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We prove that if G has isotropic rank >=1 and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra A=R[x_1,...,x_n]/I over R, the map H^1_Nis(A,G) -> H^1_Nis(R,G) induced by evaluation at x_1=...=x_n=0, is a bijection. If k has characteristic 0, then, moreover, the map H^1_et(A,G) -> H^1_et(R,G) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is"},"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":"1712.04907","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2017-12-13T18:24:03Z","cross_cats_sorted":[],"title_canon_sha256":"febf89b0b9e81529ddac486d6a3d40f3af9590921a52d1b9a541d9ab5ed06434","abstract_canon_sha256":"7d21fe10a72aea07094c4d51c9ed4a7e913fc32c9a2a71f5adcab3be7a78d0a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:09.735429Z","signature_b64":"JPkywrfkI43yoLJ7O6yEuqQdH2maXe2TTTHybDYvn0lOjVGiPJb8mYj5g+mbsRjkD0zy/cU+k46j2ZYJx506BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a7b9ff80c96aed626cca841dc5e68eadf85d6ac0945942758ab4c52c9a9ab6c","last_reissued_at":"2026-05-18T00:09:09.734844Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:09.734844Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Isotropic reductive groups over discrete Hodge algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"Anastasia Stavrova","submitted_at":"2017-12-13T18:24:03Z","abstract_excerpt":"Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra A=R[x_1,...,x_n]/I over R, the map H^1_Nis(A,G) -> H^1_Nis(R,G) induced by evaluation at x_1=...=x_n=0, is a bijection. If k has characteristic 0, then, moreover, the map H^1_et(A,G) -> H^1_et(R,G) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04907","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1712.04907","created_at":"2026-05-18T00:09:09.734949+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.04907v3","created_at":"2026-05-18T00:09:09.734949+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.04907","created_at":"2026-05-18T00:09:09.734949+00:00"},{"alias_kind":"pith_short_12","alias_value":"BJ5Z76AMS2XN","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"BJ5Z76AMS2XNMJWM","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"BJ5Z76AM","created_at":"2026-05-18T12:31:08.081275+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BJ5Z76AMS2XNMJWMVBA5YXTI5L","json":"https://pith.science/pith/BJ5Z76AMS2XNMJWMVBA5YXTI5L.json","graph_json":"https://pith.science/api/pith-number/BJ5Z76AMS2XNMJWMVBA5YXTI5L/graph.json","events_json":"https://pith.science/api/pith-number/BJ5Z76AMS2XNMJWMVBA5YXTI5L/events.json","paper":"https://pith.science/paper/BJ5Z76AM"},"agent_actions":{"view_html":"https://pith.science/pith/BJ5Z76AMS2XNMJWMVBA5YXTI5L","download_json":"https://pith.science/pith/BJ5Z76AMS2XNMJWMVBA5YXTI5L.json","view_paper":"https://pith.science/paper/BJ5Z76AM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.04907&json=true","fetch_graph":"https://pith.science/api/pith-number/BJ5Z76AMS2XNMJWMVBA5YXTI5L/graph.json","fetch_events":"https://pith.science/api/pith-number/BJ5Z76AMS2XNMJWMVBA5YXTI5L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BJ5Z76AMS2XNMJWMVBA5YXTI5L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BJ5Z76AMS2XNMJWMVBA5YXTI5L/action/storage_attestation","attest_author":"https://pith.science/pith/BJ5Z76AMS2XNMJWMVBA5YXTI5L/action/author_attestation","sign_citation":"https://pith.science/pith/BJ5Z76AMS2XNMJWMVBA5YXTI5L/action/citation_signature","submit_replication":"https://pith.science/pith/BJ5Z76AMS2XNMJWMVBA5YXTI5L/action/replication_record"}},"created_at":"2026-05-18T00:09:09.734949+00:00","updated_at":"2026-05-18T00:09:09.734949+00:00"}