{"paper":{"title":"Conjecture I for unirational algebraic groups over imperfect fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Unirational algebraic groups have trivial first Galois cohomology over fields of Kato cohomological dimension at most 1.","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alexandre Lourdeaux, Anis Zidani","submitted_at":"2026-04-06T20:22:01Z","abstract_excerpt":"Using the recent advances in the structure of algebraic groups over imperfect fields, we prove a generalization of Serre's Conjecture I and of results that revolve around it. In particular, we show that the first Galois cohomology set of any unirational algebraic group is trivial if the cohomological dimension of the field is at most 1 in Kato's sense."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we prove that the first Galois cohomology set of any unirational algebraic group is always trivial if the cohomological dimension of the field is less or equal to 1 in Kato's sense.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The recent advancements in the structure of algebraic groups over imperfect fields are sufficient to establish the triviality result for unirational groups under the stated cohomological dimension condition.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Unirational algebraic groups over fields with Kato cohomological dimension ≤1 have trivial first Galois cohomology.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Unirational algebraic groups have trivial first Galois cohomology over fields of Kato cohomological dimension at most 1.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"26dd6239f703d31e8278b03d2198416177acdd8195a73736ea98125866843ae6"},"source":{"id":"2604.05148","kind":"arxiv","version":2},"verdict":{"id":"96c069f5-c4aa-45bd-978e-1054abba1e2c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T18:59:18.480275Z","strongest_claim":"we prove that the first Galois cohomology set of any unirational algebraic group is always trivial if the cohomological dimension of the field is less or equal to 1 in Kato's sense.","one_line_summary":"Unirational algebraic groups over fields with Kato cohomological dimension ≤1 have trivial first Galois cohomology.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The recent advancements in the structure of algebraic groups over imperfect fields are sufficient to establish the triviality result for unirational groups under the stated cohomological dimension condition.","pith_extraction_headline":"Unirational algebraic groups have trivial first Galois cohomology over fields of Kato cohomological dimension at most 1."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.05148/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":2,"snapshot_sha256":"5ac02bce1ec64b5eeaab11070d884f076ae1e6fc580134fba1237dadee460d92"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}