Monoidal extensions of a CohenMacaulay unique factorization domain
Authors:
William J. Heinzer, Aihua Li, Louis J. Ratliff Jr. and David E. Rush
Journal:
Trans. Amer. Math. Soc. 354 (2002), 18111835
MSC (2000):
Primary 13A05, 13A30, 13B02, 13B22, 13C20, 13F15, 13H10
Published electronically:
January 9, 2002
MathSciNet review:
1881018
Fulltext PDF Free Access
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Abstract: Let be a Noetherian CohenMacaulay domain, , , , an sequence, = , and = . Then is CohenMacaulay, there is a natural onetoone correspondence between the sets and , and each has height . If does not have unique factorization, then some heightone 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 heightone prime ideal of , then also has height one if and only if and thus has height one for all but finitely many of the heightone 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 heightone 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 heightone prime ideal contained in at least one, but not all, prime divisors of , it holds that the heightone 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 CohenMacaulay 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 479091395
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 925210135
Email:
ratliff@newmath.ucr.edu
David E. Rush
Affiliation:
Department of Mathematics, University of California, Riverside, California 925210135
Email:
rush@newmath.ucr.edu
DOI:
http://dx.doi.org/10.1090/S0002994702029513
PII:
S 00029947(02)029513
Keywords:
Unique factorization domain,
integrally closed ideal,
monoidal transform,
CohenMacaulay 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
