{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:QEJF24NTKGKSI3A3CQ7OOLTFE6","short_pith_number":"pith:QEJF24NT","schema_version":"1.0","canonical_sha256":"81125d71b35195246c1b143ee72e6527a30d3449efa959f64aeb347bc76dc17f","source":{"kind":"arxiv","id":"1810.04999","version":1},"attestation_state":"computed","paper":{"title":"Tor as a Module over an Exterior Algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"David Eisenbud, Frank-Olaf Schreyer, Irena Peeva","submitted_at":"2018-10-11T13:14:26Z","abstract_excerpt":"Let $S$ be a regular local ring with residue field $k$ and let $M$ be a finitely generated $S$-module. Suppose that $f_1,\\dots ,f_c\\in S$ is a regular sequence that annihilates $M$, and let $E$ be an exterior algebra over $k$ generated by $c$ elements.\n  The homotopies for the $f_{i}$ on a free resolution of $M$ induce a natural structure of graded $E$-module on ${\\rm Tor}^{S}(M,k)$. In the case where $M$ is a high syzygy over the complete intersectionR:=S/(f_{1},\\dots,f_{c})$ we describe this $E$-module structure in detail, including its minimal free resolution over $E$.\n  Turning to ${\\rm Ex"},"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":"1810.04999","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-10-11T13:14:26Z","cross_cats_sorted":[],"title_canon_sha256":"1b73e1cd56e05275a2feb7e531f1dbbe5ba36e2c56064aed7185cbaab2f85904","abstract_canon_sha256":"06c8d49acd3e465c01c6614c0a4a8f4eb064c2118b6c6468fd9ad2fa213ba930"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:35.298006Z","signature_b64":"Ib+Nek0Yx/yks/0Hql3T6HHTujkiXi61ZeTpPa+gW0EGxpxZu9+bOuuM0dvokSw+fX3ZStGcgATjXZcwV2TBAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81125d71b35195246c1b143ee72e6527a30d3449efa959f64aeb347bc76dc17f","last_reissued_at":"2026-05-18T00:03:35.297424Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:35.297424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tor as a Module over an Exterior Algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"David Eisenbud, Frank-Olaf Schreyer, Irena Peeva","submitted_at":"2018-10-11T13:14:26Z","abstract_excerpt":"Let $S$ be a regular local ring with residue field $k$ and let $M$ be a finitely generated $S$-module. Suppose that $f_1,\\dots ,f_c\\in S$ is a regular sequence that annihilates $M$, and let $E$ be an exterior algebra over $k$ generated by $c$ elements.\n  The homotopies for the $f_{i}$ on a free resolution of $M$ induce a natural structure of graded $E$-module on ${\\rm Tor}^{S}(M,k)$. In the case where $M$ is a high syzygy over the complete intersectionR:=S/(f_{1},\\dots,f_{c})$ we describe this $E$-module structure in detail, including its minimal free resolution over $E$.\n  Turning to ${\\rm Ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.04999","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":"1810.04999","created_at":"2026-05-18T00:03:35.297512+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.04999v1","created_at":"2026-05-18T00:03:35.297512+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.04999","created_at":"2026-05-18T00:03:35.297512+00:00"},{"alias_kind":"pith_short_12","alias_value":"QEJF24NTKGKS","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_16","alias_value":"QEJF24NTKGKSI3A3","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_8","alias_value":"QEJF24NT","created_at":"2026-05-18T12:32:46.962924+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/QEJF24NTKGKSI3A3CQ7OOLTFE6","json":"https://pith.science/pith/QEJF24NTKGKSI3A3CQ7OOLTFE6.json","graph_json":"https://pith.science/api/pith-number/QEJF24NTKGKSI3A3CQ7OOLTFE6/graph.json","events_json":"https://pith.science/api/pith-number/QEJF24NTKGKSI3A3CQ7OOLTFE6/events.json","paper":"https://pith.science/paper/QEJF24NT"},"agent_actions":{"view_html":"https://pith.science/pith/QEJF24NTKGKSI3A3CQ7OOLTFE6","download_json":"https://pith.science/pith/QEJF24NTKGKSI3A3CQ7OOLTFE6.json","view_paper":"https://pith.science/paper/QEJF24NT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.04999&json=true","fetch_graph":"https://pith.science/api/pith-number/QEJF24NTKGKSI3A3CQ7OOLTFE6/graph.json","fetch_events":"https://pith.science/api/pith-number/QEJF24NTKGKSI3A3CQ7OOLTFE6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QEJF24NTKGKSI3A3CQ7OOLTFE6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QEJF24NTKGKSI3A3CQ7OOLTFE6/action/storage_attestation","attest_author":"https://pith.science/pith/QEJF24NTKGKSI3A3CQ7OOLTFE6/action/author_attestation","sign_citation":"https://pith.science/pith/QEJF24NTKGKSI3A3CQ7OOLTFE6/action/citation_signature","submit_replication":"https://pith.science/pith/QEJF24NTKGKSI3A3CQ7OOLTFE6/action/replication_record"}},"created_at":"2026-05-18T00:03:35.297512+00:00","updated_at":"2026-05-18T00:03:35.297512+00:00"}