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.
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.
@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, Tome 12 (2000) no. 2, pp. 489-501. http://archive.numdam.org/item/JTNB_2000__12_2_489_0/
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