TY - JOUR
T1 - Generalized Steinberg relations
AU - Efrat, Ido
N1 - Funding Information:
This work was supported by the Israel Science Foundation (Grant No. 569/21)
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2022/10/17
Y1 - 2022/10/17
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
UR - http://www.scopus.com/inward/record.url?scp=85140099590&partnerID=8YFLogxK
U2 - 10.1007/s40993-022-00386-x
DO - 10.1007/s40993-022-00386-x
M3 - Article
AN - SCOPUS:85140099590
SN - 2363-9555
VL - 8
JO - Research in Number Theory
JF - Research in Number Theory
IS - 4
M1 - 92
ER -