{"paper":{"title":"Under Spec Z","license":"","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AG","authors_text":"Bertrand Toen, Michel Vaquie","submitted_at":"2005-09-29T09:55:14Z","abstract_excerpt":"We use techniques from relative algebraic geometry and homotopical algebraic geometry in order to construct several categories of schemes defined \"under Spec Z\". We define this way the categories of N-schemes, F_1-schemes, S-schemes, S_+-schemes, and S_1-schemes, where from a very intuitive point of view N is the semi-ring of natural numbers, F_1 is the field with one element, S is the sphere ring spectrum, S_+ is the semi-ring spectrum of natural numbers and S_1 is the ring spectrum with one element. These categories of schemes are related by several base change functors, and they all possess"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0509684","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}