{"paper":{"title":"The nonexistence of sections of Stiefel varieties and stably free modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.KT"],"primary_cat":"math.AG","authors_text":"Sebastian Gant","submitted_at":"2025-09-08T22:43:58Z","abstract_excerpt":"Let $V_r(\\mathbb{A}^n)$ denote the Stiefel variety ${\\rm GL}_n/{\\rm GL}_{n-r}$ over a field. There is a natural projection $p: V_{r+\\ell}(\\mathbb{A}^n) \\to V_r(\\mathbb{A}^n)$. The question of whether this projection admits a section was asked by M. Raynaud in 1968. We focus on the case of $r \\ge 2$ and provide examples of triples $(r,n,\\ell)$ for which a section does not exist. Our results produce examples of stably free modules that do not have free summands of a given rank. To this end, we also construct a splitting of $V_2(\\mathbb{A}^n)$ in the motivic stable homotopy category over a field,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.07263","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.07263/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"}