×

Cohen-Macaulay modules over two-dimensional graph orders. (English) Zbl 0945.16013

Graph orders generalize tree orders and have been used successfully to get information on modular blocks with dihedral defect group. In the present paper the author continues his study of graph orders \(\mathcal L\) over a two-dimensional regular local ring \(\mathcal O\) [see Colloq. Math. 71, No. 2, 225-242 (1996; see Zbl 0945.16016 below)]. The indecomposable Cohen-Macaulay modules over \(\mathcal L\) are investigated. Firstly, he reduces the representation theory of \(\mathcal L\) to the case of a graph \(G=\{\overset {e}{\bullet\diagrbar\bullet}\}\) with a single edge \(e\) connecting two vertices \(v\) and \(w\). For such a graph \(G\), the order \(\mathcal L\) can be represented as an amalgamation of two regular local orders \(\Omega_v\) and \(\Omega_w\) with respect to a common factor ring \(\overline\Omega\). The paper deals with the case where \(\overline\Omega\) is regular. As the category \(\text{CM}({\mathcal L})\) of all Cohen-Macaulay modules over \(\mathcal L\) appeared to be out of reach, the concept of Cohen-Macaulay filtration of a module \(M\in\text{CM}({\mathcal L})\) is introduced. The main result provides a classification of these filtered Cohen-Macaulay modules. (We have shown that this classification actually comprises all Cohen-Macaulay modules [see Colloq. Math. (to appear)].) For some interesting examples, e.g., blocks of cyclic defect of Hecke orders and deformations of blocks with defect \(p\), the rings \(\overline\Omega\) occurring in \(\mathcal L\) are complete discrete valuation domains. If one of the \(\overline\Omega\) is not regular, the indecomposable Cohen-Macaulay \(\mathcal L\)-modules have arbitrarily large \(\mathcal O\)-rank.

MSC:

16G30 Representations of orders, lattices, algebras over commutative rings
16G50 Cohen-Macaulay modules in associative algebras
16W70 Filtered associative rings; filtrational and graded techniques
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)

Citations:

Zbl 0945.16016
PDFBibTeX XMLCite
Full Text: DOI EuDML