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We prove that for any Noetherian k-algebra A, the map of etale cohomology sets H^1(A\\otimes_k R,G)-> H^1(A\\otimes_ k K,G), induced by the inclusion of R into K, has trivial kernel. This implies the Serre-Grothendieck conjecture for such groups G.\n  The main theorem for A=k and some other results of the present paper are used significantly in arXiv"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0905.1418","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2009-05-09T18:32:53Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"0fd94a5bbf4a84aeed3b848340134a95ba1fe06cdb33a928b88ce3598dfa1acc","abstract_canon_sha256":"fca7eab737052c0843a251cb5d38f12ebdfc9096afa36a84ac5e16b2b226deac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:11.768077Z","signature_b64":"/eX/mqWUFMgxF1BIdjif8OTEC7YgCtkAanoXdGBPefdulGvZeoAoOT86B87pZ5Og/NPRRVO6DL2bFj0kjkbTBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"849faeb39aa2d39d7745930dadddfaeed268af99b3ebc58e54bf63eb410ee9a7","last_reissued_at":"2026-05-18T03:27:11.767422Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:11.767422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes:I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AG","authors_text":"A. 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