The completion problem for equivariant \(K\)-theory
For \(G\) a compact Lie group and \(X\) a compact \(G\)-space, the Atiyah-Segal theorem (Theorem 1.1) identifies the topological \(K\)-theory of the Borel space \(X_G=X\times_GE_G\) with the completion of the equivariant \(K\)-theory of \(X\) with respect to the augmentation ideal, assuming a finiteness condition on the equivariant \(K\)-groups. In this paper, the author gives an elegant answer to the same question in algebraic geometry, using Morel-Voevodsky's \(\mathbb{A}^1\)-homotopy theory [\textit{F. Morel} and \textit{V. Voevodsky}, Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)] to produce a model for the Borel space (see Section 3.1). The main theorem (Theorem 8.7) gives a positive answer to the completion problem for a connected split reductive algebraic group acting over a scheme under a condition of being filtrable (see Section 5.2), condition always satisfied for smooth projective schemes (Theorem 6.4). For non-projective schemes, the author shows that the result still holds at the level of \(K_0\) (Theorem 9.10), and more generally for singular schemes by working with equivariant \(G\)-theory (i.e. \(K\)-theory of coherent sheaves); for higher \(K\)-groups the completion theorem fails and the author gives a counterexample on complex numbers (Theorem 10.1).
scientific article; zbMATH DE number 6902331
scientific article