Twisted Witt groups of flag varieties
type: Item
instance of: scholarly article
zbMATH Open document ID: 1305.19002
author: Marcus Zibrowius
publication date: 2014-08-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1305.6492
Mathematics Subject Classification ID: 19G12
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 6334063
zbMATH Keywords: Witt groups
zbMATH Keywords: flag varieties
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: Tate cohomology
zbMATH Keywords: parabolic subgroups
MaRDI profile type: Publication
OpenAlex ID: W3098728287
arXiv ID: 1305.6492
cites work: On the algebraic \(K\)-theory of twisted flag varieties
cites work: Quadratic and Hermitian forms in additive and Abelian categories
cites work: A classification of K-local spectra
cites work: Koszul complexes and symmetric forms over the punctured affine space
cites work: Witt groups of Grassmann varieties
cites work: Triangular Witt groups. I: The 12-term localization exact sequence
cites work: Q5606652
cites work: Chow motif and higher Chow theory of \(G/P\)
cites work: Le théorème fondamental de la K-théorie hermitienne
cites work: Pairings in triangular Witt theory
cites work: Trivial Witt groups of flag varieties
cites work: On the 2-primary \(v_{1}\)-periodic homotopy groups of spaces
cites work: On a theorem of Pittie
cites work: Groupes de Grothendieck des schemas en groupes reductifs deployes
cites work: Gysin maps in Balmer-Witt theory
DOI: 10.1017/IS014003028JKT260
published in: Journal of \(K\)-Theory
title: Twisted Witt groups of flag varieties
review text: \textit{B. Calmès} and \textit{J. Fasel} [J. Pure Appl. Algebra 216, No. 2, 404--406 (2012; Zbl 1235.19002)] have shown that the twisted Witt groups of split flag varieties vanish in a large number of cases. For flag varieties over algebraically closed fields, the author sharpens their result to an if-and-only-if statement. In particular, he shows that the twisted Witt groups vanish in many previously unknown cases. In the non-zero cases, he finds that the twisted total Witt group forms a free module of rank one over the untwisted total Witt group, up to a difference in grading.NEWLINENEWLINEThe proof relies on an identification of the Witt groups of flag varieties with the Tate cohomology groups of their \(K\)-groups, whereby the verification of all assertions is eventually reduced to the computation of the (twisted) Tate cohomology of the representation ring of a parabolic subgroup.
reviewed by: L. N. Vaserstein
description: scientific article; zbMATH DE number 6334063