Almost S-GCD domains: localization, class groups, and extensions

Abstract

Let $D$ be an integral domain and let $S\subseteq D\setminus\{0\}$ be multiplicative. We study the condition that, for each nonzero $a,b\in D$, 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.