A splitting principle and the algebraic K-theory of some homogeneous varieties
type: Item
instance of: scholarly article
title: A splitting principle and the algebraic K-theory of some homogeneous varieties
zbMATH Open document ID: 0790.19003
DOI: 10.1007/BF02988479
author: Ivan Panin
published in: Journal of Soviet Mathematics
publication date: 1993-09-01T00:00:00Z
full work available at URL: https://eudml.org/doc/68268
review text: For a scheme \(X\) and a sheaf \({\mathcal A}\) of algebras let \({\mathcal M}(X,{\mathcal A})\) (resp.\ \({\mathcal M}(X,{\mathcal A}^{\text{op}})\)) denote the exact category of coherent left (resp.\ right) \({\mathcal A}\)-modules, and write \(G_ i(X,{\mathcal A})\) for the associated \(K\)-groups. Similarly, replacing coherent \({\mathcal A}\)-modules with locally free \({\mathcal O}_ X\)-modules with \({\mathcal A}\)-action, one has the \(K\)-groups \(K_ i(X,{\mathcal A})\). If \(X\) is faithfully flat of finite type over \(\mathbb{Z}\), and if \({\mathcal H}\) is a geometrically connected split reductive faithfully flat group scheme over \(\mathbb{Z}\) with right action on \(X\), then a projective \({\mathbb{Z}}\)- representation \(\rho_ i\), \(i=1,\ldots, n\), of \(\mathcal H\) and a 1-cocycle \(c\in H^ 1_{\acute{e}t}(S,{\mathcal H})\), \(S\) a finite-dimensional noetherian scheme, give rise to a twisted form \(X_ c\) of \(X\) and Azumaya algebras \({\mathcal A}_ c(\rho_ i)\). The splitting principle says that under suitable conditions, there is an isomorphism \(\displaystyle{\phi_ S:\coprod_{i=1}^ nG_ *(S,A_ c(\rho_ i))[-d(i)]{\buildrel\sim\over\longrightarrow}G_ *(X_ c)}\). Next, let \(\tilde{H}\) be a geometrically connected split reductive group scheme faithfully flat over \(\mathbb{Z}\) with free abelian \(\pi_ 1(\tilde{H}_{{\mathbf C},an})\). Fix a maximal torus \(\tilde{T}\) and a parabolic group subscheme \(\tilde{P}\) containing \(\tilde{T}\). Let \(\tilde{Z}\) be the center of \(\tilde{H}\), and write \(H=\tilde{H}/\tilde{Z}\), \(P=\tilde{P}/\tilde{Z}\), \({\mathcal F}=\tilde{H}/\tilde{P}=H/P\). From work of Atiyah, Hirzebruch, Grothendieck and Serre, it follows that one has a homomorphism \(\displaystyle{ \alpha_ S(\tilde{H},\tilde{P}):R(\tilde{P})\otimes G_ *(S)\longrightarrow G_ *({\mathcal F}_ S)}\), for any scheme \(S\), where \({\mathcal F}_ S={\mathcal F}\times S\) and \(R(\tilde{P})\) is the Grothendieck ring of the exact category of \(\math…
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14C35
zbMATH DE Number: 404565
zbMATH Keywords: algebraic group
zbMATH Keywords: Azumaya algebra
zbMATH Keywords: splitting principle
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
cites work: Q3844098
cites work: On the K-theory of Lie groups
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cites work: Higher algebraic K-theory: I
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OpenAlex ID: W1998120920
description: scientific article; zbMATH DE number 404565