The power operation in the Galois cohomology of a reductive group over a number field

Abstract

For a connected reductive group $G$ over a local or global field $K$, we define a *diamond* (or *power*) operation $$(\hat{I}¾,n)\mapsto \hat{I}{¾}^{◊n}:H^1(K,G)\times \mathbb{Z}\to H^1(K,G)$$ 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 $\hat{I}{¾}^{◊n}$ then coincides with the $n$-th power of $\hat{I}¾$ 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 $\hat{I}¾$ in $H^1(K,G)$ over local or global field, we define the period $\mathrm{per}(\hat{I}¾)$ to be the least integer $n\ge 1$ such that $\hat{I}{¾}^{◊n}=1$. We define the index $\mathrm{ind}(\hat{I}¾)$ to be the greatest common divisor of the degrees $[L:K]$ of finite extensions $L/K$ splitting $\hat{I}¾$. 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 $\mathrm{per}(\hat{I}¾)$ divides $\mathrm{ind}(\hat{I}¾)$ and that $\mathrm{ind}(\hat{I}¾)$ may be strictly greater than $\mathrm{per}(\hat{I}¾)$, but they always have the same prime factors.

V.1: 52 pages. V.3: 61 pages. V.4: 37 pages. V.5: 42 pages, with an appendix by Philippe Gille. V.6: 43 pages, with appendices by M. Borovoi and P. Gille