{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:VSVL4GCQ3DNOOHJMQD6TXW3FPV","short_pith_number":"pith:VSVL4GCQ","canonical_record":{"source":{"id":"1305.2338","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-05-10T13:35:04Z","cross_cats_sorted":[],"title_canon_sha256":"dd0fefcd05fb3ff58e0a7431976b15b96c98f67e68f97d465dc7e524eb94a31e","abstract_canon_sha256":"094e3f9369cf1a633fe9f01c727fc01d7e92d0405a38e2c04168cd06c1a2ce36"},"schema_version":"1.0"},"canonical_sha256":"acaabe1850d8dae71d2c80fd3bdb657d5bcf37d417c2453e5bb0ed92f5ab3171","source":{"kind":"arxiv","id":"1305.2338","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.2338","created_at":"2026-05-18T03:26:00Z"},{"alias_kind":"arxiv_version","alias_value":"1305.2338v1","created_at":"2026-05-18T03:26:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.2338","created_at":"2026-05-18T03:26:00Z"},{"alias_kind":"pith_short_12","alias_value":"VSVL4GCQ3DNO","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"VSVL4GCQ3DNOOHJM","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"VSVL4GCQ","created_at":"2026-05-18T12:28:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:VSVL4GCQ3DNOOHJMQD6TXW3FPV","target":"record","payload":{"canonical_record":{"source":{"id":"1305.2338","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-05-10T13:35:04Z","cross_cats_sorted":[],"title_canon_sha256":"dd0fefcd05fb3ff58e0a7431976b15b96c98f67e68f97d465dc7e524eb94a31e","abstract_canon_sha256":"094e3f9369cf1a633fe9f01c727fc01d7e92d0405a38e2c04168cd06c1a2ce36"},"schema_version":"1.0"},"canonical_sha256":"acaabe1850d8dae71d2c80fd3bdb657d5bcf37d417c2453e5bb0ed92f5ab3171","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:00.491684Z","signature_b64":"9TeCr8fbUfrhGiTbuvxSHys1UM/Itl/ha+Xp1exubPnOqp7AriroZR/M9QN+qqFBaHiucc5wWH07cNQFszwyAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"acaabe1850d8dae71d2c80fd3bdb657d5bcf37d417c2453e5bb0ed92f5ab3171","last_reissued_at":"2026-05-18T03:26:00.490916Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:00.490916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1305.2338","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":"2026-05-18T03:26:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rra6jGp6kZ3EiFkdiI7S0XWLSt1YAcPSgAdTSx5m5Dr0AoRaekAu7+f6gVhExA3q0KnPEKTWWijA0tKW6LlRBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:07:32.818059Z"},"content_sha256":"268b6eb21fe7ca57f7c64058b104fe0995eaad4d9926bd3b9458b9888037d506","schema_version":"1.0","event_id":"sha256:268b6eb21fe7ca57f7c64058b104fe0995eaad4d9926bd3b9458b9888037d506"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:VSVL4GCQ3DNOOHJMQD6TXW3FPV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the weak Lefschetz Property of graded modules over $K[x,y]$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Giuseppe Favacchio, Phong Dinh Thieu","submitted_at":"2013-05-10T13:35:04Z","abstract_excerpt":"It is known that graded cyclic modules over $S=K[x,y]$ have the Weak Lefschetz Property (WLP). This is not true for non-cyclic modules over $S$. The purpose of this note is to study which conditions on $S$-modules ensure the WLP. We give an algorithm to test the WLP for graded modules with fixed Hilbert function. In particular, we prove that indecomposable graded modules over $S$ with the Hilbert function $(h_0,h_1)$ have the WLP."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2338","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":"2026-05-18T03:26:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QTD5c1NoQ1TWqYMYEjvzO0j7yX27bxAFT2tOl6dvP5UMSrH1KU01PqY6dus4YRQwr9r68zfDchAcMTWFlE0qAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:07:32.818458Z"},"content_sha256":"578b24342362090f6fa4fc94e434b7e46e8a22c9f4c8a8d3a08423cde0da0f62","schema_version":"1.0","event_id":"sha256:578b24342362090f6fa4fc94e434b7e46e8a22c9f4c8a8d3a08423cde0da0f62"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VSVL4GCQ3DNOOHJMQD6TXW3FPV/bundle.json","state_url":"https://pith.science/pith/VSVL4GCQ3DNOOHJMQD6TXW3FPV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VSVL4GCQ3DNOOHJMQD6TXW3FPV/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":"2026-06-02T13:07:32Z","links":{"resolver":"https://pith.science/pith/VSVL4GCQ3DNOOHJMQD6TXW3FPV","bundle":"https://pith.science/pith/VSVL4GCQ3DNOOHJMQD6TXW3FPV/bundle.json","state":"https://pith.science/pith/VSVL4GCQ3DNOOHJMQD6TXW3FPV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VSVL4GCQ3DNOOHJMQD6TXW3FPV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:VSVL4GCQ3DNOOHJMQD6TXW3FPV","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":"094e3f9369cf1a633fe9f01c727fc01d7e92d0405a38e2c04168cd06c1a2ce36","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-05-10T13:35:04Z","title_canon_sha256":"dd0fefcd05fb3ff58e0a7431976b15b96c98f67e68f97d465dc7e524eb94a31e"},"schema_version":"1.0","source":{"id":"1305.2338","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.2338","created_at":"2026-05-18T03:26:00Z"},{"alias_kind":"arxiv_version","alias_value":"1305.2338v1","created_at":"2026-05-18T03:26:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.2338","created_at":"2026-05-18T03:26:00Z"},{"alias_kind":"pith_short_12","alias_value":"VSVL4GCQ3DNO","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"VSVL4GCQ3DNOOHJM","created_at":"2026-05-18T12:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"VSVL4GCQ","created_at":"2026-05-18T12:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:578b24342362090f6fa4fc94e434b7e46e8a22c9f4c8a8d3a08423cde0da0f62","target":"graph","created_at":"2026-05-18T03:26:00Z","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":"It is known that graded cyclic modules over $S=K[x,y]$ have the Weak Lefschetz Property (WLP). This is not true for non-cyclic modules over $S$. The purpose of this note is to study which conditions on $S$-modules ensure the WLP. We give an algorithm to test the WLP for graded modules with fixed Hilbert function. In particular, we prove that indecomposable graded modules over $S$ with the Hilbert function $(h_0,h_1)$ have the WLP.","authors_text":"Giuseppe Favacchio, Phong Dinh Thieu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-05-10T13:35:04Z","title":"On the weak Lefschetz Property of graded modules over $K[x,y]$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2338","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":"sha256:268b6eb21fe7ca57f7c64058b104fe0995eaad4d9926bd3b9458b9888037d506","target":"record","created_at":"2026-05-18T03:26:00Z","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":"094e3f9369cf1a633fe9f01c727fc01d7e92d0405a38e2c04168cd06c1a2ce36","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-05-10T13:35:04Z","title_canon_sha256":"dd0fefcd05fb3ff58e0a7431976b15b96c98f67e68f97d465dc7e524eb94a31e"},"schema_version":"1.0","source":{"id":"1305.2338","kind":"arxiv","version":1}},"canonical_sha256":"acaabe1850d8dae71d2c80fd3bdb657d5bcf37d417c2453e5bb0ed92f5ab3171","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"acaabe1850d8dae71d2c80fd3bdb657d5bcf37d417c2453e5bb0ed92f5ab3171","first_computed_at":"2026-05-18T03:26:00.490916Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:00.490916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9TeCr8fbUfrhGiTbuvxSHys1UM/Itl/ha+Xp1exubPnOqp7AriroZR/M9QN+qqFBaHiucc5wWH07cNQFszwyAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:00.491684Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.2338","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:268b6eb21fe7ca57f7c64058b104fe0995eaad4d9926bd3b9458b9888037d506","sha256:578b24342362090f6fa4fc94e434b7e46e8a22c9f4c8a8d3a08423cde0da0f62"],"state_sha256":"6bf7efd83b2ee70afecb1c793ecd08d2e5113e8369ab3d6e4e19766c94e332cd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zf6H1SI+uHRBNKSqB9qvYCdzpcZnzfUArZKM8hlzH67GEAn7cVKOhKwpHNq724o6msa5vCTHY1TcgQ7uc5guDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T13:07:32.820728Z","bundle_sha256":"aef1066b50a2be22a5c3c7ad429f24c124b86af77da2cb10d57ff86524589fb6"}}