Free summands of stably free modules
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913777204592640 |
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| author | Williams, Ben Gant, W. S. |
| author_facet | Williams, Ben Gant, W. S. |
| contents | Let $R$ be a commutative ring. One may ask when a general $R$-module $P$ that satisfies $P \oplus R \cong R^n$ has a free summand of a given rank. M. Raynaud translated this question into one about sections of certain maps between Stiefel varieties: if $V_r(\mathbb{A}^n)$ denotes the Stiefel variety $\textrm{GL}(n) / \textrm{GL}(n-r)$ over a field $k$, then the projection $V_r(\mathbb{A}^n) \to V_1(\mathbb{A}^n)$ has a section if and only if the following holds: any module $P$ over any $k$-algebra $R$ with the property that $P \oplus R \cong R^n$ has a free summand of rank $r-1$. Using techniques from $\mathbb{A}^1$-homotopy theory, we characterize those $n$ for which the map $V_r(\mathbb{A}^n) \to V_1(\mathbb{A}^n)$ has a section in the cases $r=3,4$ under some assumptions on the base field. We conclude that if $P \oplus R \cong R^{24m}$ and $R$ contains a field of characteristic $0$, then $P$ contains a free summand of rank $2$. If $R$ contains a quadratically closed field of characteristic $0$, or the field of real numbers, then $P$ contains a free summand of rank $3$. The analogous results hold for schemes and vector bundles over them. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_15445 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Free summands of stably free modules Williams, Ben Gant, W. S. Algebraic Geometry Commutative Algebra K-Theory and Homology 14F42, 19A13, 13C10 Let $R$ be a commutative ring. One may ask when a general $R$-module $P$ that satisfies $P \oplus R \cong R^n$ has a free summand of a given rank. M. Raynaud translated this question into one about sections of certain maps between Stiefel varieties: if $V_r(\mathbb{A}^n)$ denotes the Stiefel variety $\textrm{GL}(n) / \textrm{GL}(n-r)$ over a field $k$, then the projection $V_r(\mathbb{A}^n) \to V_1(\mathbb{A}^n)$ has a section if and only if the following holds: any module $P$ over any $k$-algebra $R$ with the property that $P \oplus R \cong R^n$ has a free summand of rank $r-1$. Using techniques from $\mathbb{A}^1$-homotopy theory, we characterize those $n$ for which the map $V_r(\mathbb{A}^n) \to V_1(\mathbb{A}^n)$ has a section in the cases $r=3,4$ under some assumptions on the base field. We conclude that if $P \oplus R \cong R^{24m}$ and $R$ contains a field of characteristic $0$, then $P$ contains a free summand of rank $2$. If $R$ contains a quadratically closed field of characteristic $0$, or the field of real numbers, then $P$ contains a free summand of rank $3$. The analogous results hold for schemes and vector bundles over them. |
| title | Free summands of stably free modules |
| topic | Algebraic Geometry Commutative Algebra K-Theory and Homology 14F42, 19A13, 13C10 |
| url | https://arxiv.org/abs/2409.15445 |