{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:FXRYDSHLTYM4AWUKXZWGG2DA5Q","short_pith_number":"pith:FXRYDSHL","schema_version":"1.0","canonical_sha256":"2de381c8eb9e19c05a8abe6c636860ec1aeafe8563cdb8bd8f6b62e0d3113ac9","source":{"kind":"arxiv","id":"1801.07305","version":4},"attestation_state":"computed","paper":{"title":"Gorenstein projective and injective dimensions over Frobenius extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"Wei Ren","submitted_at":"2018-01-22T20:23:21Z","abstract_excerpt":"Let $R\\subset A$ be a Frobenius extension of rings. We prove that: (1) for any left $A$-module $M$, $_{A}M$ is Gorenstein projective (injective) if and only if the underlying left $R$-module $_{R}M$ is Gorenstein projective (injective). (2) if $\\mathrm{G}\\text{-}\\mathrm{proj.dim}_{A}M<\\infty$, then $\\mathrm{G}\\text{-}\\mathrm{proj.dim}_{A}M = \\mathrm{G}\\text{-}\\mathrm{proj.dim}_{R}M$, the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then $\\mathrm{G}\\text{-}\\mathrm{gldim}(A)= \\mathrm{G}\\text{-}\\mathrm{gldim}(R)$."},"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":"1801.07305","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2018-01-22T20:23:21Z","cross_cats_sorted":[],"title_canon_sha256":"6c4c62255be173ccbab502fd41911f9a4a0fb24bf1a5ecf1464c82cc1c70fd90","abstract_canon_sha256":"80131568faac5eabdb32529c9ff6ef0043f7dbd846644df075554299382ee07a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:49.780696Z","signature_b64":"kNgwYsH1PQvyxPC/H/5QHHO4fxrMr9qtJ3crX14FPREazKjZyxos2/5ZCSw+A4oWilY8b9oMJXwH516P8qmcAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2de381c8eb9e19c05a8abe6c636860ec1aeafe8563cdb8bd8f6b62e0d3113ac9","last_reissued_at":"2026-05-17T23:40:49.779996Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:49.779996Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gorenstein projective and injective dimensions over Frobenius extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"Wei Ren","submitted_at":"2018-01-22T20:23:21Z","abstract_excerpt":"Let $R\\subset A$ be a Frobenius extension of rings. We prove that: (1) for any left $A$-module $M$, $_{A}M$ is Gorenstein projective (injective) if and only if the underlying left $R$-module $_{R}M$ is Gorenstein projective (injective). (2) if $\\mathrm{G}\\text{-}\\mathrm{proj.dim}_{A}M<\\infty$, then $\\mathrm{G}\\text{-}\\mathrm{proj.dim}_{A}M = \\mathrm{G}\\text{-}\\mathrm{proj.dim}_{R}M$, the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then $\\mathrm{G}\\text{-}\\mathrm{gldim}(A)= \\mathrm{G}\\text{-}\\mathrm{gldim}(R)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.07305","kind":"arxiv","version":4},"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":"1801.07305","created_at":"2026-05-17T23:40:49.780107+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.07305v4","created_at":"2026-05-17T23:40:49.780107+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.07305","created_at":"2026-05-17T23:40:49.780107+00:00"},{"alias_kind":"pith_short_12","alias_value":"FXRYDSHLTYM4","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_16","alias_value":"FXRYDSHLTYM4AWUK","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_8","alias_value":"FXRYDSHL","created_at":"2026-05-18T12:32:25.280505+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/FXRYDSHLTYM4AWUKXZWGG2DA5Q","json":"https://pith.science/pith/FXRYDSHLTYM4AWUKXZWGG2DA5Q.json","graph_json":"https://pith.science/api/pith-number/FXRYDSHLTYM4AWUKXZWGG2DA5Q/graph.json","events_json":"https://pith.science/api/pith-number/FXRYDSHLTYM4AWUKXZWGG2DA5Q/events.json","paper":"https://pith.science/paper/FXRYDSHL"},"agent_actions":{"view_html":"https://pith.science/pith/FXRYDSHLTYM4AWUKXZWGG2DA5Q","download_json":"https://pith.science/pith/FXRYDSHLTYM4AWUKXZWGG2DA5Q.json","view_paper":"https://pith.science/paper/FXRYDSHL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.07305&json=true","fetch_graph":"https://pith.science/api/pith-number/FXRYDSHLTYM4AWUKXZWGG2DA5Q/graph.json","fetch_events":"https://pith.science/api/pith-number/FXRYDSHLTYM4AWUKXZWGG2DA5Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FXRYDSHLTYM4AWUKXZWGG2DA5Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FXRYDSHLTYM4AWUKXZWGG2DA5Q/action/storage_attestation","attest_author":"https://pith.science/pith/FXRYDSHLTYM4AWUKXZWGG2DA5Q/action/author_attestation","sign_citation":"https://pith.science/pith/FXRYDSHLTYM4AWUKXZWGG2DA5Q/action/citation_signature","submit_replication":"https://pith.science/pith/FXRYDSHLTYM4AWUKXZWGG2DA5Q/action/replication_record"}},"created_at":"2026-05-17T23:40:49.780107+00:00","updated_at":"2026-05-17T23:40:49.780107+00:00"}