{"paper":{"title":"Generators of top cohomology","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Manoj Kummini, Mohit Upmanyu","submitted_at":"2025-01-08T08:54:37Z","abstract_excerpt":"Let $R$ be a commutative noetherian ring and $f: X \\to \\mathrm{Spec} R$ a proper smooth morphism, of relative dimension $n$. From Hartshorne, Residues and Duality, Springer, 1966, one knows that the trace map $\\mathrm{Tr}_f : \\mathrm{H}^n(X, \\omega_{X/R}) \\to R$ is an isomorphism when $f$ has geometrically connected fibres. We construct an exact sequence that generates $\\mathrm{Ext}_X^n(\\mathcal{O}_X, \\omega_{X/R}) = \\mathrm{H}^n(X, \\omega_{X/R})$ as an $R$-module in the following cases: \n  (1) when $R$ is a DVR and $f$ has a section;\n  (2) when $R=\\mathbb{Z}$ and $X$ is the Grassmannian $G_{2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.04357","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.04357/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}