{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:ZMBYBKPPVWCVKSFGYLAOXUNOTJ","short_pith_number":"pith:ZMBYBKPP","canonical_record":{"source":{"id":"1012.0278","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-12-01T19:34:21Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"7ec5cce86616989ddacbe354c7768a80c299843503a0739ec6c79cdc4b78995b","abstract_canon_sha256":"d2341b82ebdbeb948b0a01a698fed4df8f5d24bf18316768324556fe1519af43"},"schema_version":"1.0"},"canonical_sha256":"cb0380a9efad855548a6c2c0ebd1ae9a5f04d4262f1bff838c17f449cbd6c195","source":{"kind":"arxiv","id":"1012.0278","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.0278","created_at":"2026-05-18T04:33:28Z"},{"alias_kind":"arxiv_version","alias_value":"1012.0278v2","created_at":"2026-05-18T04:33:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0278","created_at":"2026-05-18T04:33:28Z"},{"alias_kind":"pith_short_12","alias_value":"ZMBYBKPPVWCV","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"ZMBYBKPPVWCVKSFG","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"ZMBYBKPP","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:ZMBYBKPPVWCVKSFGYLAOXUNOTJ","target":"record","payload":{"canonical_record":{"source":{"id":"1012.0278","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-12-01T19:34:21Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"7ec5cce86616989ddacbe354c7768a80c299843503a0739ec6c79cdc4b78995b","abstract_canon_sha256":"d2341b82ebdbeb948b0a01a698fed4df8f5d24bf18316768324556fe1519af43"},"schema_version":"1.0"},"canonical_sha256":"cb0380a9efad855548a6c2c0ebd1ae9a5f04d4262f1bff838c17f449cbd6c195","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:33:28.755913Z","signature_b64":"zCi0vam8Jg1cJJmGyNPrradq7QIyC2SNIUuH55CXTRBxj4uIiG9B75jS2OLVVURLhnxojSefPh1LoK6whluhCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb0380a9efad855548a6c2c0ebd1ae9a5f04d4262f1bff838c17f449cbd6c195","last_reissued_at":"2026-05-18T04:33:28.755445Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:33:28.755445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.0278","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:33:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JGX0Y/cFgz2fuKCGRjzy2e/eImVzokd0u7eVlCT4pDZh66t2h0aw2sDvJrDjFtZGRHVG2oXrj0ObdHQHg97/BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:00:59.270791Z"},"content_sha256":"a91ed1991da5447d35744f47807ef93f4d8ff60333d954dcc25a510f66fd01ea","schema_version":"1.0","event_id":"sha256:a91ed1991da5447d35744f47807ef93f4d8ff60333d954dcc25a510f66fd01ea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:ZMBYBKPPVWCVKSFGYLAOXUNOTJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Centralizer of the elementary subgroup of an isotropic reductive group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AG","authors_text":"Anastasia Stavrova, Ekaterina Kulikova","submitted_at":"2010-12-01T19:34:21Z","abstract_excerpt":"Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that, for any maximal ideal M of R, the rank of the relative root system of G_{R_M} is greater or equal than 2. We show that under this assumption the centralizer of E(R) in G(R) coincides with the abstract group-theoretic center of G(R) and with Cent(G)(R). This generalizes a result of E. Abe and J. Hurley."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0278","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:33:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8TPu5DjDtAOpxqQ7zSTkPN6OS+nHC8hXr1lJcCLlb85Pj38H0QzN1+YS8tbUWYRwfceB3ba3F5zcodvghZccDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T02:00:59.271526Z"},"content_sha256":"0dd120f7652a31430f38ec36451fba48d5c547a32b436a6b0207d79edb84a253","schema_version":"1.0","event_id":"sha256:0dd120f7652a31430f38ec36451fba48d5c547a32b436a6b0207d79edb84a253"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZMBYBKPPVWCVKSFGYLAOXUNOTJ/bundle.json","state_url":"https://pith.science/pith/ZMBYBKPPVWCVKSFGYLAOXUNOTJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZMBYBKPPVWCVKSFGYLAOXUNOTJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-11T02:00:59Z","links":{"resolver":"https://pith.science/pith/ZMBYBKPPVWCVKSFGYLAOXUNOTJ","bundle":"https://pith.science/pith/ZMBYBKPPVWCVKSFGYLAOXUNOTJ/bundle.json","state":"https://pith.science/pith/ZMBYBKPPVWCVKSFGYLAOXUNOTJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZMBYBKPPVWCVKSFGYLAOXUNOTJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:ZMBYBKPPVWCVKSFGYLAOXUNOTJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d2341b82ebdbeb948b0a01a698fed4df8f5d24bf18316768324556fe1519af43","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-12-01T19:34:21Z","title_canon_sha256":"7ec5cce86616989ddacbe354c7768a80c299843503a0739ec6c79cdc4b78995b"},"schema_version":"1.0","source":{"id":"1012.0278","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.0278","created_at":"2026-05-18T04:33:28Z"},{"alias_kind":"arxiv_version","alias_value":"1012.0278v2","created_at":"2026-05-18T04:33:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0278","created_at":"2026-05-18T04:33:28Z"},{"alias_kind":"pith_short_12","alias_value":"ZMBYBKPPVWCV","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"ZMBYBKPPVWCVKSFG","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"ZMBYBKPP","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:0dd120f7652a31430f38ec36451fba48d5c547a32b436a6b0207d79edb84a253","target":"graph","created_at":"2026-05-18T04:33:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that, for any maximal ideal M of R, the rank of the relative root system of G_{R_M} is greater or equal than 2. We show that under this assumption the centralizer of E(R) in G(R) coincides with the abstract group-theoretic center of G(R) and with Cent(G)(R). This generalizes a result of E. Abe and J. Hurley.","authors_text":"Anastasia Stavrova, Ekaterina Kulikova","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-12-01T19:34:21Z","title":"Centralizer of the elementary subgroup of an isotropic reductive group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0278","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a91ed1991da5447d35744f47807ef93f4d8ff60333d954dcc25a510f66fd01ea","target":"record","created_at":"2026-05-18T04:33:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d2341b82ebdbeb948b0a01a698fed4df8f5d24bf18316768324556fe1519af43","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-12-01T19:34:21Z","title_canon_sha256":"7ec5cce86616989ddacbe354c7768a80c299843503a0739ec6c79cdc4b78995b"},"schema_version":"1.0","source":{"id":"1012.0278","kind":"arxiv","version":2}},"canonical_sha256":"cb0380a9efad855548a6c2c0ebd1ae9a5f04d4262f1bff838c17f449cbd6c195","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb0380a9efad855548a6c2c0ebd1ae9a5f04d4262f1bff838c17f449cbd6c195","first_computed_at":"2026-05-18T04:33:28.755445Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:33:28.755445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zCi0vam8Jg1cJJmGyNPrradq7QIyC2SNIUuH55CXTRBxj4uIiG9B75jS2OLVVURLhnxojSefPh1LoK6whluhCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:33:28.755913Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.0278","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a91ed1991da5447d35744f47807ef93f4d8ff60333d954dcc25a510f66fd01ea","sha256:0dd120f7652a31430f38ec36451fba48d5c547a32b436a6b0207d79edb84a253"],"state_sha256":"a4da715e7f27b822bfee22ea889deaacdd82f82ff988071d60e87d5cb3f07cb2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tN2lgs2OtlfsKGRkAMpjfMmAEs4reitEcv0jY7mxkdDF2ltbx2a/bgfqXah95ulUFjOpkTOqwfvYaG23DPUlAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T02:00:59.275076Z","bundle_sha256":"de5a08cb9fcbe22e5423745dd21846f013caf031480528a76c095bd8677abf43"}}