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Then $\\psi$ induces a finite length endomorphism $\\overline{\\psi}\\colon S/f(\\mathfrak{m})S\\rightarrow S/f(\\mathfrak{m})S$. When $f$ is flat, under the assumption that $S$ is Cohen-Macaulay we prove an additivity formula: $h_{\\mathrm{loc}}(\\psi)=h_{\\mathrm{loc}}(\\varphi)+h_{\\mathrm{loc}}(\\overline{\\psi})$"},"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":"1404.1541","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-04-06T02:35:25Z","cross_cats_sorted":[],"title_canon_sha256":"6d6b66223be17c65675f6208f2fba6c8ba92e2c74c7b279887e3536b6d640184","abstract_canon_sha256":"33961f169dbe91eb505d7c41762bdc235e55f572085f566aed00a7f6d0a17e1f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:18.927560Z","signature_b64":"hrH114HZOQtcrTvrvTGnnxIPuTVs4BJ+OJzgnxBXrlErFeCsn+YLCwD39/I1+H6o8a6hIvsD+DiHUBXppXxlCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a188d9ec2e3eeb135e37cbec625a0e0405f51cdd4aa81568403acb64fb03b75d","last_reissued_at":"2026-05-18T02:43:18.927069Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:18.927069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On additivity of local entropy under flat extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Mahdi Majidi-Zolbanin","submitted_at":"2014-04-06T02:35:25Z","abstract_excerpt":"Let $f\\colon(R,\\mathfrak{m})\\rightarrow S$ be a local homomorphism of Noetherian local rings. Consider two endomorphisms \\textit{of finite length} (i.e., with zero-dimensional closed fibers) $\\varphi\\colon R\\rightarrow R$ and $\\psi\\colon S\\rightarrow S$, satisfying $\\psi\\circ f=f\\circ\\varphi$. Then $\\psi$ induces a finite length endomorphism $\\overline{\\psi}\\colon S/f(\\mathfrak{m})S\\rightarrow S/f(\\mathfrak{m})S$. When $f$ is flat, under the assumption that $S$ is Cohen-Macaulay we prove an additivity formula: $h_{\\mathrm{loc}}(\\psi)=h_{\\mathrm{loc}}(\\varphi)+h_{\\mathrm{loc}}(\\overline{\\psi})$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1404.1541","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":"1404.1541","created_at":"2026-05-18T02:43:18.927151+00:00"},{"alias_kind":"arxiv_version","alias_value":"1404.1541v2","created_at":"2026-05-18T02:43:18.927151+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1404.1541","created_at":"2026-05-18T02:43:18.927151+00:00"},{"alias_kind":"pith_short_12","alias_value":"UGENT3BOH3VR","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_16","alias_value":"UGENT3BOH3VRGXRX","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_8","alias_value":"UGENT3BO","created_at":"2026-05-18T12:28:52.271510+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/UGENT3BOH3VRGXRXZPWGEWQOAQ","json":"https://pith.science/pith/UGENT3BOH3VRGXRXZPWGEWQOAQ.json","graph_json":"https://pith.science/api/pith-number/UGENT3BOH3VRGXRXZPWGEWQOAQ/graph.json","events_json":"https://pith.science/api/pith-number/UGENT3BOH3VRGXRXZPWGEWQOAQ/events.json","paper":"https://pith.science/paper/UGENT3BO"},"agent_actions":{"view_html":"https://pith.science/pith/UGENT3BOH3VRGXRXZPWGEWQOAQ","download_json":"https://pith.science/pith/UGENT3BOH3VRGXRXZPWGEWQOAQ.json","view_paper":"https://pith.science/paper/UGENT3BO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1404.1541&json=true","fetch_graph":"https://pith.science/api/pith-number/UGENT3BOH3VRGXRXZPWGEWQOAQ/graph.json","fetch_events":"https://pith.science/api/pith-number/UGENT3BOH3VRGXRXZPWGEWQOAQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UGENT3BOH3VRGXRXZPWGEWQOAQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UGENT3BOH3VRGXRXZPWGEWQOAQ/action/storage_attestation","attest_author":"https://pith.science/pith/UGENT3BOH3VRGXRXZPWGEWQOAQ/action/author_attestation","sign_citation":"https://pith.science/pith/UGENT3BOH3VRGXRXZPWGEWQOAQ/action/citation_signature","submit_replication":"https://pith.science/pith/UGENT3BOH3VRGXRXZPWGEWQOAQ/action/replication_record"}},"created_at":"2026-05-18T02:43:18.927151+00:00","updated_at":"2026-05-18T02:43:18.927151+00:00"}