A categorical and algebro-geometric theory of localization

We develop a categorical and algebro-geometric treatment of localization for cohomological theories endowed with an open–closed recollement. Starting from a class on a space 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 rather a torsor of supported refinements; a canonical local term arises only once an additional uniqueness or concentration principle is imposed. We establish excision, a natural pullback map under Cartesian base change, proper pushforward, and compatibility with external products under explicit hypotheses governing the interaction between product constructions and exceptional pullback. We also prove a factorization result showing that any assignment of local terms already compatible with the localization triangle must necessarily take its values in this torsor. When supplemented by Verdier duality and the appropriate orientation data, the resulting localized classes govern local indices and yield global-to-local index formulas. Under purity and concentration, the formalism recovers the familiar Euler-denominator expressions. The later geometric examples should therefore be read as conditional realisations of the same torsorial mechanism, available only once the relevant comparison hypotheses, together with the requisite purity and concentration statements, are in force.