On the algebraic K-theory of some homogeneous varieties
type: Item
instance of: scholarly article
title: On the algebraic K-theory of some homogeneous varieties
zbMATH Open document ID: 1252.19004
author: Alexey Ananyevskiy
published in: Documenta Mathematica
publication date: 2012-05-29T00:00:00Z
full work available at URL: 07.html
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 6039008
zbMATH Keywords: homogeneous varieties
zbMATH Keywords: central simple algebras
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: algebraic groups
MaRDI profile type: Publication
review text: It is known see, e.g. [\textit{I. A. Panin}, K-Theory 8, No. 6, 541--585 (1994; Zbl 0854.19002)] that the \(K\)-theory of homogeneous projective varieties can be expressed in terms of the \(K\)-theory of certain separable algebras. In the present paper, the author provides a unified approach to the \(K\)-theory of homogeneous varieties and computes it for inner forms of \(G/H,\) where \(H \subset G\) is a pair of connected split reductive groups of the same rank (all algebraic groups are supposed to be smooth algebraic varieties over an arbitrary field \(k).\) The first main theorem asserts that, for \(r = [W(G):W(H)],\) \(W(\ldotp)\) denoting the Weyl group, there exist \(r\) vector bundles over \(G/H,\) \(V_1, \ldots, V_r,\) such that \(K_\ast (G/H) = \displaystyle\mathop\oplus^r_{i = 1} K_\ast (k) [V_i].\) Suppose moreover that \(G\) is simply connected semisimple (this is not actually a restriction) and put \(\overline G = G/Z(G)\); fix a 1-cocycle \(\gamma:Gal (k^{\mathrm{sep}/k}) \to \overline G (k^{\mathrm{sep}})\) and twist the variety \(X = G/H\) to get \(X_\gamma = (G/H)_\gamma\).NEWLINENEWLINENEWLINEThe second main theorem gives an isomorphism \(\displaystyle\mathop\oplus^r_{i = 1} K_\ast (A(\lambda_i)_\gamma) {\displaystyle\buildrel\sim \over \rightarrow} K_\ast (X_\gamma),\) where the separable algebras \(A(\lambda_i)_\gamma\) are obtained as follows : let \(R(G) = K_0(\mathrm{Rep}_k(G))\) be the representation ring of \(G\); the author shows that \(R(H)\) is a free \(R(G)\)-module of rank \(r,\) with a basis consisting of the classes of irreducible representations of \(H\); the proof of this property introduces \(r\) minimal and indecomposable (in some sense) elements \(\lambda_i \in w_i \wedge^+_G,\) one for each chamber \(w_i {\mathcal C} (G),\) \(\wedge^+_G\) denoting a cone of dominant weights; a theorem of \textit{J.-P. Serre} [Publ. Math., Inst. Hautes Étud. Sci. 34, 37--52 (1968; Zbl 0195.50802)] associates to such a \(\lambda_i\) an irreducible…
description: scientific article; zbMATH DE number 6039008