Fan cohomology and its application to equivariant K-theory of toric varieties

Mu-Wan Huang, Mark Walker and I established an explicit formula for the equivariant K-groups of affine toric varieties. We also recovered a result due to Vezzosi and Vistoli, which expresses the equivariant K-groups of a smooth toric variety in terms of the K-groups of its maximal open affine toric subvarieties. This dissertation investigates the situation when the toric variety X is neither affine nor smooth. In many cases, we compute the Cech cohomology groups of the presheaf KTq on X endowed with a topology. Using these calculations and Walker's Localization Theorem for equivariant K-theory, we give explicit formulas for the equivariant K-groups of toric varieties associated to all two dimensional fans and certain three dimensional fans.