On the cokernel of the Witt decomposition map
Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 2, pp. 489-501.

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 KG-modules and the one of finite bilinear RG-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.

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 KG-modules bilinéaires et celui des RG-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.

@article{JTNB_2000__12_2_489_0,
     author = {Gabriele Nebe},
     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},
     zbl = {0993.11020},
     mrnumber = {1823199},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2000__12_2_489_0/}
}
TY  - JOUR
AU  - Gabriele Nebe
TI  - On the cokernel of the Witt decomposition map
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2000
SP  - 489
EP  - 501
VL  - 12
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_2000__12_2_489_0/
LA  - en
ID  - JTNB_2000__12_2_489_0
ER  - 
%0 Journal Article
%A Gabriele Nebe
%T On the cokernel of the Witt decomposition map
%J Journal de théorie des nombres de Bordeaux
%D 2000
%P 489-501
%V 12
%N 2
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_2000__12_2_489_0/
%G en
%F JTNB_2000__12_2_489_0
Gabriele Nebe. On the cokernel of the Witt decomposition map. Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 2, pp. 489-501. https://jtnb.centre-mersenne.org/item/JTNB_2000__12_2_489_0/

[ARS 97] M. Auslander, I. Reiten, S.O. Smalø, Representation theory of Artin algebras. Cambridge studies in advanced math., 36Camb. Univers. Press (1997). | MR | Zbl

[CuR 81] C.W. Curtis, I. Reiner, Methods of representation theory, with applications to finite groups and orders. Vol. I. John Wiley & Sons. (1981). | MR | Zbl

[Dre 75] A.W.M. Dress, Induction and structure theorems for orthogonal representations of finite groups. Annals of Math. 102 (1975), 291-325. | MR | Zbl

[MiH 73] J. Milnor, D. Husemoller, Symmetric bilinear forms. Springer Ergebnisse 73 (1973). | MR | Zbl

[Knu 91] M.-A. Knus, Quadratic and Hermitian Forms over Rings. Springer Grundlehren 294 (1991). | MR | Zbl

[Miy 90] M. Miyamoto, Equivariant Witt Groups of Finite Groups of Odd Order. J. Algebra 133 (1990), 197-210. | MR | Zbl

[Mor 88] J.F. Morales, Maximal Hermitian forms over ZG. Comment. Math. Helv. 63 (1988), no.2, 209-225. | MR | Zbl

[Mor 90] J.F. Morales, Equivariant Witt groups. Canad. Math. Bull. 33 (1990), 207-218. | MR | Zbl

[Neb 99] G. Nebe, Orthogonale Darstellungen endlicher Gruppen und Gruppenringe, Habilitationsschrift, RWTH Aachen, Aachener Beiträge zur Mathematik26, Verlag Mainz Aachen (1999). | MR | Zbl

[Rei 75] I. Reiner, Maximal orders. Academic Press (1975). | MR | Zbl

[Scha 85] W. Scharlau, Quadratic and Hermitian Forms. Springer Grundlehren 270 (1985). | MR | Zbl

[Ser 77] J.-P. Serre, Linear Representations of Finite Groups. Springer GTM 42 (1977). | MR | Zbl

[Tho 84] J.G. Thompson, Bilinear Forms in Characteristic p and the Frobenius-Schur Indicator. Springer LNM 1185 (1984), 221-230.

[Was 82] L.C. Washington, Introduction to cyclotomic fields. Springer GTM 83 (1982). | MR | Zbl