Equivariant vector bundles, their derived category and K-theory on affine schemes

Let GG be an affine group scheme over a noetherian commutative ring RR. We show that every GG-equivariant vector bundle on an affine toric scheme over RR with GG-action is equivariantly extended from Spec ( R ) Spec(R) for several cases of RR and GG. We show that, given two affine schemes with group scheme actions, an equivalence of the equivariant derived categories implies isomorphism of the equivariant KK-theories as well as equivariant K'K′-theories