Cohen-Macaulay modules over two-dimensional graph orders

Klaus Roggenkamp

Colloquium Mathematicae (1999)

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

Abstract

top
For a split graph order ℒ over a complete local regular domain 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 : 𝒪 L ( μ ) 𝒪 L ( ν ) under the bi-action of the groups ( G l ( μ , 𝒪 L ) , G l ( ν , 𝒪 L ) ) , where 𝒪 L = 𝒪 / π for a prime π. This problem strongly depends on the nature of 𝒪 L . If 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 𝒪 L is not regular, then the category of indecomposable filtered Cohen-Macaulay modules is unbounded.

How to cite

top

Roggenkamp, 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
  1. [AuRe; 89] M. Auslander and I. Reiten, The Cohen-Macaulay type of Cohen-Macaulay rings, Adv. Math. 73 (1989), 1-23. Zbl0744.13003
  2. [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
  3. [GaRi; 79] P. Gabriel and C. Riedtmann, Group representations without groups, Comm. Math. Helv. 54 (1979), 240-287. Zbl0447.16023
  4. [Gr; 74] J. A. Green, Walking around the Brauer tree, J. Austral. Math. Soc. 17 (1974), 197-213. Zbl0299.20006
  5. [Ka; 97] M. Kauer, Derived equivalences of graph order, Ph.D. thesis, Shaker Verlag, 1998. Zbl0921.16006
  6. [KaRo; 98] M. Kauer and K. W. Roggenkamp, Higher dimensional orders, graph-orders, and derived equivalences, J. Pure Appl. Algebra, to appear. Zbl0973.16013
  7. [Mu; 88] D. Mumford, The Red Book of Varieties and Schemes, Lecture Notes in Math. 1358, Springer, 1988. Zbl0658.14001
  8. [Na; 62] M. Nagata, Local Rings, Interscience, New York, 1962. 
  9. [Re; 75] I. Reiner, Maximal Orders, Academic Press, 1975. 
  10. [Ro; 70] K. W. Roggenkamp, Lattices over Orders II, Lecture Notes in Math. 142, Springer, 1970. 
  11. [Ro; 92] K. W. Roggenkamp, Blocks with cyclic defect and Green orders, Comm. Algebra 20 (1992), 1715-1734. Zbl0748.20006
  12. [Ro; 96] K. W. Roggenkamp, Generalized Brauer tree orders, Colloq. Math. 71 (1996), 225-242. 
  13. [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
  14. [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
  15. [Ro; 98 I] K. W. Roggenkamp, The structure over [ q , q - 1 ] of Hecke orders of dihedral groups, J. Algebra, to appear. Zbl0955.16020
  16. [RoRu; 98] K. W. Roggenkamp and W. Rump, Orders in non-semisimple algebras, Comm. Algebra 27 (1999), 5267-5303. 
  17. [Ru; 98] W. Rump, Green walks in a hypergraph, Colloq. Math. 78 (1998), 133-147. Zbl0938.16008

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.