Cohen-Macaulay modules over two-dimensional graph orders

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 -