{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:U4U5A7PUFLSAAKYHO2HRSVKELE","short_pith_number":"pith:U4U5A7PU","canonical_record":{"source":{"id":"1003.3598","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-03-18T14:50:36Z","cross_cats_sorted":[],"title_canon_sha256":"1d9d16861c7836be03a942f1420af89e76f4ce4361bfecf58c8d1cb7a8d7a057","abstract_canon_sha256":"a4e372dc7c958bc6c1a2231655615f59199ef579c626aed749d642296ca37050"},"schema_version":"1.0"},"canonical_sha256":"a729d07df42ae4002b07768f195544590fca9c5feb4dc42d78c0a1340fb5d697","source":{"kind":"arxiv","id":"1003.3598","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.3598","created_at":"2026-05-18T02:56:58Z"},{"alias_kind":"arxiv_version","alias_value":"1003.3598v3","created_at":"2026-05-18T02:56:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.3598","created_at":"2026-05-18T02:56:58Z"},{"alias_kind":"pith_short_12","alias_value":"U4U5A7PUFLSA","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"U4U5A7PUFLSAAKYH","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"U4U5A7PU","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:U4U5A7PUFLSAAKYHO2HRSVKELE","target":"record","payload":{"canonical_record":{"source":{"id":"1003.3598","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-03-18T14:50:36Z","cross_cats_sorted":[],"title_canon_sha256":"1d9d16861c7836be03a942f1420af89e76f4ce4361bfecf58c8d1cb7a8d7a057","abstract_canon_sha256":"a4e372dc7c958bc6c1a2231655615f59199ef579c626aed749d642296ca37050"},"schema_version":"1.0"},"canonical_sha256":"a729d07df42ae4002b07768f195544590fca9c5feb4dc42d78c0a1340fb5d697","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:58.707609Z","signature_b64":"Mwz39eoD9dr31s1SWEQ9v1t1lKjMrqAu6wF13Pthmtxd3L50At5B/0gAWFxQR9hZCtn8UTZ/7QE5i07uAgxOCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a729d07df42ae4002b07768f195544590fca9c5feb4dc42d78c0a1340fb5d697","last_reissued_at":"2026-05-18T02:56:58.707074Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:58.707074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1003.3598","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:56:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"niWbjXn4q+voqrb5MjWPg2fM+Cvs1ebdEd0d50460lMo3ZW5sBvw4df9QHdftefxwVMdIQlVJVQY3U7rFw9eCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T17:37:08.236280Z"},"content_sha256":"6bfb2ff43a76fedab7e7fbe69b5be9761abe2b4be5a3ffb9b341b0b25a183461","schema_version":"1.0","event_id":"sha256:6bfb2ff43a76fedab7e7fbe69b5be9761abe2b4be5a3ffb9b341b0b25a183461"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:U4U5A7PUFLSAAKYHO2HRSVKELE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Reductive group schemes, the Greenberg functor, and associated algebraic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alexander Stasinski","submitted_at":"2010-03-18T14:50:36Z","abstract_excerpt":"Let $A$ be an Artinian local ring with algebraically closed residue field $k$, and let $\\mathbf{G}$ be an affine smooth group scheme over $A$. The Greenberg functor $\\mathcal{F}$ associates to $\\mathbf{G}$ a linear algebraic group $G:=(\\mathcal{F}\\mathbf{G})(k)$ over $k$, such that $G\\cong\\mathbf{G}(A)$. We prove that if $\\mathbf{G}$ is a reductive group scheme over $A$, and $\\mathbf{T}$ is a maximal torus of $\\mathbf{G}$, then $T$ is a Cartan subgroup of $G$, and every Cartan subgroup of $G$ is obtained uniquely in this way. The proof is based on establishing a Nullstellensatz analogue for sm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.3598","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:56:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MOWz9x5MBO6nQeXP3tUaFGFPFWgHn+8eRe8aCcBJ3ccTzYoP/dNaO26N4n95TUcpK0mfbhkFbaEpgpOlJERnAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T17:37:08.236653Z"},"content_sha256":"ccc25d11ae46be5a49e23fee86330380e28d75822251113069884db0e2f8dda6","schema_version":"1.0","event_id":"sha256:ccc25d11ae46be5a49e23fee86330380e28d75822251113069884db0e2f8dda6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U4U5A7PUFLSAAKYHO2HRSVKELE/bundle.json","state_url":"https://pith.science/pith/U4U5A7PUFLSAAKYHO2HRSVKELE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U4U5A7PUFLSAAKYHO2HRSVKELE/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":"2026-06-21T17:37:08Z","links":{"resolver":"https://pith.science/pith/U4U5A7PUFLSAAKYHO2HRSVKELE","bundle":"https://pith.science/pith/U4U5A7PUFLSAAKYHO2HRSVKELE/bundle.json","state":"https://pith.science/pith/U4U5A7PUFLSAAKYHO2HRSVKELE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U4U5A7PUFLSAAKYHO2HRSVKELE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:U4U5A7PUFLSAAKYHO2HRSVKELE","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":"a4e372dc7c958bc6c1a2231655615f59199ef579c626aed749d642296ca37050","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-03-18T14:50:36Z","title_canon_sha256":"1d9d16861c7836be03a942f1420af89e76f4ce4361bfecf58c8d1cb7a8d7a057"},"schema_version":"1.0","source":{"id":"1003.3598","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.3598","created_at":"2026-05-18T02:56:58Z"},{"alias_kind":"arxiv_version","alias_value":"1003.3598v3","created_at":"2026-05-18T02:56:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.3598","created_at":"2026-05-18T02:56:58Z"},{"alias_kind":"pith_short_12","alias_value":"U4U5A7PUFLSA","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"U4U5A7PUFLSAAKYH","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"U4U5A7PU","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:ccc25d11ae46be5a49e23fee86330380e28d75822251113069884db0e2f8dda6","target":"graph","created_at":"2026-05-18T02:56:58Z","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 $A$ be an Artinian local ring with algebraically closed residue field $k$, and let $\\mathbf{G}$ be an affine smooth group scheme over $A$. The Greenberg functor $\\mathcal{F}$ associates to $\\mathbf{G}$ a linear algebraic group $G:=(\\mathcal{F}\\mathbf{G})(k)$ over $k$, such that $G\\cong\\mathbf{G}(A)$. We prove that if $\\mathbf{G}$ is a reductive group scheme over $A$, and $\\mathbf{T}$ is a maximal torus of $\\mathbf{G}$, then $T$ is a Cartan subgroup of $G$, and every Cartan subgroup of $G$ is obtained uniquely in this way. The proof is based on establishing a Nullstellensatz analogue for sm","authors_text":"Alexander Stasinski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-03-18T14:50:36Z","title":"Reductive group schemes, the Greenberg functor, and associated algebraic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.3598","kind":"arxiv","version":3},"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":"sha256:6bfb2ff43a76fedab7e7fbe69b5be9761abe2b4be5a3ffb9b341b0b25a183461","target":"record","created_at":"2026-05-18T02:56:58Z","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":"a4e372dc7c958bc6c1a2231655615f59199ef579c626aed749d642296ca37050","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-03-18T14:50:36Z","title_canon_sha256":"1d9d16861c7836be03a942f1420af89e76f4ce4361bfecf58c8d1cb7a8d7a057"},"schema_version":"1.0","source":{"id":"1003.3598","kind":"arxiv","version":3}},"canonical_sha256":"a729d07df42ae4002b07768f195544590fca9c5feb4dc42d78c0a1340fb5d697","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a729d07df42ae4002b07768f195544590fca9c5feb4dc42d78c0a1340fb5d697","first_computed_at":"2026-05-18T02:56:58.707074Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:56:58.707074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Mwz39eoD9dr31s1SWEQ9v1t1lKjMrqAu6wF13Pthmtxd3L50At5B/0gAWFxQR9hZCtn8UTZ/7QE5i07uAgxOCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:56:58.707609Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.3598","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6bfb2ff43a76fedab7e7fbe69b5be9761abe2b4be5a3ffb9b341b0b25a183461","sha256:ccc25d11ae46be5a49e23fee86330380e28d75822251113069884db0e2f8dda6"],"state_sha256":"f6bd46ab5daa6a491469b8d24b6b2640210ff2ae7f5c4f7057c2e5b1ded5cdb1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QXONWzFdaoa5EQhCjysQVANRYlroxIcvZAA9rBCNOLpzgPdsWbXc0fTd5amf56FwLGFjoIYM5jbYB42tv2qhCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T17:37:08.238699Z","bundle_sha256":"759336ada287a6eb6310d08404d3643ad913620d3a30642e849fe616fdb48e22"}}