# Cohen-Macaulay modules over two-dimensional graph orders

Colloquium Mathematicae (1999)

- Volume: 82, Issue: 1, page 25-48
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topRoggenkamp, Klaus. "Cohen-Macaulay modules over two-dimensional graph orders." Colloquium Mathematicae 82.1 (1999): 25-48. <http://eudml.org/doc/210749>.

@article{Roggenkamp1999,

abstract = {For a split graph order ℒ over a complete local regular domain $\mathcal \{O\}$ 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 $ϕ : \{\{\mathcal \{O\}\}\}\{L\}^\{(μ)\} → \{\{\mathcal \{O\}\}\}\{L\}^\{(ν)\}$ under the bi-action of the groups $(Gl(μ,\{\{\mathcal \{O\}\}\}\{L\}),Gl(ν,\{\{\mathcal \{O\}\}\}\{L\}))$, where $\{\{\mathcal \{O\}\}\}\{L\} = \mathcal \{O\}/〈π〉$ for a prime π. This problem strongly depends on the nature of $\{\{\mathcal \{O\}\}\}\{L\}$. If $\{\{\mathcal \{O\}\}\}\{L\}$ 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 $\{\{\mathcal \{O\}\}\}\{L\}$ is not regular, then the category of indecomposable filtered Cohen-Macaulay modules is unbounded.},

author = {Roggenkamp, Klaus},

journal = {Colloquium Mathematicae},

keywords = {tree orders; graph orders; indecomposable Cohen-Macaulay modules; regular local orders; Cohen-Macaulay filtrations; filtered Cohen-Macaulay modules},

language = {eng},

number = {1},

pages = {25-48},

title = {Cohen-Macaulay modules over two-dimensional graph orders},

url = {http://eudml.org/doc/210749},

volume = {82},

year = {1999},

}

TY - JOUR

AU - Roggenkamp, Klaus

TI - Cohen-Macaulay modules over two-dimensional graph orders

JO - Colloquium Mathematicae

PY - 1999

VL - 82

IS - 1

SP - 25

EP - 48

AB - For a split graph order ℒ over a complete local regular domain $\mathcal {O}$ 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 $ϕ : {{\mathcal {O}}}{L}^{(μ)} → {{\mathcal {O}}}{L}^{(ν)}$ under the bi-action of the groups $(Gl(μ,{{\mathcal {O}}}{L}),Gl(ν,{{\mathcal {O}}}{L}))$, where ${{\mathcal {O}}}{L} = \mathcal {O}/〈π〉$ for a prime π. This problem strongly depends on the nature of ${{\mathcal {O}}}{L}$. If ${{\mathcal {O}}}{L}$ 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 ${{\mathcal {O}}}{L}$ is not regular, then the category of indecomposable filtered Cohen-Macaulay modules is unbounded.

LA - eng

KW - tree orders; graph orders; indecomposable Cohen-Macaulay modules; regular local orders; Cohen-Macaulay filtrations; filtered Cohen-Macaulay modules

UR - http://eudml.org/doc/210749

ER -

## References

top- [AuRe; 89] M. Auslander and I. Reiten, The Cohen-Macaulay type of Cohen-Macaulay rings, Adv. Math. 73 (1989), 1-23. Zbl0744.13003
- [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. Zbl0169.35901
- [GaRi; 79] P. Gabriel and C. Riedtmann, Group representations without groups, Comm. Math. Helv. 54 (1979), 240-287. Zbl0447.16023
- [Gr; 74] J. A. Green, Walking around the Brauer tree, J. Austral. Math. Soc. 17 (1974), 197-213. Zbl0299.20006
- [Ka; 97] M. Kauer, Derived equivalences of graph order, Ph.D. thesis, Shaker Verlag, 1998. Zbl0921.16006
- [KaRo; 98] M. Kauer and K. W. Roggenkamp, Higher dimensional orders, graph-orders, and derived equivalences, J. Pure Appl. Algebra, to appear. Zbl0973.16013
- [Mu; 88] D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Math. 1358, Springer, 1988. Zbl0658.14001
- [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. Zbl0748.20006
- [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. Zbl0977.20001
- [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. Zbl0996.16011
- [Ro; 98 I] K. W. Roggenkamp, The structure over $\mathbb{Z}[q,{q}^{-}1]$ of Hecke orders of dihedral groups, J. Algebra, to appear. Zbl0955.16020
- [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. Zbl0938.16008

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.