Milnor K-theory via commuting automorphisms

In this thesis, we give a presentation for Milnor K-theory of a field F whose generators are tuples of commuting automorphisms. This is similar to a presentation for Milnor K-theory given by the cohomology groups of Grayson. The main difference is that, in our presentation, we do not use a homotopy invariance relation, which we should not expect to hold for non-regular rings R. We go on to study this presentation for R a local ring. We conjecture that it agrees with the usual definition of Milnor K-theory for any local ring. We give some evidence towards this, including showing that the natural map Kn(R)→K ̃n(R) is injective when n = 0, 1, 2 or when R is a regular, local ring containing an infinite field. We also show a reciprocity result for K ̃n^M (R) any ring R, which, when R is a field, allows us to deduce surjectivity of the map. We prove a version of the additivity, resolution, devissage and cofinality theorems for the groups K ̃n^M (R). We also construct a comparison homomorphsim from K ̃n^M (R) to the presentation of Quillen K-theory given by Grayson