{"paper":{"title":"The coordinate biring of $\\mathbf{Spec}(\\mathbb{Z})/\\mathbb{F}_1$","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.AG","math.NT","math.QA"],"primary_cat":"math.RA","authors_text":"Lieven Le Bruyn","submitted_at":"2015-09-02T15:44:27Z","abstract_excerpt":"We propose to define $\\mathbb{F}_1$-algebras as integral bi-rings with the co-ring structure being the descent data from $\\mathbb{Z}$ to $\\mathbb{F}_1$. The coordinate bi-ring of $\\mathbf{Spec}(\\mathbb{Z})/\\mathbb{F}_1$ is then the co-ring of integral linear recursive sequences equipped with the Hadamard product.\n  We associate a noncommutative moduli space to this setting, show that it is defined over $\\mathbb{F}_1$, and has motive $\\prod_{n \\geq 0} \\frac{s-n}{2 \\pi}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00749","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":""},"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"}