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Toric varieties are varieties equipped with a torus action and constructed from cones and fans. In the joint work with Suanne Au and Mark E. Walker, we prove that the equivariant K-theory of an affine toric variety constructed from a cone can be identified with a group ring determined by the cone. When a toric variety X(Δ) is smooth, we interpret equivariant K-groups as presheaves on the associated fan space Δ. Relating the sheaf cohomology groups to equivariant K-groups via a spectral sequence, we provide another proof of a theorem of Vezzosi and Vistoli: equivariant K-theory is formed by patching equivariant K-theory of equivariant affine toric subvarieties.
This dissertation studies the sheaf cohomology groups for the equivariant K-groups tensored with Q and completed, and how they relate to the equivariant K-groups of non-smooth and non-affine toric varieties. The equivariant K-groups tensored with Q and completed coincide with the equivariant Chow rings for affine toric varieties. For a three-dimensional complete fan, we calculate the dimensions of the sheaf cohomology groups for the symmetric algebra sheaf. When the fan is given by a convex polytope, this information computes the equivariant K-groups tensored with Q and completed as extensions of sheaf cohomology groups.