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The Hilbert-Serre Theorem shows that for each finite graded $R$--module $M=\\bigoplus_{n\\in\\BZ}M_n$ the {\\it Hilbert series\\/} $\\sum_{n\\in\\BZ}(\\rank_K M_n)t^n$ is the Laurent expansion around $0$ of a rational function\n  $$ H_M(t)=\\frac{q_M(t)}{\\prod_{i=1}^e(1-t^{d_i})} $$\n with $q_M(t)\\in\\BZ[t,\\ti]$.  We demonstrate that Laurent expansions $\\left[M\\right]_z$ of $H_M(t)$ around other points $z$ of the extended complex plane $\\over"},"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":"math/9409208","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AC","submitted_at":"1994-09-23T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"d99d86ca7add4d70a243712617da74e85f56843c4ef10c95d4a272f96b385f67","abstract_canon_sha256":"4d9256178461a7d6a4278e7ed0f2d67b7620e54e40de00e5f1dee37847b5c1f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.139379Z","signature_b64":"p3co/ByCqNgbggA8NEOV5CoO7DLLY6Weu9qNTi9ojiRnzJlOBp8cWCbI6VCpaAJ9LOxHy5Hw/mpEI9nBlqaXAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"700558df71f2ebf3501159301c2a58ade7002d7462f33276c2d0b6953d71b6f7","last_reissued_at":"2026-05-18T01:05:51.138691Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.138691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Laurent coefficients and Ext of finite graded modules","license":"","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Judith D. Sally, Luchezar L. Avramov, Ragnar-Olaf Buchweitz","submitted_at":"1994-09-23T00:00:00Z","abstract_excerpt":"Let $R=\\bigoplus_{n\\ges0}R_n$ be a graded commutative ring generated over a field $K=R_0$ by homogeneous elements $x_1,\\dots,x_e$ of positive degrees $d_1,\\dots,d_e$. The Hilbert-Serre Theorem shows that for each finite graded $R$--module $M=\\bigoplus_{n\\in\\BZ}M_n$ the {\\it Hilbert series\\/} $\\sum_{n\\in\\BZ}(\\rank_K M_n)t^n$ is the Laurent expansion around $0$ of a rational function\n  $$ H_M(t)=\\frac{q_M(t)}{\\prod_{i=1}^e(1-t^{d_i})} $$\n with $q_M(t)\\in\\BZ[t,\\ti]$.  We demonstrate that Laurent expansions $\\left[M\\right]_z$ of $H_M(t)$ around other points $z$ of the extended complex plane $\\over"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9409208","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":"math/9409208","created_at":"2026-05-18T01:05:51.138791+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/9409208v1","created_at":"2026-05-18T01:05:51.138791+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9409208","created_at":"2026-05-18T01:05:51.138791+00:00"},{"alias_kind":"pith_short_12","alias_value":"OACVRX3R6LV7","created_at":"2026-05-18T12:25:47.102015+00:00"},{"alias_kind":"pith_short_16","alias_value":"OACVRX3R6LV7GUAR","created_at":"2026-05-18T12:25:47.102015+00:00"},{"alias_kind":"pith_short_8","alias_value":"OACVRX3R","created_at":"2026-05-18T12:25:47.102015+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/OACVRX3R6LV7GUARLEYBYKSYVX","json":"https://pith.science/pith/OACVRX3R6LV7GUARLEYBYKSYVX.json","graph_json":"https://pith.science/api/pith-number/OACVRX3R6LV7GUARLEYBYKSYVX/graph.json","events_json":"https://pith.science/api/pith-number/OACVRX3R6LV7GUARLEYBYKSYVX/events.json","paper":"https://pith.science/paper/OACVRX3R"},"agent_actions":{"view_html":"https://pith.science/pith/OACVRX3R6LV7GUARLEYBYKSYVX","download_json":"https://pith.science/pith/OACVRX3R6LV7GUARLEYBYKSYVX.json","view_paper":"https://pith.science/paper/OACVRX3R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/9409208&json=true","fetch_graph":"https://pith.science/api/pith-number/OACVRX3R6LV7GUARLEYBYKSYVX/graph.json","fetch_events":"https://pith.science/api/pith-number/OACVRX3R6LV7GUARLEYBYKSYVX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OACVRX3R6LV7GUARLEYBYKSYVX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OACVRX3R6LV7GUARLEYBYKSYVX/action/storage_attestation","attest_author":"https://pith.science/pith/OACVRX3R6LV7GUARLEYBYKSYVX/action/author_attestation","sign_citation":"https://pith.science/pith/OACVRX3R6LV7GUARLEYBYKSYVX/action/citation_signature","submit_replication":"https://pith.science/pith/OACVRX3R6LV7GUARLEYBYKSYVX/action/replication_record"}},"created_at":"2026-05-18T01:05:51.138791+00:00","updated_at":"2026-05-18T01:05:51.138791+00:00"}