The power operation in the Galois cohomology of a reductive group over a number field
Mikhail Borovoi, Zinovy Reichstein, Philippe Gille
TL;DR
This work introduces a diamond power operation on $H^1(K,G)$ for connected reductive groups over local or global fields, unifying torsor powers with abelian and archimedean data. It proves that the period ${\rm per}(\xi)$ divides the index ${\rm ind}(\xi)$ for such fields, and identifies cases where equality holds when the algebraic fundamental group is split; it also proves nonexistence of a functorial power operation over arbitrary fields. The construction leverages abelian Galois cohomology, the localization at infinity, and a Cartesian square, and provides explicit descriptions of the power map in terms of the algebraic fundamental group. The results connect period–index phenomena in Galois cohomology to the Brauer group, yield concrete bounds in the global setting, and raise natural questions about the limits of functorial power operations outside the local/global field context.
Abstract
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.