Gorenstein orders of finite lattice type

Let (R,m) be a commutative complete Gorenstein local ring and let Lambda be a Gorenstein order, that is to say, Lambda is a maximal Cohen-Macaulay R-module and Hom(R)(Lambda, R) is a projective A-module. The main theme of this paper is to study the representation-theoretic properties of generalized lattices, i.e. those Lambda-modules which are Gorenstein projective over R. It is proved that Lambda has only finitely many isomorphism classes of indecomposable lattices if and only if every generalized lattice is the direct sum of finitely generated ones. It is also turn out that, if R is one-dimensional, then a generalized lattice M which is not the direct sum of copies of a finite number of lattices, contains indecomposable sublattices of arbitrarily large finite (h) under bar -length, an invariant assigned to each generalized lattice which measures Hom modulo projectives. (C) 2018 Published by Elsevier Inc.