{"paper":{"title":"Localization and Affine Schemes over $\\mathbb{F}_1$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Luqiao Xu","submitted_at":"2026-07-06T09:14:13Z","abstract_excerpt":"We develop the basic notions of commutative algebra and algebraic geometry over the field with one element $\\mathbb{F}_1$, working within the Connes-Consani framework, which models $\\mathbb{F}_1$-algebras as monoid objects in the category of $\\Gamma$-sets. In this setting, $\\mathbb{F}_1$-algebras generalize commutative rings by encoding the algebraic structure functorially, using machinery originating in homotopy theory. Our main contribution is a theory of localization for $\\mathbb{F}_1$-algebras and the construction of prime spectrum $\\Spec A$ for a commutative $\\mathbb{F}_1$-algebra $A$. We"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04843","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.04843/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"}