UCLA

This dissertation takes place in three parts, all themed around motivic cohomology and Voevodsky's derived category of motives. In Chapter 1, we use Lannes's T-functor to study equivariant Chow rings. In Chapter 2, we explicitly compute certain equivariant Chow rings, namely Chow rings of classifying spaces of finite groups of order 32. In Chapter 3, we prove that isomorphisms in Voevodsky's derived category of motives are detected by a certain family of motivic homology groups