{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:2W74SXC2RWRNK7AZ52PUDPK3DG","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":"321c22c9d7ee1d0c0a3517e4edbdad4994880f170e9c7a7206aff62d4695947c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2014-08-15T16:10:23Z","title_canon_sha256":"afae585d47a9909292331b214b7f932a3b60e303600e4dfd67b45a96d46bef59"},"schema_version":"1.0","source":{"id":"1408.3580","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.3580","created_at":"2026-05-18T02:29:10Z"},{"alias_kind":"arxiv_version","alias_value":"1408.3580v2","created_at":"2026-05-18T02:29:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.3580","created_at":"2026-05-18T02:29:10Z"},{"alias_kind":"pith_short_12","alias_value":"2W74SXC2RWRN","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_16","alias_value":"2W74SXC2RWRNK7AZ","created_at":"2026-05-18T12:28:11Z"},{"alias_kind":"pith_short_8","alias_value":"2W74SXC2","created_at":"2026-05-18T12:28:11Z"}],"graph_snapshots":[{"event_id":"sha256:614d82d5aefff45d1e256e299eeae7414b68b75d0e94bea898f917f06bbcf0c3","target":"graph","created_at":"2026-05-18T02:29:10Z","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 $E$ be a directed graph, $K$ any field, and let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$. For each rational infinite path $c^\\infty$ of $E$ we explicitly construct a projective resolution of the corresponding Chen simple left $L_K(E)$-module $V_{[c^\\infty]}$. Further, when $E$ is row-finite, for each irrational infinite path $p$ of $E$ we explicitly construct a projective resolution of the corresponding Chen simple left $L_K(E)$-module $V_{[p]}$. For Chen simple modules $S,T$ we describe ${\\rm Ext}_{L_K(E)}^1(S,T)$ by presenting an explicit $K$-basis. For an","authors_text":"Alberto Tonolo, Francesca Mantese, Gene Abrams","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2014-08-15T16:10:23Z","title":"Extensions of simple modules over Leavitt path algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3580","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:ed02df57ac27c966ad81cde2ac5dd053c5e5a23ae47c80a5bda2444dc52975e8","target":"record","created_at":"2026-05-18T02:29:10Z","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":"321c22c9d7ee1d0c0a3517e4edbdad4994880f170e9c7a7206aff62d4695947c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2014-08-15T16:10:23Z","title_canon_sha256":"afae585d47a9909292331b214b7f932a3b60e303600e4dfd67b45a96d46bef59"},"schema_version":"1.0","source":{"id":"1408.3580","kind":"arxiv","version":2}},"canonical_sha256":"d5bfc95c5a8da2d57c19ee9f41bd5b19b60fe2d1f094eb1c450c89977fe2f857","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d5bfc95c5a8da2d57c19ee9f41bd5b19b60fe2d1f094eb1c450c89977fe2f857","first_computed_at":"2026-05-18T02:29:10.741452Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:29:10.741452Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JUCfmxkik4XownSxQmnx/XJkC6mqu6kYRsG0SlteIDUU1DoZfnBA+dHUBBYwxWKsU61OVJpqpG0rrzyjm+geCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:29:10.742032Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.3580","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed02df57ac27c966ad81cde2ac5dd053c5e5a23ae47c80a5bda2444dc52975e8","sha256:614d82d5aefff45d1e256e299eeae7414b68b75d0e94bea898f917f06bbcf0c3"],"state_sha256":"a670085a9e9480bbd900a33f67e7615ec855f2434fef22b6a3dc1fa77d4fa84e"}