\(K\)-theory of the Weil transfer functor
type: Item
instance of: scholarly article
title: \(K\)-theory of the Weil transfer functor
zbMATH Open document ID: 0969.19003
DOI: 10.1023/A:1007822701416
author: Seva Joukhovitski
published in: \(K\)-Theory
publication date: 2001-06-11T00:00:00Z
full work available at URL: https://eudml.org/doc/120485
review text: Let \(F\) be a field and denote by \({\mathcal A} lg(F)\) the category of separable \(F\)-algebras. Denote by \({\mathcal V}ar(F)\) the category of all quasi-projective schemes over \(F\) and by \({\mathcal {PV}}ar(F)\) the full subcategory of smooth projective schemes. For \(X\in {\mathcal V} ar(F)\) and \(A\in {\mathcal A} lg(F)\), let \({\mathcal P}(X,A)\) be the exact category 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. One denotes the \(K\)-groups by \(K_n(X,A)=K_n({\mathcal P}(X,A))\). One defines, after \textit{A. S. Merkurjev} and \textit{I. A. Panin} [``\(K\)-theory of algebraic tori and toric varieties'', \(K\)-Theory 12, No. 2, 101-143 (1997; Zbl 0882.19002)], the motivic category \({\mathcal C}= {\mathcal C}_F\) with objects the pairs \((X,A)\) with \(X\in {\mathcal {PV}}ar(F)\) and \(A\in {\mathcal A}lg(F)\), and morphisms \(\text{Mor}_{\mathcal C}((X,A),(Y,B))=K_0(X\times Y,A^{\text{op}}\otimes_FB)\). There are functors \(i:{\mathcal {PV}}ar(F)\rightarrow {\mathcal C}_F\) and \(j: {\mathcal A}lg(F)\rightarrow {\mathcal C}_F\), relating the \(K''\)'s (in the usual sense, defined via coherent, instead of locally free, modules) of schemes and algebras. Let \(L/F\) be a Galois field extension with Galois group \(G=\text{Gal}(L/F)\) and \(n=[L:F]\). For an \(L\)-linear space \(V\) and an element \(\sigma \in G\) one writes \({}^{\sigma}V\) for the \(L\)-space \(V\) with \(L\)-action given by \(x\cdot v=\sigma^{-1}(x)\cdot v\) for \(x\in L\), \(v\in V\). One also writes \(V^{\otimes G}=\bigotimes_{\sigma\in G}^{\sigma}V\), where the tensor product is taken over \(L\). The group \(G\) acts on \(V^{\otimes G}\) by permutations of tensor factors. One defines the Weil transfer of the \(L\)-linear space \(V\) as the \(F\)-linear space \(RV=R_{L/F}V=(V^ {\otimes G})^G\). Let \(\pi:X\rightarrow\text{Spec}(L)\) be a scheme over \(L\). Then for \(\sigma\…
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 19D99
Mathematics Subject Classification ID: 19A99
zbMATH DE Number: 1547219
zbMATH Keywords: Weil transfer
zbMATH Keywords: motivic category
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1547219