The nonexistence of sections of Stiefel varieties and stably free modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913130154557440 |
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| author | Gant, Sebastian |
| author_facet | Gant, Sebastian |
| contents | 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, analogous to the classical stable splitting of the Stiefel manifolds due to I. M. James. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_07263 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The nonexistence of sections of Stiefel varieties and stably free modules Gant, Sebastian Algebraic Geometry Commutative Algebra K-Theory and Homology 14F42, 19A13, 13C10 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, analogous to the classical stable splitting of the Stiefel manifolds due to I. M. James. |
| title | The nonexistence of sections of Stiefel varieties and stably free modules |
| topic | Algebraic Geometry Commutative Algebra K-Theory and Homology 14F42, 19A13, 13C10 |
| url | https://arxiv.org/abs/2509.07263 |