Tame and Wild Two by Two Matrix Orders Original Research Article

The aim of this paper is to give a criterion for a 2 × 2 matrix order Λ over a discrete valuation ring to be (lattice) tame. Here "tame" means, as in previous papers by the first author, that indecomposable Λ-lattices of any fixed rank form at most 1-parameter families. It is also shown that otherwise Λ is (lattice) wild, i.e., the classification of Λ-lattices contains in some sense that of the representations of any finite generated algebra over the residue field (see Section 1 for precise definitions). Namely, Theorem 1 asserts that there exists a unique order Λ1 such that the wild orders are just those conjugate to ist suborders (including Λ1 itself) and all other orders are tame. Furthermore, in Proposition 2 this result is reformulated in terms of some standard basis of Λ.