost abelian categories, Comm. Algebra, to appear. 
  23. [23] J.-P. Serre, Algèbre Locale Multiplicités, Lecture Notes in Math. 11, Berlin 1975. Zbl0091.03701
  24. [24] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl. 4, Gordon & Breach, New York, 1992. Zbl0818.16009
  25. [25] Y. Yoshino, Cohen-Macaulay Modules over Cohen-Macaulay Rings, London Math. Soc. Lecture Note Ser. 146, Cambridge Univ. Press, 1990. Zbl0745.13003

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