Monoidal extensions of a Cohen-Macaulay unique factorization domain

Authors:
William J. Heinzer, Aihua Li, Louis J. Ratliff Jr. and David E. Rush

Journal:
Trans. Amer. Math. Soc. **354** (2002), 1811-1835

MSC (2000):
Primary 13A05, 13A30, 13B02, 13B22, 13C20, 13F15, 13H10

DOI:
https://doi.org/10.1090/S0002-9947-02-02951-3

Published electronically:
January 9, 2002

MathSciNet review:
1881018

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Abstract: Let be a Noetherian Cohen-Macaulay domain, , , , an -sequence, = , and = . Then is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets and , and each has height . If does not have unique factorization, then some height-one prime ideals of are not principal. These primes are identified in terms of and , and we consider the question of how far from principal they can be. If is integrally closed, necessary and sufficient conditions are given for to be integrally closed, and sufficient conditions are given for to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic.

It is shown that if is a height-one prime ideal of , then also has height one if and only if and thus has height one for all but finitely many of the height-one primes of . If has unique factorization, a description is given of whether or not such a prime is a principal prime ideal, or has a principal primary ideal, in terms of properties of . A similar description is also given for the height-one prime ideals of with of height greater than one, if the prime factors of satisfy a mild condition.

If is a UFD and is a power of a prime element, then is a Krull domain with torsion class group if and only if is primary and integrally closed, and if this holds, then has finite cyclic class group. Also, if is not primary, then for each height-one prime ideal contained in at least one, but not all, prime divisors of , it holds that the height-one prime has no principal primary ideals. This applies in particular to the Rees ring . As an application of these results, it is shown how to construct for any finitely generated abelian group , a monoidal transform = such that is a UFD, is Cohen-Macaulay and integrally closed, and , the divisor class group of .

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Additional Information

**William J. Heinzer**

Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47909-1395

Email:
heinzer@math.purdue.edu

**Aihua Li**

Affiliation:
Department of Mathematics and Computer Science, Loyola University, New Orleans, Louisiana 70118

Email:
ali@loyno.edu

**Louis J. Ratliff Jr.**

Affiliation:
Department of Mathematics, University of California, Riverside, California 92521-0135

Email:
ratliff@newmath.ucr.edu

**David E. Rush**

Affiliation:
Department of Mathematics, University of California, Riverside, California 92521-0135

Email:
rush@newmath.ucr.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-02951-3

Keywords:
Unique factorization domain,
integrally closed ideal,
monoidal transform,
Cohen-Macaulay ring,
divisor class group,
Rees ring

Received by editor(s):
December 31, 2000

Published electronically:
January 9, 2002

Article copyright:
© Copyright 2002
American Mathematical Society