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Monoidal Extensions of a Locally Quasi-Unmixed Unique Factorization Domain

  • Author(s): Oeser, Paul Richard
  • Advisor(s): Rush, David E
  • et al.
Abstract

Let R be a locally quasi-unmixed domain, a, b1,&hellip, bn an

asymptotic sequence in R, I=(a, b1,&hellip, bn)R, and

S=R[b1/a,&hellip, bn/a]=R[I/a]. Then S is a locally quasi-unmixed

domain, a, b1/a,&hellip, bn/a is an asymptotic sequence in S, and

there is a one-to-one correspondence between the asymptotic primes

AssR(R/(In)a) of I and the asymptotic primes AssS(S/(aSn)a)

of aS=IS. Moreover, if a, b1,&hellip, bn is an R-sequence,

then that one-to-one correspondence extends between AssR(R/I) and

AssS(S/aS).

We give a sufficient condition for the monoidal transform

S to be a unique factorization domain, or a Krull domain whose class

group is torsion, finite, or finite cyclic. As a corollary, we give a

necessary and sufficient condition for R and its monoidal transform

to have the same class group.

In the case that R is a unique factorization domain, we examine the

height-one prime ideals of S to determine how far S is from unique

factorization. In Section 3.2, a complete description is given of which

height-one prime ideals P of S are principal or have a prinicpal primary

ideal in the case that the contraction to R of P, which we will call p, has height one. In Section 3.3, we show that if the

prime factors of a satisfy a mild condition, we may give a similar

description in the case that p has height greater than one. We give a necessary and

sufficient condition for S to be a Krull domain with finite cyclic

class group in the case that a is a power of a prime element, and we show

that this holds for the Rees ring R[1/t,It] as a monoidal transform

over R[1/t] as well. Furthermore, if a is a power of a prime element,

we show that if Rad(I) is not prime and p is a height-one prime ideal

of R contained in at least one but not all asymptotic prime divisors of I, then

the height-one prime ideal P of S, which can be represented as pR[1/a] intersected with S, has no principal primary

ideal.

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