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 $$0,1\ne z\in F$$ 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.