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 _and _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.
Mathematics Subject Classification (2000)
Primary 14H20 Secondary 13H10 13P10
Hilbert function of a local ring h-polynomial semigroup ring standard basis of a numerical semigroup monomial curves blowing-up ring Arf ring
This is a preview of subscription content, log in to check access.
V. Barucci, D.E. Dobbs and M. Fontana, Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains, Memoirs Amer. Math. Soc. 125 (1994), no. 598.Google Scholar
S. Molinelli, D.P. Patil and G. Tamone, On the Cohen-Macaulayness of the associated graded ring of certain monomial curves, Beiträge zur Algebra und Geometrie — Contributions to Algebra and Geometry 39, no. 2 (1998), 433–446.zbMATHMathSciNetGoogle Scholar
D.P. Patil and I. Sengupta, Minimal set of generators for the derivation module of certain monomial curves, Communications in Algebra 27 (1999), no. 11, 5619–5631.zbMATHCrossRefMathSciNetGoogle Scholar
D.P. Patil and B. Singh, Generators for the module of derivations and the relation ideals of certain curves, Manuscripta Math 68 (1990), no. 3, 327–335.zbMATHCrossRefMathSciNetGoogle Scholar
D.P. Patil and G. Tamone, On the type sequences and Arf rings, Annales Academiae Paedagogicae Cracoviensis, Studia Mathematica, VI 45 (2007), 35–50.MathSciNetGoogle Scholar
D.P. Patil and G. Tamone, On the type sequences of some one-dimensional rings, Universitatis Iagellonicae Acta Mathematica, Fasciculus XLV (2007), 119–130.MathSciNetGoogle Scholar
J.C. Rosales and M.B. Branco, Numerical semigroups that can be expressed as an intersection of symmetric numerical semigroups, Journal of Pure and Applied Algebra 171 (2002), no. 2-3, 303–314.zbMATHCrossRefMathSciNetGoogle Scholar
G. Scheja and U. Storch, Regular Sequences and Resultants, Research Notes in Mathematics 8 A.K. Peters, Natick, Massachusetts 2001.zbMATHGoogle Scholar