Monoidal extensions of a Cohen-Macaulay unique factorization domain

Abstract

Let A A be a Noetherian Cohen-Macaulay domain, b b , c 1 c_1 , … \dots , c g c_g an A A -sequence, J J = ( b , c 1 , … , c g ) A (b,c_1,\dots ,c_g)A , and B B = A [ J / b ] A[J/b] . Then B B is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets Ass B ⁡ ( B / b B ) \operatorname {Ass}_B(B/bB) and Ass A ⁡ ( A / J ) \operatorname {Ass}_A(A/J) , and each q q ∈ \in Ass A ⁡ ( A / J ) \operatorname {Ass}_A(A/J) has height g + 1 g+1 . If B B does not have unique factorization, then some height-one prime ideals P P of B B are not principal. These primes are identified in terms of J J and P ∩ A P \cap A , and we consider the question of how far from principal they can be. If A A is integrally closed, necessary and sufficient conditions are given for B B to be integrally closed, and sufficient conditions are given for B B to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic. It is shown that if P P is a height-one prime ideal of B B , then P ∩ A P \cap A also has height one if and only if b b ∉ \notin P P and thus P ∩ A P \cap A has height one for all but finitely many of the height-one primes P P of B B . If A A has unique factorization, a description is given of whether or not such a prime P P is a principal prime ideal, or has a principal primary ideal, in terms of properties of P ∩ A P \cap A . A similar description is also given for the height-one prime ideals P P of B B with P ∩ A P \cap A of height greater than one, if the prime factors of b b satisfy a mild condition. If A A is a UFD and b b is a power of a prime element, then B B is a Krull domain with torsion class group if and only if J J is primary and integrally closed, and if this holds, then B B has finite cyclic class group. Also, if J J is not primary, then for each height-one prime ideal p p contained in at least one, but not all, prime divisors of J J , it holds that the height-one prime p A [ 1 / b ] ∩ B pA[1/b] \cap B has no principal primary ideals. This applies in particular to the Rees ring R {\mathbf R} = = A [ 1 / t , t J ] A[1/t, tJ] . As an application of these results, it is shown how to construct for any finitely generated abelian group G G , a monoidal transform B B = A [ J / b ] A[J/b] such that A A is a UFD, B B is Cohen-Macaulay and integrally closed, and G G ≅ \cong Cl ⁡ ( B ) \operatorname {Cl}(B) , the divisor class group of B B .