{"paper":{"title":"Gorenstein projective and injective dimensions over Frobenius extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"Wei Ren","submitted_at":"2018-01-22T20:23:21Z","abstract_excerpt":"Let $R\\subset A$ be a Frobenius extension of rings. We prove that: (1) for any left $A$-module $M$, $_{A}M$ is Gorenstein projective (injective) if and only if the underlying left $R$-module $_{R}M$ is Gorenstein projective (injective). (2) if $\\mathrm{G}\\text{-}\\mathrm{proj.dim}_{A}M<\\infty$, then $\\mathrm{G}\\text{-}\\mathrm{proj.dim}_{A}M = \\mathrm{G}\\text{-}\\mathrm{proj.dim}_{R}M$, the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then $\\mathrm{G}\\text{-}\\mathrm{gldim}(A)= \\mathrm{G}\\text{-}\\mathrm{gldim}(R)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.07305","kind":"arxiv","version":4},"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"}