{"paper":{"title":"A categorical and algebro-geometric theory of localization","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Localization for theories with open-closed recollements produces a torsor of supported refinements on the closed locus rather than a single distinguished class.","cross_cats":["math-ph","math.DG","math.GT","math.MP"],"primary_cat":"math.AG","authors_text":"Mauricio Corr\\^ea, Simone Noja","submitted_at":"2026-04-04T20:00:45Z","abstract_excerpt":"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 concentra"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The existence of an open-closed recollement together with the additional uniqueness or concentration principle needed to select a canonical local term from the torsor; without it the output remains a torsor rather than a single class.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Localization for theories with open-closed recollements produces a torsor of supported refinements on the closed locus rather than a single distinguished class.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"220a13f95b64f38fada603ca26ad046670186e642771eecc7b12d9b053d6aff9"},"source":{"id":"2604.03845","kind":"arxiv","version":3},"verdict":{"id":"2c14663f-481e-44f0-8a2e-cdcb5d6d676a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T17:07:41.390575Z","strongest_claim":"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.","one_line_summary":"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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The existence of an open-closed recollement together with the additional uniqueness or concentration principle needed to select a canonical local term from the torsor; without it the output remains a torsor rather than a single class.","pith_extraction_headline":"Localization for theories with open-closed recollements produces a torsor of supported refinements on the closed locus rather than a single distinguished class."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.03845/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}