Decomposing the Diagonals of Invariant Fields
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866908505104973824 |
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| author | Saltman, David J |
| author_facet | Saltman, David J |
| contents | This work begins the process of using the decomposition of the diagonal as a tool for studying the rationality of invariant fields of finite groups $G$. Our ground field must be characteristic 0 because of the use we make of Bertini theorems. The steps we take are, first, defining and studying an "open" version of Chow zero. Second, we use this to translate our study to that of a Chow group of $G$ Galois extensions. We prove a "Sylow" property and thereby yield a connection between the invariants of $G$ and that of its Sylow subgroups. In particular, we show that if $G$ is a finite group with $p$ Sylow subgroup $P$, $V$ is a faithful $G$ module, and $F(V)^P$ has nontrivial unramified cohomology, then $F(V)^G$ is not retract rational. Finally, we prove Sylow type theorems for decomposition of the diagonal and the centers of generic division algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_18471 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decomposing the Diagonals of Invariant Fields Saltman, David J Algebraic Geometry 14E08, 12F20 This work begins the process of using the decomposition of the diagonal as a tool for studying the rationality of invariant fields of finite groups $G$. Our ground field must be characteristic 0 because of the use we make of Bertini theorems. The steps we take are, first, defining and studying an "open" version of Chow zero. Second, we use this to translate our study to that of a Chow group of $G$ Galois extensions. We prove a "Sylow" property and thereby yield a connection between the invariants of $G$ and that of its Sylow subgroups. In particular, we show that if $G$ is a finite group with $p$ Sylow subgroup $P$, $V$ is a faithful $G$ module, and $F(V)^P$ has nontrivial unramified cohomology, then $F(V)^G$ is not retract rational. Finally, we prove Sylow type theorems for decomposition of the diagonal and the centers of generic division algebras. |
| title | Decomposing the Diagonals of Invariant Fields |
| topic | Algebraic Geometry 14E08, 12F20 |
| url | https://arxiv.org/abs/2508.18471 |