Generalized Steinberg relations

Ido Efrat

doi:https://doi.org/10.1007/s40993-022-00386-x

Generalized Steinberg relations

Ido Efrat

Department of Mathematics

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

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 H¹(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 H¹(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.

Original language	American English
Article number	92
Journal	Research in Number Theory
Volume	8
Issue number	4
DOIs	https://doi.org/10.1007/s40993-022-00386-x
State	Published - 17 Oct 2022

Keywords

Galois cohomology
Kummer map
Massey products
Steinberg relations

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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https://doi.org/10.1007/s40993-022-00386-x

Cite this

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title = "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 GF acts on the group Un(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 H1(GF, μm) , we construct in a canonical way a quotient Uw of Un(Z/ m) and a cohomology element ρz in H1(GF, Uw) 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.",

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N2 - 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 GF acts on the group Un(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 H1(GF, μm) , we construct in a canonical way a quotient Uw of Un(Z/ m) and a cohomology element ρz in H1(GF, Uw) 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.

AB - 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 GF acts on the group Un(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 H1(GF, μm) , we construct in a canonical way a quotient Uw of Un(Z/ m) and a cohomology element ρz in H1(GF, Uw) 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.

KW - Galois cohomology

KW - Kummer map

KW - Massey products

KW - Steinberg relations

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JO - Research in Number Theory

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ER -

Generalized Steinberg relations

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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