A categorical and algebro-geometric theory of localization

Abstract

We develop a categorical and algebro-geometric treatment of localization for cohomological theories endowed with an open--closed recollement. Starting from a class whose restriction to the open complement vanishes, we show that the natural output of the formalism is, in general, not a distinguished localized class on the closed locus, but a torsor of supported refinements. This torsor is the secondary, pre-denominator object of localization; categorically, it gives rise to a translation groupoid of supported lifts. A canonical local term appears only after an additional uniqueness or concentration principle is imposed. We construct the torsor from the localization triangle and establish excision, Cartesian base-change pullback, proper pushforward, and compatibility with external products under explicit hypotheses. We also prove that any assignment of local terms compatible with the localization triangle factors through this torsor. With Verdier duality and orientation data, supported refinements define local indices and global-to-local index formulas. Under purity and concentration, the torsor is rigidified and the familiar Euler-denominator expressions are recovered.