## Mathematics, Department of

#### First Advisor

Mark E. Walker

#### Date of this Version

8-2009

#### Abstract

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.

## Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics. Under the Supervision of Professor Mark E. Walker. Lincoln, Nebraska: August, 2009.

Copyright (c) 2009 Mu-wan Huang.