Let be a Dedekind domain with field of fractions and a finite group. We show that, if is a ring of -adic integers, then the Witt decomposition map between the Grothendieck-Witt group of bilinear -modules and the one of finite bilinear -modules is surjective. For number fields is also surjective, if is a nilpotent group of odd order, but there are counterexamples for groups of even order.
Soit un anneau de Dedekind et son corps de fractions. Soit un groupe fini. Si est un anneau d’entiers -adiques, alors l’application de décomposition de Witt entre le groupe de Grothendieck-Witt des -modules bilinéaires et celui des -modules bilinéaires de torsion est surjective. Pour les corps de nombres , on démontre que est surjective si est un groupe nilpotent d’ordre impair, et on donne des contre-exemples pour des groupes d’ordre pair.
@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://archive.numdam.org/item/JTNB_2000__12_2_489_0/} }
Nebe, Gabriele. On the cokernel of the Witt decomposition map. Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 2, pp. 489-501. http://archive.numdam.org/item/JTNB_2000__12_2_489_0/
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