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.
Grothendieck et al.,Théorie des topos et cohomologie étale des schémas (SGA4), Lecture Notes in Mathematics, vols
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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.