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Then C is R-finitely dominated, ie, homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules, if and only if the two chain complexes C((x)) and C((1/x)) are acyclic, as has been proved by Ranicki. Here C((x)) is the tensor product over L of C with the Novikov ring R((x)) = R[[x]][1/x] (also known as the ring of formal Laurent series in x); similarly, C((1/x)) is the tensor product over L of C with the Novikov ring R((1/x)) = R[[1/x]][x]."},"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":"1108.2995","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-08-15T13:50:55Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"78d8d6d14aa59e87b8445093bea57f05afc3c078bf58fb0cc8a8eee346eaebf6","abstract_canon_sha256":"9d28d03c2351e7b61b5a100d8e73138e0d7a0747f0a95195a35b5f9f23bf853c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:16.416348Z","signature_b64":"XhmKTVO7IUMjZ6eXl/e+kF8jRP75VgIIonyNK/YZDpZhnaZst9pawpd8FH/piL0JX0Ai0sW5H7vftUJ472nTCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7c8c2b53e8b7a87edf760501224f872f0e4e1b359739b0a0519a123047987037","last_reissued_at":"2026-05-18T03:40:16.415794Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:16.415794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Finite domination and Novikov rings. 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Here C((x)) is the tensor product over L of C with the Novikov ring R((x)) = R[[x]][1/x] (also known as the ring of formal Laurent series in x); similarly, C((1/x)) is the tensor product over L of C with the Novikov ring R((1/x)) = R[[1/x]][x]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.2995","kind":"arxiv","version":4},"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":"1108.2995","created_at":"2026-05-18T03:40:16.415883+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.2995v4","created_at":"2026-05-18T03:40:16.415883+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.2995","created_at":"2026-05-18T03:40:16.415883+00:00"},{"alias_kind":"pith_short_12","alias_value":"PSGCWU7IW6UH","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_16","alias_value":"PSGCWU7IW6UH5X3W","created_at":"2026-05-18T12:26:39.201973+00:00"},{"alias_kind":"pith_short_8","alias_value":"PSGCWU7I","created_at":"2026-05-18T12:26:39.201973+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/PSGCWU7IW6UH5X3WAUASET4HF4","json":"https://pith.science/pith/PSGCWU7IW6UH5X3WAUASET4HF4.json","graph_json":"https://pith.science/api/pith-number/PSGCWU7IW6UH5X3WAUASET4HF4/graph.json","events_json":"https://pith.science/api/pith-number/PSGCWU7IW6UH5X3WAUASET4HF4/events.json","paper":"https://pith.science/paper/PSGCWU7I"},"agent_actions":{"view_html":"https://pith.science/pith/PSGCWU7IW6UH5X3WAUASET4HF4","download_json":"https://pith.science/pith/PSGCWU7IW6UH5X3WAUASET4HF4.json","view_paper":"https://pith.science/paper/PSGCWU7I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.2995&json=true","fetch_graph":"https://pith.science/api/pith-number/PSGCWU7IW6UH5X3WAUASET4HF4/graph.json","fetch_events":"https://pith.science/api/pith-number/PSGCWU7IW6UH5X3WAUASET4HF4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PSGCWU7IW6UH5X3WAUASET4HF4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PSGCWU7IW6UH5X3WAUASET4HF4/action/storage_attestation","attest_author":"https://pith.science/pith/PSGCWU7IW6UH5X3WAUASET4HF4/action/author_attestation","sign_citation":"https://pith.science/pith/PSGCWU7IW6UH5X3WAUASET4HF4/action/citation_signature","submit_replication":"https://pith.science/pith/PSGCWU7IW6UH5X3WAUASET4HF4/action/replication_record"}},"created_at":"2026-05-18T03:40:16.415883+00:00","updated_at":"2026-05-18T03:40:16.415883+00:00"}