\(K\)-theory of algebraic tori and toric varieties
type: Item
instance of: scholarly article
title: \(K\)-theory of algebraic tori and toric varieties
zbMATH Open document ID: 0882.19002
DOI: 10.1023/A:1007770500046
author: Ivan Panin
author: Alexander Merkurjev
published in: \(K\)-Theory
publication date: 1998-02-11T00:00:00Z
review text: Let \(F\) be a field. Write \({\mathcal V}ar(F)\) for the category of all varieties over \(F\), \({\mathcal {PV}}ar(F)\) for the full subcategory of smooth projective varieties over \(F\), and \({\mathcal A}lg (F)\) for the category of separable \(F\)-algebras. For \(X\in {\mathcal V}ar(F)\) and \(A\in {\mathcal A}lg (F)\) one has the exact category \({\mathcal P}(X,A)\) of left \(A\otimes_F{\mathcal O}_X\)-modules which are locally free \({\mathcal O}_X\)-modules of finite rank and morphisms of \(A\otimes_F{\mathcal O}_X\)-modules. Similarly one has the abelian category \({\mathcal M}(X,A)\) of left \(A\otimes_F{\mathcal O}_X\)-modules which are coherent \({\mathcal O}_X\)-modules and morphisms of \(A\otimes_F{\mathcal O}_X\)-modules. One can define the \(K\)-groups \(K_n(X,A):=K_n ({\mathcal P}(X,A))\) and \(K'_n(X,A):=K_n ({\mathcal M}(X,A))\), with the usual properties such as inverse image, direct image, base change, products, projection formula, pairings. For \(X\in {\mathcal {PV}}ar(F)\) and \(A\in {\mathcal A}lg(F)\) consider the pairs \((X,A)\) with morphisms given by \(\text{Mor}((X,A),(Y,B))=K_0(X\times Y,A^{\text{op}}\otimes_FB)\), where \(A^{\text{op}}\) is the opposite \(F\)-algebra. One obtains the motivic category \({\mathcal C}= {\mathcal C}_F\) with involution and tensor structure. One has functors \(i:{\mathcal {PV}}ar(F)\to {\mathcal C}\) taking \(X\) to \((X,F)\), and \(j:{\mathcal A}lg(F)\to{\mathcal C}\) taking \(A\) to \((\text{Spec}(F),A)\). For any \(V\in{\mathcal{PV}}ar(F)\) the composition \({\mathcal{PV}}ar(F)@>j>>{\mathcal C}@>{\mathbf K}_n^V>>{\mathcal A}b\) coincides with the functor \(X\mapsto K'_n(V\times X)\), and for any \(V\in{\mathcal V}ar(F)\) the composition \({\mathcal A}lg(F)@>j>>{\mathcal C}@>{\mathbf K}_n^V>>{\mathcal A}b\) coincides with \(A\mapsto K'_n(V,A)\). Let \(S\) be a reductive algebraic group over \(F\) and let \(\pi:U\rightarrow S\) be an \(S\)-torsor over the smooth projective variety \(X\). Assume there is an …
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 19D50
Mathematics Subject Classification ID: 14C35
zbMATH DE Number: 1062412
zbMATH Keywords: toric variety
zbMATH Keywords: toric model
zbMATH Keywords: Chow group
zbMATH Keywords: smooth projective varieties
zbMATH Keywords: motivic category
zbMATH Keywords: principal homogeneous space
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
full work available at URL: a:1007770500046
OpenAlex ID: W2008611168
description: scientific article; zbMATH DE number 1062412