  We study the K-theory of actions of diagonalizable group schemes on
noetherian regular separated algebraic spaces: our main result shows how to
reconstruct the K-theory ring of such an action from the K-theory rings of the
loci where the stabilizers have constant dimension. We apply this to the
calculation of the equivariant K-theory of toric varieties, and give conditions
under which the Merkurjev spectral sequence degenerates, so that the
equivariant K-theory ring determines the ordinary K-theory ring. We also prove
a very refined localization theorem for actions of this type.
