Motivic cohomology of fat points in Milnor range via formal and rigid geometries

We present a formal scheme based cycle model for the motivic cohomology of the fat points defined by the truncated polynomial rings k [ t ] / ( t m ) with m ≥ 2 , in one variable over a field k . We compute their Milnor range cycle class groups when the field has sufficiently many elements. With some aids from rigid analytic geometry and the Gersten conjecture for the Milnor K -theory resolved by M. Kerz, we prove that the resulting cycle class groups are isomorphic to the Milnor K -groups of the truncated polynomial rings, generalizing a theorem of Nesterenko-Suslin and Totaro.