Monoidal extensions of a Cohen-Macaulay unique factorization domain

William J. Heinzer, Aihua Li, Louis J. Ratliff, David E. Rush

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Let A be a Noetherian Cohen-Macaulay domain, b, c1,...,cg an A-sequence, J = (b, c1,...,cg) A, and B = A[J/b]. Then B is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets AssB(B/bB) and AssA(A/J), and each q ∈ AssA(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∩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 ∩ A also has height one if and only if b ∉ P and thus P ∩ 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 ∩ A. A similar description is also given for the height-one prime ideals P of B with P ∩ 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] ∩ B has no principal primary ideals. This applies in particular to the Rees ring 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 ≅Cl(B), the divisor class group of B.

Original languageEnglish
Pages (from-to)1811-1835
Number of pages25
JournalTransactions of the American Mathematical Society
Volume354
Issue number5
DOIs
StatePublished - 1 Jan 2002

Fingerprint

Unique factorisation domain
Cohen-Macaulay
Factorization
Torsional stress
Prime Ideal
Class Group
Krull Domain
Unique factorisation
Closed
Torsion
Divisor Class Group
If and only if
Prime factor
Sufficient Conditions
Finitely Generated Group
Noetherian
Cyclic group
One to one correspondence
Divisor
Abelian group

Keywords

  • Cohen-Macaulay ring
  • Divisor class group
  • Integrally closed ideal
  • Monoidal transform
  • Rees ring
  • Unique factorization domain

Cite this

Heinzer, William J. ; Li, Aihua ; Ratliff, Louis J. ; Rush, David E. / Monoidal extensions of a Cohen-Macaulay unique factorization domain. In: Transactions of the American Mathematical Society. 2002 ; Vol. 354, No. 5. pp. 1811-1835.
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Monoidal extensions of a Cohen-Macaulay unique factorization domain. / Heinzer, William J.; Li, Aihua; Ratliff, Louis J.; Rush, David E.

In: Transactions of the American Mathematical Society, Vol. 354, No. 5, 01.01.2002, p. 1811-1835.

Research output: Contribution to journalArticle

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AB - Let A be a Noetherian Cohen-Macaulay domain, b, c1,...,cg an A-sequence, J = (b, c1,...,cg) A, and B = A[J/b]. Then B is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets AssB(B/bB) and AssA(A/J), and each q ∈ AssA(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∩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 ∩ A also has height one if and only if b ∉ P and thus P ∩ 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 ∩ A. A similar description is also given for the height-one prime ideals P of B with P ∩ 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] ∩ B has no principal primary ideals. This applies in particular to the Rees ring 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 ≅Cl(B), the divisor class group of B.

KW - Cohen-Macaulay ring

KW - Divisor class group

KW - Integrally closed ideal

KW - Monoidal transform

KW - Rees ring

KW - Unique factorization domain

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