{"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. D. Gillam","submitted_at":"2011-03-28T20:46:45Z","abstract_excerpt":"We consider the general problem of deforming a surjective map of modules $f : E \\to F$ over a coproduct sheaf of rings $B=B_1 \\otimes_A B_2$ when the domain module $E = B_1 \\otimes_A E_2$ is obtained via extension of scalars from a $B_2$-module $E_2$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.5482","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"}