Localization, Local--Global Transfer, and Hull Theory for $C4^{\ast}$-Modules over Commutative Rings

Abstract

Let \(R\) be a commutative ring and \(M\) an \(R\)-module. We develop a localization and local-global theory for \(C4\)-modules, \(C4^{\ast}\)-modules, strongly \(C4^{\ast}\)-modules, \(C4\)-hulls, and pseudo-continuous hulls over commutative rings. The problem is structural: these notions are defined through decompositions, summand conditions, and minimal extensions, while localization changes decomposition data, support, and hull minimality. We prove forward localization theorems for the \(C4\), \(C4^{\ast}\), and strongly \(C4^{\ast}\) conditions under exact lifting hypotheses formulated through decomposition lifting, morphism lifting, and submodule lifting. We also prove converse local-global theorems under descent and patching hypotheses, showing when primewise or maximal-local \(C4^{\ast}\) behavior implies global \(C4^{\ast}\) behavior. In addition, we establish obstruction results showing that no unrestricted local-global principle can hold. We compare the localization of a global \(C4\)-hull or pseudo-continuous hull with the hull formed after localization. We show that hull commutation requires both localization stability of the hull class and envelope-type axioms for hull minimality and uniqueness, and we prove conditional patching theorems for reconstructing global hulls from compatible local hulls. Our method is purely algebraic and support-theoretic, based on summand descent, patching of local witnesses, support control, and dimension-stratified transfer on \(\operatorname{Spec} R\). As applications, we show that for commutative artinian rings these properties are detected exactly on the local factors, and that for finitely generated torsion modules over a Dedekind domain they are detected exactly on the primary components, equivalently on the localizations at maximal ideals in the support.