The period and index of a Galois cohomology class of a reductive group over a local or global field

TL;DR

This paper studies the relationship between the period and index of Galois cohomology classes of reductive groups over local and global fields, generalizing classical results for central simple algebras and providing bounds on their ratio.

Contribution

It introduces a characteristic-free proof that the index divides a power of the period for cohomology classes over global and function fields, extending previous bounds.

Findings

Index can be strictly greater than period.

Index divides period to the power d for some d.

Provides bounds on d in various field cases.

Abstract

Let $K$ be a local or global field. For a connected reductive group $G$ over $K$, in another preprint [5] we defined a power operation $$(\xi,n)\mapsto \xi^{\Diamond n}\,\colon\, H^1(K,G)\times {\mathbb Z}\to H^1(K,G)$$ of raising to power $n$ in the Galois cohomology pointed set $H^1(K,G)$. In this paper, for a cohomology class $\xi$ in $H^1(K,G)$, we compare the period ${\rm per}(\xi)$ defined to be the least integer $n\ge 1$ such that $\xi^{\Diamond n}=1$, and the index ${\rm ind}(\xi)$ defined to be the greatest common divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting $\xi$. These period and index generalize the period and index a central simple algebra over $K$. For an arbitrary reductive $K$-group $G$, we proved in [5] that ${\rm per}(\xi)$ divides ${\rm ind}(\xi)$. In this paper we show that the index may be strictly greater than the period. In [5] we proved that for any $K$, $G$, and $\xi\in H^1(K,G)$ as above, the index ${\rm ind}(\xi)$ divides ${\rm per}(\xi)^d$ for some positive integer $d$, and we gave upper bounds for $d$ in the local case and in the case of a number field. Here we give a characteristic-free proof of the fact that ${\rm ind}(\xi)$ divides ${\rm per}(\xi)^d$ for some positive integer $d$ in the global field case, and our proof gives an upper bound for $d$ that is valid also in the case of a function field.