Completions of Extremely Noncatenary Noetherian UFDs

2025/08/25 by Dugan, Eli B., Loepp, S.
#13F15 #13J10 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.18102

Abstract

Let T be a complete local ring. We present necessary and sufficient conditions for T to be the completion of a local (Noetherian) unique factorization domain A such that there exist height one prime ideals \J_k\_{k = 1}^∞ of A satisfying the following conditions: (1) J_k = J_ℓ if and only if k = ℓ, (2) there exist positive integers n ≠ m such that for each k ∈ ℕ, there are two saturated chains of prime ideals of A of the form J_k \subsetneq J⁽¹⁾_k,2 \subsetneq ⋯ \subsetneq J⁽¹⁾_{k,n - 1} \subsetneq M and J_k \subsetneq J⁽²⁾_k,2 \subsetneq ⋯ \subsetneq J⁽²⁾_{k,m - 1} \subsetneq M, where M is the maximal ideal of A, and (3) the prime ideals from condition (2) satisfy J⁽ⁱ⁾_k,a = J^(j)_ℓ,b if and only if i = j, k = ℓ, and a = b. We also find sufficient conditions for T to be the completion of a local (Noetherian) unique factorization domain B such that B/J is not catenary for all height one prime ideals J of B.

Citations

• Noncatenary splinters in prime characteristic
• Every Finite Poset is Isomorphic to a Saturated Subset of the Spectrum of a Noetherian UFD
• Characterization of Completions of Noncatenary Local Domains and Noncatenary Local UFDs

Related

• Characterization of Completions of Noncatenary Local Domains and Noncatenary Local UFDs
• Noncatenary Unique Factorization Domains
• Examples of non-Noetherian domains inside power series rings
• Completions of Countable Excellent Domains and Countable Noncatenary Domains
• Chains of Unusual Excellent Local Rings