## Department of Mathematics Facuty Scholarship and Creative Works

#### Title

Monoidal Extensions of a Cohen-Macaulay Unique Factorization Domain

Article

1-1-2002

#### 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.

#### DOI

10.1090/S0002-9947-02-02951-3

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