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We show that if $r \\geq 1$ then the Koszul homology modules\n  $H_i(Y_1,\\ldots, Y_r; \\mathcal{M})$ are $F_{\\overline{R}}$-finite modules where $\\overline{R} = R/(Y_1,\\ldots, Y_r)$ for $i = 0, \\ldots, r$. As an application if $A$ is a regular ring containing a field of characteristic $p > 0$ and $S = A[X_1,\\ldo"},"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":"2307.04473","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2023-07-10T10:53:15Z","cross_cats_sorted":[],"title_canon_sha256":"547944e1ed913c92c82f2893d2e7a586d5b9833f527017dd634dc2146bf8512c","abstract_canon_sha256":"c52029cbefcf4c53b95667b73e254a71c79f594b0319b68d837be2b2febaeafe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:29:13.975928Z","signature_b64":"ULBg1vDyJQZrBqLuw9DP/AfQPe95ne5Lfk9gAgcqzFTgPn5NsbrCe/LsaOOj5I/2y+U2NpCFjh2r9hULHRPuCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f98de6cc77f2c7a19b01122ded638245202e3ce211952d5f2bb866b790b54ce0","last_reissued_at":"2026-07-05T06:29:13.975512Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:29:13.975512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Koszul homology of $F$-finite module and applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Tony J. Puthenpurakal","submitted_at":"2023-07-10T10:53:15Z","abstract_excerpt":"Let $k$ be an infinite field of characteristic $p > 0$ and let $R = k[Y_1,\\ldots, Y_d]$ (or $R = k[[Y_1,\\ldots, Y_d]]$). Let $F \\colon \\text{Mod}(R) \\rightarrow \\text{Mod}(R)$ be the Frobenius functor and let $\\mathcal{M}$ be a $F_R$-finite module (in the sense of Lyubeznik \\cite{Lyu-2}). We show that if $r \\geq 1$ then the Koszul homology modules\n  $H_i(Y_1,\\ldots, Y_r; \\mathcal{M})$ are $F_{\\overline{R}}$-finite modules where $\\overline{R} = R/(Y_1,\\ldots, Y_r)$ for $i = 0, \\ldots, r$. As an application if $A$ is a regular ring containing a field of characteristic $p > 0$ and $S = A[X_1,\\ldo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.04473","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.04473/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2307.04473","created_at":"2026-07-05T06:29:13.975572+00:00"},{"alias_kind":"arxiv_version","alias_value":"2307.04473v1","created_at":"2026-07-05T06:29:13.975572+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.04473","created_at":"2026-07-05T06:29:13.975572+00:00"},{"alias_kind":"pith_short_12","alias_value":"7GG6NTDX6LD2","created_at":"2026-07-05T06:29:13.975572+00:00"},{"alias_kind":"pith_short_16","alias_value":"7GG6NTDX6LD2DGYB","created_at":"2026-07-05T06:29:13.975572+00:00"},{"alias_kind":"pith_short_8","alias_value":"7GG6NTDX","created_at":"2026-07-05T06:29:13.975572+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.08533","citing_title":"On the structure theorem of graded components of $\\mathcal{F}$-finite, $\\mathcal{F}$-modules over certain polynomial ring","ref_index":14,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7GG6NTDX6LD2DGYBCIW62Y4CIU","json":"https://pith.science/pith/7GG6NTDX6LD2DGYBCIW62Y4CIU.json","graph_json":"https://pith.science/api/pith-number/7GG6NTDX6LD2DGYBCIW62Y4CIU/graph.json","events_json":"https://pith.science/api/pith-number/7GG6NTDX6LD2DGYBCIW62Y4CIU/events.json","paper":"https://pith.science/paper/7GG6NTDX"},"agent_actions":{"view_html":"https://pith.science/pith/7GG6NTDX6LD2DGYBCIW62Y4CIU","download_json":"https://pith.science/pith/7GG6NTDX6LD2DGYBCIW62Y4CIU.json","view_paper":"https://pith.science/paper/7GG6NTDX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2307.04473&json=true","fetch_graph":"https://pith.science/api/pith-number/7GG6NTDX6LD2DGYBCIW62Y4CIU/graph.json","fetch_events":"https://pith.science/api/pith-number/7GG6NTDX6LD2DGYBCIW62Y4CIU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7GG6NTDX6LD2DGYBCIW62Y4CIU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7GG6NTDX6LD2DGYBCIW62Y4CIU/action/storage_attestation","attest_author":"https://pith.science/pith/7GG6NTDX6LD2DGYBCIW62Y4CIU/action/author_attestation","sign_citation":"https://pith.science/pith/7GG6NTDX6LD2DGYBCIW62Y4CIU/action/citation_signature","submit_replication":"https://pith.science/pith/7GG6NTDX6LD2DGYBCIW62Y4CIU/action/replication_record"}},"created_at":"2026-07-05T06:29:13.975572+00:00","updated_at":"2026-07-05T06:29:13.975572+00:00"}