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 ν:=ordS. Assume that IS≠S and ν∈ReesSIS. Let u:=αt with IS=αIS, where α∈S. Then V=W∩Q(R) with W=(R[It]‾)Q=(S[ISu]‾)Q′, where Q∈Min(mR[It]‾) and Q′∈Min(nS[ISu]‾). Let P,P′ be the center of W on R[It] and S[Isu], respectively. We prove that if [Sn:Rm]=1, then R[It]P=S[Isu]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 mT-adic order valuation of T with corresponding valuation v:=ordT. Let n≥1 be an integer. Assume that IT=mTn and [TmT:Rm]=1. Let P∈Min(mR[It]) such that P=Q∩R[It] with V=(R[It]‾)Q∩Q(R), where Q∈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 nd−1. In particular, R[It]P is regular if and only if n=1. We prove that k:=Rm is relatively algebraically closed in kv:=VmV. Also we determine the multiplicity of R[It]P, and we prove that if IT=mT, then R[It]P is regular if and only if [TmT:Rm]=1.