{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:2WKMYF4HDVTGMOZD4RHUSBTTRL","short_pith_number":"pith:2WKMYF4H","canonical_record":{"source":{"id":"1109.5062","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-09-23T13:01:30Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"192b4a629a12e0720cee8ca84c4bef0b2de9d2dbeeaf86d712f26583d637c0cc","abstract_canon_sha256":"b06bd088fae7dcea8786e304788cf68feaf03dab89a8d5a03dcfea230ae1ae53"},"schema_version":"1.0"},"canonical_sha256":"d594cc17871d66663b23e44f4906738af27e8745d1abed6f5fa5ccc331c3e798","source":{"kind":"arxiv","id":"1109.5062","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.5062","created_at":"2026-05-18T04:12:21Z"},{"alias_kind":"arxiv_version","alias_value":"1109.5062v1","created_at":"2026-05-18T04:12:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5062","created_at":"2026-05-18T04:12:21Z"},{"alias_kind":"pith_short_12","alias_value":"2WKMYF4HDVTG","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2WKMYF4HDVTGMOZD","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2WKMYF4H","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:2WKMYF4HDVTGMOZD4RHUSBTTRL","target":"record","payload":{"canonical_record":{"source":{"id":"1109.5062","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-09-23T13:01:30Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"192b4a629a12e0720cee8ca84c4bef0b2de9d2dbeeaf86d712f26583d637c0cc","abstract_canon_sha256":"b06bd088fae7dcea8786e304788cf68feaf03dab89a8d5a03dcfea230ae1ae53"},"schema_version":"1.0"},"canonical_sha256":"d594cc17871d66663b23e44f4906738af27e8745d1abed6f5fa5ccc331c3e798","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:12:21.857754Z","signature_b64":"2C9px67urOpaLPMoy3YDHJ567tna/TkiIEKuiLiQWJbtgFjjnNiEuqYK3HX9WvQwtNWyOqRlyQEAkFGKXbLACg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d594cc17871d66663b23e44f4906738af27e8745d1abed6f5fa5ccc331c3e798","last_reissued_at":"2026-05-18T04:12:21.857021Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:12:21.857021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.5062","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:12:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X8wmCj2SRcRCTZaqHXmK/IoAYe2Y/Tc+ORp5upCw/C1q5EDcsaiwYYEpVxzutsaje7iBTmkLTkdhPU9tvuL1Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:37:54.549908Z"},"content_sha256":"1bbc1eaf5ca4482091a276b2191e492cf3a4d13de3f8fe3c4aabe17600a37748","schema_version":"1.0","event_id":"sha256:1bbc1eaf5ca4482091a276b2191e492cf3a4d13de3f8fe3c4aabe17600a37748"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:2WKMYF4HDVTGMOZD4RHUSBTTRL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invertible unital bimodules over rings with local units, and related exact sequences of groups II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.RA","authors_text":"J. G\\'omez-Torrecillas, L. El Kaoutit","submitted_at":"2011-09-23T13:01:30Z","abstract_excerpt":"Let $R$ be a ring with a set of local units, and a homomorphism of groups $\\underline{\\Theta} : \\G \\to \\Picar{R}$ to the Picard group of $R$. We study under which conditions $\\underline{\\Theta}$ is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension $R \\subseteq S$ with the same set of local units and assuming that $\\underline{\\Theta}$ is induced by a homomorphism of groups $\\G \\to \\Inv{R}{S}$ to the group of all invertible $R$-sub-bimodules of $S$, then we construct an analogue of the Chase-Harrison-Rosen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5062","kind":"arxiv","version":1},"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":"2026-05-18T04:12:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8JyWvv/X7A9XvVCRUV2mloZJL3YCsr+h16K5orvAU4vJhAipyVRqLRfwXgkF3srBuEl29W2wLQnpqlRVG+RkAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T03:37:54.550259Z"},"content_sha256":"3a380e8443faec539239d4653600098684a6e1718bfa19b8a764e6b9c40c21af","schema_version":"1.0","event_id":"sha256:3a380e8443faec539239d4653600098684a6e1718bfa19b8a764e6b9c40c21af"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2WKMYF4HDVTGMOZD4RHUSBTTRL/bundle.json","state_url":"https://pith.science/pith/2WKMYF4HDVTGMOZD4RHUSBTTRL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2WKMYF4HDVTGMOZD4RHUSBTTRL/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":"2026-06-26T03:37:54Z","links":{"resolver":"https://pith.science/pith/2WKMYF4HDVTGMOZD4RHUSBTTRL","bundle":"https://pith.science/pith/2WKMYF4HDVTGMOZD4RHUSBTTRL/bundle.json","state":"https://pith.science/pith/2WKMYF4HDVTGMOZD4RHUSBTTRL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2WKMYF4HDVTGMOZD4RHUSBTTRL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:2WKMYF4HDVTGMOZD4RHUSBTTRL","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":"b06bd088fae7dcea8786e304788cf68feaf03dab89a8d5a03dcfea230ae1ae53","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-09-23T13:01:30Z","title_canon_sha256":"192b4a629a12e0720cee8ca84c4bef0b2de9d2dbeeaf86d712f26583d637c0cc"},"schema_version":"1.0","source":{"id":"1109.5062","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.5062","created_at":"2026-05-18T04:12:21Z"},{"alias_kind":"arxiv_version","alias_value":"1109.5062v1","created_at":"2026-05-18T04:12:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.5062","created_at":"2026-05-18T04:12:21Z"},{"alias_kind":"pith_short_12","alias_value":"2WKMYF4HDVTG","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2WKMYF4HDVTGMOZD","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2WKMYF4H","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:3a380e8443faec539239d4653600098684a6e1718bfa19b8a764e6b9c40c21af","target":"graph","created_at":"2026-05-18T04:12:21Z","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 ring with a set of local units, and a homomorphism of groups $\\underline{\\Theta} : \\G \\to \\Picar{R}$ to the Picard group of $R$. We study under which conditions $\\underline{\\Theta}$ is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension $R \\subseteq S$ with the same set of local units and assuming that $\\underline{\\Theta}$ is induced by a homomorphism of groups $\\G \\to \\Inv{R}{S}$ to the group of all invertible $R$-sub-bimodules of $S$, then we construct an analogue of the Chase-Harrison-Rosen","authors_text":"J. G\\'omez-Torrecillas, L. El Kaoutit","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-09-23T13:01:30Z","title":"Invertible unital bimodules over rings with local units, and related exact sequences of groups II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.5062","kind":"arxiv","version":1},"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":"sha256:1bbc1eaf5ca4482091a276b2191e492cf3a4d13de3f8fe3c4aabe17600a37748","target":"record","created_at":"2026-05-18T04:12:21Z","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":"b06bd088fae7dcea8786e304788cf68feaf03dab89a8d5a03dcfea230ae1ae53","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-09-23T13:01:30Z","title_canon_sha256":"192b4a629a12e0720cee8ca84c4bef0b2de9d2dbeeaf86d712f26583d637c0cc"},"schema_version":"1.0","source":{"id":"1109.5062","kind":"arxiv","version":1}},"canonical_sha256":"d594cc17871d66663b23e44f4906738af27e8745d1abed6f5fa5ccc331c3e798","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d594cc17871d66663b23e44f4906738af27e8745d1abed6f5fa5ccc331c3e798","first_computed_at":"2026-05-18T04:12:21.857021Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:12:21.857021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2C9px67urOpaLPMoy3YDHJ567tna/TkiIEKuiLiQWJbtgFjjnNiEuqYK3HX9WvQwtNWyOqRlyQEAkFGKXbLACg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:12:21.857754Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.5062","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1bbc1eaf5ca4482091a276b2191e492cf3a4d13de3f8fe3c4aabe17600a37748","sha256:3a380e8443faec539239d4653600098684a6e1718bfa19b8a764e6b9c40c21af"],"state_sha256":"865bb176ae45c5d7af0a5fa78f81e2e5e2045f505215981a5f658486d01fd017"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JdwyGOSFfxgMA+KYX0AFriZMK1vquKJcjEs09KfBuOS0X1DV9eumj+dx7Wkh82XZ1mVVve98bJmHpwd56hnXBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T03:37:54.552209Z","bundle_sha256":"b04a5fd5380d831a5d230689c2b14554271b745fca19ff94403dcdf81db4ce76"}}