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Assuming $B_1$ is flat over $A$, we show that the Atiyah class morphism $F \\to \\LL_{B/B_2} \\otimes^{\\bL} F[1]$ in the derived category $D(B)$ factors naturally through (the shift of) a morphism $\\beta : \\Ker f \\to \\LL_{B/B_2} \\otimes^{\\bL} F$. We describe the obstruction to lifting $f$ over a (square zero) extension $B_1' \\to B_1$ in terms of $\\b"},"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":"1103.5482","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-03-28T20:46:45Z","cross_cats_sorted":[],"title_canon_sha256":"937e732a27c9281d04bb1cd73595813d9aef0992fc69f3c14d484b32df4a2bf7","abstract_canon_sha256":"4c429c06a143f2b2cbb54f463784ba125129d2061464d55097a3717c0383f265"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:37.485145Z","signature_b64":"ibCqaAIO8Lf6u+EJxyKYiIUEPxnSW9ppYv6Saqv8qy0xou83fW9qVskcDJb+u3IDnQhxCneR8WR55Y5HrJf1DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a9e5b3365689281e3bf42d5621d78267a1759f14768a396235681998034439c","last_reissued_at":"2026-05-18T04:25:37.484704Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:37.484704Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Deformation of quotients on a product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"W. 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