Finitely supported *-simple complete ideals and multiplicities in a regular local ring

Let (R, m) and (S, n) be regular local rings of dim(S) = dim(R) >= 2 such that S birationally dominates R, and let V be the order valuation ring of S with corresponding valuation v := ord(s). Assume that I-S not equal S and v is an element of Rees(S) I-S. Let u := alpha t with IS = alpha I-S, where alpha is an element of S. Then V = W boolean AND Q(R) with W = (R[IT])Q = (S[I(S)u])Q', where Q is an element of Min(mR[IT]) and Q' is an element of Min(nS[I(S)u]). Let P, P' be the center of W on R[It] and S[Pu], respectively. We prove that if [S/n : R/n] = 1, then R[It]/P = S[I(S)u]/P'. Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m) of dimension d >= 2. Let T be a terminal base point of I and V be the m(T)-adic order valuation of T with corresponding valuation v := ord(T). Let n >= 1 be an integer. Assume that I-T = m(T)(n) and [T/m(T) : R/m]= 1. Let P is an element of Min(mR[It]) such that P = Q boolean AND R[It] with V = (R[It])Q boolean AND Q(R), where Q is an element of Min(mR[It]). We prove that the quotient ring R[It]/P is d-dimensional normal Cohen-Macaulay standard graded domain over k with the multiplicity n(d-1). In particular, R[It]/P is regular if and only if = 1. We prove that k := R/m relatively algebraically closed in k(v) := V/m(v) Also we determine the multiplicity of R[It]/P, and we prove that if I-T = m(T), then R[It]/P is regular if and only if [T/m(t) : R/m ] = 1. (C) 2017 Elsevier Inc. All rights reserved.