It is known that the norm map N_G for the action of a finite group G on a ring R is surjective if and only if for every elementary abelian subgroup U of G the norm map N_U is surjective. Equivalently, there exists an element x_G in R satisfying N_G(x_G) = 1 if and only for every elementary abelian subgroup U there exists an element x_U in R such that N_U(x_U) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for x_G in terms of the elements x_U. We solve this problem when the group G is abelian. The main part of the proof, which was inspired by cohomological considerations, deals with the case when G is a cyclic p-group.
Publié le : 2002-07-05
Classification:
cyclic group,
homology,
ring,
group action,
MSC 16W22, 16U99, 20C05, 20J05, 20K01,
[MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA]
@article{hal-00119910,
author = {Aljadeff, Eli and Kassel, Christian},
title = {Explicit norm one elements for ring actions of finite abelian groups},
journal = {HAL},
volume = {2002},
number = {0},
year = {2002},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00119910}
}
Aljadeff, Eli; Kassel, Christian. Explicit norm one elements for ring actions of finite abelian groups. HAL, Tome 2002 (2002) no. 0, . http://gdmltest.u-ga.fr/item/hal-00119910/