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

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.