{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:XR6WYH7J5PLOORST3WQCLS5KXS","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":"1fe386a9df050237d197196ad3e7ebfc96996e3778564a45d322f52c8057c079","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-06-30T16:03:15Z","title_canon_sha256":"351ca8c96e4b7a39839654a939d280fe82ec279c5ee87d4ffabb64c69439972c"},"schema_version":"1.0","source":{"id":"1406.7791","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.7791","created_at":"2026-05-18T02:18:40Z"},{"alias_kind":"arxiv_version","alias_value":"1406.7791v2","created_at":"2026-05-18T02:18:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.7791","created_at":"2026-05-18T02:18:40Z"},{"alias_kind":"pith_short_12","alias_value":"XR6WYH7J5PLO","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"XR6WYH7J5PLOORST","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"XR6WYH7J","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:5360acab2d28be5faec0d420a4b1753b3f0a022df0af697c76479561810a2929","target":"graph","created_at":"2026-05-18T02:18:40Z","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 R be a commutative ring and S be an R-algebra. It is well-known that if N is an injective R-module, then Hom(S,N) is an injective S-module. The converse is not true, not even if R is a commutative noetherian local ring and S is its completion, but it is close: It is a special case of our main theorem that in this setting, an R-module N with Ext^i(S,N)=0 for all i>0 is injective if Hom(S,N) is an injective S-module.","authors_text":"Fatih Koksal, Lars Winther Christensen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-06-30T16:03:15Z","title":"Injective Modules under Faithfully Flat Ring Extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.7791","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":"sha256:5c4ba9b4455ae34b9e33deb54613cda57d966b5fbd23f4891a19d6c2d2618af8","target":"record","created_at":"2026-05-18T02:18:40Z","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":"1fe386a9df050237d197196ad3e7ebfc96996e3778564a45d322f52c8057c079","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-06-30T16:03:15Z","title_canon_sha256":"351ca8c96e4b7a39839654a939d280fe82ec279c5ee87d4ffabb64c69439972c"},"schema_version":"1.0","source":{"id":"1406.7791","kind":"arxiv","version":2}},"canonical_sha256":"bc7d6c1fe9ebd6e74653dda025cbaabcad605565fc86266ec01fb7eac95027c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc7d6c1fe9ebd6e74653dda025cbaabcad605565fc86266ec01fb7eac95027c0","first_computed_at":"2026-05-18T02:18:40.922180Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:40.922180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0/CluN9H1KAPLkT5BfAOJTX7stE8RZpLRxYEIZRyDn9PFTYQoJK1bzKRYU6rtnqXz9xOo7gjUy/ddbVCTORrBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:40.922733Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.7791","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c4ba9b4455ae34b9e33deb54613cda57d966b5fbd23f4891a19d6c2d2618af8","sha256:5360acab2d28be5faec0d420a4b1753b3f0a022df0af697c76479561810a2929"],"state_sha256":"09c6565a8c3ab0ada56449c8659c0e6b5cc37dfa3655d8c28d3775f529d09a02"}