{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:QDXW4LJKTALR7MOSKTERVTGFNR","short_pith_number":"pith:QDXW4LJK","schema_version":"1.0","canonical_sha256":"80ef6e2d2a98171fb1d254c91accc56c5eca119a55a2fa02bc538ac39121b25d","source":{"kind":"arxiv","id":"1002.3652","version":2},"attestation_state":"computed","paper":{"title":"Detecting flatness over smooth bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Luchezar L. Avramov, Srikanth B. Iyengar","submitted_at":"2010-02-19T01:22:09Z","abstract_excerpt":"Given an essentially finite type morphism of schemes f: X --> Y and a positive integer d, let f^{d}: X^{d} --> Y denote the natural map from the d-fold fiber product, X^{d}, of X over Y and \\pi_i: X^{d} --> X the i'th canonical projection. When Y smooth over a field and F is a coherent sheaf on X, it is proved that F is flat over Y if (and only if) f^{d} maps the associated points of the tensor product sheaf \\otimes_{i=1}^d \\pi_i^*(F) to generic points of Y, for some d greater than or equal to dim Y. The equivalent statement in commutative algebra is an analog---but not a consequence---of a cl"},"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":"1002.3652","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-02-19T01:22:09Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"4a1386fca85a64f7d605a149ffd82d25790c178586ef263e5fb5f1c388d8bbfa","abstract_canon_sha256":"1b4b0fd67ba9e0561ef9b65cc90a6b0756d43ef3dd649b519afdfe0560aff62d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:13.125616Z","signature_b64":"4iODdXsrjASfVlACfVMfc+ub0eNrsZbetRq8UtsPtN/s4cYFGgbEX3kf71ZwIJM6X718aChX/wAEDLmt2DR1Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80ef6e2d2a98171fb1d254c91accc56c5eca119a55a2fa02bc538ac39121b25d","last_reissued_at":"2026-05-18T04:31:13.125003Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:13.125003Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Detecting flatness over smooth bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Luchezar L. Avramov, Srikanth B. Iyengar","submitted_at":"2010-02-19T01:22:09Z","abstract_excerpt":"Given an essentially finite type morphism of schemes f: X --> Y and a positive integer d, let f^{d}: X^{d} --> Y denote the natural map from the d-fold fiber product, X^{d}, of X over Y and \\pi_i: X^{d} --> X the i'th canonical projection. When Y smooth over a field and F is a coherent sheaf on X, it is proved that F is flat over Y if (and only if) f^{d} maps the associated points of the tensor product sheaf \\otimes_{i=1}^d \\pi_i^*(F) to generic points of Y, for some d greater than or equal to dim Y. The equivalent statement in commutative algebra is an analog---but not a consequence---of a cl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.3652","kind":"arxiv","version":2},"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":"1002.3652","created_at":"2026-05-18T04:31:13.125101+00:00"},{"alias_kind":"arxiv_version","alias_value":"1002.3652v2","created_at":"2026-05-18T04:31:13.125101+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.3652","created_at":"2026-05-18T04:31:13.125101+00:00"},{"alias_kind":"pith_short_12","alias_value":"QDXW4LJKTALR","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_16","alias_value":"QDXW4LJKTALR7MOS","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_8","alias_value":"QDXW4LJK","created_at":"2026-05-18T12:26:12.377268+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/QDXW4LJKTALR7MOSKTERVTGFNR","json":"https://pith.science/pith/QDXW4LJKTALR7MOSKTERVTGFNR.json","graph_json":"https://pith.science/api/pith-number/QDXW4LJKTALR7MOSKTERVTGFNR/graph.json","events_json":"https://pith.science/api/pith-number/QDXW4LJKTALR7MOSKTERVTGFNR/events.json","paper":"https://pith.science/paper/QDXW4LJK"},"agent_actions":{"view_html":"https://pith.science/pith/QDXW4LJKTALR7MOSKTERVTGFNR","download_json":"https://pith.science/pith/QDXW4LJKTALR7MOSKTERVTGFNR.json","view_paper":"https://pith.science/paper/QDXW4LJK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1002.3652&json=true","fetch_graph":"https://pith.science/api/pith-number/QDXW4LJKTALR7MOSKTERVTGFNR/graph.json","fetch_events":"https://pith.science/api/pith-number/QDXW4LJKTALR7MOSKTERVTGFNR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QDXW4LJKTALR7MOSKTERVTGFNR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QDXW4LJKTALR7MOSKTERVTGFNR/action/storage_attestation","attest_author":"https://pith.science/pith/QDXW4LJKTALR7MOSKTERVTGFNR/action/author_attestation","sign_citation":"https://pith.science/pith/QDXW4LJKTALR7MOSKTERVTGFNR/action/citation_signature","submit_replication":"https://pith.science/pith/QDXW4LJKTALR7MOSKTERVTGFNR/action/replication_record"}},"created_at":"2026-05-18T04:31:13.125101+00:00","updated_at":"2026-05-18T04:31:13.125101+00:00"}