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We prove algebraic variants and generalizations of this result in \\'etale cohomology over fields of any characteristic, where the tensor product is replaced by a certa"},"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":"1604.07004","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-04-24T09:32:12Z","cross_cats_sorted":[],"title_canon_sha256":"214ad84ea9cedf0ad2d2098b2e6dea7d5b2b7aca20b706c5ef8de12c7e9a082e","abstract_canon_sha256":"ef1a173fff296395becfec04c6ae8fad55952496aa3111f47480a8fd6126a434"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:23.342582Z","signature_b64":"oWAe1f50T+j9kmYwF24m80C2bKdl1/as73Sfbyx+wVDYbWW8Sl3oxaIPTa/S6BLpaaE3FEnbmhDCb7CbhbpHBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7abecd1d6c78154a90d8f1aa349db18734a52b75eaa7baf95a9e7ba59730716","last_reissued_at":"2026-05-18T01:16:23.341727Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:23.341727Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Around the Thom-Sebastiani theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Luc Illusie","submitted_at":"2016-04-24T09:32:12Z","abstract_excerpt":"For germs of holomorphic functions $f : (\\mathbf{C}^{m+1},0) \\to (\\mathbf{C},0)$, $g : (\\mathbf{C}^{n+1},0) \\to (\\mathbf{C},0)$ having an isolated critical point at 0 with value 0, the classical Thom-Sebastiani theorem describes the vanishing cycles group $\\Phi^{m+n+1}(f \\oplus g)$ (and its monodromy) as a tensor product $\\Phi^m(f) \\otimes \\Phi^n(g)$, where $(f \\oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n)$. We prove algebraic variants and generalizations of this result in \\'etale cohomology over fields of any characteristic, where the tensor product is replaced by a certa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07004","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":"1604.07004","created_at":"2026-05-18T01:16:23.341863+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.07004v1","created_at":"2026-05-18T01:16:23.341863+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.07004","created_at":"2026-05-18T01:16:23.341863+00:00"},{"alias_kind":"pith_short_12","alias_value":"26V6ZUOWY6AV","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"26V6ZUOWY6AVJKIN","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"26V6ZUOW","created_at":"2026-05-18T12:29:52.810259+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/26V6ZUOWY6AVJKINR4NKGSO3DB","json":"https://pith.science/pith/26V6ZUOWY6AVJKINR4NKGSO3DB.json","graph_json":"https://pith.science/api/pith-number/26V6ZUOWY6AVJKINR4NKGSO3DB/graph.json","events_json":"https://pith.science/api/pith-number/26V6ZUOWY6AVJKINR4NKGSO3DB/events.json","paper":"https://pith.science/paper/26V6ZUOW"},"agent_actions":{"view_html":"https://pith.science/pith/26V6ZUOWY6AVJKINR4NKGSO3DB","download_json":"https://pith.science/pith/26V6ZUOWY6AVJKINR4NKGSO3DB.json","view_paper":"https://pith.science/paper/26V6ZUOW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.07004&json=true","fetch_graph":"https://pith.science/api/pith-number/26V6ZUOWY6AVJKINR4NKGSO3DB/graph.json","fetch_events":"https://pith.science/api/pith-number/26V6ZUOWY6AVJKINR4NKGSO3DB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/26V6ZUOWY6AVJKINR4NKGSO3DB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/26V6ZUOWY6AVJKINR4NKGSO3DB/action/storage_attestation","attest_author":"https://pith.science/pith/26V6ZUOWY6AVJKINR4NKGSO3DB/action/author_attestation","sign_citation":"https://pith.science/pith/26V6ZUOWY6AVJKINR4NKGSO3DB/action/citation_signature","submit_replication":"https://pith.science/pith/26V6ZUOWY6AVJKINR4NKGSO3DB/action/replication_record"}},"created_at":"2026-05-18T01:16:23.341863+00:00","updated_at":"2026-05-18T01:16:23.341863+00:00"}