Generalized Steinberg Relations

Abstract

We consider a field $F$ and positive integers $n$, $m$, such that $m$ is not divisible by $\mathrm{Char}(F)$ and is prime to $n!$. The absolute Galois group $G_F$ acts on the group $\mathbb{U}_n(\mathbb{Z}/m)$ of all $(n+1)\times(n+1)$ unipotent upper-triangular matrices over $\mathbb{Z}/m$ cyclotomically. Given $0,1\neq z\in F$ and an arbitrary list $w$ of $n$ Kummer elements $(z)_F$, $(1-z)_F$ in $H^1(G_F,\mu_m)$, we construct in a canonical way a quotient $\mathbb{U}_w$ of $\mathbb{U}_n(\mathbb{Z}/m)$ and a cohomology element $\rho^z$ in $H^1(G_F,\mathbb{U}_w)$ whose projection to the superdiagonal is the prescribed list. This extends results by Wickelgren, and in the case $n=2$ recovers the Steinberg relation in Galois cohomology, proved by Tate.