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Higher algebraic \(K\)-theory of group actions with finite stabilizers. (English) Zbl 1012.19002

This paper establishes a decomposition theorem for the equivariant \(K\)-theory, \(K_* (X, G)\), of an affine group scheme, \(G\), of finite type over a field acting on a Noetherian separated regular algebraic space, \(X\). In earlier papers, the second author proved similar decomposition theorems in the cases where \(G\) is finite [A. Vistoli, Duke Math. J. 63, No. 2, 399-419 (1991; Zbl 0738.55002)] and where the action of \(G\) has finite reduced geometric stabilizers [A. Vistoli, in: Proc. Conf., Trento/Italy 1990, Lect. Notes Math. 1515, 112-133 (1992; Zbl 0784.14006)]. In the present paper the authors consider algebraic group schemes of finite type over a field \(k\) with the properties that the action of \(G\) has finite geometric stabilizers, that for essential cyclic subgroups, \(\sigma\), of \(G\) the quotient \(G/C_G(\sigma)\) is smooth, and a technical conditon on the action which they term being sufficiently rational. They show that \[ K_* (X, G) \otimes {\mathbb Z} [1/N] \cong \prod_{\sigma \in {\mathcal C}(G)} K_* (X^\sigma, C_G(\sigma)_{\text{geom}} \otimes \widetilde{R}(r))W_G(\sigma) \] where the product runs over all essential dual cyclic subgroups of \(G\), \(N\) is the least common multiple of the orders of all the essential dual cyclic subgroups of \(G_*\) and \(K_*(X, G)_{\text{geom}}\) is the localization of \(K_*(G,M)\) with respect to the multiplicative subset of the representation ring, \(R(G)\), consisting of elements whose virtual rank is a power of \(N\).

MSC:

19E08 \(K\)-theory of schemes
14L30 Group actions on varieties or schemes (quotients)
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References:

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