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This implies that K_1^G is A^1-homotopy invariant on the category of regular k-algebras, if k is perfect. If k is infinite perfect, one also deduces that K_1^G(R)-> K_1^G(K) is injective for any regular local k-algebra R with the fraction field K."},"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":"1111.4664","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-11-20T19:01:25Z","cross_cats_sorted":["math.GR","math.KT"],"title_canon_sha256":"ebb7ee045748f6d903c496ca066eabbc8bf31e3030438d88de6aa0212c003d20","abstract_canon_sha256":"6fa5abfcaac3c5b21510ad3569ace08dfa24086e1baa7947fd91b4f201e8d10f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:50.196309Z","signature_b64":"wNqV7qfg9Whv5aFubaftHQTpex1ujMGy9gxk4B6UBLhlKQj2p1zfYlQl0QHQVU5niBhhbEBQRkYYthZrje/KAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f8d5c2d81fb6c4cec318ee058120e591db27299e7fe1aebc053ca73f74069b8","last_reissued_at":"2026-05-18T03:33:50.195465Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:50.195465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homotopy invariance of non-stable K_1-functors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.KT"],"primary_cat":"math.AG","authors_text":"Anastasia Stavrova","submitted_at":"2011-11-20T19:01:25Z","abstract_excerpt":"Let G be reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank >=2. Let K_1^G be the non-stable K_1-functor associated to G (also called the Whitehead group of G in the field case). We show that K_1^G(k)=K_1^G(k[X_1,...,X_n]) for any n>= 1. This implies that K_1^G is A^1-homotopy invariant on the category of regular k-algebras, if k is perfect. If k is infinite perfect, one also deduces that K_1^G(R)-> K_1^G(K) is injective for any regular local k-algebra R with the fraction field K."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4664","kind":"arxiv","version":5},"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":"1111.4664","created_at":"2026-05-18T03:33:50.195614+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.4664v5","created_at":"2026-05-18T03:33:50.195614+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.4664","created_at":"2026-05-18T03:33:50.195614+00:00"},{"alias_kind":"pith_short_12","alias_value":"D6GVYLMB7NWE","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"D6GVYLMB7NWEZ3BR","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"D6GVYLMB","created_at":"2026-05-18T12:26:26.731475+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/D6GVYLMB7NWEZ3BRR3QFQEQOLE","json":"https://pith.science/pith/D6GVYLMB7NWEZ3BRR3QFQEQOLE.json","graph_json":"https://pith.science/api/pith-number/D6GVYLMB7NWEZ3BRR3QFQEQOLE/graph.json","events_json":"https://pith.science/api/pith-number/D6GVYLMB7NWEZ3BRR3QFQEQOLE/events.json","paper":"https://pith.science/paper/D6GVYLMB"},"agent_actions":{"view_html":"https://pith.science/pith/D6GVYLMB7NWEZ3BRR3QFQEQOLE","download_json":"https://pith.science/pith/D6GVYLMB7NWEZ3BRR3QFQEQOLE.json","view_paper":"https://pith.science/paper/D6GVYLMB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.4664&json=true","fetch_graph":"https://pith.science/api/pith-number/D6GVYLMB7NWEZ3BRR3QFQEQOLE/graph.json","fetch_events":"https://pith.science/api/pith-number/D6GVYLMB7NWEZ3BRR3QFQEQOLE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D6GVYLMB7NWEZ3BRR3QFQEQOLE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D6GVYLMB7NWEZ3BRR3QFQEQOLE/action/storage_attestation","attest_author":"https://pith.science/pith/D6GVYLMB7NWEZ3BRR3QFQEQOLE/action/author_attestation","sign_citation":"https://pith.science/pith/D6GVYLMB7NWEZ3BRR3QFQEQOLE/action/citation_signature","submit_replication":"https://pith.science/pith/D6GVYLMB7NWEZ3BRR3QFQEQOLE/action/replication_record"}},"created_at":"2026-05-18T03:33:50.195614+00:00","updated_at":"2026-05-18T03:33:50.195614+00:00"}