Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5771704 | Journal of Algebra | 2017 | 25 Pages |
Abstract
Let (R,m) and (S,n) be regular local rings of dimâ¡(S)=dimâ¡(R)â¥2 such that S birationally dominates R, and let V be the order valuation ring of S with corresponding valuation ν:=ordS. Assume that ISâ S and νâReesSIS. Let u:=αt with IS=αIS, where αâS. Then V=Wâ©Q(R) with W=(R[It]â¾)Q=(S[ISu]â¾)Qâ², where QâMin(mR[It]â¾) and Qâ²âMin(nS[ISu]â¾). Let P,Pâ² 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]Pâ². Let I be a finitely supported complete m-primary ideal of a regular local ring (R,m) of dimension dâ¥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 nâ¥1 be an integer. Assume that IT=mTn and [TmT:Rm]=1. Let PâMin(mR[It]) such that P=Qâ©R[It] with V=(R[It]â¾)Qâ©Q(R), where QâMin(mR[It]â¾). We prove that the quotient ring R[It]P is d-dimensional normal Cohen-Macaulay standard graded domain over k with the multiplicity ndâ1. 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Mee-Kyoung Kim,