{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:ACXESLC7CSBEVCCBIEA5R24LP5","short_pith_number":"pith:ACXESLC7","schema_version":"1.0","canonical_sha256":"00ae492c5f14824a88414101d8eb8b7f61ab32e428dcf337c8e4757bfa48f5e8","source":{"kind":"arxiv","id":"1412.4266","version":2},"attestation_state":"computed","paper":{"title":"Frobenius Betti numbers and modules of finite projective dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Alessandro De Stefani, Craig Huneke, Luis N\\'u\\~nez-Betancourt","submitted_at":"2014-12-13T18:18:18Z","abstract_excerpt":"Let $(R,\\mathfrak{m},K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $\\beta^F_i(M,R),$ defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\\mathfrak{m},R)=\\beta^F_0(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of $\\beta^F_i(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of "},"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":"1412.4266","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-12-13T18:18:18Z","cross_cats_sorted":[],"title_canon_sha256":"ccef077f6691c01e11ce267a1666f86dbceb6dc06f574f8482ab9d49e75f574c","abstract_canon_sha256":"63596fd5d88bac277626e2988336dee00feb0770a3c7063ec60fabd6d9ad3c4f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:48.932131Z","signature_b64":"TXaSa6Isy9C3DILqqk5IDPhXZnpiGpw5VjSGlCax/LX6URbxDqJCfi2h0uOFQ21qRrv+9wWyGlYBw9SA+vl4Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00ae492c5f14824a88414101d8eb8b7f61ab32e428dcf337c8e4757bfa48f5e8","last_reissued_at":"2026-05-18T01:33:48.931620Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:48.931620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Frobenius Betti numbers and modules of finite projective dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Alessandro De Stefani, Craig Huneke, Luis N\\'u\\~nez-Betancourt","submitted_at":"2014-12-13T18:18:18Z","abstract_excerpt":"Let $(R,\\mathfrak{m},K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $\\beta^F_i(M,R),$ defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\\mathfrak{m},R)=\\beta^F_0(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of $\\beta^F_i(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.4266","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":"1412.4266","created_at":"2026-05-18T01:33:48.931699+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.4266v2","created_at":"2026-05-18T01:33:48.931699+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.4266","created_at":"2026-05-18T01:33:48.931699+00:00"},{"alias_kind":"pith_short_12","alias_value":"ACXESLC7CSBE","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"ACXESLC7CSBEVCCB","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"ACXESLC7","created_at":"2026-05-18T12:28:19.803747+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/ACXESLC7CSBEVCCBIEA5R24LP5","json":"https://pith.science/pith/ACXESLC7CSBEVCCBIEA5R24LP5.json","graph_json":"https://pith.science/api/pith-number/ACXESLC7CSBEVCCBIEA5R24LP5/graph.json","events_json":"https://pith.science/api/pith-number/ACXESLC7CSBEVCCBIEA5R24LP5/events.json","paper":"https://pith.science/paper/ACXESLC7"},"agent_actions":{"view_html":"https://pith.science/pith/ACXESLC7CSBEVCCBIEA5R24LP5","download_json":"https://pith.science/pith/ACXESLC7CSBEVCCBIEA5R24LP5.json","view_paper":"https://pith.science/paper/ACXESLC7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.4266&json=true","fetch_graph":"https://pith.science/api/pith-number/ACXESLC7CSBEVCCBIEA5R24LP5/graph.json","fetch_events":"https://pith.science/api/pith-number/ACXESLC7CSBEVCCBIEA5R24LP5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ACXESLC7CSBEVCCBIEA5R24LP5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ACXESLC7CSBEVCCBIEA5R24LP5/action/storage_attestation","attest_author":"https://pith.science/pith/ACXESLC7CSBEVCCBIEA5R24LP5/action/author_attestation","sign_citation":"https://pith.science/pith/ACXESLC7CSBEVCCBIEA5R24LP5/action/citation_signature","submit_replication":"https://pith.science/pith/ACXESLC7CSBEVCCBIEA5R24LP5/action/replication_record"}},"created_at":"2026-05-18T01:33:48.931699+00:00","updated_at":"2026-05-18T01:33:48.931699+00:00"}