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For a bounded complex of finitely generated graded $R$-modules $L$, we prove the equality $\\text{reg}~ L=\\max_{i\\in \\mathbb Z} \\{\\text{reg}~ H_i(L)-i\\}$ given the condition $\\text{depth}~ H_i(L)\\ge \\dim H_{i+1}(L)-1$ for all $i<\\sup L$. As applications, we recover previous bounds on regularity of Tor due to Caviglia, Eisenbud-Huneke-Ulrich, among others. We also obtain strengthened res"},"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":"1311.2572","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-11-11T20:37:47Z","cross_cats_sorted":[],"title_canon_sha256":"81d8735391ee6a3d70b0cc8c17bcda7273a3718c81b4bee6710c1bf12f826b57","abstract_canon_sha256":"f21d94160152e68af580c4ef6796c8f48355d1e419b22a1a6f682334de8b2561"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:19.154116Z","signature_b64":"ArPxoiX1qRCwwc4WFd/dVlZAKjDahWFUzXH4Hmbh0ugYC+22zxM30h+cVQ6pbEvsDEZx/EsQm135EKWTvQjnBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11391910d6a1598985ec05af4ddbea8f06cd9d5e430974a7fd318ee482bfb3d2","last_reissued_at":"2026-05-18T01:32:19.153510Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:19.153510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity bounds for complexes and their homology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Hop D. Nguyen","submitted_at":"2013-11-11T20:37:47Z","abstract_excerpt":"Let $R$ be a standard graded algebra over a field $k$. We prove an Auslander-Buchsbaum formula for the absolute Castelnuovo-Mumford regularity, extending important cases of previous works of Chardin and R\\\"omer. For a bounded complex of finitely generated graded $R$-modules $L$, we prove the equality $\\text{reg}~ L=\\max_{i\\in \\mathbb Z} \\{\\text{reg}~ H_i(L)-i\\}$ given the condition $\\text{depth}~ H_i(L)\\ge \\dim H_{i+1}(L)-1$ for all $i<\\sup L$. As applications, we recover previous bounds on regularity of Tor due to Caviglia, Eisenbud-Huneke-Ulrich, among others. 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