| The higher Chow groups of zero-cycles associated to a field have been shown by Burt Totaro to be naturally isomorphic to the various Milnor K-groups of the field. In this dissertation, a definition of Milnor K-theory is proposed for smooth projective varieties over a field, and an analogue of Totaro's theorem is proven. It is shown, furthermore, that the Milnor K-groups of such varieties may be also be interpreted as cohomology groups of certain Milnor K-sheaves on these varieties.;The results described above are applied to the problem of calculating most of the higher Chow groups of zero-cycles associated to smooth varieties defined over a finite field. Combined with Steinberg's computation of the Milnor K-groups of finite fields, the abovementioned results may be used to show that most of these higher Chow groups vanish. The other groups are computed more or less explicitly, adapting techniques of Bruno Kahn to control symbols in the definition of the generalized Milnor K-groups. |
