On type sequences and Arf rings

Abstract

In this article in Section 2 we give an explicit description to compute the type sequence t1, . . . , tn of a semigroup Γ generated by an arithmetic sequence (see 2.7); we show that the i-th term ti is equal to 1 or to the type τΓ, depending on its position. In Section 3, for analytically irreducible ring R with the branch sequence R = R0 ⊊R1 ⊊ . . . ⊊Rm−1 ⊊Rm = R, starting from a result proved in [4] we give a characterization (see 3.6) of the “Arf” property using the type sequence of R and of the rings Rj, 1 ≤j ≤m −1. Further, we prove (see 3.9, 3.10) some relations among the integers ℓ∗(R) and ℓ∗(Rj), 1 ≤j ≤ m −1. These relations and a result of [6] allow us to obtain a new characterization (see 3.12) of semigroup rings of minimal multiplicity with ℓ∗(R) ≤τ(R) in terms of the Arf property, type sequences and relations between ℓ∗(R) and ℓ∗(Rj) , 1 ≤j ≤m −1.