Gorenstein orders of finite lattice type

Abstract

Let (R,m) be a commutative complete Gorenstein local ring and let Λ be a Gorenstein order, that is to say, Λ is a maximal Cohen–Macaulay R-module and HomR(Λ,R) is a projective Λ-module. The main theme of this paper is to study the representation-theoretic properties of generalized lattices, i.e. those Λ-modules which are Gorenstein projective over R. It is proved that Λ 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_-length, an invariant assigned to each generalized lattice which measures Hom modulo projectives.