Localization in recollement-equipped cohomological theories naturally produces a torsor of supported classes; a unique local term requires extra concentration or uniqueness assumptions, and the setup recovers standard Euler formulas under purity.
Vistoli,Intersection theory on algebraic stacks and on their moduli spaces, Invent
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A categorical and algebro-geometric theory of localization
Localization in recollement-equipped cohomological theories naturally produces a torsor of supported classes; a unique local term requires extra concentration or uniqueness assumptions, and the setup recovers standard Euler formulas under purity.