Generalized Steinberg Relations

TL;DR

This paper generalizes classical Steinberg relations in Galois cohomology by constructing canonical quotients and cohomology elements that extend known relations, applicable to a broader class of fields and matrices.

Contribution

It introduces a new framework for constructing quotients of unipotent matrix groups and associated cohomology elements, extending classical Steinberg relations to more general settings.

Findings

Constructs canonical quotients of unipotent matrix groups.

Defines cohomology elements with prescribed superdiagonals.

Generalizes Steinberg relations to broader algebraic contexts.

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.