Generalized Steinberg Relations

Abstract

We consider a field F and positive integers n, m, such that m is not divisible by Char(F ) and is prime to n!. The absolute Galois group G F acts on the group U n (Z/m) of all (n + 1) × (n + 1) unipotent upper-triangular matrices over Z/m cyclotomically. Given 0, 1 = z ∈ F and an arbitrary list w of n Kummer elements (z) F , (1 − z) F in H 1 (G F , µ m ), we construct in a canonical way a quotient U w of U n (Z/m) and a cohomology element ρ z in H 1 (G F , 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.