{"paper":{"title":"Equivariant Hochschild cohomology of group algebras and relative $\\operatorname{Ext}$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The Γ-equivariant Hochschild cohomology of Γ-algebras equals a kΓ-relative Ext group for any field k.","cross_cats":[],"primary_cat":"math.KT","authors_text":"Andrada Pojar, Constantin-Cosmin Todea","submitted_at":"2026-05-11T15:38:05Z","abstract_excerpt":"For a finite group $\\Gamma$, acting on a finite group $G,$ we find necessary conditions for which the first $\\Gamma_0$-equivariant Hochschild cohomology of the group algebra $kG$ is non-trivial, where $k$ is a field of characteristic $p$ dividing the order of $G$ and $\\Gamma_0$ is the stabilizer subgroup in $\\Gamma$ of some element in $G.$ For any field $k$ we show that the $\\Gamma$-equivariant Hochschild cohomology of $\\Gamma$-algebras with coefficients in a $\\Gamma$-equivariant bimodule (Jensen, 1996) is isomorphic with some $k\\Gamma$-relative $\\operatorname{Ext},$ in the context of relative"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For any field k we show that the Γ-equivariant Hochschild cohomology of Γ-algebras with coefficients in a Γ-equivariant bimodule (Jensen, 1996) is isomorphic with some kΓ-relative Ext, in the context of relative homological algebra.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The groups Γ and G are finite, the action of Γ on G is given, and for the non-triviality conditions the characteristic p of k divides the order of G, with Γ₀ the stabilizer of an element in G.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Necessary conditions are derived for non-trivial first Γ₀-equivariant Hochschild cohomology of kG, and Γ-equivariant Hochschild cohomology is shown isomorphic to kΓ-relative Ext.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Γ-equivariant Hochschild cohomology of Γ-algebras equals a kΓ-relative Ext group for any field k.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"01d37fd47e534edeb277da2f2ecea39244f39401cb118f22b2cf09b51c996a87"},"source":{"id":"2605.10733","kind":"arxiv","version":2},"verdict":{"id":"13f4fd5a-f43a-4f5c-9597-35e36ceb9e3c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T04:52:32.301205Z","strongest_claim":"For any field k we show that the Γ-equivariant Hochschild cohomology of Γ-algebras with coefficients in a Γ-equivariant bimodule (Jensen, 1996) is isomorphic with some kΓ-relative Ext, in the context of relative homological algebra.","one_line_summary":"Necessary conditions are derived for non-trivial first Γ₀-equivariant Hochschild cohomology of kG, and Γ-equivariant Hochschild cohomology is shown isomorphic to kΓ-relative Ext.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The groups Γ and G are finite, the action of Γ on G is given, and for the non-triviality conditions the characteristic p of k divides the order of G, with Γ₀ the stabilizer of an element in G.","pith_extraction_headline":"The Γ-equivariant Hochschild cohomology of Γ-algebras equals a kΓ-relative Ext group for any field k."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.10733/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T05:22:00.427324Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T14:37:24.462686Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T11:01:16.920078Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T09:01:20.110638Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"103bf821be8cbd8cef8770694f454f78e1d5d0f1eb7360999f47074474109cba"},"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"}