Type-sequences of modules

Let R be a complete and integral local k-algebra of dimension one, k an algebraically closed field of characteristic zero. In this paper the notion of type-sequence, given for rings in Barucci et al. (AMS Mem. 125 (598) (1997) Ch. II, 1), is extended to any finitely generated torsion-free R-module of rank 1. A module M, of Cohen–Macaulay type r1(M), whose type-sequence is [r1(M),1,…,1] is said to have “minimal type-sequence”, briefly m.t.s. The family of m.t.s. R-modules, which includes the canonical module, is described by means of value sets, the conductor c(M), the δ-invariant δ(M) and the C.M. type r1(M). In the case of rings the m.t.s. property is called “almost Gorenstein” (see Barucci and Fröberg, J. Algebra 188 (1997) 418–442). Inspired by analogous investigations by Barucci and Fröberg, we study in Section 3 the m.t.s. property and the reflexiveness of modules over almost Gorenstein rings.