Comparison of equivariant algebraic and topological K-theory (Q1820213)

Let k be a separably closed field, G a reduced linear algebraic scheme over k, X a separated scheme of finite type over X, and suppose G acts on X. Fix a prime power \(\ell^{\nu}\) which is invertible in k. The author's main result is a G-equivariant version of an earlier theorem of his: if the \(mod\quad \ell^{\nu}\) equivariant algebraic K-groups of the category of coherent G-modules on X are localized by inverting the Bott element \(\beta\) and then completed with respect to the augmentation ideal of the representation ring of G, the resulting groups are isomorphic to the equivariant topological K-groups of the same category: \[ G/\ell_*^{\nu}(G,X)[\beta^{-1}]^{{\hat{\;}}}_{IG}\quad \approx \quad G/\ell_*^{\nu Top}(G,X). \] If X and G are smooth over k, a similar isomorphism holds for the category of algebraic G-vector bundles on X.