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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: January 9, 2002
MathSciNet review: 1881018
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Abstract: Let $A$ be a Noetherian Cohen-Macaulay domain, $b$, $c_1$, $\dots$, $c_g$an $A$-sequence, $J$ = $(b,c_1,\dots,c_g)A$, and $B$ = $A[J/b]$. Then $B$ is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets $\mbox{{Ass}}_B(B/bB)$ and $\mbox{{Ass}}_A(A/J)$, and each $q$ $\in$ $\mbox{{Ass}}_A(A/J)$ has height $g+1$. If $B$ does not have unique factorization, then some height-one prime ideals $P$ of $B$ are not principal. These primes are identified in terms of $J$ and $P \cap A$, and we consider the question of how far from principal they can be. If $A$ is integrally closed, necessary and sufficient conditions are given for $B$ to be integrally closed, and sufficient conditions are given for $B$ to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic.

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

If $A$ is a UFD and $b$ is a power of a prime element, then $B$ is a Krull domain with torsion class group if and only if $J$ is primary and integrally closed, and if this holds, then $B$ has finite cyclic class group. Also, if $J$ is not primary, then for each height-one prime ideal $p$ contained in at least one, but not all, prime divisors of $J$, it holds that the height-one prime $pA[1/b] \cap B$ has no principal primary ideals. This applies in particular to the Rees ring ${\mathbf R}$ $=$ $A[1/t, tJ]$. As an application of these results, it is shown how to construct for any finitely generated abelian group $G$, a monoidal transform $B$ = $A[J/b]$ such that $A$ is a UFD, $B$ is Cohen-Macaulay and integrally closed, and $G$ $\cong$ $\mbox{{Cl}}(B)$, the divisor class group of $B$.


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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: http://dx.doi.org/10.1090/S0002-9947-02-02951-3
PII: S 0002-9947(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