Efficient generation of projective modules: a motivic view
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910036317437952 |
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| author | Asok, Aravind Opie, Morgan Shin, Brian Syed, Tariq |
| author_facet | Asok, Aravind Opie, Morgan Shin, Brian Syed, Tariq |
| contents | Assume $k$ is a field and $R$ is a smooth $k$-algebra of dimension $d$. If $P$ is a projective module of rank $r$, then it is well-known that $P$ can be generated by $r+d$-elements (Forster--Swan). Under suitable assumptions on $r$ and $d$, we investigate obstructions to generation of $P$ by fewer than $r+d$ elements using motivic homotopy theory. For example, we observe that a quadratic enhancement of the classical Segre class obstructs generation by $r+d-1$ elements, whether or not $k$ is algebraically closed, generalizing old results of M.P. Murthy. Along the way, we also establish efficient generation results for symplectic modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13687 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Efficient generation of projective modules: a motivic view Asok, Aravind Opie, Morgan Shin, Brian Syed, Tariq Algebraic Geometry Commutative Algebra Algebraic Topology 14F42, 13C10, 19E15 Assume $k$ is a field and $R$ is a smooth $k$-algebra of dimension $d$. If $P$ is a projective module of rank $r$, then it is well-known that $P$ can be generated by $r+d$-elements (Forster--Swan). Under suitable assumptions on $r$ and $d$, we investigate obstructions to generation of $P$ by fewer than $r+d$ elements using motivic homotopy theory. For example, we observe that a quadratic enhancement of the classical Segre class obstructs generation by $r+d-1$ elements, whether or not $k$ is algebraically closed, generalizing old results of M.P. Murthy. Along the way, we also establish efficient generation results for symplectic modules. |
| title | Efficient generation of projective modules: a motivic view |
| topic | Algebraic Geometry Commutative Algebra Algebraic Topology 14F42, 13C10, 19E15 |
| url | https://arxiv.org/abs/2510.13687 |