Localization theorem in equivariant algebraic \(K\)-theory
type: Item
instance of: scholarly article
title: Localization theorem in equivariant algebraic \(K\)-theory
zbMATH Open document ID: 0817.19003
DOI: 10.1016/0022-4049(94)90088-4
author: Yuichiro Takeda
published in: Journal of Pure and Applied Algebra
publication date: 1995-07-23T00:00:00Z
review text: In the setting of a cyclic group action on a compact manifold, the Lefschetz fixed point formula in \(K\)-theory was first established by Atiyah and Segal, using the so-called localization theorem. This last theorem asserts that the equivariant \(K\)-theory of a manifold with group action is isomorphic to that of its fixed points, after taking suitable localization. In the present article the author extends the localization theorem to the situation of equivariant algebraic \(K\)-theory \(G_ * (T,X)\) for a scheme \(X\) with a group scheme action \((T,X)\). More precisely, let \(X\) be a scheme of finite type over \(S\) and equipped with a \(T\)-action. If \(T\) is smooth over \(S\), then there is an isomorphism \(G_ * (T,X^{(p)})_ p \simeq G_ * (T,X)_ p\) induced by the inclusion, where \(X^{(p)}\) is the \(T_ p\)-fixed point scheme of \(X\). The same theorem was proved by a different method by \textit{R. W. Thomason} in ``Une formule de Lefschetz en \(K\)-théorie équivariante algébrique'' [Duke Math. J. 68, No. 3, 447-462 (1992; Zbl 0813.19002)].
reviewed by: R. Lee
Mathematics Subject Classification ID: 19E08
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 681827
zbMATH Keywords: equivariant algebraic \(K\)-theory
zbMATH Keywords: localization theorem
zbMATH Keywords: group scheme action
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90088-4
OpenAlex ID: W2017319178
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description: scientific article; zbMATH DE number 681827