### Abstract

Let A be a Noetherian Cohen-Macaulay domain, b, c_{1},...,c_{g} an A-sequence, J = (b, c_{1},...,c_{g}) A, and B = A[J/b]. Then B is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets Ass_{B}(B/bB) and Ass_{A}(A/J), and each q ∈ 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∩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 language | English |
---|---|

Pages (from-to) | 1811-1835 |

Number of pages | 25 |

Journal | Transactions of the American Mathematical Society |

Volume | 354 |

Issue number | 5 |

DOIs | |

State | Published - 1 Jan 2002 |

### Fingerprint

### Keywords

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

### Cite this

*Transactions of the American Mathematical Society*,

*354*(5), 1811-1835. https://doi.org/10.1090/S0002-9947-02-02951-3

}

*Transactions of the American Mathematical Society*, vol. 354, no. 5, pp. 1811-1835. https://doi.org/10.1090/S0002-9947-02-02951-3

**Monoidal extensions of a Cohen-Macaulay unique factorization domain.** / Heinzer, William J.; Li, Aihua; Ratliff, Louis J.; Rush, David E.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Monoidal extensions of a Cohen-Macaulay unique factorization domain

AU - Heinzer, William J.

AU - Li, Aihua

AU - Ratliff, Louis J.

AU - Rush, David E.

PY - 2002/1/1

Y1 - 2002/1/1

N2 - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=0035997620&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-02-02951-3

DO - 10.1090/S0002-9947-02-02951-3

M3 - Article

VL - 354

SP - 1811

EP - 1835

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -