On type sequences and Arf rings

Abstract

In this article in Section 2 we give an explicit description to compute the type sequence $t_1, . . ., t_n$ of a semigroup $Γ$ generated by an arithmetic sequence (see 2.7); we show that the $i$-th term $t_i$ is equal to 1 or to the type $τ_Γ$, depending on its position. In Section 3, for analytically irreducible ring $R$ with the branch sequence $R = R_0 ⊊ R_1 ⊊ . . . ⊊ R_{m−1} ⊊ R_m = Ṝ$, 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 $R_j , 1 ≤ j ≤ m − 1$. Further, we prove (see 3.9, 3.10) some relations among the integers $ℓ^*(R)$ and $ℓ^*(R_j), 1 ≤ j ≤ m − 1$. 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 $ℓ^*(R_j), 1 ≤ j ≤ m − 1$.