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For each integer i, C^i is a complex of free R-modules. For example, C^{0} is the original \"Eagon-Northcott\" complex with zero-th homology equal to the ring defined by the maximal order minors of Phi; and C^{1} is the \"Buchsbaum-Rim\" complex with zero-th homology equal to the cokernel of the transpose "},"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":"1509.04367","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-09-15T01:08:40Z","cross_cats_sorted":[],"title_canon_sha256":"69c86a06ba2fa93957c99aac5ca5d0f28eaca64f32143f4400a36316b8b9f349","abstract_canon_sha256":"1c213d1772e797556afb30353dbfc001688192bdf5c9f0260fdb1e7249aa6311"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:07.364579Z","signature_b64":"dg81lunI5ZwXtej1aLT8rgbDH8/JrZ3G0oXsgfWy8KE4jqW+VAI+u5MqPy87b37yJx1hOx43IwMEnJLxKytFAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73fc9d56d66bf8dee0af3c26560eebcb85e1c36e6927d73110e01fa8639873d4","last_reissued_at":"2026-05-18T01:33:07.364073Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:07.364073Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Canonical complexes associated to a matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Andrew R. Kustin","submitted_at":"2015-09-15T01:08:40Z","abstract_excerpt":"Let Phi be an f by g matrix with entries from a commutative Noetherian ring R, with g at most f. Recall the family of generalized Eagon-Northcott complexes {C^{i}} associated to Phi. (See, for example, Appendix A2 in \"Commutative Algebra with a view toward Algebraic Geometry\" by David Eisenbud.) For each integer i, C^i is a complex of free R-modules. For example, C^{0} is the original \"Eagon-Northcott\" complex with zero-th homology equal to the ring defined by the maximal order minors of Phi; and C^{1} is the \"Buchsbaum-Rim\" complex with zero-th homology equal to the cokernel of the transpose "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.04367","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.04367","created_at":"2026-05-18T01:33:07.364158+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.04367v1","created_at":"2026-05-18T01:33:07.364158+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.04367","created_at":"2026-05-18T01:33:07.364158+00:00"},{"alias_kind":"pith_short_12","alias_value":"OP6J2VWWNP4N","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"OP6J2VWWNP4N5YFP","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"OP6J2VWW","created_at":"2026-05-18T12:29:34.919912+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/OP6J2VWWNP4N5YFPHQTFMDXLZO","json":"https://pith.science/pith/OP6J2VWWNP4N5YFPHQTFMDXLZO.json","graph_json":"https://pith.science/api/pith-number/OP6J2VWWNP4N5YFPHQTFMDXLZO/graph.json","events_json":"https://pith.science/api/pith-number/OP6J2VWWNP4N5YFPHQTFMDXLZO/events.json","paper":"https://pith.science/paper/OP6J2VWW"},"agent_actions":{"view_html":"https://pith.science/pith/OP6J2VWWNP4N5YFPHQTFMDXLZO","download_json":"https://pith.science/pith/OP6J2VWWNP4N5YFPHQTFMDXLZO.json","view_paper":"https://pith.science/paper/OP6J2VWW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.04367&json=true","fetch_graph":"https://pith.science/api/pith-number/OP6J2VWWNP4N5YFPHQTFMDXLZO/graph.json","fetch_events":"https://pith.science/api/pith-number/OP6J2VWWNP4N5YFPHQTFMDXLZO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OP6J2VWWNP4N5YFPHQTFMDXLZO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OP6J2VWWNP4N5YFPHQTFMDXLZO/action/storage_attestation","attest_author":"https://pith.science/pith/OP6J2VWWNP4N5YFPHQTFMDXLZO/action/author_attestation","sign_citation":"https://pith.science/pith/OP6J2VWWNP4N5YFPHQTFMDXLZO/action/citation_signature","submit_replication":"https://pith.science/pith/OP6J2VWWNP4N5YFPHQTFMDXLZO/action/replication_record"}},"created_at":"2026-05-18T01:33:07.364158+00:00","updated_at":"2026-05-18T01:33:07.364158+00:00"}