Cohen-Macaulay modules over two-dimensional graph orders

Volume 82 / 1999

Klaus Roggenkamp Colloquium Mathematicum 82 (1999), 25-48 DOI: 10.4064/cm-82-1-25-48

Abstract

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.

Authors

  • Klaus Roggenkamp

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