Localization theorem in equivariant algebraic \(K\)-theory
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)].
scientific article; zbMATH DE number 681827
scientific article