On the cokernel of the Witt decomposition map

Nebe, Gabriele

Nebe, Gabriele

Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 2, pp. 489-501.

Résumé
Abstract

Soit $R$ un anneau de Dedekind et $K$ son corps de fractions. Soit $G$ un groupe fini. Si $R$ est un anneau d’entiers $p$ -adiques, alors l’application $δ$ de décomposition de Witt entre le groupe de Grothendieck-Witt des $K G$ -modules bilinéaires et celui des $R G$ -modules bilinéaires de torsion est surjective. Pour les corps de nombres $K$ , on démontre que $δ$ est surjective si $G$ est un groupe nilpotent d’ordre impair, et on donne des contre-exemples pour des groupes d’ordre pair.

Let $R$ be a Dedekind domain with field of fractions $K$ and $G$ a finite group. We show that, if $R$ is a ring of $p$ -adic integers, then the Witt decomposition map $δ$ between the Grothendieck-Witt group of bilinear $K G$ -modules and the one of finite bilinear $R G$ -modules is surjective. For number fields $K, δ$ is also surjective, if $G$ is a nilpotent group of odd order, but there are counterexamples for groups of even order.

MR Zbl

@article{JTNB_2000__12_2_489_0,
     author = {Nebe, Gabriele},
     title = {On the cokernel of the {Witt} decomposition map},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {489--501},
     publisher = {Universit\'e Bordeaux I},
     volume = {12},
     number = {2},
     year = {2000},
     mrnumber = {1823199},
     zbl = {0993.11020},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2000__12_2_489_0/}
}

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Nebe, Gabriele. On the cokernel of the Witt decomposition map. Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 2, pp. 489-501. http://www.numdam.org/item/JTNB_2000__12_2_489_0/

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