{"paper":{"title":"The K\\\"unneth Formula Of Fundamental Group Schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Lingguang Li, Niantao Tian","submitted_at":"2026-02-15T16:05:05Z","abstract_excerpt":"Let $k$ be a field, $f:X\\rightarrow S$ a proper morphism between connected schemes proper over $k$, $x\\in X(k)$ lying over $s\\in S(k)$, $X_s$ the fibre of $f$ over $s$, $\\mathcal{C}_X$, $\\mathcal{C}_{S}$, $\\mathcal{C}_{X_s}$ Tannakian categories over $X,S,X_s$ respectively, $\\pi(\\mathcal{C}_X,x)$, $\\pi(\\mathcal{C}_S,s)$, $\\pi(\\mathcal{C}_{X_s},x)$ the Tannaka group schemes respectively. We give a unified criterion for the exactness of the homotopy sequence of Tannakian fundamental group schemes $\\pi(\\mathcal{C}_{X_s},x)\\rightarrow \\pi(\\mathcal{C}_X,x)\\rightarrow \\pi(\\mathcal{C}_S,s)\\rightarrow"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.14207","kind":"arxiv","version":2},"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/2602.14207/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"}