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We prove that for any Noetherian affine scheme A over k, the kernel of the map of etale cohomology sets H^1(A\\times_k R,G)-> H^1(A\\times_ k K,G), induced by the inclusion of R into K, is trivial. If R is the semi-local ring of several points on a k-smooth scheme, then it suffices to require that k is infinite and keep the same assumption concerning G."},"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":"1204.1729","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-04-08T12:57:32Z","cross_cats_sorted":[],"title_canon_sha256":"ddf316f34c2f2ea3c21e7bb0013de2f287c257a3d2824f19b79b83fb94e1291d","abstract_canon_sha256":"14a3b638e3128733724ae71c129214a5d71fe6351d8b21b641142ef0f9561a43"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:30:42.233118Z","signature_b64":"oPswB1ZpXQxGScG+i+ky1kA5qWu+DLEnjrmzkW6bef1rfVdyWjgHbTY/OAsKNlCCSf5ronKRxventajJQ06QDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"881496910f37c33551e3a62440841f441421bd378e3fc9c7ebb76e2cb75d5333","last_reissued_at":"2026-05-18T03:30:42.232428Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:30:42.232428Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Principal bundles of reductive groups over affine schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Anastasia Stavrova, Ivan Panin","submitted_at":"2012-04-08T12:57:32Z","abstract_excerpt":"Let R be a semi-local regular domain containing an infinite perfect field k, and let K be the field of fractions of R. Let G be a reductive semi-simple simply connected R-group scheme such that each of its R-indecomposable factors is isotropic. We prove that for any Noetherian affine scheme A over k, the kernel of the map of etale cohomology sets H^1(A\\times_k R,G)-> H^1(A\\times_ k K,G), induced by the inclusion of R into K, is trivial. 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