A categorical and algebro-geometric theory of localization

We develop a categorical and algebro-geometric theory of localisation for cohomological theories with open--closed recollement.

A class whose restriction to the open complement vanishes need not determine a preferred class on the closed stratum; the localisation triangle associates with it instead a torsor of supported refinements, whose secondary indeterminacy is governed by the connecting morphism from the open complement.

We prove compatibility with excision, base change, proper pushforward, external products and local indices, and show that compatible supported constructions factor through this torsor.

Under explicit Gysin hypotheses, injectivity of Euler multiplication gives a pre-Euler canonicity criterion, making the supported refinement unique before any coefficient localisation.

Purity, concentration and Euler rigidification recover the usual Euler-denominator formulae.

We also relate the secondary boundary group to link transgression, treat equivariant algebraic K-theory as a multiplicative analogue, and introduce Milnor localisation torsors for characteristic-class defects of singularities.