Representation theory of two-dimensionalbrauer graph rings

• Autorzy: Wolfgang Rump
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Brauer graph; order; Cohen-Macaulay; Auslander-Reiten quiver

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2000
• Tom: 86
• Numer: 2
• Strony: 239-251

Daty

wydano

2000

otrzymano

1999-03-30

poprawiono

2000-02-01

Twórcy

autor

Wolfgang Rump

• Mathematisches Institut B/3, University of Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany

Bibliografia

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