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Representation theory of two-dimensional Brauer graph rings. (English) Zbl 0990.16013

Much of the successful technology in representation theory of Artin algebras has been generalized to study lattices over classical orders. Not much is known, however, for orders over ground rings of dimension two or more, although such orders occur naturally for example as Hecke algebras. Recently, K. W. Roggenkamp [J. Algebra 224, No. 2, 356-396 (2000; Zbl 0955.16020), Arch. Math. 74, No. 3, 173-182 (2000; Zbl 0960.16015), (*) Colloq. Math. 82, No. 1, 25-48 (1999; Zbl 0945.16013)] started to develop the representation theory of orders over ground rings of Krull dimension two, focussing on Hecke algebras as examples. In particular, he introduced Brauer graph orders (generalizing blocks of cyclic defect) and he constructed (see (*)) Cohen-Macaulay modules for certain orders. The present article continues this project by proving that Roggenkamp’s list of Cohen-Macaulay modules is complete. This requires new methods as the orders are not isolated singularities; thus, almost split sequences do not exist. The author makes use of his general techniques, as developed in [J. Algebra 243, No. 2, 385-408 (2001; Zbl 1002.16008), ibid. 236, No. 2, 522-548 (2001; Zbl 0982.16008)].

MSC:

16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16G50 Cohen-Macaulay modules in associative algebras
16G30 Representations of orders, lattices, algebras over commutative rings
20C08 Hecke algebras and their representations
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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