The power operation in the Galois cohomology of a reductive group over a number field
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arXiv
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| Natura: | Preprint |
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2024
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| author | Borovoi, Mikhail Reichstein, Zinovy Gille, Philippe |
| author_facet | Borovoi, Mikhail Reichstein, Zinovy Gille, Philippe |
| contents | For a connected reductive group $G$ over a local or global field $K$, we define a *diamond* (or *power*) operation $$(ξ,n)\mapsto ξ^{\Diamond n}\,\colon\, H^1(K,G)\times {\mathbb Z}\to H^1(K,G)$$ of raising to power $n$ in the Galois cohomology pointed set (this operation is new when $K$ is a number field). We show that this power operation has many good properties. When $G$ is a torus, the set $H^1(K,G)$ has a natural group structure, and $ξ^{\Diamond n}$ then coincides with the $n$-th power of $ξ$ in this group.
On the other hand, we show that a power operation on $H^1(K,G)$, functorial in $G$, which we define over local and global fields, cannot be defined for an arbitrary field $K$. Our proof of this assertion relies on the results of Appendix B written by Philippe Gille.
Using this power operation, for a cohomology class $ξ$ in $H^1(K,G)$ over local or global field, we define the period ${\rm per}(ξ)$ to be the least integer $n\ge 1$ such that $ξ^{\Diamond n}=1$. We define the index ${\rm ind}(ξ)$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $ξ$. The period and index of a cohomology class generalize the period and index a central simple algebra over $K$. For any connected reductive group $G$ defined over a local or global field $K$, we show that ${\rm per}(ξ)$ divides ${\rm ind}(ξ)$ and that ${\rm ind}(ξ)$ may be strictly greater than ${\rm per}(ξ)$, but they always have the same prime factors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07659 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The power operation in the Galois cohomology of a reductive group over a number field Borovoi, Mikhail Reichstein, Zinovy Gille, Philippe Number Theory Algebraic Geometry Group Theory Representation Theory 11E72, 20G10, 20G20, 20G25, 20G30 For a connected reductive group $G$ over a local or global field $K$, we define a *diamond* (or *power*) operation $$(ξ,n)\mapsto ξ^{\Diamond n}\,\colon\, H^1(K,G)\times {\mathbb Z}\to H^1(K,G)$$ of raising to power $n$ in the Galois cohomology pointed set (this operation is new when $K$ is a number field). We show that this power operation has many good properties. When $G$ is a torus, the set $H^1(K,G)$ has a natural group structure, and $ξ^{\Diamond n}$ then coincides with the $n$-th power of $ξ$ in this group. On the other hand, we show that a power operation on $H^1(K,G)$, functorial in $G$, which we define over local and global fields, cannot be defined for an arbitrary field $K$. Our proof of this assertion relies on the results of Appendix B written by Philippe Gille. Using this power operation, for a cohomology class $ξ$ in $H^1(K,G)$ over local or global field, we define the period ${\rm per}(ξ)$ to be the least integer $n\ge 1$ such that $ξ^{\Diamond n}=1$. We define the index ${\rm ind}(ξ)$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $ξ$. The period and index of a cohomology class generalize the period and index a central simple algebra over $K$. For any connected reductive group $G$ defined over a local or global field $K$, we show that ${\rm per}(ξ)$ divides ${\rm ind}(ξ)$ and that ${\rm ind}(ξ)$ may be strictly greater than ${\rm per}(ξ)$, but they always have the same prime factors. |
| title | The power operation in the Galois cohomology of a reductive group over a number field |
| topic | Number Theory Algebraic Geometry Group Theory Representation Theory 11E72, 20G10, 20G20, 20G25, 20G30 |
| url | https://arxiv.org/abs/2403.07659 |