Cohen-Macaulay modules over two-dimensional graph orders

For a split graph order ℒ over a complete local regular domain ${\displaystyle calO}$ 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 ${\displaystyle \varphi :o\vee \left\{\left\{calO\right\}\right\}{\left\{L\right\}}^{(\mu )}\to o\vee \left\{\left\{calO\right\}\right\}{\left\{L\right\}}^{(\nu )}}$ under the bi-action of the groups ${\displaystyle (Gl(\mu ,o\vee \left\{\left\{calO\right\}\right\}\left\{L\right\}),Gl(\nu ,o\vee \left\{\left\{calO\right\}\right\}\left\{L\right\}))}$, where ${\displaystyle o\vee \left\{\left\{calO\right\}\right\}\left\{L\right\}=cal\frac{O}{〈}\pi 〉}$ for a prime π. This problem strongly depends on the nature of ${\displaystyle o\vee \left\{\left\{calO\right\}\right\}\left\{L\right\}}$. If ${\displaystyle o\vee \left\{\left\{calO\right\}\right\}\left\{L\right\}}$ 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 ${\displaystyle o\vee \left\{\left\{calO\right\}\right\}\left\{L\right\}}$ is not regular, then the category of indecomposable filtered Cohen-Macaulay modules is unbounded.