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

Abstract

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 $$m \ge 2$$ , 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.