On the value-semigroup of a simple complete ideal in a two-dimensional regular local ring

Abstract

Let R be a two-dimensional regular local ring with maximal ideal \mathfrak m, and let \wp be a simple complete \mathfrak m-primary ideal which is residually rational. Let R_0:= R\subsetneqq ...\subsetneqq R_r be the quadratic sequence associated to \wp, let \Gamma_\wp be the value-semigroup associated to \wp, and let ((e_j(\wp))_{0\leq j\leq r} be the multiplicity sequence of \wp. We associate to \wp a sequence of natural integers, the formal characteristic sequence of \wp, and we show that the value-semigroup, the multiplicity sequence and the formal characteristic sequence are equivalent data. Furthermore, we give a new proof that \Gamma_\wp is symmetric, and give a formula for c_\wp, the conductor of \Gamma_\wp, in terms of entries of the Hamburger-Noether tableau of \wp.