On type sequences and Arf rings

Abstract

In this article in Section~2 we give an explicit description to compute the type sequence $\mathrm{t}_1,\ldots,\mathrm{t}_{n}$ of a semigroup $\Gamma$ generated by an arithmetic sequence (see 2.7); we show that the $i$-th term $\mathrm{t}_i$ is equal to $1$ or to the type $\tau_\Gamma$, depending on its position. In Section 3, for analytically irreducible ring $R$ with the branch sequence
$R=R_0 \subsetneq R_1 \subsetneq \ldots\subsetneq R_{m-1}
\subsetneq R_{m} =\overline{R}$, 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 $R_j$, $1\leq j\leq m-1$. Further, we prove (see 3.9, 3.10) some relations among the integers $\ell^*(R)$ and $\ell^*(R_j)$, $1\leq j\leq m-1$. These relations and a result of [6] allow us to obtain a new characterization (see 3.12) of semigroup rings of minimal multiplicity with $\ell^*(R)\leq \tau(R)$ in terms of the Arf property, type sequences and relations between $\ell^*(R)$ and $\ell^*(R_j)$, $1\leq j\leq m-1$.