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1999 | 82 | 1 | 25-48
Tytuł artykułu

Cohen-Macaulay modules over two-dimensional graph orders

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Języki publikacji
EN
Abstrakty
EN
For a split graph order ℒ over a complete local regular domain $\cal O$ of dimension 2 the indecomposable Cohen-Macaulay modules decompose - up to irreducible projectives - into a union of the indecomposable Cohen-Macaulay modules over graph orders of type •—• . There, the Cohen-Macaulay modules filtered by irreducible Cohen-Macaulay modules are in bijection to the homomorphisms $ϕ : \ovv{{\cal O}}{L}^{(μ)} → \ovv{{\cal O}}{L}^{(ν)}$ under the bi-action of the groups $(Gl(μ,\ovv{{\cal O}}{L}),Gl(ν,\ovv{{\cal O}}{L}))$, where $\ovv{{\cal O}}{L} = \cal{O}/〈π〉$ for a prime π. This problem strongly depends on the nature of $\ovv{{\cal O}}{L}$. If $\ovv{{\cal O}}{L}$ is regular, then the category of indecomposable filtered Cohen-Macaulay modules is bounded. This latter condition is satisfied if ℒ is the completion of the Hecke order of the dihedral group of order 2p with p an odd prime at the maximal ideal 〈q-1,p〉, and more generally of blocks of defect p of complete Hecke orders. If $\ovv{{\cal O}}{L}$ is not regular, then the category of indecomposable filtered Cohen-Macaulay modules is unbounded.
Słowa kluczowe
Rocznik
Tom
82
Numer
1
Strony
25-48
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-11-20
poprawiono
1999-03-15
Twórcy
  • Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany
Bibliografia
  • [AuRe; 89] M. Auslander and I. Reiten, The Cohen-Macaulay type of Cohen-Macaulay rings, Adv. Math. 73 (1989), 1-23.
  • [DrRo; 67] Ju. A. Drozd and A. V. Roiter, Commutative rings with a finite number of integral indecomposable representations, Izv. Akad. Nauk SSSR 31 (1967), 783-798.
  • [GaRi; 79] P. Gabriel and C. Riedtmann, Group representations without groups, Comm. Math. Helv. 54 (1979), 240-287.
  • [Gr; 74] J. A. Green, Walking around the Brauer tree, J. Austral. Math. Soc. 17 (1974), 197-213.
  • [Ka; 97] M. Kauer, Derived equivalences of graph order, Ph.D. thesis, Shaker Verlag, 1998.
  • [KaRo; 98] M. Kauer and K. W. Roggenkamp, Higher dimensional orders, graph-orders, and derived equivalences, J. Pure Appl. Algebra, to appear.
  • [Mu; 88] D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Math. 1358, Springer, 1988.
  • [Na; 62] M. Nagata, Local Rings, Interscience, New York, 1962.
  • [Re; 75] I. Reiner, Maximal Orders, Academic Press, 1975.
  • [Ro; 70] K. W. Roggenkamp, Lattices over Orders II, Lecture Notes in Math. 142, Springer, 1970.
  • [Ro; 92] K. W. Roggenkamp, Blocks with cyclic defect and Green orders, Comm. Algebra 20 (1992), 1715-1734.
  • [Ro; 96] K. W. Roggenkamp, Generalized Brauer tree orders, Colloq. Math. 71 (1996), 225-242.
  • [Ro; 97] K. W. Roggenkamp, The cell-structure of integral group rings of dihedral groups, in: Sem. Ser. Math. Algebra, Ovidius Univ., Constanţa, to appear.
  • [Ro; 98] K. W. Roggenkamp, The cell structure, the Brauer tree structure and extensions of cell-modules for Hecke orders of dihedral groups, MS, Stuttgart, 1997.
  • [Ro; 98 I] K. W. Roggenkamp, The structure over $ℤ[q,q^-1]$ of Hecke orders of dihedral groups, J. Algebra, to appear.
  • [RoRu; 98] K. W. Roggenkamp and W. Rump, Orders in non-semisimple algebras, Comm. Algebra 27 (1999), 5267-5303.
  • [Ru; 98] W. Rump, Green walks in a hypergraph, Colloq. Math. 78 (1998), 133-147.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-cmv82i1p25bwm
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