Monoidal extensions of a locally quasi-unmixed UFD

Abstract

Let R be a locally quasi-unmixed domain, a,b1,…,bn an asymptotic sequence in R, I=(a,b1,…,bn)R and S=R[b1/a,…,bn/a]=R[I/a], the monoidal transform of R with respect to I. It is shown that S is a locally quasi-unmixed domain, a,b1/a,…,bn/a is an asymptotic sequence in S and there is a one-to-one correspondence between the asymptotic primes Aˆ⁎(I) of I and the asymptotic primes Aˆ⁎(aS) of aS=IS. Moreover, if a,b1,…,bn is an R-sequence, this extends to a one-to-one correspondence between AssR(R/I) and AssS(S/aS). In the case that R is a unique factorization domain, the height one prime ideals of S are examined to determine how far S is from being a UFD. A complete description is given of which height one prime ideals P of S are principal or have a principal primary ideal in the case that P∩R has height 1. If the prime divisors of a satisfy a mild condition, a similar description is given in the case that P∩R has height >1. These are applied to give similar results for the Rees ring R[1/t,It] where t is an indeterminate.