Almost S-GCD domains: localization, class groups, and extensions
Abstract
Let D be an integral domain and let S⊆ D∖{0} be multiplicative. We study the condition that, for each nonzero a,b∈ D, some (an,bn)v is S-principal. A power estimate extends this binary condition to arbitrary finite families and permits synchronization of exponents. We describe the obstruction to descent from the AGCD domain DS by bounded S-torsion and by a finiteness condition on intersections of principal ideals. For an almost Prüfer v-multiplication domain, the almost S-GCD property, the AGCD property of DS, and torsion of the S-class group are equivalent. In Krull domains this gives a criterion in terms of the divisor classes of height-one primes meeting S, and finite rational class rank reduces the multiplicative set to one element. We prove conductor transfer without a principal-contraction hypothesis, polynomial transfer in several settings, a normalization homeomorphism away from S, and a finite-principal-cover criterion with a Picard-group obstruction. Explicit examples distinguish localization, bounded descent, and the global and sublocal conditions.
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BibTeX
@misc{hamed2026almost,
title = {{Almost S-GCD domains: localization, class groups, and extensions}},
author = {Ahmed Hamed and Hwankoo Kim and Malik Tusif Ahmed},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08340},
eprint = {2611.08340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08340},
}
RIS
TY - UNPB
TI - Almost S-GCD domains: localization, class groups, and extensions
AU - Ahmed Hamed
AU - Hwankoo Kim
AU - Malik Tusif Ahmed
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08340
UR - https://arxiv.org/abs/2611.08340
AB - Let D be an integral domain and let S⊆ D∖{0} be multiplicative. We study the condition that, for each nonzero a,b∈ D, some (a^n,b^n)_v is S-principal. A power estimate extends this binary condition to arbitrary finite families and permits synchronization of exponents. We describe the obstruction to descent from the AGCD domain D_S by bounded S-torsion and by a finiteness condition on intersections of principal ideals. For an almost Prüfer v-multiplication domain, the almost S-GCD property, the AGCD property of D_S, and torsion of the S-class group are equivalent. In Krull domains this gives a criterion in terms of the divisor classes of height-one primes meeting S, and finite rational class rank reduces the multiplicative set to one element. We prove conductor transfer without a principal-contraction hypothesis, polynomial transfer in several settings, a normalization homeomorphism away from S, and a finite-principal-cover criterion with a Picard-group obstruction. Explicit examples distinguish localization, bounded descent, and the global and sublocal conditions.
ER -