On the Blowing-up Rings, Arf Rings and Type Sequences

Abstract

In Section 1 we recall the definitions of the Hilbert functions, multiplicity, h-polynomial, blowing-up rings, standard basis, degree of singularity, Cohen-Macaulay type, type sequences and almost Gorenstein rings etc. In Section 2 we give summary of numerical invariants of monomial curves, especially monomial curves defined by arithmetic sequences and almost arithmetic sequences. In particular, we give an explicit formula for the type sequence (see (2.1)–(6)) and give a characterization of almost-Gorensteinness of the algebroid semigroup ring R = K〚Γ〛 (over a field K ) of the numerical semigroup Γ generated by an arithmetic sequence. In Section 3 we mainly study Arf rings and their type sequences. We begin with recalling definition of Arf ring and its branch sequence and give a formula (see Theorem (3.4)) for the degree of singularity of R as the sum of the lengths of quotients of the successive terms of its branch sequence as well as the sum of the first coefficients of the Hilbert-Samuel polynomials of the terms of its branch sequence. Further, we use a results proved in [3]_and [7]_to give (see Theorem (3.6)) a characterization of complete local Arf domains with algebraically residue field using the type sequence of R and type sequences of the rings in the branch sequence of R. Finally we prove that the type sequence of the blowing-up ring of a complete local Arf domain with algebraically residue field is the sequence obtained from the type sequence of R obtained by removing its first term. In Section 4 we give some examples of Arf rings and some of not Arf rings. In Example 4, we give necessary and sufficient conditions for the algebroid semigroup ring R = K〚Γ〛 of the numerical semigroup generated by an arithmetic sequence Γ over a field K to be an Arf ring.