{"paper":{"title":"Hochschild cohomology commutes with adic completion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.KT"],"primary_cat":"math.AC","authors_text":"Liran Shaul","submitted_at":"2015-05-15T19:28:16Z","abstract_excerpt":"For a flat commutative $k$-algebra $A$ such that the enveloping algebra $A\\otimes_k A$ is noetherian, given a finitely generated bimodule $M$, we show that the adic completion of the Hochschild cohomology module $HH^n(A/k,M)$ is naturally isomorphic to $HH^n(\\widehat{A}/k,\\widehat{M})$. To show this, we (1) make a detailed study of derived completion as a functor $D(A) \\to D(\\widehat{A})$ over a non-noetherian ring $A$; (2) prove a flat base change result for weakly proregular ideals; and (3) Prove that Hochschild cohomology and analytic Hochschild cohomology of complete noetherian local rings"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04172","kind":"arxiv","version":3},"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"}